Linear Circuit Analysis- DeCarlo-Lin 3e

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LINEAR

CIRCUITS

TIME DOMAIN, PHASOR, AND LAPLACE

TRANSrORM APPROACHES

T H I R D E D I T I O N

Raymond A. DeCarlo

Purdue University

Pen-Min Lin

Purdue University

Kendall Hunt

p u b l i s h i n g c o m p a n y

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^ Used under license from Shutterstock, Inc. Cover image (^^J^ikiaui

Kendall Hunft

p u b l i s h i n g c o m p a n y

www.kendallhunt.cpm Send all inquiries to: 4050 Westmark Drive Dubuque, lA 52004-1840

ISBN 978-0-7575-6499-4

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the copyright owner.

Printed in the United States of America 10 9 8 7 6 5 4 3 o n o o r ^ O

n

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TABLE OF CO N TEN TS

Preface...vii

Chapter 1 • Charge, Current, Voltage and Ohm’s Law ...1

Chapter 2 • Kirchhoff’s Current & Voltage Laws and Series-Parallel Resistive C ircu its...51

Chapter 3 • Nodal and Loop Analyses... 107

Chapter 4 • T he Operational Amplifier... 155

Chapter 5 * Linearity, Superposition, and Source Transform ation... 191

Chapter 6 • Thevenin, Norton, and Maximum Power Transfer Theorems... 227

Chapter 7 • Inductors and C apacitors... 269

Chapter 8 • First Order RL and RC Circuits...321

Chapter 9 • Second Order Linear Circuits...379

Chapter 10 • Sinusoidal Steady State Analysis by Phasor Methods ...431

Chapter 11 • Sinusoidal State State Power Calculations...499

Chapter 12 • Laplace Transform Analysis L Basics... 543

Chapter 13 • Laplace Transform Analysis II: Circuit Applications... 603

Chapter 14 • Laplace Transform Analysis III; Transfer Function Applications...683

Chapter 15 * Time Domain Circuit Response Computations: The Convolution M ethod... 763

Chapter 16 • Band-Pass Circuits and Resonance...811

Chapter 17 * Magnetically Coupled Circuits and Transformers... 883

Chapter 18 • Tw o-Ports...959

Chapter 19 • Principles o f Basic Filtering ... 1031

Chapter 20 • Brief Introduction to Fourier Series ... 1085

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PREFACE

For the last several decades, EE/ECE departments o f US universities have typically required two semesters o f linear circuits during the sophomore year for EE majors and one semester for other engineering majors. Over the same time period discrete time system concepts and computer engi neering principles have become required fare for EE undergraduates. Thus we continue to use Laplace transforms as a vehicle for understanding basic concepts such as impedance, admittance, fdtering, and magnetic circuits. Further, software programs such as PSpice, MATLAB and its tool boxes, Mathematica, Maple, and a host o f other tools have streamlined the computational drudg ery o f engineering analysis and design. MATLAB remains a working tool in this 3'''^ edition o f Linear Circuits.

In addition to a continuing extensive use o f MATLAB, we have removed much o f the more com plex material from the book and rewritten much o f the remaining book in an attempt to make the text and the examples more illustrative and accessible. More importantly, many o f the more diffi cult homework exercises have been replaced with more routine problems often with numerical answers or checks.

Our hope is that we have made the text more readable and understandable by today’s engineering undergraduates.

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C

H

A

P

T

E

R

Charge, Current, Voltage

and Ohm’s Law

CHAPTER O U TLIN E

1. Role and Importance o f Circuits in Engineering 2. Charge and Current

3. Voltage

4. Circuit Elements

5. Voltage, Current, Power, Energy, Relationships 6. Ideal Voltage and Current Sources

7. Resistance, Ohm’s Law, and Power (a Reprise)

8. V-I Characteristics o f Ideal Resistors, Constant Voltage, and Constant Current Sources

Summary

Terms and Concepts Problems

CHAPTER O B jEC TIV ES

1. Introduce and investigate three basic electrical quantities: charge, current, and voltage, and the conventions for their reference directions.

2. Define a two-terminal circuit element.

3. Define and investigate power and energy conversion in electric circuits, and demonstrate that these quantities are conserved.

4. Define independent and dependent voltage and current sources that act as energy or sig nal generators in a circuit.

5. Define Ohm’s law,

v{t) = R i{t),

for a resistor with resistance

R.

6. Investigate power dissipation in a resistor.

7. Classify memoryless circuit elements by dieir terminal voltage-current relationships. 8. Explain the difference between a device and its circuit model.

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ch ap ter 1 • Charge, Current, Voltage and O hm ’s Law

1. ROLE AND IM PORTANCE OF CIRCUITS IN ENGINEERING

Are you curious about how fuses blow? About the meaning o f different wattages on Hght bulbs? About the heating elements in an oven? And how is the presence o f your car sensed at a stoplight? Circuit theory, the focus o f this text, provides answers to all these questions.

W hen you learn basic circuit theory, you learn how to harness the power o f electricity, as is done, for example, in

• an electric motor that runs the compressor in an air conditioner or the pump in a dish washer;

• a microwave oven; • a radio, TV, or stereo; • an iPod;

• a car heater.

In this text, we define and analyze common circuit elements and describe their interaction. Our aim is to create a modular framework for analyzing circuit behavior, while simultaneously devel oping a set o f tools essential for circuit design. These skills are, o f course, crucial to every electri cal engineer. But they also have broad applicability in other fields. For instance, disciplines such as bioengineering and mechanical engineering have similar patterns o f analysis and often utilize circuit analogies.

W H A T IS A C IR C U IT ?

A circuit is an energy or signal/information processor. Each circuit consists o f interconnections o f “simple” circuit elements, or devices. Each circuit element can, in turn, be thought o f as an ener gy or signal/information processor. For example, a circuit element called a “source” produces a voltage or a current signal. This signal may serve as a power source for the circuit, or it may rep resent information. Information in the form o f voltage or current signals can be processed by the circuit to produce new signals or new/different information. In a radio transmitter, electricity powers the circuits that convert pictures, voices, or music (that is, information) into electromag netic energy. This energy then radi

ates into the atmosphere or into space from a transmitting antenna. A satellite in space can pick up this electromagnetic energy and trans mit it to locations all over the world. Similarly, a T V reception antenna or a satellite dish can pick up and direct this energy to a T V set. T h e T V contains circuits (Figure 1.1) that reconvert the information within the received signal back into pictures with

sound. FIG U RE 1.1 Cathode ray tube with surrounding circuitry for converting electrical signals into pictures.

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Chapter 1 • Charge, Current, Voltage and O hm ’s Law

2. CH A RGE AND CU RREN T

CH A R G E

Charge is an electrical property o f matter. Matter consists o f atoms. Roughly speaking, an atom contains a nucleus that is made up o f positively charged protons and neutrons (which have no charge). T he nucleus is surrounded by a cloud o f negatively charged electrons. Th e accumulated charge on 6.2415 x 10’^ electrons equals -1 coulomb (C). Thus, the charge on an electron is -1 .6 0 2 1 7 6 X 10-19 C.

Particles with opposite charges attract each other, whereas those with similar charges repel. The force o f attraction or repulsion between two charged bodies is inversely proportional to the square o f the distance between them, assuming the dimensions o f the bodies are very small compared with the distance o f separation. Two equally charged particles 1 meter (m) apart in free space have charges o f 1 C each if they repel each other with a force o f 10“^ c^ Newtons (N), where c = 3 x 10^ m/s is the speed o f light, by definition. The force is attractive if the particles have opposite charges. Notationally, Q will denote a fixed charge, and

q

or

q{t),

a time-varying charge.

Exercise.

How many electrons have a combined charge o f -5 3 .4 0 6 x 10 C?

AN SW ER; 333,3 9 1 ,5 9 7

Exercise.

Sketch the time-dependent charge profile

q{t) =

3 (l-^ ^ 0 C, ? > 0, present on a metal

plate. M ATLAB is a good tool for such sketches.

A conductor refers to a material in which electrons can move to neighboring atoms with relative ease. Metals, carbon, and acids are common conductors. Copper wire is probably the most com mon conductor. An ideal conductor offers zero resistance to electron movement. Wires are assumed to be ideal conductors, unless otherwise indicated.

Insulators oppose electron movement. Common insulators include dry air, dry wood, ceramic, glass, and plastic. An ideal insulator offers infinite opposition to electron movement.

C U R R EN T

Current refers to the net flow o f charge across any cross section o f a conductor. T he net move ment o f 1 coulomb (1 C) o f charge through a cross section o f a conductor in 1 second (1 sec) produces an electric current o f 1 ampere (1 A). The ampere is the basic unit o f electric current and equals 1 C/s.

The direction o f current flow is taken by convention as opposite to the direction o f electron flow, as illustrated in Figure 1.2. This is because early in the history o f electricity, scientists erroneously believed that current was the movement o f only positive charges, as illustrated in Figure 1.3. In metallic conductors, current consists solely o f the movement o f electrons. However, as our under standing o f device physics advanced, scientists learned that in ionized gases, in electrolytic solu

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c h ap ter 1 • Charge, Current, Voltage and O hm ’s Law

tions, and in some semiconductor materials, movement o f positive charges constitutes part or all o f the total current flow.

One Ampere of Current "

One

;

Cloud o f \

se co n d ^ ... |----

6.24x10’® 1

later

i

;

k electrons

J

Boundary

FIG U RE 1.2 A cloud o f negative charge moves past a cross section of an ideal conductor from right to left. By convention, the positive current direction is taken as left to right.

One Coulomb of positive charge Boundary One Ampere of Current One 'second later

FIGURE 1.3 In the late nineteenth cenmry, current was thought to be the movement of a positive charge past a cross section of a conduaor, giving rise to the conventional reference “direction of positive current flow.”

Both Figures 1.2 and 1.3 depict a current o f 1 A flowing from left to right. In circuit analysis, we do not distinguish between these two cases: each is represented symbolically, as in Figure 1.4(a). The arrowhead serves as a reference for determining the true direction o f the current. A positive value o f current means the current flows in the same direction as the arrow. A current o f negative value implies flow is in the opposite direction o f the arrow. For example, in both Figures 1.4a and b, a current o f 1 A flows from left to right.

1A

>

-1A

<

(a) (b)

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In Figure 1.4, the current is constant. The wall socket in a typical home is a source o f alternating current, which changes its sign periodically, as we will describe shortly. In addition, a current direc tion may not be known

a priori.

These situations require the notion o f a negative current.

E X A M P L E 1.1.

Figure 1.5 shows a slab o f material in which the following is true:

1. Positive charge carriers move from left to right at the rate o f 0.2 C/s. 2. Negative charge carriers move from right to left at the rate o f 0.48 C/s.

Given these conditions, a) Find and /^;

b) Describe the charge movement on the wire at the boundaries

A

and

B.

A

1

,

_{© o}

— 0

©

0

B Connecting wire Connecting wire Sem iconductor iVlaterial

F IG U R E 1.5 Material through which positive and negative charges move. So lu tio n

a) The current from left to right, due to the movement o f the positive charges, is 0.2 A. The current from left to right, due to the movement o f the negative charges, is 0.48 A. Therefore, /^, the total current from left to right, is 0.2 + 0.48 = 0.68 A. Since

ly

is the current from right to left, its value is then -0 .6 8 A.

b)

T he wire is a metallic conductor in which only electrons move. Therefore, at boundaries A and B, negative charges (carried by electrons) move from right to left at the rate o f 0.68 C/s.

Exercise.

In Example 1.1, suppose positive-charge carriers move from right to left at the rate o f 0.5

C/s, and negative carriers move from left to right at the rate o f 0.4 C/s. Find and AN SW ER: /, = - 0 .9 A; ^ = 0.9 A

If a net charge crosses a boundary in a short time frame o f At (in seconds), then the approxi mate current flow is

Aq

/ =

At

(1.1)

where

I,

in this case, is a constant. The instantaneous (time-dependent) current flow is the limit ing case o f Equation 1.1, i.e.,

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dq{t)

dt

(

1

.

2

) Here

q{t)

is the amount o f charge that has crossed the boundary in the time interval

[tQ, t

] . The equivalent integral counterpart o f Equation 1.2 is

q{t) = J i{r)dr

(1.3)

E X A M P L E 1.2

The charge crossing a boundary in a wire is given in Figure 1.6(a) for ? > 0. Plot the current

i{t)

through the wire.

(a)

(b)

FIG U RE 1.6 (a) Charge crossing a hypothetical boundary; (b) current flow associated with the charge plot o f (a).

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So lutio n

As per Equation 1.2, the current is the time derivative o f

q{t).

The slopes o f the straight-Une seg ments o f

q{f}

in Figure 1.6(a) determine the piecewise constant current plotted in Figure 1.6(b).

■ ■ • • l-co s(co?)

Exercise.

The charge crossing a boundary in a wire varies as

q[t) =

--- C, for

t >Q.

Compute the current flow. A N SW ER: sin(cof) A, for f > 0

Exercise.

Repeat the preceding exercise if

q{t)

=

5e

C, for

t > 0 .

A N SW ER:

A,

for f > 0

E X A M PLE 1.3

Find

q{t),

the charge transported through a cross section o f a conductor over [0, f], and also the total charge Q transported, if the current dirough the conductor is given by die waveform o f Figure 1.7(a).

-l-*-t(se c)

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So lutio n

From Equation 1.3, for

t>Q,

q { t ) = p { T ) c l T

Thus,

q{t)

is the running area under the

i{t)

versus

t

curve. Since

i{t)

is piecevv'ise constant, the integral is piecewise linear because the area either increases or decreases linearly with time, as shown in Figure 1.7(b). Since

q{t)

is constant for ^ > 3, the total charge transported is Q =

q{5) =

3 C.

Exercise.

If the current flow through a cross section o f conductor is

i{t) =

cos(120jtf) A for ? > 0

and 0 otherwise, find

q{t)

for

t>Qi.

AN SW ER:

q{t)

C for r > 0

‘

120jt

Exercise.

Suppose the current through a cross section o f conductor is given in Figure 1.8. Find

q{t)

for

t > 0 .

FIGURE 1.8

AN SW ER;

q(t)

= C for 0 < 1;

q{t) =

IC for r > I

T Y P ES OF C U R R EN T

There are two very important current types: direct current (do) and alternating current (ac). Constant current (i.e.,

dqldt

= / is constant) is called direct current, which is illustrated graphi cally m Figure 1.9(a). Figure 1.9(b) shows an alternating current, generally meaning a sinusoidal waveform, i.e., current o f the form y4sin(w? + ()>), where

A

is the peak magnitude, co is the angu lar frequency, and (|) is the phase angle o f the sine wave. W ith alternating current, the instanta neous value o f the waveform changes periodically through negative and positive values, i.e., the

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ch ap ter 1 • Charge, Current, Voltage and O h m s Law

direction o f the current flow changes regularly as indicated by the + and - values in Figure 1.9(b). Household current is ac.

Lastly, Figure 1.9(c) shows a current that is neither dc nor ac, but that nevertheless will appear in later circuit analyses. There are many other types o f waveforms. Interestingly, currents inside com puters, C D players, TV s, and other entertainment devices are typically neither dc nor ac.

i(t) (A) H 1 -3 (a) t(sec) -I---►

F IG U R E 1.9 (a) Direct current, or dc;

i{t)

=

Iq\

(b) alternating current, or ac;

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10 Chapter 1 • Charge, Current, Voltage and O hm ’s Law

Because the value o f an ac waveform changes with time, ac is measured in different ways. Suppose the instantaneous value o f the current at time

t

is A!sin(ci)i- + (j>). The term peak value refers to

K

in

K

sin(co? + (j)). The peak-to-peak value is

2K.

Another measure o f the alternating current, indicative o f its heating effect, is the root mean square (rms), or effective value. The rms or effec tive value is related to the peak value by the formula

rms = X peak-value =

Q .lO llK

(i.4)

A derivation o f Equation 1.4 with an explanation o f its meaning will be given in Chapter 11.

A special instrument called an ammeter measures current. Some ammeters read the peak value, whereas some others read the rms value. One type o f ammeter, based on the interaction between the current and a permanent magnet, reads the average value o f a current. From calculus, Fave! the average value o f any function y(^), over the time interval [0, 7] is given by

(1.5)

For a general ac waveform, the average value is zero. However, ac signals are often rectified, i.e., converted to their absolute values, in power-supply circuits. For such circuits, the average value o f the rectified signal is important. From Equation 1.5, the average value o f the absolute value o f an ac waveform over one complete cycle with

T

= 2jt/co, is

K ^

2.K

Average Value = —^\s,m{wt)\dt =

---

J sin(cot)clt

0 ^ 0

2K

T

i.e., 0 .636 X peak value.

-cos{(ot)

(O

0.5T

2,K

— = 0.636K

j t ( 1.6)

Exercise.

Suppose

i{t) -

169.7 sin(50jtr) A. Find the peak value, the peak-to-peak value, the rms

value o f

i{t),

and the average value o f

AN SW ER: 169.7, 339.4, 120, and 107.93 A, respectively

3. VO LTAG E

W hat causes current to flow? An analogous question might be. W hat causes water to flow in a pipe or a hose? W ithout pressure from either a pump or gravity, water in a pipe is still. Pressure from a water tower, a pressured bug sprayer tank, or a pump on a fire truck will force water flow In electrical circuits, the “pressure” that forces electrons to flow, i.e., produces a current in a wire or a device, is called voltage. Strictly speaking, water flows from a point o f higher pressure— say, p o in ts — to a point o f lower pressure— say, point 5 — along a pipe. Between the two points and

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Chapter 1 • Charge, Current, Voltage and O hm ’s Law 11

along a conductor will force current to flow from point

A

to point

B;

there is said to be a voltage drop from point

A

to point

B

in such cases.

Gravity forces the water to flow from a higher elevation to a lower elevation. An analogous phe nomenon occurs in an electric field, as illustrated in Figure 1.10(a). Figure 1.10(a) shows two con ducting plates separated by a vacuum. O n the top plate is a fixed amount o f positive static charge. On the bottom plate is an equal amount o f negative static charge. Suppose a small positive charge were placed between the plates. This small charge would experience a force directed toward the negatively charged bottom plate. Part o f the force is due to repulsion by the positive charges on the top plate, and part is due to the attraction by the negative charges on the bottom plate. This repulsion and attraction marks the presence o f an electric field produced by the opposite sets o f static charges on the plates.

The electric field indicated in Figure 1.10 sets up an “electric pressure” or voltage drop from the top plate to the bottom plate, which forces positive charges to flow “downhill” in the way that water flows from a water tower to your faucet. Unlike water flow, negative charges are forced “uphill” from the negatively charged bottom plate to the positively charged top plate. As men tioned in the previous section, this constitutes a net current flow caused by the bilateral flow o f positive and negative charges. The point is that current flow is induced by an electric pressure called a voltage drop.

© © © © © © © © A 0 Positive charge, q Electric Field B Force on charge q © © © © © © © © A Force on negative charge Electric Field © © © 0 © © © © (a) Negative B © charge,-q © © © © © © © © (b)

FIGURE 1.10 (a) Positive charge in a (uniform) electric field; (b) negative charge in a uniform elearic field.

As mentioned, in Figure 1.10, the positive charge ^ at ^ tends to move toward

B.

We say,

quali

tatively,

that point

A

in the electric field is at a

higher potential

than point

B.

Equivalently, point

5 is at a

lower potential

than point

A.

An analogy is now evident: a positive charge in an electric field “falls” from a higher potential point to a lower potential point, just as a ball falls from a high er elevation to a lower elevation in a gravitational field.

Note, however, that if we turn the whole setup o f Figure 1.10(a) upside down, the positive charge

q

still moves from point

A

to point

B,

an upward spatial movement. Similarly, if a negative charge

- q

is placed at

B,

as in Figure 1.10(b), then the negative charge experiences an upward-pulling force, moving from the lower potential, point

B,

to the higher potential, point

A.

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n

---

^

n

Again, consider Figure 1. 10(a). As the charge

q

moves from point ^ toward

B,

it picks up veloci

ty and gains kinetic energy. Just before

q

hits the bottom plate, the kinetic energy gained equals

the (constant) force acting on

q

multiplied by the distance traveled

in the direction o f the force.

The

kinetic energy is proportional to

q

and to the “distance traveled.” Therefore,

12 Chapter 1 ® Charge, Current, Voltage and Ohms Law —

energy converted = kinetic energy gained oc

q

1 V = 1 ^ (1.9)

n The missing proportionality constant in this relationship is defined as the potential difference or

voltage

between

A

and

B,

The term “voltage” is synonymous with “potential difference.” Mathematically,

, . , energy converted

voltage = potential difference = (1.8)

magnitude of charge

The standard unit for measuring potential difference or voltage is the volt (V).

According to

Equation 1.8, i f 1 joule {]) o f energy is convertedfrom one form to another when moving 1 C o f charge

from point K to point

B,

then the potential difference, or voltage, between

A

and

B w i VT In equation

form, with standard units of V, J, and C, we have

O

The use of terms such as “elevation diflFerence,” “energy converted,” “potential difference,” or “voltage” implies that they all have positive values. If the word “difference” is changed to “drop” (or to “rise”), then potential drop and elevation drop have either positive or negative values, as the case may be. The following four statements illustrate this point in the context of Figure 1.10:

{

The voltage

between

(or across)

A a n d

5 is 2 V.

The voltage

between

(or across)

B and A

is 2 V.

' The voltage

drop from A to B is 2W.

• The voltage

drop from B to A Is

- 2 V.

.

’

^

This discussion describes the phenomena of “voltage.” Voltage causes current flow. But what pro duces voltage or electric pressure? Voltage can be generated by chemical action, as in batteries. In a battery, chemical action causes an excess of positive charge to reside at a terminal marked with a plus sign and an equal amount of negative charge to reside at a terminal marked with a negative sign. When a device such as a headlight is connected between the terminals, the voltage causes a current to flow through the headlight, heating up the tiny wire and making it “Ught up.” Another

source of voltage/current is an electric generator in which mechanical energy used to rotate the ^ shaft of the generator is converted to electrical energy using properties of electro-magnetic fields.

All types of circuit analysis require knowledge of the potential difference between two points, say ^

A

and

B,

and specifically whether point

A

or point 5 is at a higher potential. To this end, we speak ^ of the

voltage drop

from point

A

to point

B,

conveniently denoted by a double-subscript, as

Vj^.

If the value of is positive, then point ^ is at a higher potential than point

B.

On the other

hand, if is negative, then point 5 is at a higher potential than point

A.

Since stands for the voltage drop from point

B

to point

A,

o

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The double-subscript convention is one o f three methods commonly used to unambiguously specify a voltage drop. Using this convention requires labeling all points o f interest with letters or integers so

that _’

_{KiO ^12’}

_^13 _{sense. A second, more-common convention uses + and - markings}

on two points, together with a variable or numerical labeling o f the voltage drop from the point marked + to the point marked - . Figure 1.11 illustrates this second convention, where

V

q denotes the voltage drop from

A

(marked +) to

B

(marked - ) . If

V

q is positive, then ^4 is at a higher potential than

B.

O n the other hand, if

V

q is negative, then 5 is at a higher potential than

A.

The value o f

V

q

,

togeth er with the markings + and stipulates which terminal is at a higher potential; neither alone can do this. For a general circuit element, the (+, —) markings— that is, the reference directions— can be assigned

arbitrarily.

A third method for specifying a voltage drop, using a single subscript, will be dis cussed in Chapter 2.

-I-B

V„

FIGURE 1.11 The + and - markings establish a reference direction for voltage drop. For accuracy, always place the (+, - ) markings reasonably close to the circuit element to avoid uncertainty.

The following example illustrates the use o f the double subscript and the (+, - ) markings for des ignating voltage drops.

E X A M P L E 1.4

Figure 1.12 shows a circuit consisting o f four general circuit elements, with voltage drops as indi cated. Suppose we know that = 4 V, and = 9 V. Find the values o f

V^

q and

CD-V

-I-3V

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So lutio n

T he meaiiing o f the double subscript notation and the (+, - ) markings for a voltage imply that

'DA

I V

^CZ> = - ^ Z )C = -(-2 ) = 2 V ^ 5 C = 3 V

Exercise.

In Figure 1.12, find and

Vp.^-A N S W E R :- 3 V ; - 2 V

Exercise.

T he convention o f the (+, - ) markings is commonly used as described. Figure 1.13 shows

an old 12-V automobile battery whose (+, - ) markings cannot be seen because o f the corrosion o f the terminals. A digital voltmeter (DVM ) is connected across the terminals, as shown. The display reads - 1 2 V. Figure out the (+, - ) marking o f the battery terminals.

A N SW ER: left terminal, right terminal, +

DVM

12V

battery

FIG U RE 1.13 Digital voltmeter connected to a 12-V (car) battery whose plus and minus markings have corroded away.

One final note: As with current, there are different types o f voltages— dc voltage, ac voltage, and general voltage waveforms. Figure 1.9, with the vertical axis relabeled as

v{t),

illustrates different voltage types.

4. C IR CU IT ELEM EN TS

Circuits consist o f interconnections o f circuit elements. The most basic circuit element has two terminals, and is called a two-terminal circuit element, as illustrated in Figure 1.14. A circuit

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eie-Chapter 1 • Charge, Current, Voltage and O h m s Law 15

ment called a source provides either voltage, current, or both. The battery is a very common source, providing nearly constant voltage and the usually small current needed to operate small electronic devices. Car batteries, for example, are typically 12 volts and can produce large currents during starting. The wall outlet in a home can be thought o f as a 110-volt ac source. Figure 1.14(a) shows a (battery) voltage across a general undefined circuit element. A current z(r) flows through the element. Recall from our earlier intuitive discussion that voltage is analogous to water pressure: pressure causes water to flow through pipes; voltage causes current to flow through cir cuit elements. Total water into a pipe equals total water out o f the pipe. Analogously,

the current

entering a two-terminal device must, by definition, equal the current leaving the two-terminal device.

Current

FIGURE 1.14 (a) General circuit element (connected to a battery) as an energy or signal processor:

v(i)

is the voltage developed across the circuit element, and z'(r)

is the current flowing through the circuit element; (b) practical example of a general circuit element (car headlight) connected to a car battery.

The circuit element o f Figure 1.14(a) has a specific labeling: the current i(f) flows from the plus terminal to the minus terminal through the circuit element. Such a labeling o f the voltage-current reference directions is called the passive sign convention. In contrast, the current

iij)

flows from the minus terminal to the plus terminal through the battery; this labeling is conventional for sources but not for non-source circuit elements.

In addition to sources, there are other common two-terminal circuit elements: • The resistor

• The capacitor • The inductor

For a resistor, the amount o f current flow depends on a property called resistance; the smaller the resistance, the larger the current flow for a fixed voltage across the resistor. A small-diameter pipe offers more resistance to water flow than a large-diameter pipe. Similarly, different types o f con ductors offer different resistances to current flow. A conductor that is designed to have a specific resistance is called a resistor. If the device is an ideal resistor, then

v(f)

=

Ri{i),

where i? is a con stant o f resistance. More on this shortly.

The circuit elements called the capacitor and the inductor will be described later in the text. Also, future chapters will describe the operational amplifier and the transformer that are circuit elements having more than two terminals.

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5. VO LTAG E, CURRENT, POW ER, ENERGY, RELATIONSHIPS

The relationship between voltage across and current through a two-terminal element determines whether power (and, thus, energy) is delivered or absorbed. The heating element in an electric oven can be thought o f as a resistor. The heating element absorbs electric energy and converts it into heat energy that cooks, among other things, turkey dinners.

In Figure 1.14(a), a battery is connected to a circuit element. Figure 1.14(b) concretely illustrates this with a 12-V car battery connected to a headlight. W ith reference to Figures 1.14(a) and 1.14(b), suppose

v{t) =

12 V, and

i{t)

= 5 A: 5 A o f current flows through the headlight. The head light converts electrical energy into heat and light. Power (in watts) is the rate at which the ener gy is converted. At each instant o f time, the electrical power delivered to (absorbed by) the head light is

pit) = v[t)i{t) -

12 X 5 = 60 watts. Similarly, at each instant o f time, the battery can be viewed as delivering 60 watts o f power to the headlight. Inside the battery, the stored potential energy o f the chemicals and metals undergoes a chemical reaction that produces the electrical potential difference and the current flow to the headlight: chemical energy is converted into elec trical energy that is converted into light and heat.

Figure 1.15 depicts a more general scenario: a circuit element is connected to its surrounding cir cuit at points

A

and

B.

(One, o f course, could imagine that the “remainder o f circuit” is a battery, and circuit element 1 is a headlight.) Suppose there is a

constant

voltage drop from

A

to

B,

denot ed by Also assume that a

constant

current flows from terminal

A

to terminal

B through

circuit element 1,

as shown.

FIG U RE 1.15 A general circuit in which a two-element circuit element is extracted and labeled according to the passive sign convention.

For discussion purposes, assume > 0 and > 0. During a time interval o f

T

s, (V^g x T) C o f charge moves through circuit element 1 from

A

to

B.

In “falling” from a higher potential, point

A,

to a lower potential, point

B,

the charge loses electric potential energy. The lost potential energy is con verted within element 1 into some other form o f energy— heat or light being two o f several possibil ities. According to Equation 1.8, the amount o f energy

converted {absorbed by the element)

is

y. T) >Q.

The power

absorbedhj

element 1 is, by definition, the rate at which it converts or absorbs

energy. This rate equals

^

a b

(^

ab

^ T)

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Chapter 1 • Charge, Current, Voltage and O h m s Law 17

Exercise.

In Figure 1.15, the current

^AB -

5 niA, and = 400 V. W hat is the energy absorbed

by circuit element 1 in one minute? W hat is the power absorbed by circuit element 1 ? AN SW ER: W = 1 20 J; P = 2 watts

W ith respect to Figure 1.15, for constant (direct) voltages and currents, we arrive at a very simple relationship:

P\-V

ab

I

ab 0 -1 0 )

where is the power (in W ) absorbed by the circuit element. Consequently, the energy, W , (in J), absorbed during the time interval

Tis

W^=P\xT

(1-11)

Now, let us reconsider Figure 1.15. One can think o f-/ ^ g as flowing from

A w B

through the remainder o f the circuit. In this case, -1 ^ ^ ^ ^ < 0 . This means that the remain der o f the circuit absorbs negative power or equivalently delivers | ^ 5(— | =

^

a

^AB

circuit element 1. As such, the remainder o f the circuit is said to

generate

electric energy. By definition, the electric power

generated

by the remainder o f the circuit is the rate at which it generates elec tric energy. From Equation 1.8, this rate equals

---

-

---

- ^

ab

^

ab

Observe that the rate at which the remainder o f the circuit generates power precisely equals the rate at which circuit element 1 absorbs power. This equality is called the principle o f conserva

tion o f power: total power generated equals total power absorbed. Equivalently, the sum o f the powers absorbed by all the circuit elements must add to zero, + Pq =

y^gl^B ^AB^^^AB^ ~

Exercise.

In Figure 1.15, -Pg = watts, i.e., the remainder o f the circuit absorbs - 1 0 watts o f

power. How much power does circuit element 1 absorb? A N SW ER: 10 watts

In general, whenever a two-terminal general circuit element is labeled according to the passive sign convention, as in Figure 1.15, then

P

= > 0 means the element absorbs (positive) power, whereas

P = V^

b

^

ab

^

absorbs negative power or delivers (positive) power to whatever it is connected. As a general convention, non-source circuit elements are labeled according to the passive sign convention. Usually, sources are labeled with the current leaving the terminal labeled with “+”. For such labeling o f sources, if the product o f the source voltage and the current leaving the “+” terminal is positive, then the source is delivering power to the network.

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Chapter 1 • Charge, Current, Vohage and O hm ’s Law

RULE FOR C A LC U LA TIN G A B SO R B ED PO W ER

The power absorbed by any circuit element (Figure 1.16) with terminals labeled

A

and

B

is equal to the voltage drop from

A m B

multiplied by the current through the element from

A

to 5,

+

_V

AB

FIGURE 1.16

Exercise.

Compute the power absorbed by each o f the elements in Figure 1.17.

-1A _____________ -2A _____________ 2A

>

< Z

3

10V (a) 10V (b) 10V (c) FIGURE 1.17 AN SW ER: (a) 10 W; (b) - 2 0 W; (c) 20 W

As mentioned, power is the rate o f change o f work per unit o f time. T he ability to determine the power absorbed by each circuit element is highly important because using a circuit element or some device beyond its power-handling capability could damage the device, cause a fire, or result in a serious disaster. This is why households use circuit breakers to make sure electrical wiring is not overloaded.

Exercise.

In Figure 1.18, a car heater is attached to a 12-volt D C voltage source. How much power

can the car heater absorb before the 20-amp fuse blows.

20 Amp Fuse

FIGURE 1.18 Car heater connected to a 12-volt car battery through a 20-amp fuse.

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As mentioned earlier, the calculated value o f absorbed power

P

may be negative. If the absorbed power

P

is negative, then the circuit element actually generates power or, equivalently, delivers power to the remainder o f the circuit. In any circuit, some elements will have positive absorbed powers, whereas some others will have negative absorbed powers. If one adds up the absorbed powers o f

ALL

elements, the sum is zero! This is a universal property called conservation o f

power.

PRIN C IPLE OF CO N SERVA TIO N OF PO W ER

The sum o f the powers absorbed by all elements in a circuit is zero at any instant o f time. Equivalently, the sum o f the absorbed powers equals the sum o f the generated powers at each instant o f time.

The 2"*^ edition o f this text contains a rigorous proof o f this principle. For the present, we will simply use it to solve various problems. The following example will help clarify the sign conven tions and illustrate the principle o f conservation o f power.

E X A M P L E 1.5

Light bulbs come in all sorts o f shapes, sizes, and wattages. W a t t l e measures the power consumed by a bulb. Typical wattages include 15, 25, 40, 60, 75, and 100 W. Power consumptions differ because the current required to light a higher-wattage (and brighter) bulb is larger for a fixed out let voltage: a higher-wattage bulb converts more electric energy into light energy. In Figure 1.19, the source delivers 215 watts o f power. W hat is the wattage o f the unlabeled bulb?

100V

7? ’ watts

watts watts

FIG U RE 1.19. Three bulbs connected to a 100-V battery.

So lutio n

From conservation o f power, the total power delivered by the battery equals the total power absorbed by all the bulbs. Therefore, the power absorbed by the unknown bulb is

215 - 4 0 - 100 = 75 watts

Exercise.

Determine the current / leaving the battery in Example 1.5.

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EXA M PLE 1.6

An electroplating apparatus uses electrical current to coat materials with metals such as copper or silver. In Figure 1.20, suppose a 2 2 0 -V electrical source supplies 10 A dc to the electroplating apparatus.

1 0A

Electroplating Apparatus

FIGURE 1.20 Electrical source operating an electroplating apparatus.

a) W hat is the power consumed by the apparatus?

b) If electric energy costs 10 cents per kilowatt-hour (kW h), what will it cost to operate the apparatus for a single 12-h day?

So lutio n

Step 1. From Equation 1.10, the power consumed is

/> = 220 X 10 = 2200 W, or 2.2 kW

Step 2 . According to Equation 1.11, the energy consumed per 12-h period is

2.2 X 12 = 26.4 kWh

Step 3 . Therefore, the cost to operate is

26 .4 X .01 = $2.64 / day

Exercise.

Suppose the electroplating apparatus o f Example 1.6 draws 12 A D C at the same volt

age. W hat is the cost o f operation for a single 12-h day? W hat is the cost o f operating for a 20 workday month?

AN SW ER: $3,168; $63.36

E X A M PLE 1.7

Each box in the circuit o f Figure 1.21 is a two-terminal element. Compute the power absorbed by each circuit element. W hich elements are delivering power? Verify the conservation o f power prin ciple for this circuit.

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Chapter 1 • Charge, Current, Voltage and O hm ’s 21

FIG U RE 1.21 Circuit containing several general circuit elements.

So lutio n

Step 1.

Compute power absorbed by each element.

Using either Equation 1.10 or the power con sumption rule, the power absorbed by each element is

a) b) c) d) e)

0

For element 1 For element 2 For element 3 For element 4 For element 5 For element 6 P i = 4 X 1 = 4 W P l = 8 x 2 = 1 6 W ^ 3 = 10 X 1 = 10 W 1 4x ( - 1 ) = - 1 4 W P 5 = 2 x2 = 4 W Pe = 1 0 X ( - 2 ) = - 2 0 W

Step 2 .

Verify conservation o f power.

Since P4 and Pg are negative, element 4 delivers 14 W, and element 6 delivers 20 W o f power. T he remaining four elements absorb power. Observe that the sum o f the six absorbed powers, 4 + 16 + 10 - 14 + 4 - 2 0 = 0, as expected from the principle o f conservation o f power. Equivalently, the total positive generated power, (14 + 20) = 34 W, equals the total positive absorbed power, (4 + 16 + 10 + 4) = 34 W.

Exercise.

In Figure 1.22, find the powers absorbed by elements 1, 2, and 3.

FIG U RE 1.22

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Exercise.

In Figure 1.22, suppose the current 2 A were changed to - 4 A. W hat is the new power

absorbed by element 3? A N SW ER: 56 watts

If the power absorbed by a circuit element is positive, the exact nature o f the element determines the type o f energy conversion that takes place. For example, a circuit element called a resistor (to be dis cussed shortly) converts electric energy into heat. If the circuit element is a battery that is being charged, then electric energy is converted into chemical energy within the battery. If the circuit ele ment is a dc motor turning a fan, then electrical energy is converted into mechanical energy.

N O N -D C PO W ER A N D EN ER G Y C A LC U LA TIO N S

Consider Figure 1.23, where

i{t)

is an arbitrary time-varying current entering a general two-ter minal circuit element, and

v{t)

is the time-varying voltage across the element. Because voltage and current are functions o f time, the power

p{t)

=

v{t)i{t)

is also a function o f time. For any specific value o f ^ = ?j, the value

p{t^)

indicates the power absorbed by the element at that particular time— hence, the terminology instantaneous power for

p{t).

i(t)

Circuit Elem ent Absorbing Power

p(t)

FIGURE 1.23 Calculation of absorbed power for time-varying voltages and currents for circuit ele ments labeled with the passive sign convention; here, power is

p{t)

=

v{t)i{t).

Equation 1.12 extends Equation 1.10 in the obvious way.

p{t) = v{t)i{t)

(1.12)

i.e., the instantaneous (absorbed) power

p{t),

in W, is the product o f the voltage

v{t),

in V, and the current

i{t),

in A, with labeling according to the passive sign convention. This product also makes sense from a dimensional point o f view;

, joules coulombs joules volts X amps = ;— - x =

coulomb second second

Knowing the power

p{t)

absorbed by a circuit element as a function o f

t

allows one to compute the energy

W{tQ, t)

absorbed by the element during the time interval [^q,

t > Iq], W[tQ, t)

(J) is the integral o f

p{t)

(W) with respect to

t

over [?q,

t],

i.e..

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Chapter 1 • Charge, Current, Voltage and O hm ’s Law 2 3

W(to,t)^ r p ir ) d r

where the lower limit o f the integral, could possibly be -oo. For the dc case,

p{t)

= P (a con stant). From Equation 1.13,

t

W(tQ,t) = f p ( r ) d T = P f d r = P(t-tQ) = P x T

where

T = t - t^,

as given in Equation 1.11. If, in Equation 1.13, tg = -oo, then W (-co,

t)

becomes a function only o f

t

which, for convenience, is denoted by

t

W{t)= f p ( r ) d r

L

(1.14)

W{t) =

W{—00,

t),

in joules, represents the total energy absorbed by the circuit element from the

beginning o f time to the present time rwhen

p{t)

is in watts.

Exercise,

a) Suppose the power absorbed by a circuit element over [0,oo) is

p{i)

= watts. Find

W

(0, oo). b) Now suppose the absorbed power o f the circuit element is

p{t) =

j j > 0 • for

t >

0.

A N SW ER: 4 J; (4+t) J

Since energy is the integral o f power, power is the rate o f change (derivative) o f energy. Differentiating both sides o f Equation 1.14 yields the expected equation for instantaneous power.

or, equivalently, for

t > (

q

,

dW(t)

v m o = P ( o = ^

(1.15a)

Exercise.

Suppose that for t > 0, the work done by an electronic device satisfies

W{t) =

10(1 — J- If

the voltage supplied by the device is 10 V, then for t > 0, find the power and current supplied by the device, assuming standard labeling, i.e., the passive sign convention.

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EXA M PLE 1.8

In the circuit o f Figure 1.23, the current

i{t)

and vokage

v{t)

have the waveforms graphed in Figure 1.24. Sicetch

p{t),

the instantaneous power absorbed by the circuit element, and then sketch

W(0,

t),

the energy absorbed over the interval [0,

i\.

FIGURE 1.24 (a) Current and (b) voltage profdes with respect to

t

for circuit o f Figure 1.23.

So lutio n

A simple graphical multiplication o f Figures 1.24(a) and (b) yields the sketch o f the curves in instantaneous power shown in Figure 1.25(a). From Equation 1.13 with

=

0, we have, for 0 <

t< %

,2

p(T)dr - J —

c

/

t

= —

0 and for

t> 5,

t 5 t t W ( 0 , t ) - J p ( r ) d T = J p { x ) d T + J p ( r ) d T - 5 + J ' d T - 5 + ( t - 5 ) = l 0 0 5 5

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(b)

FIGURE 1.25 (a) Profile of the instantaneous power

p{t) = v{t)i{t)

for the current and voltage wave forms of Figure 1.24; (b) associated profde of energy versus time.

6. IDEAL VO LTAG E AND CU RREN T SO URCES

Two-terminal circuit elements may be classified according to their terminal voltage-current rela tionships. The goal o f this section is to define ideal voltage and current sources via their termi nal voltage-current relationships.

The wall socket o f a typical home represents a practical voltage source. After flipping the switch on an appliance plugged into a wall socket, a current flows through the internal circuitry o f the appliance, which, for a vacuum cleaner or dishwasher, converts electrical energy into mechanical energy. For modest amounts o f current draw (below the fuse setting), the voltage nearly maintains its nominal pattern o f 120 / 2 sin(120

lit) =

169.7 sin(120

nt)

V. This practical situation is ide

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26 Chapter 1 • Charge, Current, Voltage and O h m s Law

alized in circuit analysis by the ideal voltage source symbol shown in Figure 1.26(a), a circle with a ± reference inside. The symbol is more commonly referred to as independent voltage source.

FIGURE 1.26 Equivalent representations of ideal voltage source attached to a hypothetical circuit.

The waveform or signal

v{t)

in Figure 1.26 represents the voltage produced by the source at each time

t.

The plus and minus (+, —), on the source define a reference polarity. T he reference polari ty is a labeling or reference frame for standardized voltage measurement. T he reference polarity does not mean that

v(t)

is positive. Rather, the reference polarity (+, - ) means that the voltage drop from + to - is

v{t),

whatever its value/sign. Finally, the voltage source is ideal because it maintains the given voltage

v{t),

regardless o f the current drawn from the source by the attached circuit.

V,

voltage (V)

1(A)

(b)

FIGURE 1.27 (a) Ideal battery representation of ideal voltage source; (b)

v-i

characteristic of ideal battery.

Figure 1.2 7 (a) shows a source symbol for an ideal battery. The voltage drop from the long-dash side to the short-dash side is Vg, with Vjj > 0. In commercial products, the terminal marked with a + sign corresponds to the long-dash side o f Figure 1.27(a). An ideal battery produces a constant voltage under all operating conditions, i.e., regardless o f current drawn from an attached circuit or circuit element, as indicated by the

v-i

characteristic o f Figure 1.27(b). Real batteries are not ideal but approximate the ideal case over a manufacturer-specified range o f current requirements.

Practical sources (i.e., non-ideal); voltage sources, such as commercial dc and ac generators; and real batteries deviate from the ideal in many respects. One important respect is that the terminal voltage depends on the current delivered by the source. The most common generators convert mechanical energy into electrical energy, while batteries convert chemical energy into electrical

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energy. There are two general battery categories: nonrechargeable and rechargeable. A discussion o f the dramatically advancing battery technology is beyond the scope o f this text.

Besides batteries and ideal voltage sources, devices called ideal or independent current sources maintain fixed current waveforms into a circuit, as illustrated in Figure 1.28. T he symbol o f an ideal current source is a circle with an arrow inside, indicating a reference current direction. An ideal current source produces and maintains the current

i{t)

under all operating conditions. O f course, the current

i{t)

flowing from the source can be a constant (dc), sinusoidal (ac), or any other time-varying function.

FIGURE 1.28 Equivalent ideal current sources whose current

i{t)

is maintained under all operating conditions o f the circuit.

In nature, lightning is an example o f an approximately ideal current source. W hen lightning strikes a lightning rod, the path to the ground is almost a short circuit, and very little voltage is developed between the top o f the rod and the ground. However, if lightning strikes a tree, the path o f the current to the ground is impeded by the trunk o f the tree. A large voltage then develops from the top o f the tree to the ground.

Independent sources have conventional labeling, as shown in Figure 1.29, which is different from that o f the passive sign convention. Here the source delivers power if

p{t)

=

v{t)i{t) >

0 and would absorb power i f

p{t) =

< 0. A complicated circuit called a battery charger can deliver ener gy to a drained car battery. T he car battery, although usually a source delivering power, exempli fies a source absorbing power from the charger.

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Another type o f ideal source is a dependent source. A dependent source or a controlled source produces a current or voltage that depends on a current through or voltage across some other ele ment in the circuit. Such sources model real-world devices that are used in real circuits. In the text, the symbol for a dependent source is a diamond. If a ± appears inside the diamond, it is a depend

ent voltage source, as illustrated in Figure 1.30. If an arrow appears inside the diamond, it is a

dependent current source, as illustrated in Figure 1.31. In Figure 1.30, the voltage across the dia mond-shaped source,

v{t),

depends either on a current, labeled through some other circuit device, or on the voltage across it. If the voltage across the source depends on the voltage

v^,

i.e.,

v{t)

= p then the source is called a voltage-controlled voltage source (VCVS). If the volt age across the source depends on the current z^, i.e.,

v{t) =

then the source is called a cur

rent-controlled voltage source (CCVS).

FIGURE 1.30 The right element is a voltage-controlled voltage source (VCVS) if

v{t) =

(p is here dimensionless), or a current-controlled voltage source (CCVS) if

v(t) = r^i (r^

here has units of ohm).

Exercise.

The voltage across a particular circuit element is = 5 V, and the current through the

element is 0.5 A, using the standard labeling.

a) If a V CV S (Figure 1.30) with p = 0.4 were associated with the controlled-source branch,

fmd

vit).

b) If a CCV S (Figure 1.30) with = 3 £2 were associated with the controlled branch, fmd

v{t).

ANSW ER: a) 2 V; b) 1.5 V

There is dual terminology for dependent current sources. The configuration o f Figure 1.31 shows a voltage-controlled current source (VCCS), i.e.,

i{t) = g^v^,

or a current-controlled current

source (CCCS), for which

i{t)

=

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Chapter 1 • Charge, Current, Voltage and O hm ’s Law 29 Q V_X

or

Pi: i(t) =

or

Pi - o

FIG U RE 1.31 The right element is a voltage-controlled current source (VCCS) if

i{t) =

(g^

has units o f siemens) or a current-controlled current source (CCCS) if

i{t)

= |3/^ ((3 is dimensionless).

Source voltages or currents are called excitations, inputs, or input signals. A constant voltage will nor mally be denoted by an uppercase letter, such as

V, V

q

,

V^,

and so on. A constant current will typi cally be denoted by /, /g, /p and so on. The units are volts, amperes, and so on. Smaller and larger quantities are expressed by the use o f prefixes, as defined in Standard Engineering Notation Table 1.1.

Exercise.

The voltage across a particular circuit element is = 5, and the current through the ele

ment is = 0.5 A using the standard labeling.

a) If a VCCS (Figure 1.31) with ^^ = 0.1 S were associated with the controlled-source branch, find

i{i).

b) If a CCCS (Figure 1.31) with P = 0.5 were associated with the controlled-source branch, find

i{i).

AN SW ER; a) 0.5 A; b) 0.25 A

TABLE 1.1. Engineering Notation for Large and Small Quantities

Name Prefix Value

femto i f 10-15 pico _P 10-12 nano n 10-9 micro _P 10-6 milli m 10-3 w kilo k 103 mega M lO^’ g‘ga G 109 tera T 1012

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3 0 Chapter 1 • Charge, Current, Voltage and O hm ’s Law

7. RESISTANCE, O HM 'S LAW, AND POW ER (A REPRISE)

Different materials allow electrons to move from atom to atom with different levels o f ease. Suppose the same dc voltage is applied to two conductors, one carbon and one copper, o f the same size and shape. Two different currents will flow. T he current flow depends on a property o f the conductor called resistance: the smaller the resistance, the larger the current flow for a fixed volt age. The idea is similar to water flow through different-diameter pipes (analogous to electrical con ductors): for a given pressure, a larger-diameter pipe allows a larger volume o f water to flow and, therefore, has a smaller resistance than a pipe with, say, half the diameter.

A conductor designed to have a specific resistance is called a resistor. Hence, a resistor is a device that impedes current flow. Just as dams impede water flow and provide flood control for rivers, resistors provide a means to control current flow in a circuit. Further, resistors are a good approx imate model to a wide assortment o f electric devices such as light bulbs and heating elements in ovens. Figure 1.32(a) shows the standard symbol for a resistor, where the voltage and current ref erence directions are marked in accordance with

xhie.passive sign convention.

Figure 1.32(b) pic tures a resistor connected to an ideal battery.

I

R

+ V

-(a)

FIG U RE 1.32 (a) Symbol for a resistor with reference voltage polarity and current direction consistent with the passive sign convention; (b) resistor connected to an ideal battery.

In 1827, Ohm observed that for a connection like that o f Figure 1.32(b), the direct current through the conductor/resistor is proportional to the voltage across the conductor/resistor, i.e.,

I

=

V.

Inserting a proportionality constant, one can write

1 =

—

V — GV

(1.16a)

or, equivalently,

^

V = R1

The proportionality constant

R

is the resistance o f the conductor in ohms. The resistance

R

meas ures the degree to which the device impedes current flow. For conductors/resistors, the ohm (Q) is the basic unit o f resistance.

A two-terminal device has a

1-Q

resistance i f a 1-V excitation causes

1-A o f current to flow.

In Equation 1.16(a), the proportionality constant is the reciprocal o f

R,

i.e.,

G = HR,

which is called the conductance o f the device. T he unit for conductance according to

the International System o f Units (SI) system is the siemen, S. In the United States, the older term for the unit o f conductance is the mho ^5, that is, ohm spelled backward, which is still widely used. In this text, we try to adhere to the SI system. If a device or wire has zero resistance

{R

= 0) or infinite conductance

{G

= t»), it is termed a short circuit. On the other hand, if a device or wire has infinite resistance (zero conductance), it is called an open circuit. Technically speaking,