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Introduction to Circuit Analysis and Design

Tildon H. Glisson, Jr.
Introduction to Circuit
Analysis and Design

Prof. T. H. Glisson, Jr.
3072 EB-II, Box 7911
N.C. State University
Raleigh, NC 27695
glisson@ncsu.edu
The solution manual and other supplementary material are provided to those who are using or
considering using the book as the textbook for a related course. To gain access to the solution
manual and other supplementary material, go to http://www.springer.com/978-90-481-9442-1,
click on the appropriate link under Additional Information on the right side of that page, and
follow the instructions given there.
ISBN: 978-90-481-9442-1 e-ISBN: 978-90-481-9443-8
DOI: 10.1007/978-90-481-9443-8
# Springer Science+Business Media B.V. 2011
No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any
means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written
permission from the Publisher, with the exception of any material supplied specifically for the purpose
of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
Printed on acid-free paper
springer.com

To Andy Sage, Ben O’Neal, and Sy Matthews. Each knows why.

Preface
This book is intended as the textbook for a first course in electric circuits. It has
evolved from notes that I developed over several years, while I taught the
introductory circuits course to sophomore electrical- and computer-engineering
students at North Carolina State University.
As the chapter titles indicate, the major topics are those covered in almost all books
of this kind. But selection and treatment of topics was guided not only by what is
traditional for the target course, but also by the meaning and purpose of circuit
analysis and (ultimately) design. This book reflects the views that circuit analysis is
concerned primarily with finding and interpreting the relation a circuit establishes
between excitation and response, that circuit design is concerned primarily with
creating a circuit that enforces a specified relation between excitation and response,
and that in design in particular, the devil is in the details. These perspectives led to
somewhat different approaches to some aspects of the subject and to including some
topics not treated in outwardly similar books. As a bonus, these topics give a practical
flavor to and stimulate interest in some otherwise dry subjects.
Compared with other books for the same audience, this book gives more attention
to transfer functions, input and output impedance, and two-port models; to power
dissipation and power transfer, topics whose importance in practice is far greater than
their treatment in most books would suggest; to the physical origins of capacitance
and inductance, so students can gain some understanding of the origins of stray,
residual, and parasitic capacitance and inductance, and how consideration of such
effects influences component selection and placement; and to other ways in which
physical components differ from idealized components, such as variation of
parameters with temperature and frequency. Treatment of op amps is not limited to
the ideal model, but covers topics such as output swing, output current limit, slew
rate, bias-current compensation, power dissipation, stability, and gain-bandwidth
product, all of which are important in any practical design. One unfamiliar with
these topics cannot understand an op-amp data sheet or intelligently select an op amp
for any particular application.
The book contains more than 1,100 problems and more than 800 examples. Many
of the problems are assemblages of similar problems; for example, “find the input
impedance and output impedance of each circuit in Fig. ___.” So there are really
many more than 1,100 problems. There is an adequate number of problems calling
for numerical answers, through which students can gain confidence. But consistent
with the focus on excitation-response relations, most problems ask the student to
obtain an expression for a quantity or a relation among quantities. Some problems
go further, and ask for an analysis or discussion of the influence of one or more
vii

parameters on a relation of interest. There is an adequate number of design
problems in which there are fewer specifications than free parameters, requiring
students to exercise some judgment.
Different instructors approach this subject in different but equally valid ways, so I
will not presume to suggest a syllabus. I note only that the book contains ample
material for a two-semester or three-quarter course, but can be used selectively in a
one-semester or two-quarter course. Most chapters and the book as a whole are
organized such that topics generally included in a one-semester course are covered
first.
The only prerequisite for successful study of this book is facility with differential
and integral calculus. Some exposure to electricity and magnetism, such as provided
by a freshman physics course, might be helpful, but is not required. Some problems
require use of a mathematics package such as Matlab# or Mathcad#, and some
require use of a simulation package such as Pspice# or Electronics Workbench#.
Problems requiring computer assistance are identified by the symbol :. There is no
preference for or attempt to teach any particular software package, and it is possible
to avoid these requirements altogether by simply not assigning such problems.
A companion website provides solutions to all exercises and errata and (for faculty
members that adopt the book) solutions to all problems in the book, selected
supplementary material, and a mechanism for submitting corrections, suggestions,
and comments.
I have many to thank for whatever is good about the book, and only myself to blame
for everything else. Thanks to James Kang and Bruno Osorno for their helpful
comments on an early draft; to Art Davis, John Hauser, S.C. Dutta Roy, Joel
Trussell, and Gary Ybarra for their thorough and helpful reviews; to Chris Lunsford
for his contributions to several chapters; to Hari Chandrasekar, Vinodh Kotipalli,
Misha Kumar, Christina Lee, and Satish Naidu, all excellent graduate students who
spent many hours proofreading the text and checking my solutions to examples,
exercises, and problems; and to all the companies, publishers, and individuals that
permitted me to use or adapt parts of their intellectual property. Thanks to Jeff Kahler
of Nuhertz Technologies and Siegfried Linkwitz for allowing me to use several of their
designs in examples and problems. Special thanks to Art Davis and John Hauser for
many correspondences and discussions that helped clarify or otherwise refine my
presentations of various topics. Special thanks also, to those at Springer: Mark de
Jongh, Senior Publishing Editor for Electrical Engineering; Cindy Zitter, his Senior
Assistant; and Project Manager R. Samuel Devanand, for being so capable and helpful.
Raleigh, North Carolina Tildon H. Glisson Jr.
viii Preface

Contents
1 Introduction ................................................................. 1
1.1 Electric Circuits ....................................................... 1
1.2 How to Study This Book .............................................. 3
1.3 Dimensions and Units ................................................. 4
1.4 Symbols and Notation ................................................. 9
1.5 Symbols Versus Numbers ............................................ 10
1.6 Presentation of Calculations .......................................... 10
1.7 Approximations ....................................................... 11
1.8 Precision and Tolerance .............................................. 14
1.9 Engineering Notation ................................................. 14
1.10 Problems .............................................................. 15
2 Current, Voltage, and Resistance ........................................ 19
2.1 Charge and Current ................................................... 19
2.2 Electric Field ......................................................... 21
2.3 Electric Potential and Voltage ....................................... 23
2.4 Ohm’s Law and Resistance .......................................... 25
2.5 Resistivity ............................................................ 26
2.6 Conductance and Conductivity ....................................... 26
2.7 Resistors .............................................................. 27
2.8 E Series, Tolerance, and Standard Resistance Values ............... 29
2.9 Resistor Marking ..................................................... 30
2.10 Variation of Resistivity and Resistance with Temperature .......... 32
2.11 American Wire Gauge (AWG) and Metric Wire Gauge (MWG) ... 35
2.12 DC and AC ........................................................... 36
2.13 Skin Effect and Proximity Effect .................................... 36
2.14 Concluding Remark .................................................. 42
2.15 Problems .............................................................. 42
3 Circuit Elements, Circuit Diagrams, and Kirchhoff’s Laws .......... 49
3.1 Schematics and Circuit Diagrams .................................... 49
3.2 Conductors and Connections ......................................... 50
3.3 Annotating Circuit Diagrams ......................................... 51
3.4 Series and Parallel Connections ...................................... 53
3.5 Open Circuits and Short Circuits ..................................... 53
3.6 Basic Circuit Elements: Resistors and Independent Sources ........ 54
3.7 Kirchhoff’s Current Law and Node Analysis ........................ 56
ix

3.8 Kirchhoff’s Voltage Law and Mesh Analysis ....................... 62
3.9 Voltage and Current Dividers ........................................ 66
3.10 Superposition ......................................................... 66
3.11 Problems .............................................................. 69
4 Equivalent Circuits ....................................................... 83
4.1 Terminal Characteristics ............................................... 83
4.2 Equivalent Circuits .................................................... 85
4.2.1 Resistors in Series Are Additive ................................ 85
4.2.2 Conductances in Parallel Are Additive ......................... 85
4.2.3 Voltage Sources in Series Are Additive ........................ 87
4.2.4 Current Sources in Parallel Are Additive ...................... 88
4.2.5 Elements in Series Commute, as do Elements in Parallel ..... 88
4.2.6 Finding Equivalent Resistance Using a Known Source ........ 89
4.2.7 Approximations ................................................. 90
4.3 Source Transformations ............................................... 91
4.4 Thévenin and Norton Equivalent Circuits ............................. 93
4.5 Notation: Constant and Time-Varying Current and Voltage ........ 100
4.6 Significance of Terminal Characteristics and Equivalence .......... 101
4.7 Problems .............................................................. 101
5 Work and Power ......................................................... 113
5.1 Instantaneous Power and the Passive Sign Convention ............ 114
5.2 Instantaneous Power Dissipated by a Resistor: Joule’s Law ....... 116
5.3 Conservation of Power .............................................. 119
5.4 Peak Power .......................................................... 120
5.5 Available Power ..................................................... 122
5.6 Time Averages ...................................................... 123
5.7 Average Power ...................................................... 126
5.8 Root Mean Squared (RMS) Amplitude of a Current or Voltage ... 130
5.9 Average Power Dissipated in a Resistive Load ..................... 131
5.10 Summary: Power Relations ......................................... 133
5.11 Notation .............................................................. 133
5.12 Measurement of RMS Amplitude ................................... 133
5.13 Dissipation Derating ................................................. 135
5.14 Power Dissipation in Physical Components and Circuits .......... 139
5.15 Active and Passive Devices, Loads, and Circuits ................... 141
5.16 Power Transfer and Power Transfer Efficiency ..................... 142
5.17 Superposition of Power .............................................. 147
5.18 Problems ............................................................. 148
6 Dependent Sources and Unilateral Two-Port Circuits ................ 163
6.1 Input Resistance and Output Resistance ............................. 164
6.2 Dependent Sources ................................................... 166
6.3 Linear Two-Port Models .............................................. 172
6.4 Two-Ports in Cascade ................................................ 174
6.5 Voltage, Current, and Power Transfer ................................ 174
6.6 Transfer Characteristics, Transfer Ratios, and Gain ................. 178
6.7 Power Gain ........................................................... 183
x Contents

6.8 Gains and Relative Values in Decibels (dB) ........................ 184
6.9 Design Considerations ............................................... 186
6.10 Problems ............................................................. 187
7 Operational Amplifiers I ................................................ 197
7.1 Operational Amplifier Terminals and Voltage Reference .......... 199
7.2 DC Circuit Model for an Op Amp .................................. 200
7.3 The Ideal Op Amp and Some Basic Op-Amp Circuits at DC ...... 202
7.4 Feedback and Stability of Op-Amp Circuits ........................ 207
7.5 Input Resistance and Output Resistance of Op-Amp Circuits ...... 210
7.6 Properties of Common Op-Amp Circuits ........................... 213
7.6.1 Inverting Amplifier ........................................... 213
7.6.2 Non-inverting Amplifier ...................................... 214
7.6.3 Voltage Follower ............................................. 214
7.7 Op Amp Structure and Properties ................................... 216
7.8 Output Current Limit ................................................ 216
7.9 Input Offset Voltage ................................................ 216
7.10 Input Bias Currents .................................................. 217
7.11 Power Dissipation in Op Amps and Op-Amp Circuits ............. 219
7.12 Design Considerations ............................................... 222
7.13 Problems ............................................................. 224
8 Capacitance .............................................................. 237
8.1 Capacitance .......................................................... 237
8.2 Capacitors ........................................................... 239
8.3 Terminal Characteristics of an Ideal Capacitor ..................... 242
8.4 Charge-Discharge Time Constant ................................... 246
8.5 Capacitors in Series and Parallel .................................... 252
8.6 Leakage Resistance .................................................. 253
8.7 Stray and Parasitic Capacitance; Capacitive Coupling ............. 255
8.8 Variation of Capacitance with Temperature ........................ 258
8.9 Energy Storage and Power Dissipation in a Capacitor ............. 260
8.10 Applications ......................................................... 262
8.10.1 Differentiating Circuits ...................................... 262
8.10.2 Integrating Circuits .......................................... 263
8.10.3 Bypass Capacitors ........................................... 266
8.10.4 Bypass Capacitors (Filter Capacitors) in Rectifier Circuits ... 268
8.10.5 Bypassing in Digital Systems ............................... 270
8.10.6 Coupling Capacitors ......................................... 271
8.10.7 Input Bias Current Compensation in Capacitively
Coupled Amplifiers .......................................... 276
8.10.8 Switched Capacitor Circuits ................................. 279
8.10.9 Power Dissipation in Switched-Capacitor Circuits ......... 281
8.11 Problems ............................................................. 284
9 Inductance ................................................................ 301
9.1 Magnetic Field ........................................................ 301
9.2 Self Inductance ....................................................... 302
9.3 Inductance of Air-Core Coils ......................................... 303
Contents xi

9.4 Inductors ............................................................. 304
9.5 Terminal Characteristic of an Inductor ............................. 305
9.6 Time Constant ....................................................... 307
9.7 Inductors in Series and Parallel ..................................... 312
9.8 Energy Storage and Power dissipation in an Inductor .............. 313
9.9 Parasitic Self-Inductance ............................................ 314
9.10 Reducing Ripple ..................................................... 316
9.11 Inductive Kick ....................................................... 318
9.12 Magnetically Coupled Coils and Mutual Inductance ............... 319
9.13 Parasitic Mutual Inductance ......................................... 323
9.14 Transformers ......................................................... 324
9.15 Ideal Transformers .................................................. 326
9.16 Applications of Transformers ....................................... 328
9.16.1 Source and Load Transformations: Matching Transformers .. 328
9.16.2 Step-Up and Step-Down Transformers ..................... 330
9.16.3 Isolation Transformers ...................................... 331
9.16.4 Center-Tapped Transformers and Balanced Power ......... 332
9.17 Concluding Remarks ................................................ 333
9.18 Problems ............................................................. 334
10 Complex Arithmetic and Algebra ...................................... 345
10.1 Complex Numbers ................................................... 345
10.2 Complex Arithmetic ................................................. 345
10.3 Conjugate of a Complex Number ................................... 346
10.4 Magnitude of a Complex Number .................................. 347
10.5 Arithmetic in a Complex Plane ..................................... 347
10.6 Polar Form of a Complex Number .................................. 348
10.7 Eulers Identity and Polar Arithmetic ................................ 349
10.8 The Symbols ∠ and ∡ .............................................. 351
10.9 Problems ............................................................. 351
11 Transient Analysis ....................................................... 353
11.1 Unit Step Function .................................................. 353
11.2 Notation .............................................................. 354
11.3 Initial Conditions .................................................... 354
11.4 First-Order Circuits .................................................. 355
11.5 Second-Order Circuits ............................................... 360
11.5.1 Summary: Second-Order Circuits ........................... 365
11.5.2 Dominant Time Constant (or Characteristic Root) ......... 369
11.5.3 Damping Factor ............................................. 370
11.5.4 Extrema of the Unforced Component of an Underdamped
Response ..................................................... 371
11.6 Time Invariance, Superposition, and Pulse Response .............. 373
11.7 Operator Notation ................................................... 375
11.8 Problems ............................................................. 377
12 Sinusoids, Phasors, and Impedance .................................... 383
12.1 Sinusoidal Voltages and Currents ................................... 383
12.2 Time Origin, Phase Reference, and Initial Phase ................... 385
xii Contents

12.3 Phasors ............................................................. 388
12.4 Phasor Diagrams ................................................... 390
12.5 Impedance and Generalized Ohm’s Law .......................... 391
12.6 Admittance ......................................................... 395
12.7 Impedance and Admittance Ratios in dB .......................... 396
12.8 A Fundamental Relation ........................................... 397
12.9 Circuit Reduction: Elements in Series and Parallel ............... 399
12.10 Time Domain and Frequency Domain ............................. 402
12.11 Sinusoidal and DC Steady State ................................... 404
12.12 Frequency-Domain Circuit Analysis ............................... 405
12.13 Reactance and Effective Resistance ............................... 407
12.14 Susceptance and Effective Conductance .......................... 412
12.15 Impedance and Admittance Triangles ............................. 414
12.16 Linearity and Superposition ........................................ 414
12.17 Thévenin and Norton Equivalent Circuits: Source
Transformations .................................................... 421
12.18 Checking Your Work .............................................. 431
12.19 Resonance .......................................................... 435
12.20 Quality Factors and Common Resonant Configurations .......... 438
12.21 Simulating Inductance Using Active RC Circuits ................. 444
12.22 Circuit Elements and Physical Circuit Components ............... 447
12.22.1 Resistors .................................................. 448
12.22.2 Inductors ................................................. 451
12.22.3 Capacitors ................................................ 453
12.23 Problems ........................................................... 456
13 Complex Power .......................................................... 479
13.1 Definition of Complex Power ...................................... 479
13.2 Notation ............................................................ 481
13.3 Power Calculations ................................................. 482
13.4 Reactive Power and Apparent Power .............................. 483
13.5 Conservation of Complex Power .................................. 486
13.6 Power Relations in Resonant Circuits ............................. 487
13.7 Power Factor ....................................................... 489
13.8 Power Triangle and Power-Factor Correction ..................... 491
13.9 Superposition of Complex Power .................................. 496
13.10 Power Transfer ..................................................... 496
13.10.1 Power Transfer Efficiency ............................... 497
13.10.2 Power Transfer ........................................... 500
13.10.3 Insertion Loss ............................................ 503
13.11 Impedance Matching ............................................... 504
13.11.1 Transformers ............................................. 504
13.11.2 L Sections ................................................ 506
13.12 Problems ........................................................... 510
14 Three-Phase Circuits .................................................... 521
14.1 Three-Phase Sources ................................................ 521
14.2 Power Transmission and Distribution ............................... 523
14.3 Residential Wiring ................................................... 524
Contents xiii

14.4 Three-Phase Loads .................................................. 524
14.5 Balanced Y–D and D–Y Load Transformations .................... 528
14.6 Power Calculations for Balanced Three-Phase Loads .............. 530
14.7 Power-Factor Correction for Three-Phase Loads ................... 531
14.8 Instantaneous Power Delivered to a Balanced Load ................ 534
14.9 Problems ............................................................. 535
15 Transfer Functions and Frequency-Domain Analysis ................ 539
15.1 Transfer Functions ................................................. 539
15.2 Dependence of a Transfer Function upon Source and Load ...... 544
15.3 Gain and Phase Shift ............................................... 545
15.4 Gain in Decibels (dB) .............................................. 546
15.5 Standard Form of a Transfer Function ............................. 550
15.6 Asymptotic Gain Plots: Linear Factors ............................ 555
15.7 Asymptotic Gain Plots: Quadratic Factors ........................ 562
15.8 Asymptotic Plots of Phase Shift Versus Frequency ............... 565
15.9 Filters and Bandwidth .............................................. 569
15.10 Frequency Response ............................................... 572
15.11 Problems ........................................................... 573
16 Fourier Series ............................................................ 583
16.1 Amplitude–Phase Series ........................................... 583
16.2 Exponential Series and Fourier Coefficients ...................... 585
16.3 Quadrature Series .................................................. 586
16.4 Summary: Three Forms of Fourier Series ......................... 587
16.5 Integral Formula for Fourier Coefficients ......................... 588
16.6 A Table of Fourier Coefficients .................................... 590
16.7 Modified Fourier Coefficients for Composite Waveforms ........ 595
16.8 Convergence of Fourier Series ..................................... 597
16.9 Gibbs’ Phenomenon ................................................ 599
16.10 Circuit Response to Periodic Excitation ........................... 600
16.11 Spectra and Spectral Analysis ..................................... 603
16.12 Problems ........................................................... 605
17 Operational Amplifiers II: AC Model and Applications .............. 609
17.1 AC Model for an Op Amp ......................................... 609
17.2 Linear Resistive-Feedback Amplifiers ............................. 611
17.3 Linear Reactive-Feedback Circuits ................................ 617
17.4 Output Swing ....................................................... 619
17.5 Slew Rate ........................................................... 621
17.6 Amplifiers in Cascade .............................................. 624
17.7 Capacitance Coupling .............................................. 626
17.8 Input Bias Current Compensation in Capacitance-Coupled
Amplifiers .......................................................... 627
17.9 Power Dissipation in Op Amps and Op-Amp Circuits ............ 628
17.10 Power-Conversion Efficiency ...................................... 632
17.11 Op-Amp Amplifier Circuit Design ................................ 633
17.12 Problems ........................................................... 642
xiv Contents

18 Laplace Transformation and s-Domain Circuit Analysis ............. 653
18.1 Definition of the Laplace Transformation ......................... 653
18.2 Convergence and Uniqueness ...................................... 655
18.3 One-Sided Laplace Transforms .................................... 657
18.4 Shorthand Notation ................................................ 657
18.5 The Delta Function (Unit Impulse) ................................ 657
18.6 Tables of Operational Properties and Transform Pairs ............ 659
18.7 Inverse Transforms Using Partial-Fraction Expansions ........... 663
18.7.1 Distinct Poles .............................................. 664
18.7.2 Complex-Conjugate Poles ................................. 665
18.7.3 Repeated Poles ............................................ 667
18.8 Terminal Characteristics and Equivalent Circuits ................. 669
18.9 Circuit Analysis in the s Domain .................................. 670
18.10 Checking Your Work .............................................. 675
18.11 s-Domain Transfer Functions ...................................... 678
18.12 Forced Response and Unforced Response ......................... 683
18.13 Impulse Response and Step Response ............................. 684
18.14 Relation of s-Domain to Frequency-Domain Transfer
Functions ........................................................... 689
18.15 s-Domain Models for Op Amps and Basic Op-Amp Circuits ..... 689
18.16 Circuits in Cascade ................................................. 691
18.17 Poles, Zeros, and Pole-Zero Plots .................................. 692
18.18 Stability ............................................................ 694
18.19 Pole-Zero Cancellation ............................................. 697
18.20 Dominant Poles .................................................... 697
18.21 Pole-Zero Plots and Bode Plots .................................... 700
18.22 Problems ........................................................... 703
19 Active Filters ............................................................. 723
19.1 Gain .................................................................. 723
19.2 Group Delay ......................................................... 724
19.3 A Simple Two-Pole Active Filter ................................... 725
19.4 Sallen-Key (VCVS) Filters .......................................... 727
19.5 State-Variable Biquadratic Filter .................................... 732
19.6 Modern Filter Design ................................................ 736
19.6.1 VCVS Butterworth Filters ................................... 736
19.6.2 VCVS Bessel Filters ......................................... 738
19.6.3 VCVS Chebyshev Filters .................................... 739
19.7 Problems ............................................................. 741
Appendix: Answers to Exercises ............................................. 745
Index ........................................................................... 759
Contents xv

Chapter 1
Introduction
Electrical engineering is among the largest and most
diverse professions. The Institute of Electrical and
Electronic Engineering (IEEE), which is the principal
technical and professional organization for electrical
engineers, is the largest professional organization in
the world. Within the IEEE alone, there are more than
40 technical societies, each focused on a sub-area of
electrical engineering. These are listed in Table 1.1.
Electrical engineers are employed in every major
industry, working not only with other electrical engi-
neers, but also with people from all walks of life. Their
work ranges from very applied to highly theoretical.
They work indoors and out, in government and in
private industry, alone and in large groups. They con-
tribute to the design and manufacture of virtually
every product you can name. In short, almost anyone
inclined toward engineering can find an exciting, ful-
filling, and rewarding career in electrical engineering.
You have chosen well.
1.1 Electric Circuits
The one thing that almost all electrical engineers have
in common is the fact that they design, manufacture,
maintain, teach about, or sell devices, equipment, or
systems whose operation depends primarily upon
manipulation of electricity. In such systems, electricity
is the primary means for conveying or converting
energy, transmitting or processing data, monitoring
or controlling other equipment or processes, or mea-
suring or observing various phenomena. Manipulating
electricity toward useful ends is what circuits are all
about. Thus it is no accident that almost all introduc-
tory EE courses and textbooks focus on circuits.
In the broadest sense, a circuit is something that
makes electricity do something useful. Circuits in
computers use electricity to store, retrieve, transmit,
and process data at unimaginable speed. Circuits
deliver tremendous quantities of electrical energy to
our homes and industries at bargain-basement prices.
Circuits allow us to transmit and receive hundreds of
radio and TV programs. Circuits make our automo-
biles run cleaner and more efficiently. They help us
store and prepare our food, wash our clothes, and cool
our homes. They help aircraft, ships, and space shut-
tles navigate. They help us defend our borders and
warn us of impending danger. They help monitor and
predict the weather and help physicians monitor our
health and perhaps save our lives. It is difficult to think
of anything we do in which we are not assisted by one
or more electric circuits. Certainly there are other
important subjects in EE, but without circuits, most
of those subjects would either not exist or have no
practical application. We might be able to imagine a
world without circuits, but few of us would want to
live there. All in all, study of circuits is an excellent
way to begin study of electrical engineering.
The principal quantities of interest in circuit analy-
sis and design are current, voltage, and power, none of
which is visible.1
For the most part, not even the
effects of currents and voltages in the innards of a
circuit are visible. Indeed, without a powerful micro-
scope, we cannot even see the innards of many modern
circuits. We must rely on measurements and mathe-
matics to tell us whether a circuit is working as it
should. Thus it might appear that circuit analysis and
1
Current in an arc is visible, but unless we are designing an
electric welder, we usually try to avoid arcs.
T.H. Glisson, Introduction to Circuit Analysis and Design,
DOI 10.1007/978-90-481-9443-8_1, # Springer ScienceþBusiness Media B.V. 2011
1

design is inherently more abstract than, for example,
mechanism analysis and design. We can see mechan-
isms operate, whereas most electric circuits look the
same whether operating or not. Unless something is
seriously wrong, the components of a circuit do not
jump about, make clanking noises, emit smoke, or
squirt messy fluids here and there.
But in truth, the level of abstraction of circuit analy-
sis and design relative to that of other subjects is over-
stated. It is true that we cannot see current or voltage,
but we cannot see force, pressure, or temperature,
either. Nor can we see a mechanism we have not yet
designed. Any extraordinary difficulties with circuits
probably arise more from lack of familiarity than level
of abstraction. Almost everyone has some intuitive
understanding of fluid flow and pressure and the func-
tions of pipes, pumps, and valves. But few laypersons
have an equally well-developed intuition regarding
current and voltage and the functions of circuit compo-
nents such as resistors, transistors, and capacitors.
In the beginning, analogies can be helpful. Current
is flow of electric charge and is analogous to fluid flow
(flow of molecules) in a hydraulic system. Voltage is
analogous to pressure, in that voltage can cause current
in a wire, much like pressure can cause fluid flow in a
pipe. Circuit components and circuits themselves are
described in terms of the current-voltage relationships
they establish at their terminals, just as hydraulic com-
ponents are described in terms of flow and pressure at
their terminals (inlets and outlets); for example, a
pump can be described by the flow it can provide at
one or more pressures (e.g., 3 gal/min at 100 lb/in.
2
)
and a generator can be described by the current it can
provide at the specified terminal voltage (e.g., 50 A at
120 V). If we know the flow-pressure characteristics of
each component in a hydraulic system, we can calcu-
late flow and pressure anywhere in the system. Like-
wise, if we know the current-voltage characteristics of
each component in a circuit, we can calculate current
and voltage anywhere in the circuit.
Although analogies can be helpful, they also can be
misleading if carried too far. There are circuits and
devices in which current does not behave like fluid
flow. Also, circuits can do many things that hydraulic
systems cannot (and vice versa), in which case there is
no analogy to fall back on. Ultimately, if we are to
design circuits, we must come to understand current,
voltage, and circuits on their own terms.
Table 1.1 IEEE technical societies
IEEE Aerospace and Electronic Systems Society IEEE Instrumentation and Measurement Society
IEEE Antennas and Propagation Society IEEE Lasers & Electro-Optics Society
IEEE Broadcast Technology Society IEEE Magnetics Society
IEEE Circuits and Systems Society IEEE Microwave Theory and Techniques Society
IEEE Communications Society IEEE Nanotechnology Council
IEEE Components Packaging, and Manufacturing
Technology Society
IEEE Nuclear and Plasma Sciences Society
IEEE Computational Intelligence Society IEEE Oceanic Engineering Society
IEEE Computer Society IEEE Photonics Society
IEEE Consumer Electronics Society IEEE Power Electronics Society
IEEE Control Systems Society IEEE Power and Energy Society
IEEE Council on Superconductivity IEEE Product Safety Engineering Society
IEEE Dielectrics and Electrical Insulation Society IEEE Professional Communication Society
IEEE Education Society IEEE Reliability Society
IEEE Electromagnetic Compatibility Society IEEE Robotics & Automation Society
IEEE Electron Devices Society IEEE Sensors Council
IEEE Engineering Management Society IEEE Signal Processing Society
IEEE Engineering in Medicine and Biology Society IEEE Society on Social Implications of Technology
IEEE Geoscience & Remote Sensing Society IEEE Solid-State Circuits Society
IEEE Industrial Electronics Society IEEE Systems, Man, and Cybernetics Society
IEEE Industry Applications Society IEEE Ultrasonics, Ferroelectrics, and Frequency Control
Society
IEEE Information Theory Society IEEE Vehicular Technology Society
IEEE Intelligent Transportation Systems Society
2 1 Introduction

When you have mastered the material presented in
this book, you will have a basic understanding of what
current and voltage are and how they behave. You will
be able to analyze and design a number of basic (but
important) kinds of circuits. You will have accumu-
lated a repertoire of physical laws and mathematical
procedures that are essential in study of more advanced
circuits and systems for communication, control, com-
puting, and other applications. You will have begun to
speak the language of electrical engineering, which is
the first step toward becoming an electrical engineer
and enjoying a challenging and rewarding career.
1.2 How to Study This Book
Because this is presumably one of your first engineer-
ing courses, you might be receptive to some advice
about how to study an engineering subject – and this
book, in particular. Most engineering subjects have
pretty much the same anatomy, illustrated by Fig. 1.1:
• At the core are (usually very few) undefined quan-
tities; for example, time and charge, and essential
properties of those quantities.2
• Next are definitions. These include definitions of
important quantities, such as current and voltage, in
terms of the core undefined quantities. Also
included here are important symbols, terminology,
and notation associated with the subject.
• Physical laws prescribe relations among defined
quantities; for example, in circuits, important phys-
ical laws are Ohm’s law and Kirchhoff’s laws.
• Derived results are important relations among the
defined quantities that are obtained using physical
laws, definitions, and mathematics; Thévenin’s the-
orem (Chapter 4) is an example of an important
derived result.
• Computational procedures systematically use
physical laws, derived results, and mathematics to
solve problems (e.g., to analyze or design circuits).
Examples are the node-voltage and mesh-current
methods for finding voltages and currents in circuits.
As you make your way through this book, note
whether each new thing you encounter as a definition,
a law, a derived result, or a procedure. Establish in your
mind the relation of the new thing to things learned
previously.
• You should memorize each definition, when it is
first encountered, before proceeding. If you do
not, chances are you will have difficulty under-
standing subsequent text, derivations, and exam-
ples. Definitions are given for a reason, the reason
being frequent subsequent use of the thing defined.
Special symbols comprise an important class of
defined terms and also should be memorized; for
example, in this book the letters i and v always
stand for current and voltage, respectively. Certain
dimensions and units also are defined quantities and
must be committed to memory.
• You must also memorize physical laws, because
they do not necessarily follow from anything else
presented in this book.3
Fortunately, there are rela-
tively few laws to be memorized.
• Derived results are obtained using definitions, laws,
limiting assumptions, and other derived results.
You should be sure you understand important lim-
iting assumptions before proceeding. In almost all
subjects, there are many more derived results than
physical laws. Your teacher can guide you regard-
ing which derived results you should memorize and
which you should be able to derive, yourself, pos-
sibly under slightly different conditions.
Undefined
Quantities
Definitions
Physical Laws
Derived Results
Computational
Procedures
Fig. 1.1 Anatomy of a technical subject
2
The essential property of time is perhaps best captured by Yogi
Berra’s definition: “Time is what keeps everything from hap-
pening all at once.” The essential properties of charge are given
in Chapter 2.
3
One subject’s laws can be another subject’s derived results; for
example, Ohm’s law (a law in circuits courses) is a derived
result in some more fundamental courses.
1.2 How to Study This Book 3

• A procedure is essentially a recipe for solving a class
of problems, based upon certain laws and derived
results. The class of problems to which a procedure
can be applied is defined by the conditions under
which the underlying laws and derived results apply.
For example, mesh analysis, which is a procedure
for finding voltages and currents in a circuit, is based
upon Kirchhoff’s voltage law, which in turn is appli-
cable only under certain restrictions on the natures
(e.g., wavelengths) of the voltages and currents
applied to the circuit.
Exercise 1.1. Recall from your study of phys-
ics that electric charge is conserved. In absence
of any further information, would you regard
this claim as a definition, a physical law, or a
derived result?
Viewing the content of a subject in the manner
illustrated by Fig. 1.1 can help you organize your
notes and thinking, especially at the end, when you
are preparing for a final examination. From the outset,
it can help you plan your outline and identify what is
truly important or general and what is less important or
specific as you proceed through the text. Before each
exam, for example, you might be sure that you can
answer the following questions:
• What quantities have been defined or described and
what are the definitions or defining properties?
Include symbols, terms, and basic units.
• What fundamental laws have been presented and what
relations do they establish among defined quantities?
• What derived results have been obtained and under
what conditions do they apply? What are the dimen-
sions and units of derived quantities and how are they
related to the dimensions and units of quantities
defined or derived previously? What laws and math-
ematical operations are needed to derive the results?
• What procedures have been presented and under
what conditions are they useful? What laws, derived
results, and mathematical operations do they use?
• What is a paradigm example4
for each law, derived
result, and procedure?
If you can do all that, you can be confident that you
have a good grasp of the material. The only thing left to
do is to hone your skills through practice; e.g., by
reworking homework problems and other problems
that are like those your teacher assigned or recom-
mended. As you work through examples and problems,
attempt to discern what definitions, laws, derived
results, and procedures the example or problem is
intended to reinforce.
1.3 Dimensions and Units
A quantity is something that is quantifiable; i.e.,
something that can be counted, measured, or calcu-
lated. In everyday life and especially in engineering
and science, we deal with many different quantities;
for example, time, length, weight, temperature, pres-
sure, speed, area, and cost. A quantity has a numerical
value and a dimension and is expressed by a number
and a unit. Both the numerical value and the unit are
necessary. One dollar and 1 yen have the same numer-
ical value (one) but have quite different values in the
marketplace.
A quantity can be expressed in various units; for
example, length can be expressed in inches, meters,
miles, furlongs, yards, angstroms, rods, light-years,
and other units. No matter how it is expressed, length
remains length. Length is what is being measured or
specified. The unit describes how (on what scale)
length is expressed. Length is a dimension and the
meter is a unit of length. Similarly, time is a dimension
and the second is a unit of time.
Exercise 1.2. Which of the following are quan-
tities (dimensioned or dimensionless), which are
units, and which are neither? (a) Time, (b) cen-
trifugal force, (c) electron-volt, (d) Newton-
meter, (e) light-year, (f) age, (g) checking
account balance, (h) purple, (i) longitude, (j)
relative humidity, (k) north pole, (l) virtue, (m)
valor, (n) micron.
There are many different quantities, but only seven
basic dimensions (basic quantities). The dimensions
of all other known quantities can be expressed in terms
of these seven and are called derived dimensions.
4
A paradigm example is one that illustrates application of a law,
derived result, or procedure in the simplest possible setting,
uncluttered by need for other laws, results, or procedures.
4 1 Introduction

Derived dimensions and units also are called com-
pound dimensions and compound units because they
are products and quotients of basic dimensions and
units; for example, the dimension of area is length
squared and the dimension of speed is length/time.
The dimensions of area and speed are derived (com-
pound) dimensions.
Choosing the seven basic dimensions, the asso-
ciated basic units, and names of units for important
derived quantities establishes a system of units. The
official system of units for engineering and science is
the SI system,5
which was adopted by a number of
very important people at a long meeting in Paris in
1960. In the SI system, the seven basic dimensions are
length, time, mass, electric current, temperature,
amount of substance, and luminous intensity.6
So far
as we know, none of the seven can be expressed in
terms of the other six. Of the seven basic quantities,
only five are used in this book. These five and the
associated SI units are given in Table 1.2.
Note that electric current is the basic electrical quan-
tity in the SI system. Electric charge would serve as well,
and because current in a circuit is flow of charge (charge
per unit time), charge would seem to be the more logical
choice. But there is a good reason for choosing current,
rather than charge, as a fundamental quantity. It turns out
that there are two kinds of current: Conduction current,
which is flow of charge, and displacement current,
which can exist where no charge is present, as in a
time-varying electromagnetic field (e.g., a radio wave).
Thus, from an engineering perspective, current appears
to be a more basic quantity than charge.
In any particular discipline, many quantities of
interest are derived quantities; for example, in EE,
we deal not only with current and time, but also with
voltage, work (or energy), power, and many other
quantities. Repeated use of compound units for all
these quantities, such as kg m2
s2
for energy, is cum-
bersome. Compound units for important derived quan-
tities are given their own names, most of which are
chosen in honor of great engineers or scientists some-
how associated with the quantity. Table 1.3 gives the
names, symbols, and SI units for quantities (both basic
and derived) used in this book, in alphabetical order.
Table 1.3 also gives compound equivalents for derived
units. Compound equivalents are useful in checking
expressions for dimensional correctness (see below).
Quantities used only occasionally in this book are not
included in the table but are defined and discussed
appropriately where they are introduced.
There are two kinds of dimensionless quantities:
(1) Those whose definitions do not involve any dimen-
sioned quantities, such as atomic number and valence;
and (2) those defined in terms of dimensioned quan-
tities, but in such a way that all of the basic units
involved cancel, such as the fine-structure constant
and an angle. We use angles extensively in this book,
so a little discussion of their nature is in order.
Fundamentally, an angle is the quotient of two
lengths (arc length/radius). The dimensions of length
cancel from the quotient, so an angle is dimensionless
(the compound unit of an angle is unity). Most dimen-
sionless quantities are expressed without units. Angles
are exceptions because an angle can be specified in
either radians (rad) or degrees (deg), where p rad ¼
180 deg. Although dimensionless, radians and degrees
are different scales of measurement. In absence of any
prior agreement, we must attach the correct one of
those units to the numerical specification of an angle.
Except in Chapter 14, we specify angles in radians, so
we do not attach the dimensionless unit rad to any such
specification. In this book, specifying an angle as (e.g.)
y ¼ 1:47 means y ¼ 1:47 radians.
Exercise 1.3. Give a simpler unit for each of
the following: (a) J s–1
, (b) V A s, (c) C s–1
, (d)
A2
O, (e) O H F2
s2
, (f) kg m2
s2
.
Table 1.2 Basic SI quantities and units
Quantity (basic dimension) Name of SI unit SI abbreviation
(symbol) for unit
Length Meter m
Time Second s
Mass Kilogram kg
Electric current Ampere A
Temperature Kelvin K
5
SI System is redundant, because SI stands for Systeme Inter-
nationale, which (obviously) is French for international system.
Nonetheless, “the SI system” is what everyone calls it.
6
Actually, which seven dimensions are defined as basic is a
matter of choice. Any seven independent dimensions would
do, but history and common sense have ruled in favor of
the seven given here. There is much wider agreement on
which seven quantities are basic than on the standard units for
those seven.
1.3 Dimensions and Units 5

There are several important conventions and rules
regarding dimensioned quantities:
• The symbol for a quantity stands for both the
numerical value and the associated unit, as in x ¼
10 m and t ¼ 5 s. Constructions such as “x meters”
and “t seconds” are redundant.
• Reserved symbols for quantities (e.g., t for time)
carry subscripts where more than one such quantity
is of interest. For example, times at which various
events occur in a circuit might be designated t0, t1,
t2, and so on. The presence of a subscript does not
alter the dimension or unit of a quantity.
• Terms that are equated, compared (using or ),
added, or subtracted must have the same dimension
and, if numerical, must be expressed in the same
unit.
• In particular, all equations must be dimensionally
homogeneous; that is, all terms in an equation must
have the same dimension.
• The dimension of the product of quantities is the
product of the dimensions of the quantities.
• The dimension of the quotient of quantities is the
quotient of the dimensions of the quantities. The
quotient of like quantities is dimensionless.
Table 1.3 Symbols and SI units. Names of basic dimensions are in bold type
Quantity Symbol(s) SI unit (symbol) Equivalent unit(s)
Admittance Y siemen (S) O1
, A V1
Angle y, f, a, b, F, Y radian (rad) None
Angular frequency o radians per second (rad/s) s1
Capacitance C farad (F) AsV1
: CV1
Charge q coulomb (C) A·s
Complex frequency s radians per second (rad/s) s1
Complex power S volt·ampere (VA) J s1
, W
Conductance G, g siemens (S) O1
, A V1
Conductivity s siemens per meter (S m1
) Om
ð Þ1
Current i, I ampere (A) Basic dimension; Cs1
; VO1
Distance or length x, y, z, h, d, l meter (m) Basic dimension
Electric potential F volt (V) J C1
Electric field strength E newtons per coulomb or volts per meter NC1
or Vm1
Energy w joule (J) V C, N m, W s
Force f, F newton (N) kg m s2
, V C m1
Frequency f hertz (Hz) s1
Impedance Z ohm (O) V A1
Inductance L henry (H) V s A1
; WbA1
Magnetic flux f weber (Wb) Vs
Mass m kilogram (kg) Basic dimension
Period or interval T second (s) S
Permeability m henries per meter (H m1
) V s A1
m1
Permittivity e farads per meter (F m1
) A s V1
m1
Pole p radians per second s1
Power p, P watt (W) J s1
, V A
Reactance X ohm (O) V A1
Reactive power Q volt ampere reactive (VAR) J s1
, V A, W
Resistance R, r ohm (O) V A1
Resistivity r ohm meter (O m) V A1
m
Speed or velocity u meters per second (m s1
) m s1
Susceptance B siemen (S) O1
, A V1
Temperature T kelvin (K) Basic dimension
Time t second (s) Basic dimension
Time constant t second (s) S
Voltage v,V volt (V) J C1
, A O
Work w joule (J) V C, N m, W s
Zero z radians per second s
6 1 Introduction

• The dimension of a differential is that of the asso-
ciated variable; e.g., the dimension of dt is that of t.
• Dimensionally, a derivative is a quotient and an
integral is a product; e.g., the dimension of dq/dt
is that of q/t and the dimension of
R
i dt is that of i·t.
• Definite limits of integration must have the dimen-
sion of the differential; for example, in
Rb
a fðxÞ dx,
both a and b must have the dimension of x.
A similar condition applies to limits in general.
• The units of infinite limits of integration 1
ð Þ
are understood to be those of the differential; e.g.,
in
Rt1
1 iðtÞ dt it is understood that 1 stands for
an exceedingly large and negative time. A similar
condition applies to limits in general.
• With very few exceptions, arguments of functions
must be dimensionless; for example, we cannot take
the logarithm of 50 kg. The reason why is that ln (x)
can be expressed as an infinite power series, and we
cannot (e.g.) add kg to kg2
to kg3
. . . .7
• Finally, voltage, current, power, and time, if zero or
infinite, are zero or infinite in any system of units.
We do not attach units to those quantities when
they are zero or infinite; for example, we say
t!1, not t!1 s, because if time approaches infin-
ity, it does so whether measured in seconds, min-
utes, or weeks.8
Similarly, we say v ¼ 0, not v ¼
0 V, because if a voltage equals zero, it is zero in
any system of units.
It is poor practice to write statements such as “v ¼
v0et
, where time t is expressed in milliseconds.” Such
practice makes it impossible to check relations for
dimensional consistency and thereby robs us of an
opportunity to catch careless errors. Further, such
practice robs us of many opportunities for developing
insight; e.g., by hiding the all-important time constant
in the exponential relation above. It is far better to
adopt the convention that t always stands for time in
seconds and to write the statement above as
“v ¼ v0et=t
, where t ¼ 1 ms.”
Units and dimensions are not only essential parts of
the values of physical quantities, but also provide
insight into the meaning and reasonableness of derived
results. Sometimes they even guide derivations of
new relations. They also provide a powerful means of
checking correctness of equations and answers. An
equation that is dimensionally inconsistent is incor-
rect. There is no reason to proceed with a solution until
the error is found and corrected.
Make it a habit to check all derived equations and
relations for dimensional consistency. An equation is
dimensionally consistent if all terms have the same
unit. We use SI units exclusively, so you may check
units instead of dimensions, which is somewhat easier,
because it is easier to remember the units of quantities
than it is to remember the (usually compound) dimen-
sions for the quantities. We use SI b
ð Þ to denote the SI
unit of a quantity b, where
SI b
ð Þ ¼
the SI unit of b; if b is dimensioned;
1; if b is dimensionless;
(
(1.1)
and
SI ab
ð Þ ¼ SI a
ð Þ SI b
ð Þ; SI
a
b

¼
SI a
ð Þ
SI b
ð Þ
: (1.2)
Example 1.1. We wish to show that the dimen-
sion of the quantity RC (resistance times capac-
itance) is that of time. From Table 1.3,
SI RC
ð Þ ¼ SI R
ð ÞSI C
ð Þ ¼ VA1

A sV1

¼ s:
Thus the unit of the quantity RC is seconds
(s) and the dimension of RC is time.
Example 1.2. We wish to show that the
expression
i ¼ C
dv
dt
is dimensionally consistent, where i is current,
C is capacitance, v is voltage, and t is time.
7
Of course, the coefficients of a Taylor series can be dimen-
sioned, and one could define different series for (e.g.) the loga-
rithm of different physical quantities. Such an approach would
be cumbersome, at best.
8
Some mathematical software (e.g., Mathcad) that allow units to
be used insist that they be attached to all quantities, even those
having zero magnitude. For example, if you ask Mathcad to
compare 10 s to 0, you will get an error. You must compare
10 s to 0 s.
1.3 Dimensions and Units 7

Current i stands alone on the left, so we seek to
reduce the unit of the term on the right to the
ampere. From Table 1.3
SI C
dv
dt

¼ SI C
ð Þ
SI v
ð Þ
SI t
ð Þ
¼ AsV1
V
s

¼ A:
Thus the equation is dimensionally consis-
tent.
Exercise 1.4. Show that the relation
i ¼
1
L
ðt
1
vðt0
Þdt0
where i is current, L is inductance, v is voltage,
and t is time, is dimensionally consistent.
Exercise 1.5. Check each relation for dimen-
sional consistency.
ðaÞ L
di
dt
þ Ri
1
RC
ðt
1
i t0
ð Þdt0
¼ 5v1;
ðbÞ
V
I
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 þ oL
ð Þ2
q
1 þ oRC
:
As an aside, the SI system includes no unit for
weight. Weight is force due to gravity and is approxi-
mately a fixed multiple of mass everywhere on the
earth’s surface, given by f ¼ mg, where m is mass and
g is acceleration due to gravity. Thus, on the earth’s
surface, mass serves as well as weight to express how
heavy one thing is relative to others; that is, if one
thing has twice the mass of another, it also has twice
the weight of the other (in the same location). Conse-
quently, there is really no need for a quantity called
weight. Mass serves just as well. Conversions from
mass in kilograms to weight in pounds build in the
acceleration due to gravity (approximately 9.8 m s–2
)
and are really conversions from mass to force on the
mass due to gravity. The conversion factor works out
to be about 0.454 kg lb–1
, so 1 kg (mass) of butter
weighs about 2.2 lb.9
The size (scale) of a unit often is inappropriate to a
particular problem or even to an entire discipline; for
example, the meter is much too large a unit to use in
atomic physics and much too small a unit to use in
astrophysics. Even though the SI system is the official
system of units, other units that have proved useful
remain in use (e.g., the angstrom and the light-year).
Also, engineers must communicate effectively with
those in other professions and trades, not all of whom
have embraced the SI system so warmly, and it often is
necessary to convert one unit of measure to another (e.g.,
meters to inches). We use the SI system almost exclu-
sively in this book so we omit a table of conversion
factors. Such tables can be found in most standard
handbooks.10
In the SI system, very large multiples and very
small fractions of units are made more manageable
by using named prefixes that denote multiplication by
a power of ten. The power of ten denoted by most
prefixes is a positive or negative whole multiple of
three. Those that are not are not widely used in elec-
trical engineering, except for the centimeter (0.01 m)
and angstrom (1010
m). Table 1.4 gives selected SI
prefixes.
Table 1.4 SI prefixes
Prefix Abbreviation Value
Femto f 1015
Pico p 1012
Nano n 109
Micro m 106
Milli m 103
Centi c 102
Kilo k 103
Mega M 106
Giga G 109
Tera T 1012
Peta P 1015
9
The abbreviation lb stands for libra, an ancient Roman unit of
weight which was presumably the predecessor to the English
pound. See Yunus A. Cengel and Michael Boles, Thermodynam-
ics: An Engineering Approach (3rd Ed.), McGraw-Hill, 1998.
10
An especially complete table of conversion factors is given in
the Handbook of Chemistry and Physics (76th Ed.), edited by
David R. Lide, CRC Press, New York, 1995.
8 1 Introduction

There are three basic rules regarding use of
prefixes:
• Prefixes are not to be repeated or used with other
prefixes; e.g., use pF, not mmF and ns, not mms.
• Where a prefix is used with a symbol, the prefix and
the symbol become one quantity that can be raised
to a power or used in a fraction without parenth-
eses; for example, 1 cm2
¼ 104
m2
and 4 m/ms ¼
4000 m s1
.
• Prefixes are never used alone, even for purely
numerical quantities.
These rules are designed to eliminate ambiguity;
for example, although the symbol m for meter and the
prefix m for milli are identical, there is no possibility
for confusion because m for milli is never used alone
or repeated (mm always stands for millimeter, never
milli-milli).
Nonetheless, there remain possibilities for confusion.
For example, does mV mean millivolt or meter-volt? In
this case, we can avoid confusion by writing mV for
millivolt and Vm for volt-meter. But what about Vms?
Does it mean volt-millisecond or volt-meter-second? In
such cases we can use spaces, hyphens, dots, or par-
entheses to avoid ambiguity. For example, Vms means
volt-millisecond and Vm s, Vm-s, Vm s, and (Vm)s all
mean volt-meter-second.
There is no need (at present) to memorize any entries
in the tables above. You will learn many of them
through repeated use in subsequent sections. But you
should learn the three rules above and become profi-
cient at checking dimensions (units) using the proce-
dure illustrated in the examples above. Again, you
should form the habit of checking units with reasonable
frequency as you work through a problem. Doing so
will help you catch many careless errors and help you
avoid the fruitless labor of solving incorrect equations.
Exercise 1.6. Express each of the following
without prefixes: (a) 100 mAs1
(b) 10 V ms1
,
(c) 100 kV A, (d) 10 mm ms2
.
Exercise 1.7. Using prefixes, express each
of the following in at least two other ways:
(a) 5 mVs1
, (b) 25 kAms, (c) 100 mJms1
.
Exercise 1.8. Specify whether each m stands
for meter or milli in each of the following:
(a) Vms1
, (b) m Ams1
, (c) Nm·s1
,
(d) m2
s1
, (e) mms1
.
1.4 Symbols and Notation
Symbols for quantities and units used in this book are
those in Table 1.3. Because the alphabet is finite, some
symbols have more than one meaning. To the extent
possible, we avoid conflicts as follows:
• Symbols for quantities are always in italics and
symbols for units are always in roman; e.g., C
(italic) is the symbol for capacitance (a quantity)
and C (roman) is the symbol for the coulomb (the
SI unit of charge). This convention is not entirely
satisfactory because it is difficult to distinguish
italic and roman letters in handwritten notes and
assignments. Where conflicts arise (and confusion
is possible), we suggest using an underscore to
denote roman (e.g., C for coulomb).
• Symbols for complex representations of currents
and voltages are printed under a tilde; for example,
I˜and ~
V denote complex quantities called phasors,
defined in Chapter 12, and used to represent sinu-
soidal currents and voltages. We use a tilde in these
cases because V and I (no tilde) denote associated
real quantities (the magnitudes of the corresponding
complex quantities).
• We rely on context when there is no tolerable
alternative; for example, the symbols for period
and temperature are identical, and we must rely
on context to tell us what T stands for.
• The principal quantities of interest in circuit analy-
sis and design are time, current, voltage, work, and
power. In this book, t always stands for time and
i and v and their italic capitals always stand for
current and voltage, respectively.
• To avoid clutter, explicit time dependence often is
omitted; for example, v and i alone denote v(t) and
i(t), respectively and dv/dt stands for dv(t)/dt.
The mathematical notation used in the text is gen-
erally that used in standard prerequisite mathematics
1.4 Symbols and Notation 9

courses. Table 1.5 gives symbols and abbreviations
used throughout the text. For example, we might
write that 3x þ 4 ¼ 10 ) x ¼ 2, which reads as
“three x plus four equals ten implies that (or yields) x
equals two.”
Equations are numbered only for cross-reference.
A numbered equation is not necessarily an important
equation.
Boldface type is used when an important quantity
or term is introduced or defined. Italic type is used to
highlight secondary definitions, identify important
terms, and emphasize important comments.
As is standard in technical works, we use a number
of Greek letters. Table 1.6 gives the Greek alphabet and
the usual meaning (if any) assigned to each Greek letter
in this book. For the most part, these choices reflect the
meanings assigned to symbols in practice. Some (such
as d) have multiple meanings, whereas others are not
used. Also, the usual meaning is not necessarily the
only meaning used in this book. We occasionally assign
a meaning not given in Table 1.6 for temporary use in a
particular development or section.
1.5 Symbols Versus Numbers
In general, symbolic relations among quantities are
much more valuable than particular numerical values.
For example, Newton’s law f ¼ ma is much more
valuable and informative than would be a table of
particular values for force, mass, and acceleration.
The symbolic relation f ¼ ma tells us that acceleration
is proportional to force, whereas knowing only that a
1 kg mass subjected to a 2 N force experiences an
acceleration of 2 ms2
is of little general value. In
engineering and especially in conceptual design, rela-
tions among quantities are of far more interest than
particular numerical values for those quantities. Indeed,
much of electrical engineering focuses on designing
things (such as circuits) that create desirable relations
among physical quantities, such as voltages.
Resist the urge to prematurely replace quantities by
their numerical values, even when those values are
known and fixed. Instead, form the habit of working
with symbols until the desired relation is obtained,
checking units along the way. Only when you are
certain you have a correct relation should you seek a
particular numerical answer (with correct units), if that
is what is called for in the problem. Jumping to num-
bers too quickly eliminates the possibility of checking
dimensions, greatly limits the implications that can be
drawn from the solution, and multiplies the number of
different problems that must be solved by a large
factor. To encourage you to use symbols and symbolic
relations and to help you become proficient in that
regard, many of the problems in this book ask you to
obtain relationships among or (symbolic) expressions
for quantities.
Also, put your years of mathematics study to good use.
Assign symbols to quantities that occur frequently in a
development. For example, if you find the quantity oL/R
occurs repeatedly in a particular development, you might
define Q ¼ oL/R to simplify further manipulations.
Where possible, use a symbol that suggests the dimension
of the quantity; for example, if the quantity RC occurs
repeatedly in a particular analysis, you might let t0 or t
denote RC, because the dimension of RC is time.
1.6 Presentation of Calculations
Intermediate steps in the calculation of a value from a
symbolic expression can be presented in basically two
ways: The first (and generally best for students) is to
exhibit both the value and the unit of each quantity in
the expression. The second is to simply exhibit values,
attaching the correct unit to only the final expression.
For example, suppose we must calculate the value of
Z
j j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R2 þ 2pf L
ð Þ2
q
;
where R ¼ 1 kO, f ¼ 10 kHz, and L ¼ 10 mH. Any
of the following presentations would be considered
Table 1.5 Notation used in the text
Symbol Meaning
! Approaches
ffi Is approximately equal to
Is identically equal to
) Implies or yields
, Implies and is implied by
iff if and only if
e.g. For example
i.e. Namely, or that is, or in other words
10 1 Introduction

acceptable by most publishers of technical journals
and textbooks:
Z
j j¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1;000O
½ 2
þ 2p 104
Hz

102
H

2
q
¼1;181O;
Z
j j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 kO
½ 2
þ 2p 10 kHz
ð Þ 10 mH
ð Þ
½ 2
q
¼ 1:18 kO;
Z
j j ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1; 000
½ 2
þ 2p 104

102

2
q
¼ 1; 181 O ¼ 1:18 kO
:
In the first two of the three presentations above, the
units of the quantities involved are shown throughout.
In the third, the unit is attached to only the ultimate
result, and the unit (O) transcends the equal sign. All
three presentations are considered dimensionally cor-
rect. Which of these presentations you use will depend
upon your instructor’s preference, your confidence in
the dimensional correctness of the symbolic expres-
sion, and your familiarity with units and prefixes (e.g.,
you might recognize that the k in kHz and the m in
mH in the second of the two presentations cancel).
Generally, it is good to use the first form in the
beginning, progressing to the third as you gain famil-
iarity with various expressions and the quantities
involved.
1.7 Approximations
Approximations are valuable and even inherent in
engineering analysis and design. They are valuable
because they can simplify computations and aid
insight. In many cases, engineers employ without hes-
itation approximations that can introduce errors as
large as 10%. Approximations are inherent because
circuit parameters are imprecise. For example, the
actual capacitance of a 10 nF capacitor might be
anywhere in the range 10 nF 20%. As a reminder,
we use the symbol ffi to denote is approximately equal
to; e.g., p ffi 3:14.
Opportunities for approximation arise repeatedly in
circuit analysis and design. Almost all approximations
are based in some way upon the basic approximation:
x1
j j x2
j j ) x1 þ x2 ffi x1: (1.3)
Table 1.6 Greek alphabet and common uses for characters
Lower-case Name Usual meaning Upper-case Usual meaning
a Alpha Peaking factor A Not used
b Beta Tolerance B Not used
w Chi Not used X Not used
d Delta skin depth; damping factor; loss angle D Increment prefix; e.g., Dx means a small
increase in a quantity x
e Epsilon Error, permittivity, energy E Not used
f Phi Angle F Angle, electric potential
g Gamma Ripple factor G Not used
Eta Efficiency H Not used
i Iota Not used I Not used
k Kappa Not used K Not used
l Lambda Wavelength L Not used
m Mu Permeability; voltage gain M Not used
n Nu Not used N Not used
o Omicron Not used O Not used
p Pi 3.14159. . . P Product
y Theta Angle Y Angle
r Rho Resistivity P Not used
s Sigma Conductivity; damping ratio S Sum
t Tau Time constant T Not used
u Upsilon Not used U Not used
o Omega Angular frequency O Ohm (unit)
x Xi Not used X Not used
c Psi Angle C Angle
z Zeta Not used Z Not used
1.7 Approximations 11

In many cases, the basic approximation (1.3) is
acceptable if x1
j j 10 x2
j j (error ffi 10%) because, as
it turns out, values of circuit parameters often are even
more uncertain than that.
Example 1.3. Assume R1
j j R2
j j and find
the approximate value of
G ¼
1
R1
þ
1
R2
:
Solution:
G ¼
1
R1
þ
1
R2
¼
R1 þ R2
R1 R2
ffi
R1
R1 R2
¼
1
R2
:
Or
R1
j j R2
j j )
1
R1
1
R2
)
1
R1
þ
1
R2
ffi
1
R2
:
Example 1.4. Assume x
j j 1 and find the
approximate value of y ¼ sinðxÞ.
Solution: Using Taylor’s expansion gives
y ¼ sinðxÞ ¼ x
x3
3!
þ
x5
5!
;
which can be written
y ¼ sinðxÞ ¼ x 1
x2
3!
þ
x4
5!
:
Because x
j j 1,
1
x2
3!
þ
x4
5!
ffi 1;
and so
y ¼ sinðxÞ ffi x; x
j j 1:
Example 1.5. Certain irrational numbers such
as p and
ffiffiffi
2
p
arise often in expressions for quan-
tities of interest in circuit analysis. Faced with
such an expression, and needing to make a
quick mental or pencil-and-paper calculation,
we might use
p ¼ 3:0000 þ 0:1416 ffi 3 ðerror 5%Þ;
ffiffiffi
2
p
¼ 1:000 þ 0:4142 ffi 1:5 ðerror ffi 6%Þ:
Certainly the wide availability of pocket calcula-
tors and personal computers has greatly reduced need
for hand computation and therefore the need for
approximations in numerical computations. On the
other hand, ability to make approximations that sim-
plify symbolic expressions is valuable, because such
simplifications often facilitate insight; e.g., in identi-
fying which components of a circuit are critical to
performance. Judicious use of approximations can
foster insights that even the most careful computer
calculation cannot reveal.
Example 1.6. It is found that a performance
measure called voltage gain for a certain elec-
tronic circuit is given by
Av ¼
b RC RL
RB þ rb þ RS
ð Þ RC þ RL
ð Þ
;
where all quantities on the right are circuit para-
meters. It is known that RB rb, RB RS, and
RL RC. Using reasonable approximations,
simplify the expression on the right and identify
the circuit parameters that are critical to obtain-
ing a specified voltage gain.
Solution: Using (1.3), we have
RB rb ) RB þ rb þ RS ffi RB þ RS;
RB RS ) RB þ RS ffi RB;
RL RC ) RL þ RC ffi RL:
Thus
12 1 Introduction

Av ¼
b RC RL
RB þ rb þ RS
ð Þ RC þ RL
ð Þ
ffi
b RC RL
RB RC þ RL
ð Þ
ffi
b RC RL
RB RL
¼
b RC
RB
:
Under the conditions given, the critical para-
meters are b, RC, and RB. The parameters RS, RL
and rb have relatively little influence on voltage
gain.
Exercise 1.9. The voltage gain of a certain
circuit is given by
Av ¼
a
1 þ a b
;
where a and b are circuit parameters. Assume
a b
j j 1 and give an approximate expression
for Av. If this assumption holds, which circuit
parameter determines the voltage gain of the
circuit?
Some care is required in applying (1.3) to quantities
raised to powers. For example, even if x
j j 1, it is not
necessarily true that 1 þ x
ð Þn
ffi 1, especially if n
j j is
large. In such cases, the binomial series expansion
1þx
ð Þn
¼1þnxþ
n n1
ð Þ
2!
x2
þ
n n1
ð Þ n2
ð Þ
3!
x3
þ
often justifies the approximation
1 þ x
ð Þn
ffi 1 þ n x: (1.4)
The approximation (1.4) is justified if the quadratic
term is much smaller (in magnitude) than the linear
term; i.e., if
n 1
ð Þ x
j j 2: (1.5)
For example, the true value of (1.05)4
is
1.21550625. Equation (1.4) gives
1:05
ð Þ4
¼ 1 þ 0:05
ð Þ4
ffi 1 þ ð4Þð0:05Þ ¼ 1:2;
which is within 1.3% of the true value.
An asymptotic approximation to a function is
the limiting form of the function for either exceed-
ingly large or exceedingly small values of the inde-
pendent variable. As an example, functions of the
form
fðxÞ ¼ 1 ex
are encountered frequently in electrical engineering.
Such functions arise in analysis of circuits containing
capacitors and inductors, where (usually) x 0 and
in analysis of semiconductor devices, such as diodes
and transistors where (often) x 0. Thus two asymp-
totic approximations of this function are important:
x 0; jxj 1 ) 1 ex
ffi 1;
x 1 ) 1 ex
ffi ex
: (1.6)
In some cases, these approximations are deemed
valid if x
j j 3, because e3
ffi 20 is 20 times larger
than one (5% error) and e3
ffi 0:05 is only about 5%
of one. In others, we might require x
j j 5 or more, in
which case the error in (1.6) is less than 1%.
Exercise 1.10. An asymptotic approximation
used extensively in a subsequent chapter is
log x þ 1
ð Þ ffi log x
ð Þ; x 1:
For what values of x is the error in this
approximation less than 10%?
Approximations for elementary functions can
be obtained from Taylor series expansions. For
example,
exp x
ð Þ ¼ 1 þ x þ
x2
2!
þ ffi 1 þ x; x
j j 1
cos x
ð Þ ¼ 1
x2
2!
þ
x4
4!
ffi 1; x
j j 1
sin x
ð Þ ¼ x
x3
3!
þ
x5
5!
ffi x; x
j j 1
ln x þ 1
ð Þ ¼ x
x2
2
þ
x3
3
ffi
0; x
j j 1
ln x
ð Þ; x 1
( )
(1.7)
1.7 Approximations 13

Exercise 1.11. Without using a calculator or
computer, give approximate values for (a)
1
2:02
, (b) ln 1 þ e5

, (c) sin 0:501p
ð Þ, (d)
2 þ e0:02
, (e)
1
5
þ
1
0:02
:
Exercise 1.12. A particular current in an elec-
tric circuit is found to be
i1 ¼
R2
R1 þ R2
i0:
Find approximations for i1 (a) if R2 R1
and (b) as R1 becomes arbitrarily large (i.e., as
R1 ! 1).
You should cultivate your ability to make reason-
able approximations, partly because approximations
will help you determine whether a particular answer
or expression is reasonable, but mainly because
approximations often foster useful insight; e.g., as to
which components of a circuit are critical to accept-
able performance.
1.8 Precision and Tolerance
Values of circuit components are never known with
infinite precision. Rather, there is a precision asso-
ciated with every such value; for example, the value
of a resistance might be specified as 1 kO 5%, mean-
ing that the actual value is somewhere between 950.0
and 1,050.0 O. Thus, there also are precisions asso-
ciated with calculated values of functions of circuit
parameters.
Example 1.7. In the expression I¼V/R, the
value of V is known to be in the range
V0 2% and the value of R is known to be in
the range R0 10%. What is the precision of
the calculated value of I?
Solution: Let V0, R0 denote the nominal (zero-
error) values for V and V, and let I0 ¼ V0/R0
denote the zero-error value for I. The maximum
error occurs if V has its maximum value and
R has its smallest value. The minimum error
occurs if V has its minimum value and R has its
maximum value. Thus
Imax ¼
Vmax
Rmin
¼
1:02V0
0:90R0
¼ 1:13I0;
Imin ¼
Vmin
Rmax
¼
0:98V0
1:10R0
¼ 0:89I0:
It follows that 0:89I0 I 1:13I0, or
I ¼ I0 þ 13% or 11%, which might be writ-
ten I ¼ I0 þ 13%/11%.
In electrical engineering, limits on precision (e.g.,
the þ13% and 11% in Example 1.7) are called tol-
erances. For example, if the resistance of a resistor is
known with precision 5%, we say that the tolerance
of the resistor is 5%. Tolerances can be symmetric,
as in 5%, but often are not. Capacitances often have
asymmetric tolerances; e.g., C ¼ C0 þ 20%= 10%.
1.9 Engineering Notation
In engineering notation (a special kind of scientific
notation), values are typically written using the appro-
priate number of significant digits and a multiplier of
ten raised to a power that is a whole multiple of three;
for example, 1.56 106
and 67.0 103
. Usually, the
factor 10n
is then replaced by a prefix on whatever unit
is associated with the value written (see Table 1.4). For
example, we would write 62.63 103
V(volts) as
62.63 kV (kilovolts). Using engineering notation can
avoid ambiguity associated with trailing zeros on a
whole number. For example, whereas the significance
of the trailing zero in 320 is ambiguous, it is clear that
3.2 102
has two significant digits. Similarly, 3200 is
ambiguous, but 3.20 103
has three significant digits.
Exercise 1.13. Write each number using engi-
neering notation. Assume trailing zeros are not
significant. (a) 2015, (b) 16,380,000, (c) 0.759,
(d) 0.000462, (e) 47.92 102
.
14 1 Introduction

Circuit components are available only in certain
standard sizes and tolerances; for example, standard
carbon-film resistors11
are readily available in toler-
ances of 10% and 5%, meaning that the actual
resistance is within those percentages of the specified
value. One can obtain high-precision components, such
as metal-film resistors having tolerances of 0.1% or
less; however, normal variations in ambient tempera-
ture and humidity or heating during soldering can cause
the actual values to drift by 1% or 2% or more from the
specified values. In general, actual values of circuit
parameters are seldom within even 1% of specified
values. Thus, when analyzing or designing circuits, it is
almost always unnecessary and sometimes misleading
to specify more than three significant figures in the
numerical value of any quantity. Using too many sig-
nificant figures can also cause you to miss opportunities
for laborsaving approximations; for example, by lead-
ing you to believe that two resistors having resistances
of 1:01256 kO and 1:01274 kO are different.
It would be troublesome and distracting to give the
precision of every computed value throughout our
introduction to linear circuits, so unless otherwise stated
in a problem or example, we shall for the most part treat
values as if they are known exactly. Again, this is a
relatively harmless practice in instructional material, so
long as you are forewarned that numerical results are
often specified more precisely than is truly justified. In
actual practice, however, it is usually essential to spec-
ify the precision of circuit parameters, based upon the
accuracy required of various currents and voltages.
1.10 Problems
Section numbers in shaded boxes indicate prerequisite
sections for problems that follow.
Section 1.2 is prerequisite for the following
problems.
P 1.1 Pick one of the IEEE technical societies from
the list on page 1.1 that sounds interesting to you. Visit
the IEEE website (www.ieee.org) to find out more
about the society. Briefly summarize the interests and
purpose of the society, and give an example of an item
that is related to the society’s work.
P 1.2 Use the procedure described in Section 1.2 to
outline the content of this chapter; i.e., list (or give a
reference to) the defined quantities (e.g., symbols,
special notation), physical laws, derived results, and
procedures (if any).
problems.
P 1.3 The symbols in each quantity below are as
defined in Table 1.2. Determine the SI unit of the
quantity.
(a)
1
C
ðt
1
iðt0
Þdt0
, (b) VI cos y
ð Þ, (c)
dq
dt
, (d) Ri2
,
(e)
v2
R
, (f)
Rv
~
Z
, (g)
L
R
di
dt
, (h) Cv.
P 1.4 Check each equation for dimensional
consistency.
(a)
R1R2
R1 þ R2
i þ R1C
dv
dt
¼ 0,
(b)
1
C
ðt
1
iðt0
Þdt0
þ L
di
dt
þ Ri ¼ v,
(c) C
dv
dt
þ
v
R
þ
1
L
ðt
1
vðt0
Þdt0
¼ i,
(d) R1R2C1C2
d2
v
dt2
þ
R1
L
dv
dt
þ R2i ¼ I,
(e)
~
Z1
~
Z2
R þ ~
Z1
~
I þ
~
V
~
Y
¼ ~
V0,
(f) LC
d2
v
dt2
þ RC
dv
dt
þ v ¼ RC
di
dt
.
P 1.5 Each quantity below is dimensionless.
Express the SI unit of the variable a in terms of the
SI units for current (A), voltage (V), and time (s).
ðaÞ
a V
RL
ðbÞ
I V
a L C
ðcÞ a V2 dv
dt
ðdÞ
a R I
L
di
dt

ðeÞ
1
a
ðt0
0
vðtÞ dt ðfÞ
a
ffiffiffiffiffiffiffi
L C
p
R C
ðgÞ
a p t
C v2
ðhÞ a2
C
dv
dt

L
di
dt

2
6
6
4
3
7
7
5
2
11
Resistance is defined in Chapter 2.
1.10 Problems 15

P 1.6 Express the speed of light in furlongs per
fortnight. (Refer to your dictionary for definitions of
furlong and fortnight.)
P 1.7 What is the SI equivalent of one light-year?
P 1.8 Express 1 kV/MHz in V/Hz.
P 1.9 Express the speed of light in m/ms.
P 1.10 Obtain the dimension and SI unit of
ffiffiffiffiffiffi
LC
p
,
where L is inductance and C is capacitance.
P 1.11 Obtain the dimension and SI unit of L/R,
where L is inductance and R is resistance.
P 1.12 Obtain the dimension and unit of ~
I Z, where ~
I
is current and Z is impedance.
P 1.13 Obtain the dimension and unit of
ffiffiffiffiffi
me
p
,
where m is permeability and e is permittivity.
P 1.14 Express the charge of an electron in m C, n C,
and p C.
P 1.15 Electrical utilities (and power-system engi-
neers) often express energy in kWh (kilowatt-hours).
Express 1 kWh in J (joules).
P 1.16 Find the mass, in kilograms, of a man that
weighs 200 lb.
P 1.17 List as many units as you can for each of
the following quantities and name disciplines in which
such units would be used: (a) length, (b) area, (c)
volume, (d) energy.
P 1.18 The fine-structure constant is a ¼
1
2 m0 c qe
2
h1
, where m0 is the permeability of a vacuum,
c is the speed of light in a vacuum, qe is the charge of an
electron, and h is Planck’s constant. Show that the fine-
structure constant is dimensionless.
P 1.19 The Stefan-Boltzmann constant is
s ¼ ðp2
=60Þ k4
h3
c2
, where k is the Boltzmann
constant, h is Planck’s constant, and c is the speed of
light in a vacuum. (a) Express the unit of the Stefan-
Boltzmann constant in terms of basic SI units. (b)
Show that the unit of the Stefan-Boltzmann constant
can be expressed as W m2
K4
.
P 1.20 The SI unit for resistivity is the O m, but this
unit is inconvenient (too large) in many cases, so
resistivity is often expressed in mO cm. Find the factor
that converts O m to mO cm.
P 1.21 Show that
i t
ð Þ ¼
1
L
ðt
1
vðt0
Þdt0
is dimensionally consistent.
P 1.22 Show that 1Vm1
¼ 1N C1
.
problems.
P 1.23 For each function given below: (i) Find
the small-x and large-x linear asymptotes. (ii) If
the asymptotes intersect, calculate the values of the
function and either asymptote at that point. (iii) Sketch
a graph of the function, using the asymptotes as guides.
(a) fðxÞ ¼ tan1
x
ð Þ, (b) fðxÞ ¼
1
1 þ x2
,
(c) fðxÞ ¼
1
x
, (d) fðxÞ ¼
0; x 0;
ex
; x 0;
(
(e) fðxÞ ¼
0; x 0;
1 ex
; x 0:
(
P 1.24 A certain voltage Vout is expressed in terms
of another voltage Vn by
Vout ¼
Rout mR1
Rout þ R1
Vn;
where R1, Rout are resistances and m is dimensionless.
It is known that R1 Rout and m 1. Find a reason-
able approximation for Vout.
P 1.25 A certain voltage VL in an electric circuit is
found to be
VL ¼
RL
RT þ RL
VT;
(a) Find a reasonable approximation for VL if RL RT.
(b) Find a reasonable approximation for VL if RT RL.
(c) Find VL as RT becomes arbitrarily large (i.e., as
RT ! 1).
P 1.26 A performance measure called current gain
for a particular circuit is found to be
Ai ¼
RinRoutRSg
Rout þ RL
ð Þ Rin þ RS
ð Þ
;
where all quantities on the right are circuit parameters.
(a) Find a reasonable approximation for Ai if
Rout RL.
(b) Find a reasonable approximation for Ai if
Rin RS.
(c) Find a reasonable approximation for Ai if both
Rout RL and Rin RS.
16 1 Introduction

(d) The current gain Ai is dimensionless. What is the
SI unit of g?
P 1.27 Give asymptotic approximations to
log
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ f=f0
ð Þ2
q
for (a) 0 f f0 and
(b) f f0 0.
P 1.28 Given that R1 R2 R3, obtain an
approximation to
1
R2
þ
R1 þ R3
R1R3
.
P 1.29 Given that R1 R2 R3, obtain an
approximation to
R1R2R3
R1R2 þ R1R3 þ R2R3
.
P 1.30 Let
Gð fÞ ¼ log 1 þ f=f0
ð Þ2
h i
þ log 1 þ f=f1
ð Þ2
h i
log 1 þ f=f2
ð Þ2
h i ;
where 0 f0 f1 f2. Give asymptotic approxima-
tions for G(f) for (a) f0 f f1, (b) f1 f f2, and
(c) f f2
P 1.31 Let
H f
ð Þ ¼
K
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ f=f1
ð Þ2
q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ f=f0
ð Þ2
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ f=f2
ð Þ2
q ; f0 f1 f:
Give asymptotic approximations for H(f) for (a)
f0 f f1, (b) f1 f f2, and (c) f f2
P 1.32 Show that if x and y are both positive, then
1
x
þ
1
y

1
x þ y
:
problems.
P 1.33 Estimate the precision of the indicated
quantity:
(a)
R1
R2
, where the precision of each R is 5%
(b)
R1R2
R1 þ R2
, where the precision of each R is 5%
(c) f=f0
ð Þ2
, where the precision of f0 is 1%. Does
the magnitude of the error depend upon the vari-
able f?
(d) 20 log½1 þ f=f0
ð Þ2
, where f is known exactly,
f f0, and the precision of f0 is 5%. Does
the magnitude of the error depend upon the
variable f ?
(e) expðt=RCÞ for t ¼ 20 ms, where R ¼ 1 kO 5%
and C ¼ 20 nF 20%.
(f)
R1R2R3
R1R2 þ R1R3 þ R2R3
, where the precision of
each R is 1%
problems.
P 1.34 Express each of the following using engi-
neering notation. Omit ambiguous zeros.
(a) 14,600 V, (b) 9,870,000 W, (c) 48:3 108
, (d) Five
million people, (e) 0.0056, (f) 0.0001, (g) 0.000708.
P 1.35 Express each of the following using unit
prefixes; for example, as in 47; 000 O ¼ 47k O.
(a) 12 million volts, (b) 4:2 106
A, (c) 56 105
W,
(d) 0.050 V, (e) 0.0002 A, (f) 4:8 63:4 109:0m,
(g) 0:0506 g 2048, (h) 10 109
Hz, (i) 34:7
1013
O.
P 1.36 Express each of the following using engi-
neering notation, without using unit prefixes; for
example, 123 kW ¼ 123 103
W.
(a) 55GHz, (b) 12 ns, (c) 100 MW, (d) 2:2 kOs,
(e) 100nF, (f) 14:6 ms, (g) 12mm, (h) 47 mF, (i) 12pF
(j) 108 mO cm, (k) 75 kV A, (l) 100 T O, (m) 1 fA.
1.10 Problems 17

Chapter 2
Current, Voltage, and Resistance
In this chapter, we define current and voltage, which
are among the principal quantities of interest in circuit
analysis. We present Ohm’s law, which is one of the
fundamental laws of electrical engineering, and which
is the defining relation for resistance and conductance.
2.1 Charge and Current
Charge is a property of electrons and protons, which
are building blocks of atoms. The SI unit of electric
charge is the coulomb (C).1
A quantity of charge is
represented by the symbol q and is in general a func-
tion of time.
Physicists are still hard at work trying to determine
what charge is made of, but we do not need to await
the results. We already know enough about how
charge behaves to make it do many useful things. We
accept charge as an undefined quantity whose proper-
ties are well understood. For electrical engineers, the
important properties of charge are the following:
• There are two kinds of charge: The negative charge
of an electron and the positive charge of a proton.
The charge of an electron is qe ﬃ 1:602 1019
C:
The charge of a proton is qp ¼ qe:
• Like charges (charges having the same sign) repel
each other and unlike charges attract each other.
The force on a point charge2
q1 due to the presence
of another point charge q2 is given (in a vacuum) by
Coulomb’s law
fðxÞ ¼
q1q2
4 p er e0 x2
¼ k
q1 q2
x2
; (2.1)
where x is the distance between the charges and
k ¼ 4 p er e0
ð Þ1
is a constant whose value depends
upon the medium. In free space, k is approximately
9 109
N m2
C2
: The force on q1 given by (2.1)
is attractive (directed toward q2) if q1 and q2 have
opposite signs and repulsive (directed away from
q2) if q1 and q2 have the same sign. The fact that
charges exert forces on one another allows us to
establish conditions under which some charges
make other charges do useful things.
• The charge of an electron (or proton) is the smal-
lest unit of charge and all charges are whole multi-
ples of that fundamental unit. This fact is
unimportant in ordinary circuit analysis, where so
many electrons and/or ions are involved in the
processes of interest that we can treat quantity of
charge as a real variable–one that can assume any
value. But it might be important in future ultra-
small devices, where relatively few electrons will
be involved in processes of interest.
• Charge can be neither created nor destroyed; that is,
total charge is conserved.3
If we enclose a volume
1
After the French physicist Charles Augustin de Coulomb
(1736–1806).
2
In practical applications, two charges can be considered point
charges if their radii are much smaller than their separation.
3
Total charge is the algebraic sum of charge. Thus, the net
charge of a hydrogen atom, which consists of one electron and
one proton, is zero.
T.H. Glisson, Introduction to Circuit Analysis and Design,
DOI 10.1007/978-90-481-9443-8_2, # Springer ScienceþBusiness Media B.V. 2011
19

with a surface and monitor the charge passing
through the surface, any increase or decrease in
the net charge enclosed in the volume will be
accounted for by passage of charge through the
surface. It is good that charge is conserved, because
if charge could suddenly appear (or disappear) here
and there, it would be difficult to keep track of and
control.
• A moving charge creates a magnetic field and a
magnetic field exerts a force on a moving charge.
These facts make electric generators, motors, and a
host of other devices possible.
Because of the properties above, charge can be
made to do useful work. We can use electric charge
to move enormous amounts of energy easily from one
place to another, transmit information from one place
to another, perform computations at unimaginable
speed, convert mechanical energy to electrical form
(and vice versa), and do many other useful things.
To do work, charge must move under the influence
of a force. Consequently, moving charge usually is of
more interest to electrical engineers than charge itself.
Motion or flow of charge is called current, denoted by
i. By convention, the positive direction of current is
opposite to the direction of electron flow.4
Current i
through a plane (e.g., a cross section of a wire) is
defined by5
i ¼
dq
dt
; (2.2)
where dq is a quantity of charge passing through the
plane in time dt and i is the current through the plane in
the direction of the flow of positive charge. Alterna-
tively, if q is a quantity of charge in an enclosed space,
then (2.2) gives the current passing into the space
through the enclosing surface. The SI unit of current
is the ampere (A),6
where 1 A equals 1 C s1
.
Example 2.1. A steady flow of 1020
electrons per
second is directed from left to right through a
cross section of a wire, as illustrated in Fig. 2.1.
The current from left to right through the cross
section is constant and equals
i¼ 1020 electrons
second

1:6021019 coulombs
electron

¼16:02 Cs1
¼16:02A
:
The current from left to right is negative
because negative charge is moving from
left to right. The current from right to left
is positive and equals +16.02 A.
Exercise 2.1. An electric current is estab-
lished in a tank of salt water, as shown in
Fig. 2.2. The positively charged sodium ions
are drawn left to the negative electrode and the
negatively charged chlorine ions are drawn
right to the positive electrode. Each ion is
singly charged and N ions reach each electrode
during each interval T. Give an expression for
the current I. Show that the expression is
dimensionally consistent.
electron flow
positive current
Fig. 2.1 The direction of positive current is opposite to the
direction of electron flow. See Example 2.1
4
We have Ben Franklin to thank for this unfortunate accident of
history. He decided to call the static charge induced on glass by
rubbing the glass with silk positive and stated that the direction
of current is from positive to negative (from glass to silk). Had
he made either of these choices the other way, the positive
direction of current would be in the direction of electron flow.
5
Explicit time-dependence often is omitted for the sake of brev-
ity. Thus, in (2.2) q stands for q(t) and i for i(t). Also, current
defined by (2.2) is called conduction current. When you study
electromagnetism, you will learn about another kind of current
called displacement current, that does not involve flow of
charge. Conduction current is the current of interest in ordinary
circuit analysis, where we are relatively unconcerned with the
inner workings of circuit elements.
6
After the French Mathematician and Physicist André Marie
Ampére (1775–1836), who discovered basic laws of electro-
magnetism.
20 2 Current, Voltage, and Resistance

2.2 Electric Field
In this section, we review the definition of work and
define the electric field established by a charge, draw-
ing on an analogy with gravity and the gravitational
field. This section is preparation for the next, where
voltage is defined in terms of work done on a charge
by an electric field.
Recall from physics that potential energy is capac-
ity for doing work and that the work w done by a force
f(x) on an object as the object moves from x1 to x2 is
given by7
w ¼
ðx2
x1
fðxÞ dx: (2.3)
The work done by the force is positive if the force is
in the direction of motion and negative if the force is
opposite to the direction of motion. Another way to
say this is that the force does work on the object if the
force and motion are in the same direction and work is
done on whatever provides the force if the force and
motion are in opposite directions. The SI unit of work
is the joule (J),8
where (in mechanical units) 1 J equals
1 N-m.
Work and energy are expressed in the same unit (J)
and are physically the same thing. The distinction
between work and energy is semantic, but useful.
Energy describes the state of something; for example,
we might speak of the potential energy of a mass that
has been raised to some height or the kinetic energy of
a moving mass. Work is a quantity of energy trans-
ferred from one thing to another or converted from
one form to another; for example, w in (2.3) is work
because it is a quantity of energy transferred from
whatever supplies the force f to an object acted on by
the force. Similarly, a quantity of electrical energy
converted to mechanical form by an electric motor is
called work.
An electric field and its interaction with charge are
analogous to a gravitational field and its interaction
with mass. On or near the earth’s surface, the gravita-
tional force f on a mass m is nearly independent of
position, is directed toward the center of the earth, and
is given by f ¼ mg, where g is the acceleration due to
gravity (approximately 9.8 m s–2
at sea level). For
purposes of ordinary (non-relativistic) mechanics, we
can define gravitational field strength F (a vector) near
the earth’s surface as the force per unit mass due to
gravity; that is, as the gravitational force on a mass
divided by the mass. Thus, the magnitude of the field
strength F is F ¼ f=m ¼ mg
ð Þ=m ¼ g and its direction
is toward the center of the earth.
A gravitational field possesses potential energy
because it does work on a mass that is allowed to
fall. The work done on the mass by the gravitational
field as the mass falls from height h to the ground is
w ¼ force x distance ¼ mgh. To put this another way,
the work w ¼ mgh is the amount of energy transferred
from the field to the mass as the mass falls from height
h. The work the gravitational field can do on the mass
(the energy that can be transferred from the field to the
mass) increases with the height of the mass. Thus,
energy is stored in the gravitational field when a
mass is raised and energy is released by the field
(transferred to the mass) when the mass falls.9
The
gravitational potential at a distance h above (but near)
the earth’s surface is defined by ’ðhÞ ¼ g h and equals
the work done per unit mass as a mass falls from or is
I
Battery
Fig. 2.2 See Exercise 2.1
7
To avoid using vector notation, we assume the force f(x) is
directed along the x-axis, in either the same direction as the
motion or opposite to the direction of motion.
8
After the English physicist James Prescott Joule (1818–1889).
9
In this discussion, we are concerned only with energy
exchanged between the gravitational field and a single mass.
There may be other forces acting on the mass in addition to that
due to gravity, but they do not affect the energy exchange
between the mass and the field; for example, we can throw a
mass toward the ground, in which case additional energy is
transferred to the mass, but the energy transferred to the mass
from the field is unaffected by the throwing.
2.2 Electric Field 21

raised to a height h. Thus the work w done is given by
w ¼ m ’ðhÞ: If the mass falls, work is done by the field
(energy is transferred from the field to the mass). If the
mass is raised, work is done on the field (energy is
transferred from the mass to the field).
Example 2.2. An object having mass m ¼
2 kg is raised from the ground to a height h ¼
10 m and released. The potential energy stored
in the field when the object is raised is
w ¼ mgh ﬃ 2kg
ð Þ 9:8ms2

10m
ð Þ ¼ 196 J:
When the object is released, this energy is
given back to the object as kinetic energy. The
kinetic energy of the object when it hits the
ground is 196 J. When the object is raised, 196
J are stored in the field. When the object is
released, the field gives up the stored energy by
doing work w ¼ 196 J on the object.
Exercise 2.2. An object having mass m is
tossed up. When it strikes the ground, its veloc-
ity (downward) is u. Give an expression for the
height attained by the object. Show that the
expression is dimensionally consistent.
Below, by analogy with gravitation, we define elec-
tric field strength and electric potential. We illustrate
these definitions using the field and potential due to a
point charge. The stated definitions are general, but
the expressions given apply to a point charge only.
Consider the situation shown in Fig. 2.3, where
positive point charges q1 and q2 are separated by a
distance x. We assume the charge q2 is fixed and we
can somehow move the charge q1 about.
By Coulomb’s law (2.1), the force f(x) on q1 due to
the presence of q2 is given by
f x
ð Þ ¼ k
q1 q2
x2
;
where k ¼ 9 109
Nm2
C2
: Because like charges
repel, the force given by (2.1) is repulsive; that is,
the force is positive if the charges have the same
sign. A positive (repulsive) force on q1 is directed
away from q2 along a line from q2 through q1, as
shown in Fig. 2.3.
The force given by (2.1) is attributed to an electric
field produced by the charge q2. The electric field
strength E (vector) at a point is defined as the force
per unit charge on a positive charge at that point.
Thus, the electric field strength at a distance x from
an isolated point charge q2 is given by
E x
ð Þ ¼ k
q2
x2
: (2.4)
The coulomb force given by (2.1) and the electric
field strength given by (2.4) for a point charge depend
only upon distance from the charge; that is, electric
field strength is directed radially outward from a posi-
tive charge, as illustrated in Fig. 2.4. If q2 were nega-
tive, the electric field strength given by (2.4) would
be negative, meaning the force on a positive charge
is directed radially inward toward q2. Note that (2.4)
gives electric field strength a distance x from a point
charge. It is not a general result. Other charge distri-
butions lead to different expressions for electric field
strength.
If the charge q1 in Fig. 2.3 is free to move, and if
both charges have the same sign, then the charge q1
will accelerate away from q2, taking energy from the
electric field as it accelerates. On the other hand, if we
somehow force the charge q1 toward the charge q2, we
q2 q1
x
f
Fig. 2.3 Coulomb force
E E
Fig. 2.4 Electric field due to a point charge. Electric field lines
of force emanate from positive charge and terminate on negative
charge

store potential energy in the field because we increase
the capacity of the field to do work on the charge q1. If
a positive charge moves in the direction of electric
field strength, then the motion and the force are in the
same direction and energy is transferred from the field
to the charge. If a positive charge moves opposite to
the field, energy is transferred to the field from what-
ever moves the charge.
Recall that a mass moving toward the earth’s sur-
face (in the direction of the gravitational field) takes
energy from the gravitational field and a mass moving
away from the earth’s surface (opposite to the direc-
tion of the gravitational field) gives energy to the
gravitational field. Similarly, a positive charge moving
in the direction of an electric field takes energy from
the field and a positive charge moving opposite to the
direction of the field gives energy to the field. Currents
in electric circuits are due primarily to electron flow,
but we may treat a flow of electrons in one direction as
a flow of positively charged particles in the opposite
direction. Thus energy is taken from an electric field
by a current moving in the direction of the field and
energy is given to the field by current moving opposite
to the direction of the field.
Continuing the point-charge example, we can cal-
culate the work done on the charge q1 by the field
using (2.3), where the force f(x) is the product of the
charge q1 and the electric field strength (the force per
unit charge). From (2.4), the force on the charge is
given by
fðxÞ ¼ EðxÞ q1 ¼ k
q1 q2
x2
:
Suppose both charges have the same sign, q2 is
stationary, and the charge q1 moves from point a to
point b on a line drawn from point a to q2. From (2.3),
the work done on the charge by the field is given by
w ¼
ðb
a
fðxÞ dx ¼ k q1 q2
ðb
a
dx
x2
¼ k q1 q2
1
x
b
a
¼ k q1 q2
1
b

1
a

: (2.5)
If a b; the charge q1 moved away from q2 (in the
direction of the field), the work w is positive, and the
field has done work on the charge. If a b; the charge
q1 moved toward q2 (opposite to the direction of the
field), w is negative, and work has been done on the
field. In the first case, energy is removed from the field
(transferred to the charge). In the second, energy is
stored in the field.
Example 2.3. In Fig. 2.3, the movable charge
q1 ¼ 1 mC and the fixed charge q2 ¼ 5 mC. If
q1 moves from a ¼ 10 m to b ¼ 20 m (away
from q2), the work done by the field on q1 is
given by (2.5) and equals
w ¼ k q1 q2
1
b

1
a

ﬃ 45 103
N m2
1
20m

1
10m

¼ 2; 250 J
which is approximately the work done in lift-
ing a 225 kg mass to a height of 1 m (a 225 kg
mass weighs about 496 lb).
Exercise 2.3. Point charges q, –q are a dis-
tance d apart. Obtain an expression for the
charge q in terms of the separation d and the
force f each charge exerts on the other. Show
that the expression is dimensionally consistent.
2.3 Electric Potential and Voltage
We continue with the point-charge example. From the
discussion above, the potential energy of the field due
to q2 depends upon the position of q1 relative to that of
q2. The closer is q1 to q2, the greater is the potential
energy of the field and the farther is q1 from q2, the
smaller is the potential energy of the field.
The electric potential at a point is denoted by F
and is defined as the work done on an electric field (or
by an external force) per unit charge when a positive
charge is brought from infinity (where the electric field
strength is assumed to be zero) to the point.10
Thus the
electric potential at a point is numerically (but not
10
Note that electric potential is not the same as potential energy.
The unit of electric potential is that of energy divided by charge.
2.3 Electric Potential and Voltage 23

dimensionally) equal to the work required of an exter-
nal force to bring a one-coulomb positive charge from
infinity to the point. The work done and thus the
potential can be positive or negative, depending upon
whether the charge is forced to move against the
electric field or is allowed to drift in the direction of
the field. The SI unit of electric potential is the volt
(V),11
where 1 V equals 1 J C–1
.
Equation (2.5) gives the work done by the field on a
point charge when the charge is moved from a to b in
the neighborhood of another point charge. The work
done on the field is just the negative of the work done
by the field. Changing the sign of the right side of (2.5),
dividing by q1, and taking the limit as a ! 1, we find
that the electric potential (the work done on the field
per unit charge) in bringing a charge from x ! 1 to
x ¼ b is given by
FðbÞ ¼
k q2
b
: (2.6)
Point b can be any point (any distance from q2), so
the electric potential at distance x from a point charge
q2 is given by
FðxÞ ¼
k q2
x
: (2.7)
For a point charge, the electric field strength given
by (2.4) and the electric potential given by (2.7) are
related as
EðxÞ ¼
FðxÞ
x
: (2.8)
Electric field strength is defined as force per unit
charge, so electric field strength can be expressed in
units of newtons per coulomb (N C–1
). Equation (2.8)
for electric field strength is specific to a point charge,
but illustrates the fact that electric field strength also
can be expressed in volts per meter (V m–1
), which is
the preferred unit in electrical engineering (both are SI
units).
Example 2.4. From (2.4) the electric field
strength at a point 10 m from a 5 mC point
charge is
E ¼
kq
x2
ﬃ
9109
Nm2
C2

5106
C

10m
ð Þ2
¼ 450NC1
¼ 450Vm1
:
From (2.7), the electric potential at the point
is
F ¼
k q
x
ﬃ
9 109
N m2
C2

5 106
C

ð10 mÞ
¼ 4:50 kV:
Exercise 2.4. Equal point charges are fixed in
space as shown in Fig. 2.5 and exert a force
f on one another. Find an expression for the
electric potential due to the charges at the point
x1 x0 0 in terms of the distances x0; x1: and
the force f Show that the expression is dimen-
sionally consistent. Hint: The total electric
potential due to two point charges is the sum of
the potentials due to each charge individually.
What causes water to flow through a pipe is a pres-
sure difference from one end of the pipe to the other. It
is pressure difference that causes flow, not absolute
pressure at either end. Similarly, what causes charge to
flow is a potential difference. In electric circuits, we
are interested in potential differences, not absolute
potentials.
The voltage at point a with respect to point b,
denoted by vab; is the potential difference
vab ¼ FðaÞ FðbÞ; (2.9)
x0 x1
0
x
q
q q
11
After Count Alessandro Volta (1745–1827), an Italian inven-
tor who discovered hydrolysis and invented the battery and the
electric condenser (now called a capacitor).

where F a
ð Þ is the potential at point a and F b
ð Þ is the
potential at point b The SI unit for electric potential is
the volt, so the SI unit of voltage is also the volt (V).
Note that the voltage vab is the work done on the field per
unit charge in moving a charge from point b to point a.
The difference of two potentials can be formed in
two ways and the order of the subscripts a and b is
significant. Two points are required to specify (or
calculate, or measure) a voltage and one of the points
must be identified as the reference. By convention, the
second subscript on the voltage defined by (2.9) iden-
tifies the reference point; e.g., in (2.9) the potential
F b
ð Þ is the reference potential. To measure vab with a
voltmeter, we would touch the black (–) probe to the
reference point b and the red (+) probe to point a. If we
made the measurement the other way around, we
would be measuring the negative of vab because, by
comparison with (2.9),
vba ¼ FðbÞ FðaÞ ¼ vab: (2.10)
Example 2.5. From (2.9) and (2.7), the volt-
age at a point a with respect to another point b
in the neighborhood of an isolated point charge
q2, is given by
vab ¼
k q2
xa

k q2
xb
¼ k q2
1
xa

1
xb

;
where xa is the distance from point a to q2 and
xb is the distance from point b to q2. If q2 is
positive, the voltage vab is positive if a is
nearer q2 than b (if xa xb) and negative if a
is farther from q2 than b (if xa xb).
Exercise 2.5. The point a is at a distance xa
from an isolated point charge q, as shown in
Fig. 2.6. Let vab denote the voltage between a
and another point b. Express the distance d in
terms of q, xa, and vab. Show that the expres-
sion is dimensionally consistent.
If a voltage vab is positive, the potential energy of a
positive charge is greater at the point a than it is at
point b. In that case, a positive charge experiences a
voltage drop in moving from a to b and a voltage rise
in moving from b to a.12
If vab is negative, the reverse
is true. Thus current from a to b goes through a voltage
drop if vab is positive and a voltage rise if vab is
negative. Figure 2.7 illustrates these definitions.
2.4 Ohm’s Law and Resistance
Ohm’s law states that the voltage across a piece of
material is proportional to the current through the
material; for example, if the object in Fig. 2.8 repre-
sents a sample of conducting material, the voltage v
across the sample is related to the current i through the
sample according to13
v ¼ R i; (2.11)
where R is called the resistance of the sample. The SI
unit of resistance is the ohm (O),14
where 1 O equals 1
V/A (1 VA1
). Despite its simplicity, Ohm’s law is
one of the most important laws in all of electrical
engineering.
b
a
voltage drop
voltage rise
Φ(a) Φ(b)
vab 0
Fig. 2.7 Definitions of voltage rise and voltage drop
v
+ –
i
Fig. 2.8 Voltage polarity and current direction in Ohm’s law
q
xa d
a b
12
The terms rise and drop arise from analogy with gravity,
where a mass gains potential energy if we raise it and loses
potential energy if we lower (or drop) it.
13
In general (and usually), both voltage and current vary with
time.
14
After the German physicist Georg Simon Ohm (1787–1854).
2.4 Ohm’s Law and Resistance 25

In Ohm’s law (2.11) the positive direction of
current is in the direction of a voltage drop (from a
higher to a lower potential), as shown in the figure. If
either the assumed voltage polarity or the assumed
current direction is opposite to that shown in
Fig. 2.8, the right side of (2.11) must be negated.
Example 2.6. In Fig. 2.8, the current i is equal
to 5 A in the direction shown and the resistance
of the sample is 100 O. From (2.11), the volt-
age v is 500 V.
Exercise 2.6. Refer to Fig. 2.9. Find the cur-
rent i2 if the voltage v is 10 V and the resistance
of the sample is 1 kO.
2.5 Resistivity
The resistance of a sample of material depends upon
the kind and state of the material (e.g., copper at 20
C)
and upon the size and shape of the sample. The effects
of the kind and state of the material can be separated
from those of size and shape. A material can be char-
acterized by its resistivity, denoted by r, which is
independent of size and shape. The resistance of a
homogeneous sample of material having resistivity r,
length l, and uniform cross-sectional area A is given by
R ¼
r l
A
: (2.12)
The SI unit of resistivity is the ohm meter (O m).15
Depending upon the material, resistivity can depend
strongly or weakly upon the state of the material (e.g.,
temperature and pressure).
Example 2.7. The resistivity of pure copper at
20
C is 1:72 108
O m: At 20
C, the resis-
tance of a pure copper cylindrical wire having
diameter d ¼ 0:2 cm and length l ¼ 5km is
R ¼
rl
A
ffi
1:72 108
Om

5000 m
ð Þ
p
ð Þ 0:001m
ð Þ2
ffi 27:4 O:
If the current in the wire is 15 A, the voltage
drop from end to end is
v ¼ R i ffi ð27:4 OÞð15AÞ ffi 411V:
Exercise 2.7. A certain conductor on a printed
circuit board has a rectangular cross section.
The conductor has width w0 and is built from a
material having resistivity r0: Suppose we
want to build the conductor from a different
material having resistivity r1: Find an expres-
sion for the new width w1 if we want to main-
tain the length, height, and resistance of the
original conductor
Many materials are classified according to their
resistivity as insulators (very high resistivity) or
conductors (very low resistivity). Materials having
resistivities between these extremes are called semi-
conductors. Glass and air are good insulators. Most
metals are good conductors.16
Silicon containing cer-
tain impurities is a semiconductor.
2.6 Conductance and Conductivity
The reciprocal of resistance is called conductance and
is denoted by G:
G ¼
1
R
: (2.13)
v
R
+ – i2
15
In reference works, resistivity often is specified in mO cm
(10–8
O m).
16
Metals are good conductors because the outer electrons of
metal atoms are only loosely held and are relatively free to
“drift” through the solid under the influence of an applied
voltage.

The unit of conductance is the siemen (S), where
1 S ¼ 1 O1
. The reciprocal of resistivity is called
conductivity, denoted by s:
s ¼
1
r
: (2.14)
The unit of conductivity is siemens per meter
(S m1
). From (2.12)–(2.14), the conductance of a
homogeneous piece of material having conductivity s,
length l and uniform cross-sectional area A is given by
G ¼
s A
l
: (2.15)
Using conductance, Ohm’s law is written
i ¼ G v: (2.16)
Conductance often is used symbolically in place of
resistance when writing circuit equations that would
otherwise contain many terms of the form v=R; how-
ever, in specifications of values, resistance is almost
always the quantity specified.
2.7 Resistors
Resistance is introduced (intentionally) in circuits
using components called resistors. Resistors are
made of various materials and in various configura-
tions. A composition resistor is essentially a cylindri-
cal core of material consisting of very fine carbon or
metallic granules imbedded in a non-conducting or
semi-conducting material and incased in plastic. The
resistance is determined by the concentration of car-
bon or metal in the core material. There are two main
types of film resistors. One kind is made by depositing
a film of metal or carbon on a non-conducting cylin-
drical core and then cutting away some of the material
to leave a spiral ribbon of the material. The other kind
consists of a thin film of (usually) metal or certain
metal oxides on a planar surface. Many chip resistors
and surface-mount resistors are of this kind, as are
resistors in integrated circuits. A wirewound resistor
is just what the name suggests: A wire wound on a
non-conducting core. The resistance of a wirewound
resistor is determined by the resistivity, cross-sectional
area, and length of the wire. If adjacent windings are in
contact, the wire is insulated.
Resistors are fixed (have fixed values) or variable.
Variable resistors can be continuously variable, like
potentiometers and rheostats, or have contacts at only
certain angular or linear positions. A continuously
variable resistor incorporates a tap or a slide that can
be positioned and repositioned along a strip of con-
ducting material or wirewound core. Those having
only a limited set of possible values sometimes are
called encoders because they provide a precise relation
between position and resistance. Figures 2.10 and 2.11
show assorted fixed and variable resistors.
Years ago, carbon-composition resistors were the
most common (numerous) kinds of resistors. Nowa-
days, the most numerous resistors are film resistors,
and carbon-composition resistors have almost disap-
peared. Chip resistors, most small axial-lead resistors,
and resistors in integrated circuits are constructed of
thin carbon or metallic films (often, nichrome or tan-
talum nitride). In chip and integrated resistors, the film
is deposited on an insulating substrate and joined to
contact pads or conductors at each end, as illustrated in
Fig. 2.12. The resistance of a thin-film resistor is given
by the usual relation
R ¼
rl
A
¼
rl
w h
: (2.17)
where w and h are the width and thickness, respec-
tively, of the strip, r is the resistivity of the film, and l
is the length of the strip.
It is conventional and convenient to define the
sheet resistance of a film resistor as
r
h
¼ sheet resistance: (2.18)
Sheet resistance has the dimension of resistance
and usually is expressed in either ohms or ohms per
square. With this definition (2.17) becomes
R ¼
r
h

l
w

¼ sheet resistance
ð Þ number of squares
ð Þ:
(2.19)
The number of squares in a film resistor is dimen-
sionless and is obtained by dividing the length of the
film into squares having sides equal to the width of the
film, as illustrated by Fig. 2.13.
2.7 Resistors 27

The resistance of a thin-film resistor is determined
by the sheet resistance, which is a function of the
material and film thickness, and the number of squares.
The resistance is independent of the size of a square.
For example, a film 2 mm wide by 10 mm long has the
same resistance as one 200 nm wide and 1mm long, as
illustrated by Fig. 2.14, if both have the same sheet
resistance.
Tantalum nitride (TaN) is widely used for thin-film
resistors, partly because the resistivity of TaN can
be varied over a wide range by varying the composi-
tion. A common composition has a resistivity of about
250 mO cm at 25
C: A common thickness for thin-film
resistors is 50 nm. A 50 nm film of 250 mO cm TaN
has a sheet resistance of 50 O=sq; which proves to be a
convenient value.
Fig. 2.11 Variable Resistors
(not to scale) (Photos courtesy
Rapid Electronics, Ltd.)
Fig. 2.10 Fixed resistors (not
to scale): (a) Aluminum-
encased wirewound, (b)
power wirewound, (c) metal-
oxide, (d) carbon-film, (e)
metal-film, (f) carbon
composition, (g) surface-
mount chip (Photographs
courtesy Rapid Electronics,
Ltd.)

Exercise 2.8. The resistivity of a certain com-
position of nichrome is108 mO cm at 25
C:
What is the thickness of a layer of this material
that will have a sheet resistance of 75 O=sq at
25
C?
Exercise 2.9. The length of the film in a 500 O
thin film resistor is fixed while its width is
halved. What is the new resistance?
Exercise 2.10. The resistivity of a certain
material at 25
C is 95 mO cm: (a) What is the
sheet resistance of a 100 nm thick film of the
material? (b) What is the resistance at 25
C of
a thin film resistor made of 100 nm film that is
75 nm wide and 1mm long?
The natural tolerances of thin film resistors are at
best about 10%; however, a thin film resistor can
be trimmed to a very precise value by monitoring the
resistance while using a laser to cut one or more
notches in the film, as illustrated by Fig. 2.15. It is
difficult to specify the width and depth of the notch
because of variations in the thickness and even the
composition of the film (which is why the natural
tolerances are relatively poor). Ability to trim thin
film resistors to such precise values is important to
design and construction of high-precision electronic
circuits.
2.8 E Series, Tolerance, and Standard
Resistance Values
It is impossible for parts suppliers to stock resistors in
every possible value, so a convention for defining a
reasonable set of standard values is necessary. The
convention used is based on what are called E series,
which are approximately logarithmic divisions of a
decade. Table 2.1 gives the values of the E12, E24,
E48, E96, and E192 series, which are the most com-
monly used series for resistors. The E12 series divides
1 decade into 12 values, the E24 series divides 1
decade into 24 values, and so on. The E3 and E6 series
are seldom used for resistors and are not shown here.
The E3 series contains every other value from the E6
series, which contains every other value from the E12
series, which contains every other value from the E24
series. Similarly, the E48 series contains every other
value from the E96 series, which contains every other
value from the E192 series. The E24 series does not
contain every other value from the E48 series, because
E48 values have three significant digits, whereas E24
values have only two. But one can say that the E24
series contains rounded-off values from the E48 series.
The resistance of a resistor from any particular E
series consists of a number from the series, multiplied
by an integer power of ten. For example, the value 133
appears in the E48 series, so one can readily obtain
resistors having nominal resistances of (e.g.) 133 O;
13:3 kO; and 1:33 MO: Every standard (off-the-shelf)
resistor value equals the product of a number from an E
series and an integer power of ten, but not all such
values are necessarily available from a higher or lower
series. For example, one can buy 10% and 5%
(E12 and E24) resistors having values of 33, 330 O,
3.3 kO, and so on, but none of these values are available
top view
w
contact pads
nichrome film
l
h
edge view
Fig. 2.12 Thin-film resistor
1
2
sq
1 sq 1 sq 1 sq
Fig. 2.13 Thin-film resistor squares
2.8 E Series, Tolerance, and Standard Resistance Values 29

in the higher-precision (E48, E96, and E192) series. As
another example, resistors having values 383 10n
O;
with n ¼ 1; 0; 1; 2; are available in the E48, E96,
and E192 series, but not in the E12 or E24 series.
A standard tolerance is associated with each value
from an E series. Standard tolerances for off-the-shelf
resistors are 20% (E6 – rarely used, nowadays),
10% (E12), 5% (E24), and 1% (E96). Stan-
dard tolerances for high-precision resistors (obtainable
from some large suppliers and manufacturers) are
0:5%; 0:1%; and 0:05%: Resistors having
tolerances as tight as 0:001% are available in
selected resistances from certain manufacturers. How-
ever, high-precision resistors are not bulk-produced in
every value from any particular E series, but typically
are produced in only certain values having wide appli-
cation (e.g., in precision instrumentation and measure-
ment systems).
Some manufacturers can produce special-order
resistors having virtually any nominal value, with tol-
erances of better than 0.05%. Ordering a batch of such
resistors might be justified if a particular value and
tight tolerance are required for a circuit being pro-
duced in quantity.
Example 2.8. A calculation indicates that a
resistor having resistance R ¼ 61:7kO is needed
in a certain circuit. Refer to Table 2.1 and give
the nearest standard value from each series.
Solution: The E12 value nearest to the speci-
fied value is 56 kO. The E24 value nearest to
the specified value is 62 kO. The value from
the E48, E96, and E192 series nearest to the
specified value is 61.9 kO.
Example 2.9. One kind of ammeter measures
the current in a wire by measuring the voltage
across a precision resistor called a shunt
connected in series with the wire. In a particular
case, the shunt has resistance R ¼ 0:05 O
0:05% and the voltage across the shunt is
V ¼ 1:872 mV: What is the current I through
the shunt?
Solution: The current is in the range
Imin I Imax; where
Imin ¼
V
Rmax
¼
1:872 mV
0:05 O
ð Þ 1:0005
ð Þ
¼ 37:421 mA;
Imax ¼
V
Rmin
¼
1:872 mV
0:05 O
ð Þ 0:9995
ð Þ
¼ 37:459 mA:
Exercise 2.11. A calculation indicates that a
resistor having resistance R ¼ 50 kO 1% is
needed in a certain application. What value
would you specify and from which series?
2.9 Resistor Marking
Axial-lead composition, coated wire-wound, carbon-
film, and metal-film resistors are color-coded for resis-
tance and tolerance by either 4-band or 5-band codes
10μm
2μm
2μm
5 squares
1 μm = 1000nm
200nm
5 squares
200nm
Fig. 2.14 Illustrating
calculation of sheet resistance
Fig. 2.15 Trimmed thin-film resistor

as defined in Table 2.2. A 4-band code is used for
resistors from the E24 and lower series. A 5-band
code is used for resistors from the E48 and higher
series. The first band is closer to one end of the resistor
than the last band is to the other. The first two bands
denote the first two significant digits of the resistance.
In a 4-band code, the third band is the multiplier,
which is an integer power of ten, and the fourth band
denotes the tolerance. In a 5-band code, the third
band denotes the third significant digit, the fourth
band denotes the multiplier, and the fifth band is the
tolerance.
Example 2.10. Give the resistance of resistors
marked as follows:
(a) Red, violet, yellow, green; (b) red, vio-
let, brown, orange, violet.
Answers: (a) 270 kO 0:5%; (b) 271 kO
0:1%.
Exercise 2.12. Give the resistance of resistors
marked as follows:
(a) Yellow, orange, red, gold; (b) yellow,
green, white, orange, violet.
In addition to resistance and tolerance, properies of
resistors include power-dissipation rating, self-heating
coefficient (described in Chapter 5), maximum
operating temperature, and temperature coefficient of
resistance (described in the next section). Most manu-
facturers and vendors provide that information in data-
sheets for their offerings.
Table 2.2 Resistor color codes
Color Value Multiplier Precision
Black 0 100
Brown 1 101
1%
Red 2 102
2%
Orange 3 103
Yellow 4 104
Green 5 105
0.5%
Blue 6 106
0.25%
Voilet 7 107
0.1%
Grey 8 108
0.05%
White 9 109
Gold 101
5%
Silver 102
10%
None 20%
Table 2.1 E series for standard resistor values
E192 E96 E48 E24 E12
0:5% 1% 2% 5% 10%
100 178 316 562 100 316 100 10 10
101 180 320 569 102 324 105 11 12
102 182 324 576 105 332 110 12 15
104 184 328 583 107 340 115 13 18
105 187 332 590 110 348 121 15 22
106 189 336 597 113 357 127 16 27
107 191 340 604 115 365 133 18 33
109 193 344 612 118 374 140 20 39
110 196 348 619 121 383 147 22 47
111 198 352 626 124 392 154 24 56
113 200 357 634 127 402 162 27 68
114 203 361 642 130 412 169 30 82
115 205 365 649 133 422 178 33
117 208 370 657 137 432 187 36
118 210 374 665 140 442 196 39
120 213 379 673 143 453 205 43
121 215 383 681 147 464 215 47
123 218 388 690 150 475 226 51
124 221 392 698 154 487 237 56
126 223 397 706 158 499 249 62
127 226 402 715 162 511 261 68
129 229 407 723 165 523 274 75
130 232 412 732 169 536 287 82
132 234 417 741 174 549 301 91
133 237 422 750 178 562 316
135 240 427 759 182 576 332
137 243 432 768 187 590 348
138 246 437 777 191 604 365
140 249 442 787 196 619 383
142 252 448 796 200 634 402
143 255 453 806 205 649 422
145 258 459 816 210 665 442
147 261 464 825 215 681 464
149 264 470 835 221 698 487
150 267 475 845 226 715 511
152 271 481 856 232 732 536
154 274 487 866 237 750 562
156 277 493 876 243 768 590
158 280 499 887 249 787 619
160 284 505 898 255 806 649
162 287 511 909 261 825 681
164 291 517 919 267 845 715
165 294 523 931 274 866 750
167 298 530 942 280 887 787
169 301 536 953 287 909 825
172 305 542 965 294 931 866
174 309 549 976 301 953 909
176 312 556 988 309 976 953
2.9 Resistor Marking 31

Exercise 2.13. Using a search engine of your
choosing, do a search on “resistors” and locate
several manufacturers’ web sites. Visit each
site and examine the technical data they pro-
vide on their offerings. Identify the technolo-
gies that provide (a) the lowest temperature
coefficient of resistance, (b) the highest pre-
cision, and (c) the highest power-dissipation
ratings.
2.10 Variation of Resistivity and
Resistance with Temperature
Resistivities of materials and thus resistances of con-
ductors and resistors vary with temperature.17
Tem-
perature dependence of resistivities of certain metals is
of interest because conductors in electric circuits are
metallic. Figure 2.16 shows resistivity as a function
of temperature for aluminum (Al), copper (Cu), and
gold (Au).18
The scales on the axes are logarithmic.
The vertical (resistivity) scale is logðr=r0Þ; where
r0 ¼ 108
O m (1 mO cm). The horizontal (tempera-
ture) scale is logðT=T0Þ; where T0 ¼ 1K (one kelvin).
For example, for logðT=T0Þ ¼ 2; we have T=T0 ¼
100 and the temperature is T ¼ 100 T0 ¼ 100 K: At
that temperature, the resistivity of copper is given by
logðr=r0Þ ¼ 0:5; so r ¼ r0 100:5
¼ 0:316 mOcm:
The vertical reference line indicates T ¼ 298 K ¼
25
C (77
F), where the values are rAl=r0 ¼ 2:709;
rCu=r0 ¼ 1:712; and rAu=r0 ¼ 2:255:
Figure 2.16 suggests that resistivity is very nearly
a linear function of temperature for temperatures
above about 102:25
K ffi 178 K; which is approxi-
mately 95
C: Operating temperatures at or above
room temperature are common for linear circuits, so
we are particularly interested in temperatures above
about 298 K ¼ 20
C or so. Figure 2.17 shows graphs
of resistivity versus temperature (both K and
C)
for silver (Ag), aluminum (Al), gold (Au), copper
(Cu), and tungsten (W). For 200 K T 900 K
(25
C T 673
C), resistivity is very nearly a linear
function of temperature. Actually, the linear range for
these metals extends from about 100
C to almost
1000
C.
For a sufficiently small change DT in temperature,
the derivative of resistivity with respect to temperature
can be approximated by
dr T
ð Þ
dT
ffi
rðT þ DTÞ r T
ð Þ
DT
;
which gives, for the variation of resistivity with
temperature,
rðT þ DTÞ ffi r T
ð Þ þ
dr T
ð Þ
dT
DT
¼ r T
ð Þ 1 þ a T
ð Þ DT
½ ; (2.20)
where
a T
ð Þ ¼
1
r T
ð Þ
dr T
ð Þ
dT
(2.21)
is the temperature coefficient of resistivity at tem-
perature T. In engineering applications of (2.20), tem-
perature T often is expressed in degree Celsius
and resistivities of materials and the associated tem-
perature coefficients are commonly specified at
T25 ¼ 25
C ¼ 298 K. In such applications it is usual
to specialize (2.20) and write
^
rT ¼ r25 1 þ a25 T T25
ð Þ
½ ; (2.22)
where T25 ¼ 25
C ¼ 298 K; the parameters r25; a25
are the resistivity and temperature coefficient, respec-
tively; and ^
rT is an approximation to the true
(measured) resistivity at temperature T. For many
metals (2.22) can be used to estimate resistivity over
a wide range – from about 100
C to almost 1000
C.
The numerical value of a temperature coefficient is
the same whether expressed in
C1
or K1
, because
the quantity T T25 in (2.22) is the same, whether
both temperatures are expressed in degree Celsius or
Kelvin. In practice (e.g., on component data sheets),
temperature coefficients are commonly expressed in
parts per million (ppm), which is 106
K times the
coefficient expressed in K1
(or
C1
):
17
Resistivity can also vary with pressure, strain, strength and
direction of an applied magnetic field, and other things. The
operation of various transducers (e.g., strain gauges) is based
upon such effects.
18
Handbook of Chemistry and Physics (76th Ed.), edited by
David R. Lide, CRC Press, 1995.

a ppm
ð Þ ¼ a K1

106
K: (2.23)
Table 2.3 gives resistivities and temperature coeffi-
cients at 25
C for selected metals. Be aware that such
data are available from many sources, and temperature
coefficients in particular can vary significantly from
one source to the next. Also, the values in Table 2.3
are for pure metals, whereas wires and cables used in
electrical components and apparatus often are alloys
whose resistivities and temperature coefficients can
differ significantly from those for the dominant pure
metal. The data given in Table 2.3 are intended for use
only in examples and problems given in this book, not
for actual engineering design.
Figure 2.18 shows graphs of the percent error
versus temperature for resistivities calculated using
(2.22), calculated as
error T
ð Þ ¼ 100
^
rT rT
rT
(2.24)
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3
–4
–3.5
–3
–2.5
–2
–1.5
–1
–0.5
0
0.5
1
Al
Cu
Au
T = 298 K
log
ρ
0
ρ
T
T0
log
Fig. 2.16 Graphs of
resistivity versus temperature
(log-log scale) for pure
aluminum (Al), copper (Cu),
and gold (Au), where
r0 ¼ 108
Om ¼ 1 mOcm
and T0 ¼ 1 K: (See
Footnote 18)
0
5
10
15
20
25
T
r (μΩ cm)
200 300 400 500 600 700 800 900
–73 27 127 227 327 427 527 627
W
Al Au
Cu
Ag
(°C)
(K)
Fig. 2.17 Resistivities of
selected metals (See
Footnote 18)
2.10 Variation of Resistivity and Resistance with Temperature 33

For temperatures between 298 K (25
C) and 898 K
(625
C), the maximum error (magnitude) is less than
5%. For that range of temperatures, the maximum
error (magnitude) for copper (a popular material for
wires) is approximately 2%.
Exercise 2.14. Refer to Fig. 2.18. Why is the
error at 298 K equal to zero for all four linear
approximations?
Example 2.11. A copper wire is heated from
25
C to 100
C. What is the percent increase in
resistivity?
Solution: From Table 2.3, the resistivity and
temperature coefficient for copper at 25
C are
r25 ¼ 1:71 mO cm and a25 ¼4;138ppm; respec-
tively. From (2.22), the resistivity at 100
C is
r100 ¼ r25 1 þ a25 T T25
ð Þ
½
¼ 1:71 mO cm
ð Þ½1 þ 4;138 106
C1

100
C 25
C
ð Þ
¼ 2:24 mO cm:
The percent increase is
100
r100 r25
r25
ﬃ 31%:
Exercise 2.15. The resistivity of a certain
metal is 1:6 108
O m at 10
C and
2:0 108
O m at 70
C. Estimate the temper-
ature coefficient of resistivity for the metal at
10
C.
The resistivities of materials used to make resis-
tors also vary with temperature. Such variation is
approximately linear over the range of operating
temperatures for any particular resistor. In practice,
it usually is more convenient to deal directly with
resistance than resistivity. For any particular resistor,
resistance is proportional to resistivity, so (2.22)
gives
RT ¼ R25 1 þ a25 T T25
ð Þ
½ : (2.25)
In view of (2.25), the temperature coefficient of
resistivity is also called the temperature coefficient
of resistance, usually called just the temperature
coefficient or abbreviated as TCR.
Temperature coefficients for resistors vary con-
siderably. Generally, carbon composition and car-
bon-film resistors have temperature coefficients on
the order of 103
K1
; whereas wirewound and
some metal-film and metal-oxide resistors have tem-
perature coefficients on the order of 105
K1
: In
general, resistances of wirewound, metal-film, and
metal-oxide resistors are much less sensitive to tem-
perature than carbon-film and carbon-composition
resistors.
The resistivities of many materials approach
zero (the materials become superconductors) at
temperatures approaching 0 K. The resistivity of
a superconductor is less than 1022
O cm; whereas
the lowest (room temperature) resistivity found in
metals is on the order of 106
O cm: Thus resistiv-
ities for superconductors are about a factor of
10-16
smaller than those of materials ordinarily
used for conductors in electric circuits. Some
materials become superconductors at temperatures
near 77 K (the boiling point of nitrogen), which
are much easier to attain and maintain than those
near absolute zero. As of this writing, use of
superconducting materials is limited to a few
specialized applications.
Table 2.3 Resistivities and temperature coefficients for
selected metals19
Metal Resistivity
at 25
C
mO cm
ð Þ
Temperature
coefficient
at 25
C (K1
)
Temperature
coefficient at
25
C (ppm)
Silver (Ag) 1.62 4:053 103 4053
Aluminum (Al) 2.71 4:438 103 4438
Gold (Au) 2.26 4:022 103 4022
Copper (Cu) 1.71 4:138 103 4138
Tungsten (W) 5.39 4:849 103 4849
19
Ibid. The temperature coefficients were obtained from the
slopes of linear-least-squares fits to the resistivity data given
there.

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Without a word the guard led me on a distance of a hundred feet
and called another soldier. A short talk ensued, and the second man
motioned me to follow him through a trail in the brush. We went on
for ten minutes, then came to a clearing hemmed in by a cliff and
several high rocks.
Here were over a hundred soldiers on foot and twice as many on
horseback. In the midst of the latter was the Cuban general I had
asked to see—the gallant soldier who had fought so hard in the
cause of Cuban liberty.

CHAPTER XIV.
GENERAL CALIXTO GARCIA.
My first view of General Calixto Garcia was a disappointing one.
For some reason, probably from the reports I had heard concerning
his bravery, I had expected to see a man of great proportions and
commanding aspect. Instead, I saw an elderly gentleman of fair
figure, with mild eyes and almost white mustache and beard, the
latter trimmed close. But the eyes, though mild, were searching, and
as he turned them upon me I felt he was reading me through and
through.
He was evidently surprised to see a boy, and an American at
that. He spoke but little English, but an interpreter was close at
hand, who immediately demanded to know who I was, where I had
come from, and what I wanted.
“My name is Mark Carter, and I have journeyed all the way from
Santiago de Cuba,” I replied. “I heard that my father and his friend,
Señor Guerez, had joined General Garcia’s forces.”
“You are Señor Carter’s son!” exclaimed the Cuban officer, and
turned quickly to General Garcia. The two conversed for several
minutes, and then the under-officer turned again to me.
“General Garcia bids you welcome,” he said, and at the same
time the great Cuban leader smiled and extended his hand, which I
found as hard and horny as that of any tiller of the soil. “He knows
your father and Señor Guerez well.”
“And where are they now?” I asked quickly.

“They were with the army two days ago, but both went off to
escort the ladies of Señor Guerez' family to a place of safety. The
señor was going to take his wife and daughters to an old convent up
a river some miles from here.”
This was rather disheartening news, yet I had to be content. I
asked if my father was well.
“Very well, although hardly able to walk, on account of a leg he
broke some time ago.”
“And have you seen Alano Guerez? He is about my own age, and
was with me up to this morning,” I went on, and briefly related my
adventures on the road, to which the officer listened with much
interest.
“We have seen nothing of him,” was the reply I received. “But he
may be somewhere around here.”
The officer wished to know about the Spanish detachment we
had met, and I told him all I knew, which was not much, as I had
not understood the Spanish spoken and Alano had not interpreted it
for me. But even the little I had to say seemed to be highly
important, and the officer immediately reported the condition of
affairs to General Garcia.
By this time some of the soldiers who had taken part in the fight
at the foot of the plateau came back, bringing with them several
wounded men, including the negro whose wound I had bound up.
The disabled ones were placed in a temporary hospital, which
already sheltered a dozen others, and General Garcia rode off with
his horsemen, leaving the foot soldiers to spread out along the
southeastern slope of the mountain.
Left to myself, I hardly knew what to do. A black, who could
speak a few words of “Englis',” told me I could go where I wanted,
but must look out for a shot from the enemy; and I wandered over
to the hospital and to the side of the fellow I had formerly assisted.

The hospital, so called, consisted of nothing more than a square
of canvas stretched over the tops of a number of stunted trees.
From one tree to another hammocks, made of native grass, were
slung, and in these, and on piles of brush on the ground, rested the
wounded ones. Only one regular doctor was in attendance, and as
his surgical skill and instruments were both limited, the sufferings of
the poor fellows were indeed great.
“Him brudder me—you help him,” said the black who spoke
“Englis',” as he pointed to the fellow whose wound I had dressed.
“Jorge Nullus no forget you—verra good you.”
“Is your name Jorge Nullus?”
“Yeas, señor—him brudder Christoval.”
“Where did you learn English?”
“Me in Florida once—dree year ago—stay seex months—no like
him there—too hard work,” and Jorge Nullus shrugged his shoulders.
“You verra nice leetle man, señor,” and he smiled broadly at his open
compliment.
“Do you know Señor Guerez?” I questioned quickly.
“Me hear of him—dat’s all.”
“Do you know where the old convent on the river is?” I
continued.
The Cuban nodded. “Yeas—been dare many times—bring 'taters,
onions, to Father Anuncio.”
“Could you take me there—if General Garcia would let you go?”
“Yeas, señor. But Spaniards all around—maybe shoot—bang!—
dead,” and he pointed to his wounded brother. The brother
demanded to know what we were talking about, and the two
conversed for several minutes. Then Jorge turned again to me.

“GENERAL GARCIA, THE
GALLANT SOLDIER WHO
HAD FOUGHT SO HARD IN
THE CAUSE OF CUBAN
LIBERTY.”
“Christoval say me take you; you verra good leetle man, señor.
We go now, you say go.”
“Will you be allowed to go?”
“Yeas—General Garcia no stop me—he know me all right,” and
the negro grinned and showed his teeth.

I was tempted to start at once, but decided to wait until morning,
in the hope of finding Alano. In spite of the fact that I knew my
chum would be doubly cautious, now we were separated, I felt
decidedly anxious about him. The Spanish troops were on every
side, and the soldiers would not hesitate to shoot him down should
they learn who he was.
The night passed in comparative quietness. Toward morning we
heard distant firing to the northwest, and at five o’clock a messenger
dashed into camp with the order to move on to the next mountain, a
distance of two miles. Through Jorge I learned that the Spaniards
had been outwitted and driven back to the place from whence they
had come.
There now seemed nothing for me to do but to push on to the
convent on the river, in the hope of there joining my father. We
were, so I was told, but a few miles from Guantanamo, but the route
to the convent would not take us near the town.
Jorge’s brother felt much better, so the negro went off with a
light heart, especially after I had made it plain to him that my father
would reward him for any trouble he took on my account. I told him
about Alano, and before leaving camp we walked around among the
sentries in the hope of gaining some information concerning him.
But it was all useless.
“Maybe he went on to Father Anuncio’s,” said my negro guide,
and this gave me a grain of comfort.
The soldiers and Jorge and myself left the camp at about the
same time, but we did not take the same road, and soon my guide
and I found ourselves on a lonely mountain trail overlooking a valley
thick with brush and trees. The sun shone brightly, but the air was
clear and there was a fine breeze blowing, and this made it much
cooler than it would otherwise have been.
I missed the horse, and wondered if Alano still had the animal he
had captured. It might be possible he had ridden straight on to

Guantanamo, and was now bound from there up the river. If that
was so, we might meet on the river road.
“Werry bad road now,” said Jorge, as we came to a halt on the
mountain side. “Be careful how you step, Señor Mark.”
He pointed ahead, to where a narrow trail led around a sharp
turn. Here the way was rocky and sloped dangerously toward the
valley. He went on ahead, and I followed close at his heels.
“No horse come dis way,” observed Jorge, as he came to another
turn. “Give me your hand—dis way. Now den, jump!”
We had reached a spot where a tiny mountain stream had
washed away a portion of the trail. I took his hand, and we prepared
to take the leap.
Just then the near-by crack of a rifle rang out on the morning air.
Whether or not the shot was intended for us I cannot say, but the
sound startled me greatly and I stumbled and fell. Jorge tried to
grab me, but failed, and down I shot head first into the trees and
bushes growing twenty feet below the trail!

CHAPTER XV.
A PRISONER OF WAR.
By instinct more than reason, I put out both hands as I fell, and
this movement saved me from a severe blow on the head. My hands
crashed through the branches of a tree, bumped up against the
trunk, and then I bounced off into the midst of a clump of brush and
wild peppers.
“Hi, yah!” I heard Jorge cry out, but from my present position I
could not see him. “Is you killed?” he went on.
“No, but I’m pretty well shook up and scratched up,” I answered.
“Take care—somebody shoot,” he went on.
I concluded I was pretty well out of sight, and I kept quiet and
tried to get back the breath which had been completely knocked out
of me. A few minutes later I heard a crashing through the brush,
and my guide stood beside me.
“Lucky you no killed,” he observed. “Bad spot dat.”
He searched around and soon found a hollow containing some
water, with which I bathed the scratches on my face and hands. In
the meantime he gazed around anxiously in the direction from which
he imagined the shot had come.
“Maybe no shoot at us,” he said, quarter of an hour later. “Me
find out.”
With his ever-ready machete he cut down a young tree and
trimmed the top branches off, leaving the stumps sticking out about

six inches on every side. On the top of the tree he stuck his hat, and
then, having no coat, asked me for mine, which he buttoned about
the tree a short distance under the hat, placing a fluttering
handkerchief between the two.
With this rude dummy, or scarecrow, he crawled up the side of
the gully until almost on a level with the trail. Then he hoisted the
figure up cautiously and moved it forward.
No shot was fired, and after waiting a bit Jorge grew bolder and
climbed up to the trail himself. Here he spent a long time in viewing
the surroundings, and finally called to me.
“Him no shoot at us. Maybe only hunter. Come up.”
Not without some misgivings, I followed directions. To gain the
trail again was no easy matter, but he helped me by lowering the
end of the tree and pulling me up. Once more we proceeded on our
way, but with eyes and ears on guard in case anybody in the shape
of an enemy should appear.
By noon Jorge calculated we had covered eight miles, which was
considered a good distance through the mountains, and I was glad
enough to sit down in a convenient hollow and rest. He had brought
along a good stock of provisions, with which the rebel camp had
happened to be liberally provided, and we made a meal of bread,
crackers, and cold meat, washed down with black coffee, cooked
over a fire of dead and dried grass.
“We past the worst of the road now,” remarked Jorge, as we
again moved on. “Easy walkin' by sundown.”
He was right, for about four o’clock we struck an opening among
the mountains where there was a broad and well-defined road
leading past several plantations. The plantations were occupied by a
number of Cubans and blacks, who eyed me curiously and called out
queries to Jorge, who answered them cheerfully.

The plantations left behind, we crossed a brook which my guide
said ran into the river, and took to a path running along a belt of oak
and ebony trees, with here and there a clump of plantains. We had
gone but a short distance when we crossed another trail, and Jorge
called a halt and pointed to the soft ground.
The hoofprints of half a dozen horses were plainly visible, and as
they were still fresh we concluded they had been made that very
day, and perhaps that afternoon.
“Who do you think the horsemen are, Jorge?” I asked.
He shrugged his shoulders.
“Can’t say—maybe soon tell—me see,” and on he went, with his
eyes bent on the ground.
For my part, I thought it best to keep a watch to the right and
the left. We went on slowly until the evening shadows began to fall.
Then Jorge was about to speak, when I motioned him to be silent.
“There is something moving in yonder brush,” I said, pointing
with my hand. “I think I saw a horse.”
We left the road and proceeded in the direction, moving along
slowly and silently. I had been right; there was not one horse, but
half a dozen, tethered to several stunted trees.
No human beings were present, but from a distance we presently
heard the murmur of voices, and a minute later two Spanish soldiers
came into view. Jorge drew his pistol, but I restrained him.
The soldiers had evidently come up to see if the horses were still
safe. Satisfied on this point, one passed to the other a roll of tobacco
for a bite, and both began to converse in a low but earnest tone.
Jorge listened; and, as the talk ran on, his face grew dark and
full of hatred. The backs of the two Spaniards were toward us, and
my guide drew his machete and motioned as if to stab them both.

I shook my head, horrified at the very thought. This did not suit
Jorge, and he drew me back where we might talk without being
overheard.
“What is the use of attacking them?” I said. “Let us be on our
way.”
“Them men fight General Garcia’s men—maybe hurt my brudder,”
grunted Jorge wrathfully. “They say they have prisoner—kill him
soon.”
“A prisoner?”
“Yes.”
“Where?”
“At camp down by river. They kill udder prisoner, now rob dis one
an' kill too. Bad men—no good soldiers.”
I agreed with him on this point. Yet I was not satisfied that he
should go back and attack the pair while they were off their guard.
“It would not be fair,” I said, “and, besides, the noise may bring
more soldiers down upon us. I wish we could do something for their
prisoner, whoever he is.”
We talked the matter over, and, seeing the soldiers depart,
concluded to follow them. We proceeded as silently as two shadows,
and during the walk Jorge overheard one soldier tell the other that
the prisoner was to be shot at sunrise.
A turn in the path brought us to a broad and roughly flowing
stream. Here a temporary camp had been pitched. Half a dozen
dirty-looking Spaniards were lolling on the ground, smoking and
playing cards. From their talk Jorge said they were waiting for some
of their former comrades to join them, when all were to travel back
to where the Spanish commander, Captain Campona, had been left.

“There ees the prisoner,” said Jorge, in a whisper, and pointed
along the river shore to where rested a decaying tree, half in and
half out of the water. The prisoner was strapped with rawhides to
one of the tree branches, and it was—my chum Alano!

CHAPTER XVI.
A RESCUE UNDER DIFFICULTIES.
Mere words cannot express my astonishment and alarm when I
saw who the prisoner tied to the tree was. As I gazed at Alano my
heart leaped into my throat, and like lightning I remembered what
Jorge had told me the Spaniards had said, that the prisoner was to
be shot at sunrise.
Alano shot! I felt an icy chill creep over me. My own chum! No,
no, it must not be! In my excitement I almost cried aloud. Noting
how strangely I was affected, my guide placed his hand over my
mouth and drew me back into a thicket.
“It is Alano Guerez!” I whispered, as soon as I was calm enough
to speak—“Señor Guerez' son!”
“Ah, yah!” ejaculated Jorge. “I see he is but a boy. Perros!
[Dogs!]”
“We must save Alano,” I went on. “If he was shot, I—I would
never forgive myself.”
Jorge shrugged his shoulders. “How?” he asked laconically. “Too
many for us.”
“Perhaps we can do something when it grows darker.”
The guide drew down the corners of his mouth. Then, as he
gazed at the river, his big black eyes brightened.
“Yeas, when it is darker we try. But must be careful.”

“Perhaps we can get to him by the way of the river.”
Jorge smiled grimly. Catching me by the arm he led me along the
bank, overgrown with grass and rushes. Not far away was something
that looked like a half-submerged log covered with mud. Taking a
stone he threw it, and the “log” roused up and flopped angrily into
the stream.
“Alligators!” I cried, with a shiver. “No, we won’t be able to get to
him by way of the river. But we must do something.”
“We cross river, and I tell you what we do,” replied my guide.
Crossing was not an easy matter, as neither of us cared to
attempt swimming or fording with alligators in the vicinity. But by
passing along the bank we presently discovered a spot where half a
dozen rocks afforded a footing, and over we went in the semi-
darkness, for the sun was now setting.
As we hurried down the course of the stream again, Jorge cut
several cedar and pine branches which appeared to be particularly
dry. Then he handed me a number of matches, of which, fortunately,
he had an entire box.
“We will put one pile of branches here,” he said, “and another
further down, and one further yet. Den I go back to camp. You
watch tree over there. When you see light wait few minutes, den
light all dree fires.”
“But how will that help us?”
“Soldiers see fires, want to know who is dar—don’t watch Alano
—me go in and help him. After you make fires you run back to
where we cross on stones.”
Jorge’s plan was not particularly clear to me, yet I agreed to it,
and off he sped in the gloom. Left to myself, I made my way
cautiously to the water’s edge, there to await the signal he had
mentioned.

It was a hot night and the air was filled with myriads of
mosquitoes, gnats, flies, and other pests. From the woods behind
me came the occasional cry of a night bird, otherwise all was silent.
Frogs as big as one’s two hands sat on the rocks near by, on the
watch for anything in the shape of a meal which might come their
way.
But bad as the pests around me were, I gave them scant
consideration. My whole mind was concentrated upon Alano and
what Jorge proposed to do. Silently I prayed to Heaven that the
guide might be successful in rescuing my chum.
About half an hour went by,—it seemed an extra long wait to me,
—when suddenly I saw a flash of fire, in the very top of a tree
growing behind the Spaniards' camp. The flash lasted but a second,
then died out instantly.
Arising from my seat, I ran to the furthest pile of boughs and
waited while I mentally counted off a hundred and eighty seconds,
three minutes. Then I struck a match, ignited the heaped-up mass,
and ran to the second pile.
In less than ten minutes the three fires, situated about three
hundred feet apart, were burning fiercely, and then I ran at topmost
speed for the spot where the river had been crossed. I had just
reached the locality when I heard a shout ring out, followed by two
musket shots.
A painful, anxious two minutes followed. Were Alano and Jorge
safe? was the question I asked myself. I strained my eyes to pierce
the gloom which hung like a pall over the water.
Footsteps on the rocks greeted my ears. Someone was coming,
someone with a heavy burden on his back. Once or twice the
approaching person slipped on the rocks and I heard a low cry of
warning.
“Mark!”

It was the voice of Alano, and my heart gave a joyful bound. In
another second my Cuban chum appeared in view, carrying on his
manly back the form of Jorge.
“Alano,” I ejaculated excitedly, “what is the matter with him?”
“He has been shot in the leg,” was the reply. “Come on, help me
carry him and get to cover. I am afraid they are on my track!”
“Run into the woods!” groaned Jorge. “Den we take to trees—
dat’s best.”
As Alano was almost exhausted, I insisted that the guide be
transferred to my back, and this was speedily done, and on we
went, away from the river and directly into the forest. Of course,
with such a burden I could not go far, and scarcely a hundred yards
were traversed when I came to a halt, at the foot of a giant
mahogany tree.
Not without a good deal of difficulty Jorge was raised up into the
branches of the tree, and we followed.
“Still now and listen!” cried Jorge, with a half-suppressed groan.
With strained ears we sat in the mahogany tree for fully half an
hour without speaking. We heard the Spaniards cross the river and
move cautiously in the direction of the three fires, and presently
they returned to their own camp.
“Thank fortune, we have outwitted them!” murmured Alano, the
first to break the silence. “You poor fellow!” he went on to Jorge;
“you saved my life.”
He asked about the wound which had been received, and was
surprised, and so was I, to learn that it was but slight, and what had
caused the guide’s inability to run had been a large thorn which had
cut through his shoe into his heel. By the light of a match the thorn
was forced out with the end of Jorge’s machete, and the foot was
bound up in a bit of rag torn from my coat sleeve, for I must admit

that rough usage had reduced my clothing to a decidedly dilapidated
condition.
As we could not sleep very well in the tree without hammocks,
we descended to the ground and made our way to a bit of upland,
where there was a small clearing. Here we felt safe from discovery
and lay down to rest. But before retiring Alano thanked Jorge
warmly for what he had done, and thanked me also.
“I thought you were a goner,” he said to me. “How did you
escape when the horse balked and threw you into the stream?”
I told him, and then asked him to relate his own adventures,
which he did. After leaving me, he said, his horse had taken the bit
in his teeth and gone on for fully a mile. When the animal had come
to a halt he had found himself on a side trail, with no idea where he
was.
His first thought was to return to the stream where the mishap
had occurred, his second to find General Garcia. But Providence had
willed otherwise, for he had become completely tangled up in the
woods and had wandered around until nightfall. In the morning he
had mounted his horse and struck a mountain path, only to fall into
the hands of the Spanish soldiers two hours later. These soldiers
were a most villainous lot, and, after robbing him of all he
possessed, had decided to take his life, that he might not complain
of them to their superior officer.
“From what I heard them say,” he concluded, “I imagine they
have a very strict and good man for their leader—a man who
believes in carrying on war in the right kind of a way, and not in
such a guerrilla fashion as these chaps adopt.”
“I don’t want any war, guerrilla fashion or otherwise,” I said
warmly. “I’ve seen quite enough of it already.”
“And so have I,” said my Cuban chum.

Of course he was greatly interested to learn that his father was
on the way to place his mother and sisters in the old convent on the
river. He said that he had seen the place several years before.
“It is a tumbled-down institution, and Father Anuncio lives there
—a very old and a very pious man who is both a priest and a doctor.
I shouldn’t wonder if the old building has been fitted up as a sort of
fort. You see, the Spaniards couldn’t get any cannon to it very well,
to batter it down, and if they didn’t have any cannon the Cubans
could hold it against them with ease.”
“Unless they undermined it,” I said.
“Our people would be too sharp for that,” laughed my Cuban
chum. “They are in this fight to win.”
Jorge now advised us to quit talking, that our enemies might not
detect us, and we lay down to rest as previously mentioned. I was
utterly worn out, and it did not take me long to reach the land of
dreams, and my companions quickly followed suit.
In the morning our guide’s heel was rather sore, yet with true
pluck he announced his readiness to go on. A rather slim and hasty
breakfast was had, and we set off on a course which Jorge
announced must bring us to the river by noon.

CHAPTER XVII.
A TREACHEROUS STREAM TO CROSS.
I must mention that now that we had gained the high ground of
the mountains the air was much cooler and clearer than it was in the
valleys, and, consequently, traveling was less fatiguing.
Jorge went ahead, limping rather painfully at times, but never
uttering a word of complaint. Next to him came Alano, while I
brought up in the rear. It is needless to state that all of us had our
eyes and ears wide open for a sight or sound of friend or enemy.
The road was a hard one for the most part, although here and
there would be found a hollow in which the mud was from a few
inches to several feet deep. Jorge always warned us of these spots,
but on several occasions I stepped into the innocent-looking mud
only to find that it was all I could do to get clear of the dark, glue-
like paste.
It was but eleven o’clock when we came in sight of the river,
which at this point was from thirty to forty feet wide. Looking up and
down the water-course, we saw that it wound its way in and out
among the hills in serpentine fashion. The bottom was mostly of
rough stones, and the stream was barely three to four feet deep.
“How will we get over?—by swimming?” I questioned, as we
came to a halt on a bank that was twenty feet above the current.
“Find good place by de rocks,” said Jorge. “Must be careful.
Water werry swift.”

I could see that he was right by the way the water dashed
against the rocks. Our guide led the way along the bank for a
distance of several hundred feet and began to climb down by the aid
of the brush and roots.
“That doesn’t look pleasant,” remarked Alano, as he hesitated.
“Just look at that stream!”
Picking up a dry bit of wood he threw it into the water. In a few
seconds it was hurried along out of our sight.
Nevertheless, we followed Jorge down to the water’s edge.
Before us was a series of rocks, which, had the stream been a bit
lower, would have afforded an excellent fording-place.
“De river higher dan I think,” said our guide. “You take off boots,
hey?”
“That we will,” I answered, and soon had my boots slung around
my neck. Alano followed my example, and with extreme caution we
waded down and out to the first rock.
“Any alligators?” I cried, coming to a pause.
“No 'gators here,” answered Jorge. “Water too swift—'gators no
like dat.”
This was comforting news, and on I went again, until I was up to
my knees. The water felt very refreshing, and I proposed to Alano
that we take advantage of our situation and have a bath.
“I feel tremendously dirty, and it will brace us up. We needn’t
lose more than ten minutes.”
My Cuban chum was willing, and we decided to take our bath
from the opposite shore. Jorge declined to go swimming and said he
would try his luck at fishing, declaring that the river held some
excellent specimens of the finny tribe.

We had now reached the middle of the stream. I was two yards
behind Alano, while Jorge was some distance ahead. We were
crossing in a diagonal fashion, as the fording rocks ran in that
direction.
Suddenly Alano muttered an exclamation in Spanish. “It’s mighty
swift out here!” he cried. “Look out, Mark, or——”
He did not finish. I saw him slip and go down, and the next
instant his body was rolling over and over as it was being carried
along by the rushing current.
“Jorge, Alano is gone!” I yelled, and took a hasty step to catch
hold of my chum’s coat. The movement was a fatal one for me, and
down I went precisely as Alano had done. The water entered my
eyes and mouth, and for the moment I was blinded and bewildered.
I felt my feet touch bottom, but in the deeper water to obtain a
footing was out of the question.
When my head came up I found myself at Alano’s side. I saw he
had a slight cut on the forehead and was completely dazed. I caught
him by the arm until he opened his eyes and instinctively struck out.
“We’re lost, Mark!” he spluttered.
“Not yet,” I returned. “Strike out for the shore.”
With all the strength at our command we struck out. To make
any headway against that boiling current was well-nigh impossible,
and on and on we went, until I was almost exhausted. Alano was
about to sink when he gave a cry.
“The bottom!” he announced, and I put down both feet, to find
the stream less than three feet deep. With our feet down, we were
now able to turn shoreward; and five minutes later Jorge had us
both by the hands and was helping us out.
“Well, we wanted a bath and we got it,” were Alano’s first words.
“Have you had enough, Mark?”

“More than sufficient,” I replied, with a shudder. “Ugh, but that is
a treacherous stream, and no mistake!”
“You lucky boys,” said Jorge. “Horse get in and roll over, he lose
his life.”
We stopped long enough to wring out our clothing and put on
our boots, and then followed our guide again. Half an hour later we
reached a sheltered spot and here took dinner. By the time the
repast was ended our light summer suits were almost dried. Luckily,
through it all each of us had retained his hat.
“We haven’t had the fish Jorge promised us,” said Alano, as we
were preparing to resume our journey. “A bit of something baked
wouldn’t go bad.”
“Fish to-night,” said the guide.
“Have you a line and hook, Jorge?” I asked.
“Yes, always carry him,” he answered; and, upon further
questioning, I learned that to carry a fishing outfit was as common
among the rebels as to carry a pistol or the ever-ready machete.
They had to supply themselves with food, and it was often easier
and safer to fish in the mountain streams than to shoot game or
cattle.
We made a camp that night under the shelter of a clump of
grenadillo trees; and, as Jorge had promised, he tried his luck at
fishing in a little pool under some rocks. He remained at his lines,
two in number, for nearly an hour, and in that time caught four fish—
three of an eel-like nature and a perch. These were cooked for
supper, and tasted delicious.
“When will we reach the old convent?” I asked, as we were about
to turn in.
“Reach him by to-morrow afternoon maybe, if no storm come,”
said Jorge.

“Do you think there will be a storm?”
The guide shrugged his shoulders.
“Maybe—time for storm now.”
The fire had been put out as soon as the fish were baked, that it
might not attract the attention of any Spaniards who might be in the
neighborhood. At eight o’clock we turned in, making our beds on a
number of cedar boughs, which were easy to obtain in this
mountainous locality. We had no coverings but our coats, but found
these sufficient under the shelter of the grenadillos.
How long I slept I did not know. I awoke with a start and raised
up. All was silent. I gazed around in the gloom, and saw that Alano
and our guide slumbered soundly.
“I must have been dreaming,” I muttered to myself, when a
rustle in the brush behind me caused me to leap to my feet. There
was another rustle, and then came what I imagined was a half-
subdued growl of rage.
Fearful that we were on the point of being attacked by some wild
animal, I bent over my companions and shook them.
“Wake up! Wake up!” I cried. “There are wild beasts about!
Quick, and get your pistols ready!”
And then I looked toward the bushes again, to see an ugly, hairy
head thrust forward and a pair of glaring eyes fastened full upon
me!

CHAPTER XVIII.
ALONE.
“What is it?” cried Alano, as he scrambled to his feet.
“I don’t know!” I yelled. “Look! look!”
As I spoke I pulled out my pistol. By this time Jorge was also
aroused.
“Que ha dicho V.? [What did you say?]” he demanded, leaping up
and catching at his machete.
“An animal—a bear, or something!” I went on. “There he is!”
I raised my pistol, and at the same time our guide looked as I
had directed. I was about to pull the trigger of my weapon when he
stopped me.
“No shoot! Puerco!” he cried, and gave a laugh. Leaping forward,
he made after the animal, which turned to run away. But Jorge was
too quick for him. Presently there was a grunt and a prolonged
squeal, and then I understood what my wild beast was—nothing but
a wild pig! In a couple of minutes Jorge came back to camp
dragging the tough little porker by the hind legs. He had killed the
animal in true butcher’s style.
“We have pork to-morrow,” he grinned, for Cuban negroes are as
fond of pig meat as their Northern brothers. Taking a short rope
from one of his pockets, he attached it to the pig’s hind legs and
hung the body up on a convenient tree branch.

The incident had upset my nerves, and for the balance of the
night I slept only by fits and starts, and I was glad when dawn came
and the rising sun began to gild the tops of the surrounding hills.
The sight was a beautiful one, and I gazed at it for some time, while
Jorge prepared some pork chops over a tiny fire he had kindled.
“We carry what pork we can,” he said. “No use to leave it behind.
Father Anuncio very glad to get pig, so sweet!” and once again Jorge
grinned. After breakfast the guide cut up the balance of the animal,
wrapped the parts in wet palm leaves, and gave us each our share
to carry.
Our involuntary bath had done me good, and I stepped out
feeling brighter and better than I had for several days. I was
becoming acclimated, and I was glad of it, for had I been taken
down with a fever I do not know what I would have done.
Alano was as eager as myself to reach the old convent on the
river, and we kept close upon Jorge’s heels as our guide strode off
down the mountain side toward a forest of sapodillas and plantains.
“I trust we find everybody safe and sound,” I remarked. “The fact
that your father thought it best to conduct your mother and sisters
to the convent would seem to indicate he was disturbed about their
safety.”
“I am hoping he did it only to be clear to join the rebel army,”
replied Alano. “I hope both your father and mine are in the ranks,
and that we are allowed to join too.”
I did not wish to discourage my Cuban chum on this point, yet I
had my own ideas on the subject. I was not anxious to join any
army, at least not while both sides to the controversy were
conducting the contest in this guerrilla-like fashion. I was quite sure,
from what I had heard from various sources, that up to that date no
regular battle had been fought in the eastern portion of Cuba,
although the western branch of the rebel army, under General
Gomez, was doing much regular and effective work.

The reasons for this were twofold. In the first place, General
Gomez' forces were composed mainly of white men, while a large
portion of the soldiers under General Garcia were black. Nearly all of
the Americans who came to Cuba to fight for Cuban liberty, came by
way of Havana or Jibacoa and joined General Gomez, and these
fellows brought with them a large stock of arms and ammunition. It
was said that there were three armed men in the West to every man
who had even a pistol in the East. Many of the negroes were armed
only with their machetes, which they tied to their wrists with
rawhides, that they might not lose this sole weapon while on the
march or in a skirmish. To shoot off a cartridge in a pistol without
doing some effective work with it was considered under General
Garcia and his brother officers almost a crime.
The guerrilla warfare in the mountains I felt could be kept up for
a long time, perhaps indefinitely. The Spanish troops had sought to
surround General Garcia a dozen times, only to discover, when too
late, that he and his men had left the vicinity. The Cuban forces
moved almost always at night, and often detachments of soldiers
were sent off on swift horses to build false campfires dozens of miles
away from the real resting-place of the army.
In the valley we crossed through a large coffee plantation. In the
center was a low, square house with several outbuildings. The house
was closed tightly, and so were the other buildings, yet as we drew
close I fancied I heard sounds from within.
I notified Jorge, and a halt ensued. Hardly had we stopped than
the door of the house flew open and out rushed half a dozen well-
dressed Spanish soldiers.
“Halte!” came the command, but instead of halting we turned
and fled—I in one direction, and Alano and our guide in another.
Bang! bang! went a couple of guns, and I heard the bullets clipping
through the trees. Surprised and alarmed, I kept on, past a field of
coffee and into a belt of palms. Several of the soldiers came after

me, and I heard them shouting to me to stop and promising all sorts
of punishment if I did not heed their command.
But I did not intend to stop, and only ran the faster, past the
palms and into a mass of brushwood growing to a height of ten or
twelve feet. At first the bushes were several feet apart, and I went
on with ease; but soon the growth was more dense, and numerous
vines barred the way; and at last I sank down in a hollow, unable to
go another step, and thoroughly winded.
I remained in the hollow at least half an hour, trying to get back
my breath and listening intently to the movements of my pursuers.
The soldiers passed within fifty feet of me, but that was as close as
they got, and presently they went off; and that was the last I heard
of them.
In the excitement of the chase I had dropped my pig meat, and
now I discovered that nearly all of my other traps were gone,
including my pistol, which had left my hand during a nasty trip-up
over a hidden tree root. The trip-up had given me a big bump on the
temple and nearly knocked me unconscious.
Crawling around, I found a pool of water, in which I bathed my
forehead, and then I set about finding out what had become of
Alano and Jorge. I moved with extreme caution, having no desire to
be surprised by the enemy, who might be lying in ambush for me.
Moving onward in the brush I soon discovered was no light
undertaking, and it was fully an hour before I found my way out to
where the vines grew less profusely. The spot where I emerged was
not the same as that at which I had entered the undergrowth, and
on gazing around I was dismayed to find that the whole topography
of the country looked different.
I was lost!
The thought rushed upon me all in an instant, and I half groaned
aloud as I realized my situation. I must be all of a mile from the

plantation, and where my friends were I had not the remotest idea.
The sun beat down hotly in the valley, and it was not long before
I was both dry and hungry. I searched around for another pool, but
could not find any, and had to content myself with the taste of a wild
orange, far from palatable.
Noon came and went and found me still tramping around the
valley looking for Alano and Jorge. In my passage through the
bushes my already ragged clothing was torn still more, until I felt
certain that any half-decent scarecrow could discount me greatly in
appearance.
At four o’clock, utterly worn out, I threw myself on the ground in
a little clearing and gave myself up to my bitter reflections. I felt that
I was hopelessly lost. Moreover, I was tremendously hungry, with
nothing in sight with which to satisfy the cravings of my appetite.
Night, too, was approaching. What was to be done?

CHAPTER XIX.
THE CAVE IN THE MOUNTAIN.
I lay in the clearing in the valley for all of half an hour. Then,
somewhat rested, I arose, unable to endure the thought that night
would find me in the wilds alone and unarmed.
I could well remember how the sun had stood when I had
separated from my companions, and now, using the sun as a guide,
I endeavored once more to trace my steps to the path leading down
to the river. Once the stream was gained, I resolved to search up
and down its banks until the old convent was sighted.
My course led me up the side of a small mountain, which I
climbed with great difficulty, on account of the loose stones and dirt,
which more than once caused my ankle to give a dangerous twist. A
sprained ankle would have capped the climax of my misfortunes.
Just as the sun was beginning to set behind the peaks to the
westward of me, I reached a little plateau which divided a ridge from
the mountain proper. Here I rested for a few minutes and obtained a
refreshing drink at a spring under some rocks. Then I went on, in
some manner satisfied that I was on the right path at last.
But, alas! hardly had I taken a score of steps than I stepped on a
bit of ground which appeared solid enough, but which proved to be
nothing but a mass of dead brushwood lying over a veritable chasm.
The whole mass gave way, and with a lurch I was hurled forward
into black space.
As I went down I put out my hands to save myself. But, though I
caught hold of several roots and bits of rocks, this did not avail; and

I did not stop descending until I struck a stone flooring twenty feet
below the top of the opening. Fortunately the floor was covered with
a large mass of half-decayed brush, otherwise the fall must have
been a serious if not a fatal one.
As I went down, on hands and knees, a lot of loose branches,
dirt, and small stones rolled on top of me, and for the minute I had
a vision of being buried alive. But the downfall soon ceased; and,
finding no bones broken, I crawled from under the load and
surveyed the situation.
I felt that I was now worse off than ever. The well-hole—I can
call it nothing else—was about ten feet in diameter, and the walls
were almost smooth. The top of the opening was far out of my
reach, and, as for a means of escape, there seemed to be none.
However, I was not to be daunted thus easily, and, striking a
match and lighting a cedar branch, I set about looking for some spot
where I might climb up. But the spot did not present itself.
But something else did, and that was an opening leading directly
into the mountain. On pulling at a projecting rock, I felt it quiver,
and had just time to leap back, when it fell at my feet. Behind the
rock was a pitch-black hole, into which I thrust the lighted branch
curiously. There was a cave beyond—how large was yet to be
discovered.
I had no desire to explore any cave at that moment, my one idea
being to get out of the well-hole and proceed on my way. But
getting out of the hole was impossible, and I was forced to remain
where I was, much to my disgust and alarm.
Jorge had been right about the coming storm. At an hour after
sunset I heard the distant rumble of thunder, and soon a lively
breeze blew through the trees and brush on the mountain side. A
few flashes of lightning followed, and then came a heavy downpour
of rain.

Not wishing to be soaked, I retreated to the cave I had
discovered, although with caution, for I had no desire to take
another tumble into a deeper hole. But the floor of the cavern
appeared to be quite level, and with rising curiosity I took up my
lighted cedar branch, whirled it around to make it blaze up, and
started on a tour of investigation and discovery.
That I should not miss my way back, I lit a pile of small brush at
the mouth of the opening. Then I advanced down a stony corridor,
irregular in shape, but about fifty feet wide by half as high.
The opening appeared to be a split in the mountain, perhaps
made ages before by volcanic action. I felt certain there was an
opening above, for in several spots the rain came down, forming
small pools and streams of water.
Suddenly the idea struck me to watch which way the water ran,
and I did so and learned that its course was in the very direction I
was walking. Moreover the tiny streams merged one into another,
until, several hundred feet further on, they formed quite a water
course.
“If only this stream flows into the main river!” I thought, and on
the spur of the moment resolved to follow it as far as I was able,
satisfied that if it led to nowhere in particular I could retrace my
steps to its source.
I now found the cave growing narrower, and presently it grew
less than a dozen feet in width, and the stream covered the entire
bottom to the depth of several inches. Throwing my boots over my
shoulders, I began wading, feeling sure of one step ere I trusted
myself to take another.
It took me fully ten minutes to proceed a hundred feet in this
fashion. The stream was now not over six feet wide and all of a foot
deep.

Making sure that my torch was in no danger of going out, I
continued to advance, but now more slowly than ever, for in the
distance I could hear the water as it fell over a number of rocks.
There was a bend ahead; and this passed, I fervently hoped to
emerge into the open air, on the opposite side of the mountain and
close to the bank of the river for which I was seeking.
At the bend the water deepened to my knees, and I paused to
roll up my trousers, in the meantime resting the torch against the
wall, which afforded a convenient slope for that purpose.
I had just finished arranging my trouser-legs to my satisfaction,
when a rumble of thunder, echoing and re-echoing throughout the
cavern, made me jump. My movement caused the cedar branch to
roll from the rocks, and it slipped with a hiss into the stream. I made
a frantic clutch for it, and, in my eagerness to save it from going out
or getting too wet, I fell on it in the very middle of the stream.
With a splutter I arose to find myself in utter darkness. Moreover,
the cedar branch was thoroughly soaked, and it would take a good
many matches to light it again. And what was still worse, every
match my pocket contained was soaked as badly as the torch.
I must confess that I was utterly downcast over my mishap, and
if there had been any dry ground handy I would have thrown myself
down upon it in abject despair. But there was only water around,
and, disconsolate as I was, I felt I must either go forward or
backward.
How I became turned about I do not know, but certain it is that,
in essaying to return to the spot from whence I had come, I
continued on down the stream. I did not notice the mistake I had
made until fifty yards had been passed and I brought up against an
overhanging rock with my shoulder. Putting up my hands, I was
dismayed to discover that the passage-way was just high enough to
clear my head.

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