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The Fields of Electronics: Understanding Electronics Using Basic Physics
The Fields of Electronics: Understanding Electronics Using Basic Physics
Ralph Morrison
A practical new approach that brings together circuit theory and field theory for the practicing engineer To put it frankly, the traditional education of most engineers and scientists leaves them often unprepared to handle many of the practical problems they encounter. The Fields of Electronics: Understanding Electronics Using Basic Physics offers a highly original correction to this state of affairs. Most engineers learn circuit theory and field theory separately. Electromagnetic field theory is an important part of basic physics, but because it is a very mathematical subject, the connection to everyday problems is not emphasized. Circuit theory, on the other hand, is by its nature very practical. However, circuit theory cannot describe the nature of a facility, the interconnection of many pieces of hardware, or the power grid that interfaces each piece of hardware. The Fields of Electronics offers a unique approach that brings the physics and the circuit theory together into a seamless whole for today's practicing engineers. With a clear focus on the realworld problems confronting the practitioner in the field, the book thoroughly details the principles that apply to: * Capacitors, inductors, resistors, and transformers * Utility power and circuit concepts * Grounding and shielding * Radiation * Analog and digital signals * Facilities and sites Written with very little mathematics, and requiring only some background in electronics, this book provides an eminently useful new way to understand the subject of electronics that will simplify the work of every novice, experienced engineer, and scientist.
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Anno:
2002
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1st
Editore:
WileyInterscience
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english
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201
ISBN 10:
0471222909
ISBN 13:
9780471222903
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THE FIELDS OF ELECTRONICS THE FIELDS OF ELECTRONICS Understanding Electronics Using Basic Physics Ralph Morrison A WileyInterscience Publication JOHN WILEY & SONS, INC. " This book is printed on acidfree paper. ! c 2002 by John Wiley & Sons, Inc., New York. All rights reserved. Copyright ! Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate percopy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 7508400, fax (978) 7504744. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 101580012, (212) 8506011, fax (212) 8506008, EMail: PERMREQ@WILEY.COM. For ordering and customer service, call 1800CALLWILEY. Library of Congress CataloginginPublication Data Is Available ISBN 0471222909 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 CONTENTS Preface xi 1 The Electric Field 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 Introduction 1 Charge 2 Electrical Forces on Charged Bodies Electric Field 4 Work 5 Voltage 6 Charges on Surfaces 6 Equipotential Surfaces 8 Field Units 8 Batteries—A Voltage Source 10 Current 11 Resistors 12 Resistors in Series or Parallel 13 E Field and Current Flow 15 Problems 15 Energy Transfer 16 Resistor Dissipation 17 Problems 17 Electric Field Energy 18 Ground and Ground Planes 19 Induced Charges 20 Forces and Energy 20 Problems 21 Review 21 3 2 Capacitors, Magnetic Fields, and Transformers 2.1 2.2 Dielectrics 23 Displacement Field 23 23 vi 2.3 2.4 2.5 2.6 2.7 2.8 ; 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 CONTENTS Capacitance 25 Capacitance of Two Parallel Plates 25 Capacitance in Space 26 Current Flow in Capacitors 27 RC Time Constant 28 Problems 29 Shields 30 Magnetic Field 31 Solenoids 32 Ampère’s Law 32 Problems 34 Magnetic Circuit 34 Induction or B Field 34 Magnetic Circuit without a Gap 36 Magnetic Circuit with a Gap 38 Transformer Action 39 Magnetic Field Energy 40 Inductors 41 L=R Time Constant 42 Mutual Inductance 43 Problems 44 Review 44 3 Utility Power and Circuit Concepts 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 Sine Waves 46 Reactance and Impedance 47 Problems 50 Resonance 50 Phase 52 Parallel RL and RC Circuits 53 Problems 54 RMS Values 54 Problems 55 Transmission Lines 56 Poynting’s Vector 57 Transmission Line over an Equipotential Surface 58 Transmission Lines and Sine Waves 59 Coaxial Transmission 61 Utility Power Distribution 62 46 vii CONTENTS 3.16 3.17 3.18 3.19 3.20 3.21 Earth as a Conductor 64 Power Transformers in Electronic Hardware 65 Electrostatic Shields in Electronic Hardware 67 Where to Connect the Metal Box 69 Problems 73 Review 74 4 A Few More Tools 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 Introduction 75 Resistivity 75 Inductance of Isolated Conductors 76 Ohms per Square 77 Problems 77 Radiation 77 HalfDipole Antennas 78 Current Loop Radiators 80 Field Energy in Space 82 Problems 82 Reflection 83 Skin Effect 84 Problems 84 Surface Currents 85 Ground Planes and Fields 86 Apertures 86 Multiple Apertures 87 Waveguides 88 Attenuation of Fields by a Conductive Enclosure Gaskets 89 Honeycombs 89 Wave Coupling into Circuits 90 Problems 91 Square Waves 91 Harmonic Content in Utility Power 94 Spikes and Pulses 95 Transformers 96 Eddy Currents 98 Ferrite Materials 99 75 88 viii 4.30 4.31 CONTENTS Problems 99 Review 100 5 Analog Design 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21 5.22 5.23 5.24 5.25 5.26 5.27 5.28 5.29 Introduction 101 Analog Signals 101 CommonMode Interference 102 CommonMode Rejection in Instrumentation Problems 107 Voltage Measurement: Oscilloscopes 107 Microphones 108 Resistors 109 Guard Rings 110 Capacitors 111 Problems 112 Feedback Processes 113 Problems 115 Miller Effect 115 Inductors 116 Transformers 117 Problems 119 Isolation Transformers 120 Solenoids and Relays 121 Problems 122 Power Line Filters 123 Request for Energy 124 Filter and Energy Requests 125 Power Line Filters above 1 MHz 125 Mounting the Filter 125 Optical Isolators 127 Hall Effect 127 Surface Effects 127 Review 127 6 Digital Design and Mixed Analog/Digital Design 6.1 6.2 6.3 6.4 Introduction 129 Logic and Transmission Lines 129 Decoupling Capacitors 130 Ground Planes 131 101 106 129 ix CONTENTS 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 6.16 6.17 6.18 6.19 6.20 6.21 6.22 6.23 6.24 6.25 6.26 6.27 6.28 6.29 6.30 6.31 6.32 6.33 Power Planes 132 Decoupling Power Geometries 132 Ground Plane Islands 133 Radiation from Loops 133 Problems 133 Leaving the Board 134 Ribbon Cable and CommonMode Coupling Braided Cable Shields 135 Transfer Impedance 137 Mechanical Cable Terminations 138 Problems 138 Mounting Power Transistors 139 Electrostatic Discharge 139 ESD Precautions 141 Zapping 141 Product Testing: Radiation 142 Military Testing 142 Chattering Relay Test 143 Euro Standards 143 LISN 144 Sniffers 144 Simple Antenna 145 Peripherals 145 Problems 146 Lightning 146 Problems 147 Mixing Analog and Digital Design 147 Ground Bounce 148 Review 148 7 Facilities and Sites 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Introduction 149 Utility Power 149 Floating Utility Power 151 Isolated Grounds 152 SinglePoint Grounding 153 Ground Planes 155 Alternative Ground Planes 156 Power Centers 157 135 149 x 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 CONTENTS Lightning Protection 158 Surge Suppression 160 Racks 161 Magnet Fields around Distribution Transformers 161 Monitor Fields 162 Motor Controllers 162 Screen Rooms 163 Review 164 Appendix I: Solutions to Problems 165 Appendix II: Glossary of Common Terms 177 Appendix III: Abbreviations 183 Index 187 PREFACE This book provides a new way to understand the subject of electronics. The central theme is that all electrical phenomena can be explained in terms of electric and magnetic fields. Beginning students place their faith in their early instruction. They assume that the way they have been educated is the best way. Any departure from this format just adds complications. This book is a departure—hopefully, one that helps. There are many engineers and scientists struggling to function in the real world. Their education did not prepare them for handling most of the practical problems they encounter. The practitioner in trouble with grounds, noise, and interference feels that something is missing in his education. The new engineer has a very difficult time ordering, specifying, or using hardware correctly. Facilities and power distribution are a mystery. Surprisingly, all these areas are accessible once the correct viewpoint is taken. This book has been written to provide a better introduction to the field of electronics so that the parts that are often omitted can be put into perspective. The book uses very little mathematics. It helps to have some background in electronics, but it is not necessary. The beginning student may need some help from an instructor to fill in some of the blanks. The practicing engineer will be able to read this book with ease. Field phenomena are often felt to be the domain of the physicist. In a sense this is correct. Unfortunately, without a fieldbased understanding, many electronic processes must remain mysteries. It is not necessary to solve difficult problems to have an appreciation of how things work. It is only necessary to appreciate the fundamentals and understand the true nature of the world. To illustrate the problem, consider an electric field that is constant everywhere. Place a conducting loop of wire at some crazy angle in this field and ask a question: What is the shape of the new field? This is a very difficult problem even with a great deal of computing power. Now, have the field change sinusoidally and consider current flow and skin effect and the problem really gets difficult. The ideas are important, but the exact answer is not worth worrying about. Connecting wires and components to form circuits is standard practice. These conductors modify the fields around them. This is the same nasty problem, and again it does not need an exact solution. What is needed is an understanding of what actually takes place. Circuit theory does not consider this type of problem. xii PREFACE Most students in electronics spend a great deal of time with circuit theory. The viewpoint of circuit theory is to treat lumpedparameter models. Circuit theory provides an excellent way to predict the behavior of a group of components. The mathematics is very straightforward. Field theory, on the other hand, provides very little in terms of simple answers. Most practical problems cannot be approached by field theory, and yet circuit theory and field theory are inseparable. Circuit theory has no way to handle component size or orientation. Circuit theory, with its zeroohm connections, avoids any reference to loop area, commonimpedance coupling, or commonmode coupling. It fails to reference radiated energy from any source. Circuit theory has its successes, but it also has its failures. Field theory has its place, too, and yet it fails, as there is no convenient methodology. Educators are oriented toward problemsolving courses. Circuit theory fits this model, as it lends itself to solving many practical problems. Electricity and magnetism courses are more difficult, and only very simple geometries can be approached. The mathematics of vector fields, complex variable, and partial differential equations are not for the faint of heart. This leaves the practicing engineer with one solution. Drop physics and concentrate on circuit theory to provide answers. The circuit diagram of a building or the grounding diagram of a power grid is of no help in analyzing interference. These diagrams can be attempted, but they fail to provide a useful approach. They do not fit the textbook models, as they are not lumpedparameter circuits. The engineer is at a loss. This book allows the student to solve problems by means of simple ratios. In each area, typical practical problems are solved in the text. The student is expected to use this information to work the problem sets. The answers are all worked out in Appendix I. This makes it possible for the engineer or technician out of school to use the book for selfstudy. It also makes it possible to use the text in school, where problems can be assigned. The teacher can modify the parameters in the problems so that the student must work out the details rather than copy the answers. This book is not intended to teach circuit theory. It is not a substitute for teaching physics. It is a tool that can be used to connect the two subjects. There is a need to establish an elementary understanding in both areas so that the reader can understand the things that occur in the real world. This is done in the early chapters. The problems that are discussed throughout the book occur frequently. Exact solutions are not attempted. The simplifications that are applied are brought out in the text. These simple approaches provide insight into what can be done to handle practical situations. If students want to study physics or expand their knowledge of circuit theory, many texts and courses are available. This book takes the liberty of choosing important features from both areas in order to provide students with a different view of the electrical world—a view from the bridge between electrical behavior and physics. Redwood City, CA February 14, 2001 Ralph Morrison The Fields of Electronics: Understanding Electronics Using Basic Physics Ralph Morrison c 2002 John Wiley & Sons, Inc. Copyright ! ISBNs: 0471222909 (Hardback); 0471433934 (Electronic) 1 The Electric Field 1.1 INTRODUCTION This book is written to bring together two topics: circuit theory and field theory. Electromagnetic field theory is an important part of basic physics. In school it is usually taught as a separate course. Because physics is a very mathematical subject, the connection to everyday problems is not emphasized. Circuit theory, by its very nature, is very practical. It provides a methodology that connects with the many problems that students will encounter in practice. It is natural for most technical people to reinforce circuit concepts and push basic physics into the background. Circuit theory is not a match for describing the nature of a facility, the interconnection of many pieces of hardware, or the power grid that interfaces each piece of hardware. In circuit theory the emphasis is on components, not on such items as facilities or power distribution. A building or a power grid is not a topic for discussion in a physics course, as these areas are far too complex to consider. Basic physics can handle only very simple geometries, not buildings. Given an interference problem, an engineer defaults to circuit theory and circuit diagrams, as this is where he or she is usually most successful. The circuits that might be considered for a facility usually do not communicate well and bring little understanding to the problem. The fact that there are no actual circuit components to consider is just one of the problems. Circuit theory is a very powerful tool. If the right circuits are considered, the answers can be meaningful. In this book we place the concepts of fields into every aspect of circuit behavior. Every component functions because of internal or external fields. A facility has its own fields, and these fields enter into every circuit. When all the fields are considered, many problem areas become clearer. A solution may require changing the geometry of a system to limit the influence of the extraneous fields. Circuit theory is still used, but the influence of the environment becomes a part of the design. In effect, field theory brings geometry into circuit design. Experienced designers understand how important geometry can be to circuit performance. Fields are fundamental even in static circuits, and this is where the first chapter starts. All circuits function through the motion of field energy, and this idea must be considered at all circuit speeds. This includes batteries, utility power, audio, radio frequencies, and microwaves. Fields are needed to operate every circuit component, and conductors are needed to bring fields to each 1 2 THE ELECTRIC FIELD component. This means that the flow of field energy, to every component describes performance. The environment also includes field energy and this energy cannot be ignored. Understanding this fact makes it possible to design practical products. Today’s circuits operate at very high speeds. The demand to process vast amounts of data in very short periods is ever present. To understand highspeed problems, it is necessary to start slowly. The fields involved in all electrical phenomena are the same. In the first chapter we treat static charge and the concept of voltage. These very elementary ideas lay the foundation for understanding circuit behavior at all speeds. In later chapters, when the fields are changing more rapidly, the problems of radiation are discussed. All circuits, including the lowly flashlight, are explained using the same physics. This is where the book starts: fields, batteries, and resistors. 1.2 CHARGE In very dry weather, rubbing a comb through one’s hair will cause static electricity. The rubbing has removed some of the electrons from the surface of the comb. This group of electrons is said to be a charge. Since electrons are negative charges, the comb is left positively charged. Thus the absence of electrons is also considered a charge. In a clothes drier, where clothes are rubbed together and against the walls of the tumbler, charges are moved from one surface to another. This condition can reach a point where the electrical pressures in the dryer space remove outer electrons from air molecules. This process is known as ionization. The motion of electrons between molecules causes a glow that can be seen in dim light. The same thing happens in the atmosphere when falling raindrops strip outer electrons from air molecules. Raindrops carry these electrons to earth, leaving a net positive charge. This ionization in the air builds in intensity until there is breakdown or lightning. The electrons now have a path to return to the clouds where they originated. Normally, the surface charges on an object are balanced by opposite charges located inside the atoms (protons). This means that on average, physical objects are neutrally charged. When electrons are moved from one body to another, the object receiving electrons is charged negatively and the object giving up the charge is said to be charged positively. A steady charged condition is not normally found in nature. In time, any accumulation of charges will dissipate and a neutral condition will return. The idea of having a positive charge as a counterpart to the negative charge (a group of electrons) is appealing. In the real world a positive charge is usually the absence of electrons. It really makes no difference if we use the concept of positive charges as opposed to the absence of negative charges. In a semiconductor, electrons move inside a crystal lattice. When they move, they leave a hole (a vacant space). In effect, negative charges move one direction and holes move in the opposite direction. The holes behave very much like ELECTRICAL FORCES ON CHARGED BODIES 3 positive charges. An example in real life might be people seated in an auditorium. Assume that a row has an empty seat at the right end. If the people move one at a time to sit in the empty seat, the people move right but the empty seat moves left. The number of electrons on the surface of any metal or insulator is extremely large. For most electrical activity the percentage of electrons that are moved is infinitesimal, yet the effects can easily be observed and measured. The letter Q is used to represent positive charge—usually a depletion of electrons. The unit of charge is the coulomb (C). In some cases this unit is extremely large. A more practical unit is the microcoulomb (¹C) one millionth of a coulomb. One electron has a charge of "1:6 # 10"19 C.$ 1.3 ELECTRICAL FORCES ON CHARGED BODIES It is relatively easy to perform tests on charged objects. Procedures exist that can remove or add charges to objects. Rubbing a hardrubber wand with a silk cloth is one technique. Touching this charged rod to small insulators that hang on a string can transfer charge onto the balls. Two balls will repel each other if they both have positive or negative charges. When the charges are of opposite sign, the balls will attract. These forces are between charges, not “between” the matter in the ball. The larger the charge that is added, the greater the force. If the insulating balls are replaced by very small, lightweight metallized spheres hanging on insulating threads, the results are the same (Figure 1.1). The amazing thing here is that there is a force acting at a distance. The forces exist whether the spheres are in air or in a vacuum. On a perfect insulator, the forces cannot move the charges around on the object. A nearly ideal insulator would be glass. On a metal sphere the excess charges spread out over the surface as like charges repel each other. This is the same force that repelled the two spheres in Figure 1.1. These charges cannot leave the sphere, as there FIGURE 1.1 Charged metallized spheres. $ The abbreviations used in this book are listed in Appendix III at the back of this book. 4 THE ELECTRIC FIELD is no available conductive path. This force between charges is one of the fundamental forces in nature. It is one of the forces that hold all molecules together in all matter.$ Gravity is another fundamental force that acts at a distance. It is a weak force because it takes the mass of the entire earth to attract a person with a force equal to his or her weight. 1.4 ELECTRIC FIELD When forces exist at a distance, it is common practice to say that a force field exists in space. In this case, the force field is called an electric field or E field. This field is represented by field lines drawn between charged objects. These symbolic lines connect units of positive charge (the absence of electrons) with units of negative charge. When more charges are involved, convention says that there are more lines (Figure 1.2). The nature of the field is determined by placing a small test charge in the field. Note that the test charge must be small enough not to change the nature of the field that it is measuring. (This test charge is truly hypothetical. It may not be realizable except as a thought experiment. It does take a bit of faith to accept this idea.) The test charge experiences forces that have both magnitude and direction. The lines are drawn so that a small arrow on the line points in the direction of the force. After the lines are drawn, it can be determined that the forces are greatest near the charged objects where the lines get close together. The E field exists through all space, not just on the lines. Thus these lines are FIGURE 1.2 Electric field lines between oppositely charged spheres. $ There are forces between the outer electrons and the protons in the atom’s nucleus. When molecules are formed there are binding forces between atoms that are controlled by the electrons in the outer shells of the atoms. These same forces help to bind molecules together in solids and liquids. WORK 5 only a representation. At every point in space, the force has a magnitude and a direction. This is properly known as a vector field.$ The spheres in Figure 1.2 are relatively close together. If one of the spheres is moved very far away, the E field on the remaining sphere still exists. The lines leaving the near sphere will be evenly spaced around the sphere. This means that the charges are spaced uniformly on the sphere’s surface. This uniform spacing is unique to a sphere. On any other conductor shape the charges will arrange themselves so that the resulting field stores the least amount of energy. The idea of field energy storage is discussed in more detail in later sections. 1.5 WORK In physics, the definition of work is force times distance, f # d, where f and d are in the same direction. A good example of mechanical work involves lifting a bottle of water into a storage tank. If the tank is 25 feet (ft) high and the water weighs 1 pound (lb), then the work expended per bottle of water is 25 ftlb. In the intervening space the work is 1 ftlb for every foot in elevation. In the case of a test charge in an electric field, work is done in moving this charge between the two charged bodies. A force is required to move the test charge along any field line. The force for short distances along this line is nearly constant. The work over any short interval on this line is the E field intensity times this short distance. The total work along the entire path is the sum of all the bits of work. The work done in moving the bottle of water is stored as potential energy. When the water is released, it can do work as it falls: for example, it could turn a turbine. The same thing happens when charges are moved in a field and added to a conductor. The work that is done on the charge is stored and is available to do work when it is released. It will turn out that this work is actually stored in the electric field. Work in this case is the process of adding to or subtracting from the electric field. Once the energy is stored, it can be used at a later time. This use of stored energy is an important topic in the book. In the case of the bottle of water, the path taken by the bottle does not change the amount of work that must be done. The same thing is true of the unit charge. No matter what path is taken, the work required to move the unit charge between the charged bodies is the same.† This type of field is said to be conservative. Gravity is also a field phenomenon. The gravitational field and the electric field are both examples of a conservative field. Later we discuss the magnetic field, which is not conservative. $ The E field is often represented by a line with an arrow. The length of the line represents the intensity and the arrow shows the direction of the force on a test charge. This arrow is only a representation of the intensity and direction at a point in space. † In calculating work, the force and the distance moved must be in the same direction. If other paths are taken, the angle between the force and direction of motion must be a part of the calculation. 6 THE ELECTRIC FIELD Free electrons in a vacuum are accelerated by an electric field. This is analogous to a mass above Earth accelerated by gravity. In a conductor the electrons are also accelerated, but they keep bumping into molecules. This means that on average they do not accelerate. This motion of charge is a current and it takes a continuous E field on the inside of the conductor to keep charges moving at an average velocity. In all the discussions above, the fields are static and it is assumed that the charges are not moving (the exception being the test charge). 1.6 VOLTAGE The fundamental definition of voltage relates to the work required to move a unit of charge between two points. In this case the unit of charge is our test charge. By convention, the unit of charge is positive. The amount of work does not in any way require a reference level. To lift water 25 ft, the amount of work required is the same whether this work is done at sea level or at 5000 ft. (This assumes that the gravitational force is constant.) The same is true in the electric field. The work we are interested in involves moving the test charge between the two bodies. The work required is measured by the potential difference. It is correct to say that the work per unit charge is the voltage difference. The words voltage and potential are thus used interchangeably. Any point can be selected to be the zero of potential. If a remote point is selected, work may be required to get the test charge to the first body. If this work is 10 volts (V), then the work required to get to the second body may be 5 more volts. The potential difference between the two bodies is simply 5 V. There is no place that can be called the absolute zero of potential. It is misleading to believe that such a point exists. It will be obvious as we proceed that potential differences are our main concern. When the force is positive and the test charge is positive, positive work is done in moving this charge. This work is actually stored in the E field as potential energy. When the charge is allowed to return to its starting point, a bit of potential energy is removed from the field. The abbreviation mV stands for millivolt (0.001 V), ¹V stands for microvolt (0.000001 V), and kV stands for kilovolt (1000 V). The range of values that is encountered in practice is large. Writing lots of zeros before or after the decimal point is really an inconvenience. The circuit symbol for a source of voltage is a circle with the letter V in the center. 1.7 CHARGES ON SURFACES An E field exerts forces on charges. If these charges are on a conductive surface, they will try to move apart. The small metallized spheres we used in Figure 1.2 held charges, which generated an E field. These charges were CHARGES ON SURFACES 7 distributed over the conducting surface. Since the charges were at balance and not moving, we conclude that there cannot be a component of the E field (a force) directed along the surface of the sphere. If there were a tangential E field, there would be current flow. This is impossible because we have postulated a static situation. This means that any E field that touches the conductive surface must have a direction that is perpendicular to the surface. These Efield lines must terminate or originate on surface charges. This E field cannot move these charges, as the electrons cannot jump off the surface into the surrounding space. Also note that an E field cannot exist inside the metal, or charges would be moving to the surface. Again remember that this is a conductive material, and an E field would imply a current moving to the surface from within. These arguments lead us to three important conclusions: 1. For there to be a voltage difference, charges must be present. These charges result in an E field. 2. In electric circuits with potential differences, charges exist on the surfaces of all conductors. In a static situation, the E field touching a conductive surface has a direction perpendicular to the surface. The field does not extend into the surface. In Figure 1.2 the field lines terminate on charges at the surface of the spheres. Note that most of the lines terminate on the facing sides of the spheres. This means that the charges do not spread out evenly. 3. Charge distributions are not necessarily uniform on a conductive surface. In a static situation, the potential along the surface is constant. This means that the work required to bring a test charge to the conductive surface is the same for all points on the surface. In Figure 1.3 the field pattern for two conductors over a conductive plane is shown. Conductor 1 is at a potential of 1 V and conductor 2 is at a potential FIGURE 1.3 Field pattern of three conductors. 8 THE ELECTRIC FIELD of "2 V. This means it takes 1 V of work to move a unit charge from the conductive plane to the surface of the first conductor. By convention the field lines have arrows showing that they start on positive charges and terminate on negative charges. Consider the conductive plane as the reference conductor and consider it to be at 0 V. It takes 2 V of work to move the unit charge from conductor 2 back to the conductive plane. A conducting plane is sometimes called a ground plane or a reference plane. It is important to note that the ground plane has areas with positive and negative charge accumulations on its surface. Also remember that no work is required to move a unit charge along this reference surface. The entire surface is at one potential, which is defined as zero. In this example the two conductors could be round wires used to connect points in a circuit. The voltages on the conductors might represent signals at one point in time. The reference conductor could be a metal chassis or a metallized surface on a printed circuit board. 1.8 EQUIPOTENTIAL SURFACES In Figures 1.2 and 1.3 the geometry is simple and it is easy to draw the Efield lines. In most circuits, the conductor geometries are far too complex to consider drawing field patterns. This does not stop nature, as the fields do exist. When there are voltages, there are charge distribution patterns and there are fields. This fundamental idea is often forgotten. So far we have discussed the E or force field. The next step is to discuss the associated equipotential surfaces. When a unit test charge is moved from one surface to another, the work required is the potential difference. As the test charge is moved, it is possible to note points of constant work (constant potential). A plot of all points that are at the same potential is an equipotential surface. This is equivalent to climbing a mountain and noting points of equal elevation. In Figure 1.4, two spheres are shown with intermediate equipotential surfaces. Of course, the conducting spheres themselves are equipotential surfaces. In the space between the spheres, these equipotential surfaces are everywhere perpendicular to field lines. Moving a test charge along these new surfaces requires no work. The figure shows that the equipotential surfaces are close together near the spheres. This is the same thing as saying that the mountain is getting steeper as we near the summit. The work required to move the test charge a unit of distance is greatest near the surfaces. This is where the field lines are closest together. This is where the field is said to have its highest gradient. 1.9 FIELD UNITS In the previous figures the Efield lines are curved and not equally spaced. This implies that the intensity of the E field changes over all space. As noted 9 FIELD UNITS FIGURE 1.4 Equipotential surfaces perpendicular to field lines. FIGURE 1.5 E field between parallel conductive plates. earlier, where the E field lines get closer together, the force on a test charge increases. A simpler field pattern results when charges are placed on two parallel conductive planes as in Figure 1.5. The E field in the central area are straight lines. This means that the force on a test charge is constant at any point between the two surfaces. If the distance is 0.1 meter (m) and the potential difference is 10 V, the E field times 0.1 m must equal 10 V. In other words, the E field must be expressed as 100 V=m. In equation form, 100 V=m # 0:1 m = 10 V. Thus the E field has units of volts per meter. Two parallel conducting surfaces form what is known as a capacitor. More will be said about capacitors in later sections. 10 THE ELECTRIC FIELD 1.10 BATTERIES—A VOLTAGE SOURCE Energy can be stored chemically. When there is a chemical reaction, energy is released. In an explosion, this energy can be released as heat, light, and mechanical motion. In some arrangements, chemical energy can be released electrically. A battery is an arrangement of chemicals that react when the active components are allowed to circulate their electrons in an external circuit. The energy that is stored chemically is potential energy that is available to do electrical work. In rechargeable batteries the chemistry is reversible and energy can be put back into the battery. The terminals of the battery present a voltage to the world. This is electrical pressure trying to move electrons so that the chemicals in the battery can attain a lower energy state. This is analogous to water pressure in a water tank where the water is trying to get to a lower energy state. This water pressure is no different from the voltage between two oppositely charged conductors in space. There is an E field between the terminals of the battery. If this is a 12V battery, it takes 12 V of work to move a unit of charge between the two terminals. This work is independent of the path taken by the test charge. This includes a path through the heart of the battery. The E field cannot be seen, but it is there. This field extends right into the battery, where the atoms are under pressure to release their external electrons. In Figure 1.2 the static charges can be removed and the E field disappears. In the case of the battery, the E field and the associated charges on the conductors will persist until the battery is dead. When charge is allowed to flow through a circuit connected to the terminals, the battery replaces this charge and maintains the electrical pressure. A battery is thus a voltage source that does not sag. It is like being connected to the city water supply. No matter how much water you draw, the water pressure is the same. The E field around battery terminals is shown in Figure 1.6. The positive terminal is called the anode and the negative terminal the cathode. The charges that are allowed to flow from a battery to a circuit release stored chemical energy. The voltage and associated charge flow constitutes the electrical energy that is flowing from the battery. A connected circuit can convert this energy to heat, light, or sound. In some cases it is radiated. An example of radiation might be a cell phone transmission. In most common circuit applications the energy is converted to heat. Of course, it is possible to store some of this energy in E fields within a circuit. More will be said about this later. Batteries are usually formed from basic cells. A typical flashlight battery is such a single cell. The singlecell voltage in most size A and D batteries is 1.1 to 1.5 V. Different battery materials develop different voltages. To obtain higher battery voltages, basic cells are placed in series. The cell connections are made internally, and the connections are not available for external connections. A 12V automobile lead acid battery is constructed with six such cells in series. Each internal plate forms a cell that develops a voltage of about 2.0 V. Batteries can be connected in series to increase the available voltage. This 11 CURRENT FIGURE 1.6 Battery and its associated fields. series arrangement will work even if the batteries have different voltages. Batteries cannot be paralleled unless the batteries themselves are identical. This parallel arrangement can be used to provide additional current capacity. When considerable power is involved, very careful monitoring of the batteries is necessary. 1.11 CURRENT The motion of charge is current. The unit of current is the ampere. The letter symbol for current is A or sometimes I. A source of current is often represented by the letter I in a circle. The smallest charge is an electron. In most practical circuits the number of electrons that constitute current flow is so large that it makes little sense to consider the individual electrons. There are cases, 12 THE ELECTRIC FIELD however, where individual electrons are counted, such as in a photomultiplier. For our discussion, current flow is continuous and the effect of individual electrons is not considered. A steady current is a continuous stream of charges that flow past an area. A coulomb of charge passing by in 1 second is defined as 1 ampere (A). In other words, a coulomb per second is an ampere. In equation form, Q=t = A. In the power industry, an ampere is a small unit. In an electronics circuit it is a big unit. For this reason, smaller units of current are a convenience. The abbreviation mA stands for milliampere (0.001 A) and the abbreviation ¹A stands for microampere (0.000001 A). The positive terminal of a battery is a source of current flowing out of the battery. This direction is a convention only. The actual flow of negative charges is in the opposite direction. This may seem confusing at first, but it is how the world of electricity developed. Historically, there was an assumption that moving charges are positive and the convention has persisted. Electrons are attracted to the positive battery terminal, but by convention, current flows out of this terminal. Current does not ordinarily flow in air. It can flow easily in conductors such as copper or iron. Plastics and glass are examples of very poor conductors. Conductors for electrical wiring are available in many configurations, all the way from power lines to circuit traces on a printed circuit board. When conductors are attached to a battery, the field across the terminals is extended out on the conductors. This means that charges now exist on the surface of these added conductors. These charges move out on the conductors looking for a path that will allow them to work their way from the anode to the cathode (current is seeking a path to flow from the cathode to the anode). A direct conducting path between the cathode and anode will destroy the battery. This direct path simply shorts out the battery. The circuits that are normally connected to the terminals will drain charge at a rate that the battery can supply for a useful period. A car battery might be able to supply 1 A for 60 hours, for example. A flashlight battery might be able to supply 100 mA for 10 hours. 1.12 RESISTORS A resistor is a controlled limited conductor. When electrical pressure is applied across its terminals, a limited current will flow. The water pipe analogy can serve to illustrate the point. Consider a water hose connected to a cylinder full of packed sand. The amount of water that could flow through the cylinder will depend on the length of the cylinder, the crosssectional area, the size of the grains of sand, and the water pressure. This cylinder is in effect a water flow restrictor. The electrical form of this restrictor is a resistor. One type of resistor is made from a mixture of powdered carbon and a nonconductive plastic filler. This mixture in compressed form constitutes a resistor. The resistance can be controlled by varying the ratio of filler to carbon. This controlled mixture RESISTORS IN SERIES OR PARALLEL 13 FIGURE 1.7 E field along a resistor. is fused together under pressure to form the resistive element. This element is encased in an inert housing that is marked with a color code. Conductive leads that contact the resistive element at both ends are molded into the body of the resistor. This briefly describes a carbon resistor, which is just one of many resistor types commercially available. The letter symbol for a resistor is a capital R. The electrical symbol for a resistor is shown in Figure 1.7. When an electric field is impressed across a resistor, a limited current flows. For a practical resistor, if the electrical pressure is doubled, the current flow is doubled. Figure 1.7 shows how the E field distributes itself along and through the resistor. The field, in effect, pushes charges along over the entire resistive path. Note that there is an E field inside the resistor. Also note that some of the field bypasses the resistor. This field will be important in later discussions. Resistance to current flow has units of ohms. The symbol for ohm is the capital Greek letter omega (). A resistance of 1 ohm will limit the current flow to 1 ampere when the electrical pressure is 1 volt. Typical circuit resistors are usually much greater than 1 . A 1000 resistor will limit the current flow to 1 mA for an electrical pressure of 1 V. The linear relationship between resistance and current flow is known as Ohm’s law. Double the voltage and the current flow doubles. Double the resistance and the current flow halves. In equation form, the relationship is given as I = V=R, where I is in amperes, V is in volts, and R is in ohms. 1.13 RESISTORS IN SERIES OR PARALLEL When resistors are placed in series, the total resistance simply adds. A 1000 resistor in series with 2000 is simply 3000 . When resistors are placed in parallel across a voltage source, the total current that flows is the sum of the two individual currents. Using this new current and the applied voltage, an equivalent resistance can be calculated. For example, a voltage of 10 V across a 1000 resistor results in a current of 10 mA. Ten volts across a 2000 resistor results in a current of 5 mA. For these resistors in parallel, 14 THE ELECTRIC FIELD the total current is thus 15 mA. The equivalent resistance value is thus the voltage divided by the current (expressed in amperes). The answer is simply 10 divided by 0.015, or 666.6 . Another way to look at resistors is to consider how well they conduct. Resistors resist current flow and thus their name. It also makes sense to measure a resistor’s ability to conduct current in terms of conductance. It is easy to see that the higher the resistance, the lower the conductance. This simply means that resistance is the reciprocal of conductance. The unit of conductance is the siemens (S). A resistance of 2 is said to have a conductance of 0.5 S. A 1000 resistor has a conductance of 0.001 S, which can also be written as 1 mS. Using the concept of conductance, when resistors are placed in parallel their conductances add. In the preceding example, the 1000 resistor has a conductance of 1 mS. The 2000 resistor has a conductance of 0.5 mS. The combined conductance is simply the sum of the two conductances, or 1.5 mS. This can be converted to ohms simply by taking the reciprocal of the conductance in units of siemens. Our 1.5 mS can be written as 0.015 S. The reciprocal of 0.015 S is 666.6 , the same answer as before. When two equal resistors are placed in series, the voltage across each resistor is onehalf the total voltage. This means that the E field is reconfigured around the resistors so that the work required to move a unit charge around or through each resistor is one half the total voltage. If the resistors are placed at right angles to each other, the E field takes on a new shape to make this happen. A typical field pattern is shown in Figure 1.8. FIGURE 1.8 E field around two series resistors mounted at right angles. PROBLEMS 15 1.14 E FIELD AND CURRENT FLOW The E field in Figure 1.7 enters the heart of the resistor to push the charges through. The battery and the resistor form a very simple circuit. When there is current flow, even in a conductor, some E field is needed to push the charges along. In effect, the conductors that connect the resistor are also resistors, but their resistance is very low. Typical connecting conductors might have resistances below 0.001 . In Figure 1.7 the E field appears perpendicular to the conductors. To push the charges along, a small component of the E field exists inside the connecting wires in the direction of current flow. This means that the E field is not exactly perpendicular to the conductor and it leans forward just a little. There are still charges on the outside surface of the conductors, as most of the E field is perpendicular to the surface. The E field component that is inside the conductor can be calculated as follows. If the current is 10 mA and the resistance is 1 m, Ohm’s law requires a voltage drop of 0.000010 V. If the path length is 0.1 m, the E field is 0.00010 V=m. Compare this with the E field between the conductors if the conductor spacing is 0.1 m. The E field is approximately 10 V per 0.1 m, or 100 V=m. In other words, the component of the E field in the conductor is one millionth of the E field between the conductors. Of course, this would change if a larger current were to flow. Charges do not move in a practical conductor unless there is an E field to push them along. In a vacuum, electrons are unimpeded and accelerate across the available space. They give up their energy on impact at the end of their journey. This is what happens in a vacuum tube when the electrons strike the plate. Without a vacuum, the moving charges constantly collide with atoms. On average, they attain a fixed velocity. This steady flow of charge is called a dc current. The abbreviation dc stands for direct current. A steady or dc current that flows in a conductor flows uniformly though the entire conductor.$ It does not flow on the surface. In a resistor the charges flow uniformly in the resistive element. 1.15 PROBLEMS 1. A 10V battery is connected to two conductors. What is the Efield intensity when the conductors are separated by the following distances: 1 m, 10 cm, 1 cm, and 1 mm? 2. A 10k resistor element is 1 cm long. The E field is 20 V=m. What current flows in the resistor? 3. A 2000 resistor is placed in series with a 3000 resistor. What is the total resistance? What is the parallel resistance? What is the parallel conductance expressed in siemens? $ The charges that move on the inside of the conductors at dc are distinct from the static charges that reside on the surface as the result of external E fields. 16 THE ELECTRIC FIELD 4. A conductor has a resistance of 0.002 =cm. How much current flows if the E field parallel to the conductor is 0.0001 V=m? 5. Two conductors carrying signals over a ground plane are at 2 V and "3 V. How much work is required to move a test charge between the two conductors? 6. Two resistors with conductances of 1 mS and 3 mS are placed in series. What is the total resistance? What is the resistance if they are placed in parallel? 7. Four resistors of 1000 each are arranged in a square. What is the resistance across either diagonal? If 10 V is placed across one diagonal, what is the voltage across the other diagonal? 8. In problem 7, assume that one of the resistors increases by 1%. If 10 V is placed across one diagonal, what is the voltage across the other diagonal? 9. A 12V battery sags 0.1 V when supplying 10 A. What is the internal resistance of the battery? (Hint: The internal resistance is a series resistor.) 10. The battery in problem 9 is being charged at 2 A. Assume that the internal battery voltage stays at 12 V. What voltage must be supplied by the charger? 1.16 ENERGY TRANSFER The charges that move in a resistor through an E field convert their potential energy to heat. This happens because the molecules that are hit by these moving charges take on a higher average velocity. This increased motion of molecules is simply heat. The energy is first stored chemically in the battery. Some of this energy is moved into the E field. Charges moving in this field take energy from the field and convert it to energy of motion. This motion is transferred to the molecules of the resistor as heat. The energy that is removed from the E field is resupplied by the battery. The battery heats the resistor, but there are obviously several intervening steps involved. This transfer of energy via the E field is not the entire story. How energy moves along conductors is explained in detail in Section 3.10. The important idea here is that the energy heating the resistor involves the E field, which was generated by the charges emanating from the battery. The work to move a unit charge between two points in a field is simply the voltage V. When many units of charge are involved, the total charge is simply Q. When a charge Q moves through a potential difference, the total work involved is the product of charge # voltage, QV. Assume that this work is stored as energy. This energy is the product of QV and has units of joules (J). The rate at which work is done is a measure of power. This is simply charge # voltage % time. When 1.0 J of work is done in 1 second (s), the power PROBLEMS 17 is said to be 1 watt (W). In equation form, power equals QV=t. From Section 1.11, current is simply charge per unit time, or Q=t. By letting Q=t = A, the power becomes volts # amperes, or P = VA. If the voltage is 10 V and the resistor is 1000 , the current is 10 mA. The product of voltage # current is 10 # 0:01, or 0.1 W. This means that 0.1 J of energy is dissipated in the resistor in each second. 1.17 RESISTOR DISSIPATION When current flows in a resistor, it dissipates heat. The temperature rise will depend on its radiating surface and on how the resistor is cooled. In most commercial applications it is a good idea to limit dissipation to about onehalf the rating. In practice, most resistors are used far below their ratings. It is convenient to use resistors of one size and ignore the issue of ratings except where it is obviously a problem. In an integrated circuit, the resistors are individually designed. The power dissipated in a resistor is simply IV, where I is in amperes and V is in volts. For example, the current that flows in a 100 resistor with 10 V across its terminals is 0.1 A. The dissipation in watts is 10 V # 0:1 A, or 1 W. Three parameters are involved: resistance, current, and resistance. When two parameters are known, the power dissipated can be calculated. Power is I 2 R or V2 =R or VI. 1.18 PROBLEMS 1. Ten volts is impressed across a 100 and 200 resistor in parallel. What is the power dissipated in each resistor? What is the total power dissipated? 2. Ten volts is impressed across a 100 and a 200 resistor in series. What is the power dissipated in each resistor? What is the total power dissipated? 3. A charge of 4 C flows across a potential of 4 V in 8 s. What is the energy dissipated in joules? What was the power level during this interval of time? 4. Four 14 W 100 resistors are available to dissipate energy. How many configurations will dissipate equal power in each resistor? What configuration will accept the greatest voltage? What configuration will accept the smallest voltage? 5. In problem 4, assume that the maximum dissipation per resistor is 18 W. What are the voltages applied to each configuration? How many resistance values are available? 6. A switch connects a 10V battery for 2 s and disconnects it for 3 s. If this switching cycle is repeated every 5 s, what is the average power dissipated in a 40 resistor? What wattage rating would you select for the resistor? 18 THE ELECTRIC FIELD 7. A 24V battery is reversed in polarity every 2 s. What is the power dissipated in a 48 resistor? 8. How many joules does a 12V car battery store if its rating is 60 ampereshours? 9. In problem 8, a resistor of what value will discharge the battery in 24 hours? Assume a steady voltage. What wattage will be dissipated in the resistor? 10. A headlamp on a car dissipates 200 W. What is the current flow in the headlamp? Assume a 12V battery. What is the current if the battery is 6 V? 1.19 ELECTRIC FIELD ENERGY When charges are distributed onto conductors, electric fields exist. Consider Figure 1.5, where there are two parallel plates relatively close together. If opposite charges are placed on the two plates, an E field will exist between the plates. The field lines start on the top surface and terminate on the bottom surface. When charge is moved from the bottom surface to the top surface, work must be done on the charge. This work is volts per unit charge. If the edges are avoided, the E field has the same intensity at all points between the plates. Thus V is equal to Ed, where d is the spacing in meters. When charge is moved across the distance d, the work done on the charge does not make any physical change to the plates. The only change that can be observed is that the E field is increased. The work done in moving the new charge must therefore be stored in the increased E field. The E field increases over the total volume between the plates. It is correct to say that an E field stores energy per unit volume of space. In a static situation the E field does not exist inside conductors. This means that the energy can exist only in the space between conductors. No energy is stored in the conductors. In most conductor configurations the Efield intensity varies in space. To calculate the total field energy, the space must be divided into very small volumes where the E field is relatively constant. The total field energy is the sum of the energies in all the small volumes. In general, this calculation is very difficult to make. The important thing here is the concept of field energy stored in space. The E field is proportional to the charge on the plates. If the plate area is one square meter, a coulomb of charge produces a field of 1 V=m. To move charge between the plates requires work. To calculate the total work, the charges must be moved in small increments. The first elements of charge require very little work. As the field increases, the work per unit charge also increases. It turns out that the total work required to move a net charge between the two plates is 12 EV. But V is equal to Ed. Therefore, the energy in the field is 12 E 2 d. Remember, this is the energy stored for plates 1 m on a side. For an area GROUND AND GROUND PLANES 19 A, the energy stored is proportional to area or 12 "20 Ad. Since Ad is volume, the energy in joules is 12 "0 E 2 V, where in this term V stands for volume. The proportionality factor "0 is needed to make the units come out correctly. The value of "0 is discussed later. The important thing to understand is when there is an E field, there is energy storage. This fact is often ignored and can be a source of trouble. In Section 1.16 the idea of power was introduced. Energy flowing over some time period represents power. Energy cannot be moved in zero time, as this takes infinite power. If 0.01 J is dissipated in 10 milliseconds (ms), the power level is 1 W for that period of time. If this same energy is dissipated in 1 ms, the power level during this shorter time is 10 W. It is important to realize that E field energy does not simply disappear. It takes time for this energy to be moved or dissipated. This field energy can be used in several ways. For example, the energy can be transferred to another circuit for storage, it can be used to heat other components, it can be used to create sound, or it can be radiated out of the area. Often, all of these processes are involved. 1.20 GROUND AND GROUND PLANES In Figure 1.3 the concept of a ground plane was introduced. Both positive and negative charges were distributed along this conducting surface. The words ground and ground plane are a part of the language of electronics and electrical engineering and they are often a source of trouble. The word ground is often associated with an adjective that describes function or use. Expressions such as computer ground, analog ground, clean ground, signal ground, and isolated ground are frequently encountered. At this point these adjectives only confuse the issue, and they will be avoided. The definition we will use is that a ground plane is a conductive surface referenced to zero volts. The earth is an electrical conductor. Connections to earth are required for lighting protection and electrical safety. The earth is considered a form of ground plane, although it obviously is not flat. A ground plane may or may not be connected to earth. An example might be in an aircraft. The framework might be considered a ground plane, but it definitely is not connected to earth. A facility built on a lava bed is insulated from earth because lava is an insulator. The building steel might still be considered a ground. Electronic systems may have many reference conductors. Each reference conductor serves a specific purpose. Some of these reference conductors may be earthed (connected to earth), and others may be defined by the circuit itself. In any case the concept is to have a conductive surface in the circuit that is considered to be at a zero reference potential. The idealized ground plane we have used so far is a source of positive or negative charge. This surface can supply any amount of charge without difficulty and it remains at the zero of potential. For the ideal ground plane, no work is required to move charges along its surface. It is an equipotential surface. 20 THE ELECTRIC FIELD FIGURE 1.9 Threeconductor arrangement. 1.21 INDUCED CHARGES The conductors in Figure 1.9 show field lines that terminate on charges. Conductor 3 is grounded (connected to the ground plane) by a small wire. It is at zero potential and it contains a charge of "Q3 . If the small connecting wire is cut, the charge on the conductor cannot leave. This charge is said to be induced on the conductor. Work is required to move this conductor in the existing E field. Wherever it is moved, the total charge on its surface remains Q3 . If conductor 3 were initially far removed from conductor 1, it would have no charge on its surface. As it is moved near conductor 1, charges distribute themselves on the surface, but the total charge must remain zero. When the small wire is connected to conductor 3, a charge "Q3 will flow in the wire to this conductor. This is the induced charge shown in Figure 1.9. 1.22 FORCES AND ENERGY The electric field is a force field. These forces are between charges. When charges reside on a mass, the electrical forces are transferred to the mass (see Figure 1.1). Normally, the masses are constrained so that there is no detectable motion. This force field stores potential energy. When a body is moved in an E field, a new field configuration results. This configuration stores a different amount of field energy. The difference is simply the work done in moving the body to new positions. In other words, the forces on a body are related to the change in energy level that results from the motion. The direction of these forces can be determined by noting the direction that causes the greatest change in energy storage. This concept is simple, but the calculation to determine the force and direction on any one conductor is usually very REVIEW 21 difficult. Forces on conductors in an E field are not generally important in an electronic circuit. The concept of field energy storage is quite important, however. An application of electrostatic force occurs in tweeters used in audio systems. The voltage applied between plates moves the plates, which in turn moves the air. The moving air is the sound we hear. In a cathode ray tube the beam of electrons that writes on the front surface is often deflected by electric fields. The electrons that boil off a heated cathode are accelerated across the viewing tube and give up their energy to phosphors on the surface. These phosphors give up their energy by emitting light that we see. 1.23 PROBLEMS 1. The E field in a volume 10 cm # 10 cm # 0:1 cm is 1000 V=m. What is the energy stored? (Hint: All dimensions must be in meters.) 2. In problem 1, assume that the 10 cm # 10 cm surfaces are conductors. How much charge resides on these surfaces? 3. Assume that a round conductor runs parallel to and above a ground plane. Sketch the E field lines from the conductor to the ground plane. Assume that the potential of the conductor is 5 V. Draw the equipotential surfaces at 1 and 3 V. 4. The force on a conductor is the change in energy stored divided by the distance moved assuming that the distance is very small. Use this fact to calculate the force between the plates in problem 1. (Hint: Calculate the change in energy if the plates are moved 0.01 cm. Use units of meters. The value of "0 is 8:854 # 10"12 . The answer will be in kilograms.) 5. A current flows in a 10 resistor. What is the average power if the current is 1 A? What is the average power if 10 A flows for 0.1 s every second? What is the average power if 100 A flows for 0.01 s every second? 6. In problem 5, what is the peak power in each case? 1.24 REVIEW The presence of charge implies that there is an electric force field. The work required to move a unit charge in this E field is a measure of potential difference or voltage difference. If there is a potential difference between two conductors, there will be charges on their surfaces. Potential difference is measured in units of volts. There is no absolute zero of potential or zero of voltage. Any conductor can be used as a zero reference conductor. A ground is a conductor that can be used as a reference conductor. A reference conductor may or may not be connected to earth. 22 THE ELECTRIC FIELD One source of potential is a battery where chemical action moves charges in an external circuit. A battery can be used to force a steady flow of charge through a resistor. The charges in the resistor are accelerated in the E field. These charges collide with molecules, which increase their average molecular velocity. This motion is heat. A steady flow of charge is a current measured in amperes. The relationship between resistance, voltage, and current is Ohm’s law. An electric field stores energy. This energy cannot be created or dissipated in zero time. This energy is stored in space, not in conductors. This energy is most dense where the E field is most intense. When energy is taken from an E field, it may be replenished by a voltage source such as a battery. In this case, energy moves from chemical, to E field, to motion of charge, to heat. The Fields of Electronics: Understanding Electronics Using Basic Physics Ralph Morrison c 2002 John Wiley & Sons, Inc. Copyright ! ISBNs: 0471222909 (Hardback); 0471433934 (Electronic) 2 Capacitors, Magnetic Fields, and Transformers 2.1 DIELECTRICS The E field considered in Chapter 1 involved charge on conductors and the fields in the surrounding space. Figure 2.1 shows a dielectric filling the space between two parallel conducting plates. The term dielectric refers to the insulating material used in capacitors. Typical materials are Mylar, mica, and polypropylene. When a voltage V is applied between the plates, the work to move a unit charge between the plates is just V. The presence of the dielectric increases the charge that is present on the surface of the conductors, and thus the charge per unit voltage is increased. The ratio of charge on the plates with and without the dielectric is known as the relative dielectric constant, "R . The higher the dielectric constant, the more charge is stored on the plates for a given voltage. The relative dielectric constant for air is 1.0. In Chapter 1 the Efield lines were drawn so that they started on positive charges and terminated on negative charges. When a dielectric is introduced into an existing E field, the field is reduced in the dielectric. This means that there is less field energy stored in this volume of space. The Efield discontinuity at the boundary does not imply a charge on the dielectric surface. FIGURE 2.1 Parallel conducting plates with a dielectric. 2.2 DISPLACEMENT FIELD It is interesting to see what happens when a dielectric occupies part of the space between the two parallel conducting plates. This situation is shown in 23 24 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS FIGURE 2.2 Air and a dielectric between parallel plates. Figure 2.2. The equipotential surfaces are no longer uniformly spaced across the space. Most of the potential difference appears across the air space. The ratio of the E field in air to the E field in the dielectric is the relative dielectric constant "R . It is convenient to consider a new field, one that is not dependent on the dielectric. This field is called the displacement field or D field. This D field is generated by charges and it is not a function of the dielectric. To make the units come out correctly, the D field in air is the E field times the dielectric constant of free space. This constant, "0 , is also known as the permittivity of free space. This constant has the value 8:854 " 10#12 . The D field has units of charge per unit area (coulombs per square meter). At the interface between air and the dielectric, the D field has the same intensity on both sides of the interface. In Figure 2.2 it is assumed that the dielectric surface is free of charge. If a surface charge did exist, a new D field would start at this surface. The energy stored in a field is proportional to the E field, not the D field. This is because the E field is still the force field.$ A direct measure of the E field inside a dielectric is not practical, so the forces on charges in the dielectric must be inferred. In Figure 2.2 most of the energy is stored in the air space. In Figure 2.1 the energy is all stored in the dielectric, as there is no air space. When a dielectric is introduced into an existing uniform E field, the field reconfigures itself to store the least possible energy. It is just like water in a puddle. When an object is removed from the puddle, the water level drops and evens out until the water stores the least potential energy. This field pattern when a dielectric is present is shown in Figure 2.3. Liquid dielectrics are often used in highvoltage transformers or in highvoltage power switches. The presence of the dielectric reduces the E field in critical areas and this helps to limit arcing. The liquid can also help in conducting heat out of a big transformer. $ The work done in moving charges is stored in the field. The work is equal to the force times distance. The force is proportional to the Efield intensity (see Section 1.19). CAPACITANCE OF TWO PARALLEL PLATES 25 FIGURE 2.3 Dielectric inserted into a uniform E field. 2.3 CAPACITANCE The two plates in Figure 2.1 form a capacitor. A capacitor is an electrical component that can store electric field energy. In circuit diagrams the letter symbol C is used to represent a capacitor. (The electrical symbol for a capacitor is identified in Figure 2.5.) A capacitor C is said to have a capacitance. The unit of capacitance is the farad (F). One farad of capacitance means that a charge of 1 coulomb can be stored for 1 volt of potential difference. A typical capacitor has capacitance measured in microfarads (¹F). This is one millionth of a farad. A full farad of capacitance is possible, but it is not the usual circuit component. Capacitors cover the practical range from a few picofarads to thousands of microfarads. A picofarad (pF) is one millionth of a microfarad. The statement C = 10 ¹F means that capacitor C has a capacitance of 10 ¹F. It is interesting to note that capacitor values cover a range of over nine orders of magnitude. This is greater than the ratio of an inch to the distance around our earth. Capacitors come in many shapes and forms, one of which is a foil capacitor. It is made from layers of metalized paper rolled into a tight cylinder. Smaller capacitors are often just parallel layers of metallized dielectrics. Whatever the manufacturing technique, the basic capacitor is a version of Figure 2.1. 2.4 CAPACITANCE OF TWO PARALLEL PLATES Capacitance is the ratio of charge stored to the voltage applied. It is a geometric property of conductors and dielectrics. For materials in common use, the capacitance does not change with voltage level. There are many parameters involved in selecting a commercial capacitor. Among the factors are voltage 26 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS rating, shape, temperature stability, dielectric losses, and accuracy. The charge on the plates of a capacitor is the D field times the area A in square meters. Q is therefore "0 "R EA. The voltage across the plates is the E field times the plate spacing d. Substituting E = V=d in the equation for Q, it is easy to see that C = Q=V = "R "0 A=d, where the units for A and d are meters. A typical dielectric might have a relative dielectric constant of 10. For a plate area of 100 cm2 and a spacing of 0.01 cm, the capacitance is 0:00885 ¹F. 2.5 CAPACITANCE IN SPACE The space between a group of conductors can hold a complex electric field. This field is a function of the voltages on all the conductors. A specific portion of the field cannot be allocated to individual pairs of conductors, as a change in the potential for any one conductor modifies the entire field. Also, any change in field alters the charge distribution in all conductors that are embedded in the field. If there are electrical paths between conductors, charges will adjust to modify the field further. If there are no electrical paths, the charges will simply redistribute themselves on the various surfaces. In either case, the new field will reconfigure itself to store the least amount of field energy. The concept of mutual capacitance is introduced to handle this complex situation. To measure a mutual capacitance, all conductors are grounded except one. This means that they are all connected to the reference potential, zero volts. It is convenient if a large conductive surface can be used as the zero of potential. The mutual capacitance between the ungrounded conductor and any other conductor is the ratio of charge on the grounded conductor to the voltage on the first conductor. This physical arrangement is shown in Figure 2.4. The selfcapacitance of a conductor is the ratio of charge to voltage on that conductor with all other conductors grounded. The notation for mutual capacitance is C12 . The subscripts indicate the two conductors that are involved FIGURE 2.4 Mutual capacitance. CURRENT FLOW IN CAPACITORS 27 in the ratio of charge to voltage. A selfcapacitance would be written C11 . A capacitor as a component has selfcapacitance or simply, capacitance. Mutual capacitances are always negative, as the induced charge is always opposite in sign to the applied voltage. Like selfcapacitance, mutual capacitance is a geometric quality. In practice, an individual mutual capacitance is measured dynamically. This simply means that the applied voltage is varied and the resulting induced current flow is monitored. Because mutual capacitances are often very small, the techniques for measurement can be somewhat complicated. The difficulty arises because the measurement conductors modify the very geometry of the circuit being measured. Field energy does not distribute itself so that it can be partitioned by mutual capacitances. The field at any one point has energy associated with every mutual capacitance. The stored energy can be calculated in terms of mutual capacitances, but the methods used would take us far from our main task. Suffice it to say that there is a complex field pattern between conductors in every circuit. The hope is that the field energy stored in components is much greater than the field energy stored between components. In highspeed circuits, mutual effects are an important consideration. For slower circuits the effect of mutual capacitance can be ignored. Exceptions can occur in active circuits. The transistors in an active circuit can multiply the effect of mutual capacitance. Circuit performance, and in some cases circuit stability, can depend on limiting certain mutual capacitances (see Section 5.14). 2.6 CURRENT FLOW IN CAPACITORS Figure 2.5 shows a series circuit consisting of a capacitor, a switch, a series resistor, and a voltage source V. This is called a series RC circuit. At the moment the switch closes, there is no charge on the plates of the capacitor. This means that there can be no voltage across the capacitor terminals. At the moment of switch closure, all the battery voltage appears across the resistor. This voltage across the resistor means that there is a current flow equal to V=R. This current flow will begin to place charge on the capacitor plates. As the charge builds up on the plates, the E field will increase between the plates. FIGURE 2.5 Series RC circuit. 28 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS This E field implies a voltage across the plates of the capacitor. The sum of the voltages across the resistor and the capacitor must equal the impressed voltage. As the voltage across the plates of the capacitor rises, the voltage across the resistor drops reducing the current flow. Eventually after a long period of time, the charge stops flowing and the capacitor is said to be fully charged. The full charge is given by the formula Q = CV, where C is in farads, Q is in coulombs, and V is in volts. When the resistor is made smaller, a higher initial current will flow. This means that the capacitor charges more rapidly. A capacitor cannot be charged in zero time, as this requires infinite power. When the resistor is reduced to zero, there seems to be a conflict. At first glance the charging current would have to be infinite. On closer examination every battery has some internal resistance. The conductors of the circuit have some resistance, and even the plates of the capacitor have resistance. This means that in a practical circuit a capacitor is always charged through some finite series resistance. The charge time might be microseconds, but it can never be zero. There are other limitations to rapidly changing current flow, which will be discussed later. Charge seems to flows around this simple circuit, but does it flow through the dielectric? The dielectric is an insulator, and yet somehow charge is moving onto one plate and off the other. The only answer that makes sense is to consider a changing D field to be the equivalent of current. The more rapidly this field changes, the more current flows. When the capacitor is fully charged, the D field is "0 "R V=d, where "R is the relative dielectric constant. A changing D field is, in effect, current flow in space. This current is called a displacement current. This idea makes it possible to consider a current flowing around the circuit, including the space between the plates. The current flowing through a capacitor is proportional to the changing voltage that appears across the plates. The larger the capacitance, the greater the current. The faster the field changes, the greater the current. A capacitor opposes current flow, but it would be incorrect to say that it is a resistor. For a steady voltage the opposition to current flow is infinite. The opposition to current flow is discussed in Section 3.2. 2.7 RC TIME CONSTANT After the switch is closed in Figure 2.5, the voltage across the capacitor begins to rise. The voltage plotted against time is described by an exponential curve where the exponent is #t=RC. The letter t is time in seconds. This charging curve is shown in Figure 2.6. The product RC has units of time when R is in ohms and C is in farads. When t = RC, the voltage reaches 63% of its final value V. This value of t is called a time constant. If the capacitor starts out with a voltage V and a switch connects a resistor R across its terminals, the voltage will begin to drop. This voltage also follows an exponential waveform. In one time constant the voltage drops to 63% of 29 PROBLEMS FIGURE 2.6 Exponential charging curve of a capacitor. its initial value. It is easy to see that if two different voltages were alternately connected to the RC circuit, current would flow at all times. If R = 100,000 and C is 10 ¹F, the time constant is 1 s. If the battery potential is 10 V, the voltage across the capacitor will rise to 6.3 V in 1 s. After the capacitor is fully charged, a 100,000 resistor will discharge the capacitor to 3.7 V in 1 s. A 10,000 resistor would discharge the capacitor to 3.7 V in 0.1 s. If the voltage were 100 V, the 10,000 resistor would discharge the capacitor to 37 V in 0.1 s. The concept of time constant appears often in science. The current that flows in the circuit of Figure 2.6 dissipates heat in the resistor. The potential energy that is finally stored in the capacitor is available to do external work at a later time. When the capacitor is discharged through a resistor, this potential energy is converted to heat. A capacitor stores field energy, it never dissipates energy.$ 2.8 PROBLEMS 1. Two metallized sheets separated by an insulator are rolled into a cylinder to form a capacitor. The sheets are 3 cm wide by 1 m long. If the spacing between conductors is 0.03 cm and the insulator has a dielectric constant of 10, what is the capacitance of the capacitor? 2. Ten volts is placed across the terminals of the capacitor in problem 1. What charge is stored in the capacitor? 3. What is the D field in problem 2? 4. A 10V battery is in series with a switch, a resistor of 1000 , and a 10¹F capacitor. At the moment the battery is connected to the RC circuit, what is the value of the current? 5. What is the RC time constant in problem 4? $ Practical capacitors can overheat from excess current flow. Manufacturers often list an equivalent series resistance so that the dissipation can be calculated. 30 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS FIGURE 2.7 Shielded conductor. 6. A 10¹F capacitor is charged to 10 V. A 10,000 resistor is placed across the capacitor terminals. What is the voltage after one time constant? What is the voltage after two time constants? 7. A resistor discharges a 10¹F capacitor to 37% of its initial voltage in 2 s. What is the resistor value? 8. Draw the current curve in problem 7. Assume an initial voltage of 10 V. 9. In problem 8, divide up the discharge current into 10 equal periods of time. Limit the maximum time to a period of one time constant. Estimate the energy dissipated in the resistor. How does this compare with the energy supplied from the capacitor? (Hint: The energy stored in the capacitor is 1 CV2 , where C is in farads and V is in volts.) 2 2.9 SHIELDS The E field around a conductor extends to all nearby conductors, and in theory it extends far out into space. It is posssible to limit the E field around a conductor by changing the geometry. Figure 2.7 shows a grounded conductor surrounding the conductor in question. This conductor is called a shield. The field lines leave the center conductor and terminate on the inside surface of the shield. This geometry limits the E field to the space between the two conductors. In this example there is no E field or charge on the outside of the shield. If an external field did exist, it would terminate on the outer walls of the shield. This field would cause a charge distribution on the external surface. The external field cannot enter the space between the two conductors. The internal conductor is said to be shielded from external influences. Shields are not perfect, and small amounts of field can exit or enter directly through the shield. Fields can also enter at the ends of the shield, and this can be a significant problem. This problem is discussed in Section 6.14. The shield and its center conductor are called a shielded cable. If a voltage is placed between the shield and the center conductor, the E field is confined to the inside of the cable. In most applications the shield is grounded, but this is not a requirement for there to be shielding. Shielding is not limited to cables. A conducting box can be a shield for a circuit. If the box surrounds the circuit, the E field activity in the box is 31 MAGNETIC FIELD confined to the box and external E fields cannot enter. A totally sealed box is impractical, as circuits must be ventilated and leads must be brought in and out of the box. Even if it is not perfect, shielding is a very important tool in electronic design. There are many techniques that get around the imperfect shield or box. Rapidly changing signals pose separate shielding problems. This is a topic for later chapters. The shield of a typical cable is a conducting braid that surrounds a center conductor. This braid is made of many small tinned wires that allow for flexibility and an electrical connection at the ends. Braided cable has many small holes, and some of the internal E field can find its way out of the cable. This E field terminates on induced charges on external conductors. This leakage of the E field is described correctly as a mutual capacitance. If the E field varies, the induced charges must change. This changing charge implies current flow, which can be a source of a signal in an external circuit. The signal in the cable is said to be coupled to an external conductor through a mutual capacitance. This capacitance is also called a leakage capacitance. In most examples, leakage capacitances are on the order of a few picofarads per foot of cable. 2.10 MAGNETIC FIELD The second electrical field found in nature is the magnetic field. Perhaps the most common magnetic field is the one that surrounds the earth. This field makes it possible to navigate through the use of a compass. Early experimenters found that a similar magnetic field was created near conductors that carried current. This means that there is both an E field and a magnetic field whenever charges are moving along a conductor. Remember that charges will move along a conductor only if there is an internal electric field. The magnetic field around a conductor carrying current is shown in Figure 2.8. This same field would exist if the current were a beam of electrons in a vacuum. The field resulting from a current is called an H field. This field is a force field, but this time the force is not on static charges. Evidence of the force exists when a steady current is passed through a piece of paper containing a sprinkling of iron filings. The filings tend to align themselves FIGURE 2.8 Magnetic field around a current. 32 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS with the field and form circles. A compass needle will also align itself with this magnetic field. In Figure 2.8, the H field continues out into the surrounding space. The field lines represent the alignment direction of a small compass. Note that the field lines or magnetic flux form closed circles. By convention the number of lines that are drawn is proportional to the current flow. These flux circles are spaced along the length of the current path. If the path is twice as long, the amount of flux is doubled. The electrons that make up atoms have spin. In some materials the spin axis of electrons in adjacent atoms tends to line up in the same direction. It is this alignment that gives a material its magnetic properties. Each spinning electron generates a small magnetic field. The result of having many electrons spinning in the same direction is a magnetized material. Permanent magnets are materials that retain this alignment. Magnetic materials such as iron or steel are attracted to permanent magnets. Once in contact with a strong magnetic field these materials will also retain some magnetism. An example might be a magnetized iron nail or a steel screwdriver. When an iron bar or keeper is attracted to a permanent magnet, a considerable force may be required to pull the bar loose. This indicates that significant energy can be stored in a magnetic field. There is no magnetic particle equivalent to the charge that can be used to measure the magnetic field force and direction. If such a particle did exist, it would be called a monopole (a single magnetic pole). Physics books use a hypothetical element of current to characterize the magnetic field. This is used because the forces between two elements of current are well understood. The current element must be small enough so that it does not modify the field it is measuring. Just like the E field, the magnetic field exerts a force at a distance. Where the flux lines are close together, the magnetic field intensity is greatest. 2.11 SOLENOIDS When the current path consists of many turns of wire along a cylinder, the Hfield intensity in the center of the turns is proportional to the number of turns. This conductor geometry is called a solenoid. Note that the Hfield lines are no longer circles. Flux lines are still loops that close on themselves. A solenoid and its associated H field is shown in Figure 2.9. The Hfield intensity is greatest and nearly constant everywhere inside the solenoid. Outside the solenoid the Hfield intensity drops off significantly. 2.12 AMPÈRE’S LAW In the Efield case, a force exerted over a distance represented work and the work per unit charge was defined as voltage. In the magnetic case, the force AMPÈRE’S LAW 33 FIGURE 2.9 Simple solenoid and its associated H field. exerted over a distance also represents work, but here the work must be done on a unit segment of current or on that imaginary monopole. The product of Hfield intensity " distance turns out to be a measure of current. In the Efield case the Efield intensity times the distance between the plates yielded the voltage across the plates. In the Hfield case the Hfield intensity times the distance around the flux path yields the current that creates the flux. An external circuit supplies the current that creates the magnetic field. In ordinary circuits, energy must be dissipated continuously to sustain a steady current. This energy is lost in the resistance of conductors that make up the current path. The positive work done in moving a current element through a magnetic field reduces the amount of energy that must be supplied by the external circuit. In an ideal circuit where the resistances are zero, the positive work done on the current element would increase the current in the loop. In Figure 2.8, the Hfield intensity is constant along any flux path. The product of H " the total path length is a measure of current in the nearby conductor. This fact is known as Ampère’s law. The path length is 2¼r, and therefore 2¼rH = I. This means that H = I=2¼r, where r is in meters and I is current in amperes. The units of H are thus amperes per meter. Compare this with the units for the E field, which are volts per meter. Ampère’s law can be applied to the solenoid. In this case the Hfield intensity changes along the flux path. This can be seen in Figure 2.9 where the Hfield lines are not uniformly spaced. In the Efield case, the product of E " distance had to be considered over short distances. The H field must be treated in the same way. The product of H " distance must be made over short distances, where H is nearly constant. If these products are summed around the flux loop, the answer will be the current enclosed. In the case of a solenoid with n turns, the answer will be nI, where I is the current in the solenoid. In equation form, H = nI=2¼r, where I is in amperes and r is in meters. 34 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS 2.13 PROBLEMS 1. A conductor carries a 3A steady current. What is the Hfield intensity 10 cm away from the center of the conductor? 2. One hundred turns of wire are evenly spaced around a toroid. What is the H field in the center of the toroid if the radius is 5 cm and the current is 0.1 A? 3. Solve problem 2 where the radius of the toroid is 10 cm. 4. Is the H field in the toroid different if the turns are square? 5. A conductor 2 cm in diameter carries a dc current of 10 A. What is the H field inside the conductor at a depth of 0.5 cm? What is the H field at the center of the conductor? 2.14 MAGNETIC CIRCUIT Most magnetic applications require the use of specially alloyed magnetic materials where iron is usually one of the components. These alloys play a role in magnetics just as a dielectric plays a role in electrostatics. In the Efield case, the dielectric reduced the intensity of the E field in the dielectric. In the magnetic field case the magnetic material reduces the intensity of the H field in the magnetic material. The ability to reduce the H field is called relative permeability. If the H field is reduced by a factor of 100, the relative permeability of the material is 100. Ampère’s law still holds when magnetic materials are introduced in the flux path. If the current is held constant, the H field adjusts so that it is reduced in the magnetic material and increased in the remaining air space. If the magnetic material forms a loop with a small air gap, the H field will be very intense in the air gap. The iron and the gap form what is known as a magnetic circuit. A typical magnetic circuit is shown in Figure 2.10. In this example the length of the gap is 0.1 cm, the length of the magnetic path is 10 cm, the relative permeability is 1000, and the current is 0.1 A. The flux path in the iron is 0.099 m and in the gap it is 0.0001 m long. The H field in the iron is 1=1000 the H field in the air gap. Ampère’s law requires that the Hfield intensity times the path length must add up to the total current. This means that (H=1000) " 0:099 + H " 0:0001 = 0:1 A. The value for H is 502.5 A=m. This is the field strength in the air gap. The Hfield intensity in the iron is 1=1000 of this value, or 0.5025 A=m. 2.15 INDUCTION OR B FIELD The H field is reduced in a material with permeability. This parallels the E field case where the E field is reduced in a dielectric. It is convenient to 35 INDUCTION OR B FIELD FIGURE 2.10 Magnetic circuit. introduce a second magnetic field that is not changed by permeability. This field is called the B field or induction field. The ratio of the B field to the H field is the permeability of the material. The units of H are amperes per meter and the units for B are teslas. Another common unit for the B field is the gauss (G); 10,000 gauss is equal to 1 tesla. In equation form, B = ¹R ¹0 H, where ¹R is the relative permeability and ¹0 is the permeability of free space. The value of ¹0 is 4¼ " 10#7 . This constant is needed to convert amperes per meter to teslas. Note that ¹R for air is unity. The B field is the true force field on moving charges. This force is perpendicular to both the Bfield direction and the direction of the moving charge. When charged particles from the sun get near the earth, they encounter the earth’s magnetic field. The B field causes the charges to spiral in the magnetic field as they near the earth. This spiraling releases energy in the form of radiation. This is familiar to most people as the aurora borealis. This colorful display is a polar phenomenon, as this is where the magnetic field is most intense. A strong duality exists between the electric field and the magnetic field. When the D field changes in space, it is equivalent to current flow. When the B field changes in space, it turns out to be equivalent to voltage in space. This can be observed by placing a loop of wire in a changing magnetic field. The voltage that is induced in this loop is proportional to the amount of B flux that crosses the loop and how rapidly this flux is changing.$ A larger loop area captures more flux, and this increases the voltage across the terminals of the loop. If several turns couple to the same flux, the induced voltage will be proportional to the number of turns. This circuit is shown in Figure 2.11, where two turns of wire couple to the changing Bfield flux. The changing $ The term flux is used interchangeably with the term field. It is sometimes a help to visualize a portion of the field as lines of flux crossing a given area in space. 36 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS FIGURE 2.11 Voltage coupling from a changing B field. magnetic field forces charges to move to the surface of the added coil. The coil end that is positive has positive charges and the negative end has negative charges. These charges cannot move unless a circuit path is provided across the coil ends. The voltage that is coupled to the loop of wire means that there is an E field whenever there is a changing B field. The voltage induced into a loop by a magnetic field is given by the rate of change of flux expressed in webers per second. This is known as Faraday’s law. To obtain the flux in webers, the B field in teslas must be multiplied by the crosssectional area of the loop in square meters. To obtain the B field in teslas, the H field must be divided by the permeability of free space and the relative permeability. Faraday’s law in equation form is V = n " (the rate of change of magnetic induction flux per second), where the magnetic flux is in units of maxwells, V is in volts, and n is the number of turns. It is important to recognize that Faraday’s law works two ways. If there is a changing magnetic flux crossing an open conductive loop, a voltage can be sensed at the ends of the loop. Conversely, if a voltage is placed across a conductive loop, there must be a corresponding changing flux crossing the loop. This changing flux implies current flow supplied by the voltage source. 2.16 MAGNETIC CIRCUIT WITHOUT A GAP Figure 2.12 shows a simple transformer that has a core without a gap. The magnetizing properties of this material are considered ideal. This means that B and H are always proportional to each other. Practical magnetic materials have hysteresis, which means that there is nonlinear relationship between the H and B fields. The current required to establish the H field is called a magnetizing current. If the magnetic material were ideal (an infinite permeability), the magnetizing current would be zero. Two coils are wrapped around the magnetic path so that any magnetic flux in the path threads through both coils. When a steady voltage is impressed across the first coil, the flux that threads through that coil must change at a given rate. A steady voltage implies a constantly changing flux and a constantly changing flux, implies a steady voltage. This statement is Faraday’s law. This rate of change in magnetic flux requires a corresponding rate of change in the B field. This rate of change in the B field requires a correspond MAGNETIC CIRCUIT WITHOUT A GAP 37 FIGURE 2.12 Magnetic circuit without a gap. ing rate of change in the H field. This further requires a rate of change in the current flowing in the coil. The Bfield intensity in the magnetic path will increase until the core saturates (the iron cannot handle any more flux). At this point the permeability drops and the current supporting the H field will increase sharply. To be practical, the B field must be kept within bounds. This means that the impressed voltage can last only a limited period of time. In this magnetic circuit the only function of the magnetizing current is to establish the B field. Here is an example of how this magnetic circuit works. The magnetic path length in Figure 2.12 is 0.1 m long. The maximum Bfield intensity in this material might be 1 tesla (T). Assume the following parameters: The relative permeability is 10,000, the voltage impressed across the first coil is 10 V, and the coil has 1000 turns. The voltage per turn is 0.01 V. Using Faraday’s law, this voltage requires that the flux coupling to each turn must change at 0.01 weber (Wb) per second. (One turn and 1 V supports 1 Wb=s.) The Bfield intensity is simply this flux divided by the crosssectional area. If the crosssectional area is 0.001 m2 , the B field must change at the rate of 10 T=s. To find the H field, the B field must be divided by both the permeability of free space (4¼ " 10#7 ) and the relative permeability of the iron (10,000). The rate of change of the H field is 104 =4¼ amperes per meter per second. Because the path length is 0.1 m and the number of turns is 1000, the current in the coil must change at 1=4¼ amperes per second. In 0.01 s the current in the coil will go from zero to 0.796 mA, a small magnetizing current. Remember that the B field is changing at 10 T=s. At the end of 0.01 s the B field has reached an intensity of 0.1 T. If the voltage is left connected to the coil for a full 0.1 s, the B field will increase to 1 T and the magnetic material will just saturate. At this point the H field has also increased and the magnetizing current is up to 7.96 mA. If the magnetic material saturates and the permeability drops to 38 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS FIGURE 2.13 Waveforms associated with impressing a voltage on a coil in a magnetic circuit. 100, the H field must increase by another factor of 100. This means that the magnetizing current will rise. It is obvious that the permeability must remain high if the magnetizing current is to remain low. See Figure 2.13 for the waveforms in this example. 2.17 MAGNETIC CIRCUIT WITH A GAP Figure 2.14 shows the previous magnetic circuit with a gap of 0.1 cm. The total path length is 0.1 m. The maximum B field before core saturation is again 1 T. A voltage of 10 V is applied across 1000 turns, as before. The magnetic material has a relative permeability of 10,000. Faraday’s law requires that the flux crossing each coil must change at 0.01 Wb=s. The Bfield intensity is simply the flux change divided by the crosssectional area of the TRANSFORMER ACTION 39 FIGURE 2.14 Magnetic circuit with a gap. magnetic material. As before, this area is 0.001 m2 and the Bfield intensity must change at 10 T=s. The H field intensity in the air gap is the B field divided by the permeability of free space or 4¼ " 10#7 . The Hfield intensity in the iron is reduced further by the relative permeability. The H field in the air gap must change at 7:96 " 106 A=m per second and the H field in the iron must change at 7:96 " 102 A=m per second. The current flowing in one turn can be calculated by multiplying by the proper magnetic path length. The changing current required to support the H field in the gap is 7:96 " 106 times 0.001 m, or 7:96 " 103 A=s. In the iron the path length is 0.1 m and the magnetizing current changes at 79.6 A=s. The total requirement is the sum of the two changing currents, or 8039 A=s per turn. Because there are 1000 turns, the current in the coil changes at 8.039 A=s. In 0.01 s the current rises to 0.080 A. Note how the presence of the gap has increased the required current. This added current is storing field energy in the gap. 2.18 TRANSFORMER ACTION The second coil in Figure 2.12 surrounds the magnetic circuit with a different number of turns. The voltage per turn on this coil is identical to the voltage per turn on coil 1. If the number of turns on the second coil is double, the voltage is double. Using the previous example, if the second coil has 2000 turns, the steady voltage across the coil would be 20 V. The two coils and the core would be called a stepup transformer. If the second coil had 500 turns, the terminal voltage would be 5 V and this would be called a stepdown transformer. In our ideal transformer the resistance of the coils is considered zero. What happens when a resistor is placed across the ends of the second coil? A current must flow that obeys Ohm’s law. Assume that coil 2 has 2000 turns. If 10 V is impressed on coil 1, there will be 20 V across coil 2. If there is a 2000 40 CAPACITORS, MAGNETIC FIELDS, AND TRANSFORMERS resistor across coil 2, a current of 10 mA will flow. This current flows steadily and does not increase in time like the magnetizing current. This current causes an H field in the iron that seems to modify the ratio of Bfield to Hfield intensity. This does not happen. A current flows in coil 1 that exactly cancels this added H field. If 10 mA flows in coil 2 with 2000 turns, a current of 20 mA must flow in coil 1, as it only has 1000 turns. In effect, there is a balance of ampere turns (At). Coil 2 has 10 mA in 2000 turns, or 20 At. This is balanced by coil 1, which has 20 mA in 1000 turns, or the same 20 At. Coil 1 is called the primary coil and is connected to the source of voltage. Coil 2 is called a secondary coil and has a voltage that is determined by the number of turns and the voltage per turn impressed on the primary coil. Of course, there can be more than two coils wound around a magnetic circuit. The ampere turns of all secondary coils must add up to the ampere turns on the primary coil. The magnetizing current is independ