Electric Circuits, 9th Edition P21 pdf

Assigning the reference direction of the current in the direction of the volt-age drop across the terminals of the inductor, as shown in Fig.. 6.1b; that is, the current reference is in

Section 6.3 describes techniques used to simplify circuits with series or parallel combinations of capacitors or inductors

Energy can be stored in both magnetic and electric fields Hence you should not be too surprised to learn that inductors and capacitors are capable of storing energy For example, energy can be stored in an induc-tor and then released to fire a spark plug Energy can be sinduc-tored in a capac-itor and then released to fire a flashbulb In ideal inductors and capaccapac-itors, only as much energy can be extracted as has been stored Because

induc-tors and capaciinduc-tors cannot generate energy, they are classified as passive elements

In Sections 6.4 and 6.5 we consider the situation in which two circuits are linked by a magnetic field and thus are said to be magnetically cou-pled In this case, the voltage induced in the second circuit can be related

to the time-varying current in the first circuit by a parameter known as

mutual inductance The practical significance of magnetic coupling

unfolds as we study the relationships between current, voltage, power, and several new parameters specific to mutual inductance We introduce these relationships here and then describe their utility in a device called a trans-former in Chapters 9 and 10

6.1 The Inductor

Inductance is the circuit parameter used to describe an inductor Inductance

is symbolized by the letter L, is measured in henrys (H), and is represented graphically as a coiled wire—a reminder that inductance is a consequence

of a conductor linking a magnetic field Figure 6.1(a) shows an inductor

Assigning the reference direction of the current in the direction of the volt-age drop across the terminals of the inductor, as shown in Fig 6.1(b), yields

The inductor v - i equation • v = L—, (6.1)

dt

where v is measured in volts, L in henrys, i in amperes, and t in seconds

Equation 6.1 reflects the passive sign convention shown in Fig 6.1(b); that

is, the current reference is in the direction of the voltage drop across the inductor If the current reference is in the direction of the voltage rise,

Eq 6.1 is written with a minus sign

Note from Eq 6.1 that the voltage across the terminals of an inductor

is proportional to the time rate of change of the current in the inductor

We can make two important observations here First, if the current is con-stant, the voltage across the ideal inductor is zero Thus the inductor behaves as a short circuit in the presence of a constant, or dc, current

Second, current cannot change instantaneously in an inductor; that is, the current cannot change by a finite amount in zero time Equation 6.1 tells

us that this change would require an infinite voltage, and infinite voltages are not possible For example, when someone opens the switch on an inductive circuit in an actual system, the current initially continues to flow

in the air across the switch, a phenomenon called arcing The arc across

the switch prevents the current from dropping to zero instantaneously

Switching inductive circuits is an important engineering problem, because arcing and voltage surges must be controlled to prevent equipment dam-age The first step to understanding the nature of this problem is to master the introductory material presented in this and the following two chapters

Example 6.1 illustrates the application of Eq 6.1 to a simple circuit

L

(a)

(b) Figure 6.1 • (a) The graphic symbol for an inductor

with an inductance of L henrys (b) Assigning reference

voltage and current to the inductor, following the

pas-sive sign convention

Example 6.1 Determining the Voltage, Given the Current, at the Terminals of an Inductor

The independent current source in the circuit

shown in Fig 6.2 generates zero current for t < 0

and a pulse 10/e~5'A, for t > 0

/ < 0

/ = 0,

100 mH

i = 10/<T5'A, t > 0

Figure 6.2 • The circuit for Example 6.1

a) Sketch the current waveform

b) At what instant of time is the current maximum?

c) Express the voltage across the terminals of the

100 mH inductor as a function of time

d) Sketch the voltage waveform

e) Are the voltage and the current at a maximum at

the same time?

f) At what instant of time does the voltage change

polarity?

g) Is there ever an instantaneous change in voltage

across the inductor? If so, at what time?

Solution

a) Figure 6.3 shows the current waveform

b) di/dt = 10(-5te~5' + e~ 5c ) = 10e- 5'

( 1 - 5/) A/s; di/dt = 0 when t = 1 s (See Fig 6.3.)

c) t; = Ldi/dt = (0.1)10e_5'(l - 5/) = e~ St

(1-5/) V , / > 0;v = 0 , / < 0

d) Figure 6.4 shows the voltage waveform

e) No; the voltage is proportional to di/dt, not i

f) At 0.2 s, which corresponds to the moment when

di/dt is passing through zero and changing sign

g) Yes, at t - 0 Note that the voltage can change

instantaneously across the terminals of an inductor

Figure 6.3 A The current waveform for Example 6.1

Figure 6.4 A The voltage waveform for Example 6.1

Current in an Inductor in Terms of the Voltage

Across the Inductor

Equation 6.1 expresses the voltage across the terminals of an inductor as a

function of the current in the inductor Also desirable is the ability to

express the current as a function of the voltage To find i as a function of v,

we start by multiplying both sides of Eq 6.1 by a differential time dt:

Multiplying the rate at which i varies with /by a differential change in time

generates a differential change in /, so we write Eq 6.2 as

We next integrate both sides of Eq 6.3 For convenience, we interchange the two sides of the equation and write

L dx

Jt(tn)

Note that we use x and r as the variables of integration, whereas i and /

become limits on the integrals Then, from Eq 6.4,

where /(/) is the current corresponding to /, and /(/0) is the value of the inductor current when we initiate the integration, namely, /0 In many practical applications, /0 is zero and Eq 6.5 becomes

if

wo

/(/) = — / v dr + /(0) (6.6)

Equations 6.1 and 6.5 both give the relationship between the voltage and current at the terminals of an inductor Equation 6.1 expresses the voltage as a function of current, whereas Eq 6.5 expresses the current as a function of voltage In both equations the reference direction for the cur-rent is in the direction of the voltage drop across the terminals Note that /(/()) carries its own algebraic sign If the initial current is in the same direc-tion as the reference direcdirec-tion for /, it is a positive quantity If the initial

current is in the opposite direction, it is a negative quantity Example 6.2

illustrates the application of Eq 6.5

Example 6.2 Determining the Current, Given the Voltage, at the Terminals of an Inductor

The voltage pulse applied to the 100 mH inductor

shown in Fig 6.5 is 0 for t < 0 and is given by the

expression

v{t) = 20/e"10' V

for / > 0 Also assume i = 0 for / < 0

a) Sketch the voltage as a function of time

b) Find the inductor current as a function of time

c) Sketch the current as a function of time

b) The current in the inductor is 0 at / = 0 Therefore, the current for / > 0 is

hil : I 20T<T1 07/T + 0

200

-10r

100 -(10T + 1)

= 2(1 - \0te~ U)t - e~ mt ) A, / > 0

c) Figure 6.7 shows the current as a function of time

Solution

a) The voltage as a function of time is shown in

Fig 6.6

y = 0, t<0

i i < 100 mH

v = 20te-mV, r > 0

Figure 6.5 A The circuit for Example 6.2

Figure 6.6 A The voltage waveform for Example 6.2

0 0.1 0.2 0.3

Figure 6.7 • The current waveform for Example 6.2

Note in Example 6.2 that i approaches a constant value of 2 A as t

increases We say more about this result after discussing the energy stored

in an inductor

Power and Energy in the Inductor

The power and energy relationships for an inductor can be derived

directly from the current and voltage relationships If the current

refer-ence is in the direction of the voltage drop across the terminals of the

inductor, the power is

Remember that power is in watts, voltage is in volts, and current is in

amperes If we express the inductor voltage as a function of the inductor

current, Eq 6.7 becomes

r di

(6.8) A Power in an inductor

We can also express the current in terms of the voltage:

1

/; = v

l^k

v (IT + /(/•()) (6.9)

Equation 6.8 is useful in expressing the energy stored in the inductor

Power is the time rate of expending energy, so

dw di

p = —^ = Li —

Multiplying both sides of Eq 6.10 by a differential time gives the

differen-tial relationship

Both sides of Eq 6.11 are integrated with the understanding that the

ref-erence for zero energy corresponds to zero current in the inductor Thus

dx = L I y dy,

Jo

w = -Lr

As before, we use different symbols of integration to avoid confusion with the limits placed on the integrals In Eq 6.12, the energy is in joules, inductance is in henrys, and current is in amperes To illustrate the appli-cation of Eqs 6.7 and 6.12, we return to Examples 6.1 and 6.2 by means of Example 6.3

a) For Example 6.1, plot i, v.p, and w versus time

Line up the plots vertically to allow easy

assess-ment of each variable's behavior

b) In what time interval is energy being stored in

the inductor?

c) In what time interval is energy being extracted

from the inductor?

d) What is the maximum energy stored in the

inductor?

e) Evaluate the integrals

0.2

0.2

and comment on their significance

f) Repeat (a)-(c) for Example 6.2

g) In Example 6.2, why is there a sustained current

in the inductor as the voltage approaches zero?

Solution

a) The plots of /, v,p, and w follow directly from the

expressions for i and v obtained in Example 6.1

and are shown in Fig 6.8 In particular, p - vi,

and w = (f)Ii2

b) An increasing energy curve indicates that energy

is being stored Thus energy is being stored in the

time interval 0 to 0.2 s Note that this

corre-sponds to the interval when p > 0

c) A decreasing energy curve indicates that energy

is being extracted Thus energy is being extracted

in the time interval 0.2 s to oo Note that this

cor-responds to the interval when p < 0

d) From Eq 6.12 we see that energy is at a maximum

when current is at a maximum; glancing at the

graphs confirms this From Example 6.1, maximum

current = 0.736 A Therefore, w = 27.07 mJ

e) From Example 6.1, / = 10fe"5'A and v = e_5t(l - 50 V Therefore,

p = vi = lOte' 101 - 50t 2 e~ m W

0.2 0.4 0.6 0.8 1.0

Figure 6.8 A The variables /', v, p, and w versus / for

Example 6.1

Thus

0.2

p dt = 10

, - 1 Of

ion (-10? - 1)

0.2

0

-K)r

100 ( - 1 0 / - 1)

g) The application of the voltage pulse stores energy in the inductor Because the inductor is ideal, this energy cannot dissipate after the volt-age subsides to zero Therefore, a sustained cur-rent circulates in the circuit A lossless inductor obviously is an ideal circuit element Practical inductors require a resistor in the circuit model (More about this later.)

-2 _

Q2e~z = 27.07 mJ,

p dt = 10

0.2

-l()f

100 (-10* - 1)

, t 2 e~ m 2

0,2

-10/

100 (-10/ - 1) 0.2

= - 0 2 e- 2 = -27.07 mJ

Based on the definition of p, the area under

the plot of p versus t represents the energy

expended over the interval of integration

Hence the integration of the power between

0 and 0.2 s represents the energy stored in the

inductor during this time interval The integral

of p over the interval 0.2 s - oo is the energy

extracted Note that in this time interval, all

the energy originally stored is removed; that is,

after the current peak has passed, no energy is

stored in the inductor

f) The plots of v, i, p, and w follow directly from

the expressions for v and i given in Example

6.2 and are shown in Fig 6.9 Note that in this

case the power is always positive, and hence

energy is always being stored during the

volt-age pulse

0 0.1 0.2 0.3 0.4 0.5 0.(

/(A) 2.0

(s)

0 0.1 0.2 0.3 0.4 0.5 0.6 p(mW)

600

(s)

0 0.1 0.2 0.3 0.4 0.5 0.6

•w (mJ)

f ( s )

200

100

0

-0.1

I

0.2

1

0.3

1

0.4

1

0.5

1 0.6 r(s)

Figure 6.9 • The variables v, i, p, and w versus t for

Example 6.2

I / A S S E S S M E N T PROBLEM

Objective 1—Know and be able to use the equations for voltage, current, power, and energy in an inductor

6.1 The current source in the circuit shown

gener-ates the current pulse

ig(t) = 8e-™' - 8e~im* A, t > 0

Find (a) v(0); (b) the instant of time, greater

than zero, when the voltage v passes through

zero; (c) the expression for the power delivered

to the inductor; (d) the instant when the power

delivered to the inductor is maximum; (e) the

maximum power; (f) the instant of time when

the stored energy is maximum; and (g) the

max-imum energy stored in the inductor

NOTE: Also try Chapter Problems 6.1 and 6.4

Answer:

M 4 m

(a) 28.8 V;

(b) 1.54 ms;

(c) -76.8<T600' + 384e-1500' -307.2<T2400fW, t > 0;

(d) 411.05 jus;

(e) 32.72 W;

(f) 1.54 ms;

(g) 28.57 mJ

6.2 The Capacitor

(a)

C

+ v

/ (b)

Figure 6.10 • (a) The circuit symbol for a capacitor,

(b) Assigning reference voltage and current to the

capacitor, following the passive sign convention

The circuit parameter of capacitance is represented by the letter C is

measured in farads (F), and is symbolized graphically by two short paral-lel conductive plates, as shown in Fig 6.10(a) Because the farad is an extremely large quantity of capacitance, practical capacitor values usually lie in the picofarad (pF) to microfarad (/xF) range

The graphic symbol for a capacitor is a reminder that capacitance occurs whenever electrical conductors are separated by a dielectric, or insulating, material This condition implies that electric charge is not transported through the capacitor Although applying a voltage to the terminals of the capacitor cannot move a charge through the dielectric, it can displace a charge within the dielectric As the voltage varies with time, the displacement of charge also varies with time, causing what is

known as the displacement current

At the terminals, the displacement current is indistinguishable from a conduction current The current is proportional to the rate at which the voltage across the capacitor varies with time, or, mathematically

where /' is measured in amperes, C in farads, v in volts, and t in seconds

Equation 6.13 reflects the passive sign convention shown in Fig 6.10(b); that is, the current reference is in the direction of the voltage drop across the capacitor If the current reference is in the direction of the voltage rise,

Eq 6.13 is written with a minus sign

Two important observations follow from Eq 6.13 First, voltage cannot

change instantaneously across the terminals of a capacitor Equation 6.13

indicates that such a change would produce infinite current, a physical

impossibility Second, if the voltage across the terminals is constant, the

capacitor current is zero The reason is that a conduction current cannot be

established in the dielectric material of the capacitor Only a time-varying

voltage can produce a displacement current.Thus a capacitor behaves as an

open circuit in the presence of a constant voltage

Equation 6.13 gives the capacitor current as a function of the

capaci-tor voltage Expressing the voltage as a function of the current is also

use-ful To do so, we multiply both sides of Eq 6.13 by a differential time dt

and then integrate the resulting differentials:

•'•(') i ft

i dt = C dv or I dx = — I i dr

h(t ( >) c Jh

Carrying out the integration of the left-hand side of the second

equa-tion gives

v(t) = - / idr + v(t0) (6.14) < Capacitor v - i equation

In many practical applications of Eq 6.14, the initial time is zero; that is,

t{) = 0 Thus Eq 6.14 becomes

We can easily derive the power and energy relationships for the capacitor

From the definition of power,

p = vi = O dv

It* (6.16) -4 Capacitor power equation

or

P = l

C , , i dr + v(t

Combining the definition of energy with Eq 6.16 yields

dw = Cv dv,

from which

f>W A »

/ dx = C I y dy,

Jo J[)

or

2

(6.18)

In the derivation of Eq 6.18, the reference for zero energy corresponds to zero voltage

Examples 6.4 and 6.5 illustrate the application of the current, voltage, power, and energy relationships for a capacitor

Example 6.4 Determining Current, Voltage, Power, and Energy for a Capacitor

The voltage pulse described by the following

equa-tions is impressed across the terminals of a 0.5 /xF

capacitor:

Solution

a) From Eq 6.13,

0, / < 0 s;

{ At V, 0 s < t < 1 s;

4 ^ ^ V, / > l s

(0.5 X Kr6)(0) = 0, t < 0s;

(0.5 X 10"6)(4) = 2 yuA, 0 s < / < 1 s; (0.5 x 10_6)(-4e"<'~1)) = - 2 < r( , _ 1V A , t > 1 s

a) Derive the expressions for the capacitor current,

power, and energy

b) Sketch the voltage, current, power, and energy as

functions of time Line up the plots vertically

c) Specify the interval of time when energy is being

stored in the capacitor

d) Specify the interval of time when energy is being

delivered by the capacitor

e) Evaluate the integrals

p dt and / p dt

and comment on their significance

P =

w

The expression for the power is derived from Eq.6.16:

0, (4/)(2) = 8/ /iW,

( 4 < , - ( / - i ) ) ( _ 2 £ > - ( ' - i ) ) =

t < 0 s;

0 s < t < 1 s;

-8e_ 2 ('_ 1Vw, t > 1 s

The energy expression follows directly from Eq.6.18:

0 t < 0 s;

1(0.5)16/2 = 4f2/xJ, 0 s < t < 1 s;

±(0.5) 16e-2('_1) = Ae-^-VfiJ, t > 1 s

b) Figure 6.11 shows the voltage, current, power,

and energy as functions of time

c) Energy is being stored in the capacitor whenever

the power is positive Hence energy is being

stored in the interval 0 - 1 s

d) Energy is being delivered by the capacitor

when-ever the power is negative Thus energy is being

delivered for all / greater than 1 s

e) The integral of p dt is the energy associated with

the time interval corresponding to the limits on

the integral Thus the first integral represents the

energy stored in the capacitor between 0 and 1 s,

whereas the second integral represents the

energy returned, or delivered, by the capacitor in

the interval 1 s to oo:

l , i

p dt = / 8t dt = At 2 4/AJ,

pdt= I ( - 8 ^ - ^ = (-8) —

The voltage applied to the capacitor returns to

zero as time increases without limit, so the energy

returned by this ideal capacitor must equal the

energy stored

P 0*w)

4

2

0

-1

- 1

3

1

4

l

5

1

6 Us)

Figure 6.11 A The variables v, i, p, and w versus t for

Example 6.4

Example 6.5 Finding v, p, and w Induced by a Triangular Current Pulse for a Capacitor

An uncharged 0.2 /AF capacitor is driven by a

trian-gular current pulse The current pulse is described by

m = <

0, f s O ;

5000* A, 0 < t < 20 /AS;

0.2 - 5000* A, 20 < t < 40 jus;

a) Derive the expressions for the capacitor voltage,

power, and energy for each of the four time

inter-vals needed to describe the current

b) Plot i, v, p, and w versus t Align the plots as

spec-ified in the previous examples

c) Why does a voltage remain on the capacitor after

the current returns to zero?

Solution

a) For t < 0, v, p, and w all are zero

ForO < t •& 20/is,

v 5 : : 10h I (5000r) dr + 0 = 12.5 X 10V V,

/o

vi = 62.5 X l O ' V W ,

w = -Cv2 = 15.625 X l O ' V j

2 For 20 /AS < r < 40 /AS,

v = 5 X 106 / (0.2 - 5 0 0 0 T ) ^ T + 5