Electric Circuits, 9th Edition P55 pot

545 Know the RL and RC circuit configurations that act as low-pass filters and be able to design RL and RC circuit component values to meet a specified cutoff frequency.. Know the RL

Figure P13.61

h(t)'

10

(a)

*(0"

4

h(ty

40

0 10 t

(b)

0 1

(c)

13.62 A rectangular voltage pulse i>,- = [«(/) - u(t - 1)] V

is applied to the circuit in Fig P13.62 Use the

con-volution integral to find v n

>•', in:

13.63 Interchange the inductor and resistor in

Problem 13.62 and again use the convolution

inte-gral to find v Q

13.65 a) Repeat Problem 13.64, given that the resistor in

the circuit in Fig PI3.50(a) is decreased to 10 kll b) Does decreasing the resistor increase or decrease the memory of the circuit?

c) Which circuit comes closer to transmitting a replica of the input voltage?

13.66 a) Assume the voltage impulse response of a

circuit is

lOtT4', t > 0

Use the convolution integral to find the output voltage if the input signal is 10*<(0 V

b) Repeat (a) if the voltage impulse response is

0, t < 0;

= \ 10(1 - 20, 0 < t < 0.5 s;

0, t > 0.5 s

c) Plot the output voltage versus time for (a) and

(b) for 0 < t < 1 s

13.67 The voltage impulse response of a circuit is shown in

Fig P13.67(a) The input signal to the circuit is the rectangular voltage pulse shown in Fig P13.67(b) a) Derive the equations for the output voltage Note the range of time for which each equation

is applicable

b) Sketch v 0 for - 1 < t < 34 s

13.64 a) Use the convolution integral to find the output

voltage of the circuit in Fig P13.50(a) if the

input voltage is the rectangular pulse shown in

Fig P13.64

b) Sketch v 0 (t) versus t for the time interval

0 < t < 10 ms

Figure P13.64

»i(V)

lo

r(ms)

13.68 Assume the voltage impulse response of a circuit

can be modeled by the triangular waveform shown

in Fig P13.68.The voltage input signal to this circuit

is the step function 10«(^) V

a) Use the convolution integral to derive the

expressions for the output voltage

b) Sketch the output voltage over the interval

0 to 15 s

c) Repeat parts (a) and (b) if the area under the

voltage impulse response stays the same but the

width of the impulse response narrows to 4 s

d) Which output waveform is closer to replicating

the input waveform: (b) or (c)? Explain

Figure P13.68

A(0 (V)

13.69 a) Find the impulse response of the circuit shown

in Fig P13.69(a) if v g is the input signal and v 0 is

the output signal

b) Given that v q has the waveform shown in

Fig P13.69(b), use the convolution integral to

find v a

c) Does v a have the same waveform as v g l Why or

why not?

Figure P13.69

4H

v g (V)

75

0

-75

13.70 a) Find the impulse response of the circuit seen in

Fig PI3.70 if v g is the input signal and v n is the

output signal

b) Assume that the voltage source has the wave-form shown in Fig P13.69(b) Use the

convolu-tion integral to find v (r c) Sketch % for 0 < / < 2 s

d) Does v a have the same waveform as v„l Why or

why not?

Figure P13.70

13.71 The sinusoidal voltage pulse shown in Fig P13.71(a)

is applied to the circuit shown in Fig P13.71(b) Use

the convolution integral to find the value of v () at

t = 75 ms

Figure P13.71

5H

v, 160

n-77/20 ir/io r(s)

(b)

13.72 U s e the convolution integral to find v ( , in the circuit

seen in Fig P13.72 if v t = 75u{t) V

Figure P13.72

40 0

J - T Y Y V

4

16 H

13.73 The current source in the circuit shown in Fig P13.73(a) is generating the waveform shown in Fig PI 3.73(b) Use the convolution integral to find

v„ at t = 5 ms

Figure P13.73

i g (mA)

10

H — I — I —

1 2 3 4

- 2 0 4 (b)

0.4 /itF

(a)

Figure P13.75

i„ {fiA)

50

0

-50 (b)

100 200 /(ms)

(a)

13.74 The input voltage in the circuit seen in Fig PI 3.74 is

V; = 5[u(t) - u(t - 0.5)] V

a) Use the convolution integral to find v a

b) Sketch v a for 0 < t < 1 s

Figure P13.74

2 0

100 mF

13.76 a) Show that if y(t) = h(() * x(t), then Y{s)

H(s)X(s)

b) Use the result given in (a) to find /(f) if

F(s) = s(s + a)"

Section 13.7 13.77 The transfer function for a linear time-invariant

circuit is

13.75 a) Use the convolution integral to find v 0 in the

cir-cuit in Fig P13.75(a) if i s is the pulse shown in

Fig PI3.75(b)

b) Use the convolution integral to find i 0

c) Show that your solutions for v v and i ( , are

consis-tent by calculating v a and i n at 100" ms,

100+ ms, 200" ms, and 200+ ms

H(s) V 0 4(s + 3)

If v K = 40 cos 3/ V, what is the steady-state

expres-sion for v a 'l

13.78 When an input voltage of 30u(t) V is applied to a

circuit, the response is known to be

-80(H)/ . .111)0,

v a = (50*-*™" - 20e-™*")u{t) V

What will the steady-state response be if

v = 120 cos 6000/ V?

13.79 The op amp in the circuit seen in Fig P13.79 is ideal

PSPKE a) Find the transfer function VJV„

MULTISIM ' "' S

b) Find v a if v g = 0.6//(0 V

c) Find the steady-state expression for v„ if

v g = 2 cos 10,000/: V

Figure P13.79

13.82 The inductor L x in the circuit shown in Fig P13.82

is carrying an initial current of p A at the instant the switch opens Find (a) v(t); (b) /-i(/); (c) i 2 (t)',

and (d) A(r), where A(f) is the total flux linkage in the circuit

Figure P13.82

y„*15kll

13.80 The operational amplifier in the circuit seen in

PSPICE pig P13.80 is ideal and is operating within its

lin-MULTISIM

ear region

a) Calculate the transfer function V„/V R

b) If v g = 2cos400f V, what is the steady-state

expression for v (> '~!

13.83 a) Let R - » oo in the circuit shown in Fig P13.82,

and use the solutions derived in Problem 13.82

to find v(t), ii(t), and i 2 {t)

b) Let R = oo in the circuit shown in Fig P13.82

and use the Laplace transform method to find

-u(f), ii(t), and i 2 {t)

13.84 There is no energy stored in the circuit in Fig P13.84

at the time the impulsive voltage is applied

a) Find v (> (t) for t > 0

b) Does your solution make sense in terms of known circuit behavior? Explain

Figure P13.80

y r > 5 2 0 k n

Section 13.8

13.81 Show that after V^C C coulombs are transferred from

C] to C2 in the circuit shown in Fig 13.47, the

volt-age across each capacitor is C\V {) f(C\ + C2) (Hint:

Use the conservation-of-charge principle.)

Figure P13.84

200 H 4 mH

13.85 The parallel combination of R 2 and C 2 in the circuit shown in Fig P13.85 represents the input circuit to

a cathode-ray oscilloscope (CRO) The parallel

combination of i?j and C\ is a circuit model of a

compensating lead that is used to connect the CRO

to the source There is no energy stored in C\ or C 2

at the time when the 10 V source is connected to the CRO via the compensating lead The circuit values are Q = 4 pF, C2 = 16 pF, R l = 1.25 Mft, and

R 2 = 5 MH

a) Find v a

b) Find i 0

c) Repeat (a) and (b) given Cj is changed to 64 p F

Figure P13.85 13.89 Tliere is no energy stored in the circuit in Fig P13.89

at the time the impulsive current is applied

a) Find v () for t > ()+ b) D o e s your solution m a k e sense in terms of

k n o w n circuit behavior? Explain

Figure P13.89

250 nF

13.86 Show t h a t if R\C\ = RiC 2 in t h e circuit shown in

Fig P13.85, v (> will b e a scaled replica of t h e

s o u r c e voltage

13.87 T h e switch in the circuit in Fig PI3.87 has b e e n

closed for a long time T h e switch opens at t — 0

C o m p u t e (a) «,((T); (b) / , ( 0 + ) ; (c) / 2 (<T); (d) / 2 ((T);

(e) i!(r); (f) / 2( 0 ; and (g) v{t)

Figure P13.87

t = 0

1 = u v

+ :8 mH

::•! 0 jr1 v(t) k\\

4 k O | 1 6 k H

13.90 T h e voltage source in the circuit in E x a m p l e 13.1 is

changed to a unit impulse; that is, v g = 8(t)

a) H o w much energy does the impulsive voltage source store in the capacitor?

b) H o w m u c h energy does it store in the inductor?

c) U s e the transfer function t o find v a (t)

d) Show that t h e r e s p o n s e found in (c) is identical

to the response g e n e r a t e d by first charging the capacitor to 1000 V and then releasing the charge to the circuit, as shown in Fig P13.90

Figure P13.90

looon

k - 1 - *

^ r> < i-

1000 V

13.88 The switch in the circuit in Fig P13.88 has been in

position a for a long time A t t = 0, the switch

moves to position b C o m p u t e (a) ^ ( O - ) ; (b) y?(0 _ );

(c) v 3 (0-); (d) i(t); (e) ^ ( 0+) ; (f) v 2 (0 + ){ and

( g ) ^ 3 ( 0 + )

Figure P13.88

—'VW-20kfl

1 0 0 v ( - )

A

0.5 ^ F ;

2.0/XF;

r = 0

+ +

13.91 T h e r e is n o energy stored in the circuit in Fig P13.91

at the time the impulse voltage is applied

a) Find i { for t > 0+

b) Find i 2 for t > 0+

c) Find v a for t > 0+

d) D o your solutions for i u / 2, and v (} m a k e sense in terms of k n o w n circuit behavior? Explain

i(0

:¾

Figure P13.91

0.5 H

1>"~

Sections 13.1-13.8

13.92 Assume the line-to-neutral voltage Y 0 in the 60 Hz

m o m c i r c u i t of Fig- 13.59 is 120 /CT V (rms) Load R CI is

absorbing 1200 W; load R b is absorbing 1800 W; and

load X a is absorbing 350 magnetizing VAR The

inductive reactance of the line (X{) is 1 fl Assume

V<, does not change after the switch opens

a) Calculate the initial value of i 2 (t) and i[ 0 (t)

b) Find V0, v () (t), and v () (Q + ) using the s-domain

circuit of Fig 13.60

c) Test the steady-state component of v a using

pha-sor domain analysis

d) Using a computer program of your choice, plot

v 0 vs t for 0 £ t < 20 ms

13.93 Assume the switch in the circuit in Fig 13.59

'ERSPECTIVE ° P e n s a l t n e instant the sinusoidal steady-state

voltage v a is zero and going positive, i.e.,

v 0 = 120V2~sinl207rtV

a) Find v 0 {t) for t > 0

b) Using a computer program of your choice, plot

v 0 (t) vs t for 0 < t < 20 ms

c) Compare the disturbance in the voltage in

part (a) with that obtained in part (c) of

Problem 13.92

13.94 The purpose of this problem is to show that the

•ERSPEcnvE l in e- t ° -n e u t r al voltage in the circuit in Fig 13.59

can go directly into steady state if the load R h is disconnected from the circuit at precisely the

right time Let v 0 = V m cos( 12077/ - 0°) V, where

V m = 120 V2 Assume v g does not change after R b

is disconnected

a) Find the value of 6 (in degrees) so that v 0 goes directly into steady-state operation when the

load R f) is disconnected

b) For the value of 6 found in part (a), find %(t) for

t > 0

c) Using a computer program of your choice, plot

on a single graph, for - 1 0 ms ^ t ^ 10 ms,

v a (t) before and after load R b is disconnected

C H A P T E R C O N T E N T S

14.1 Some Preliminaries p 524

14.2 Low-Pass Filters p 526

14.3 High-Pass Filters p 532

14.4 Bandpass Filters p 536

14.5 Bandreject Filters p 545

Know the RL and RC circuit configurations that

act as low-pass filters and be able to design

RL and RC circuit component values to meet a

specified cutoff frequency

Know the RL and RC circuit configurations that

act as high-pass filters and be able to design

RL and RC circuit component values to meet a

specified cutoff frequency

Know the RLC circuit configurations that act as

bandpass filters, understand the definition of

and relationship among the center frequency,

cutoff frequencies, bandwidth, and quality

factor of a bandpass filter, and be able to

design RLC circuit component values to meet

design specifications

Know the RLC circuit configurations that act as

bandreject filters, understand the definition of

and relationship among the center frequency,

cutoff frequencies, bandwidth, and quality

factor of a bandreject filter, and be able to

design RLC circuit component values to meet

design specifications

522

Introduction to Frequency Selective Circuits

Up to this point in our analysis of circuits with sinusoidal

sources, the source frequency was held constant In this chapter,

we analyze the effect of varying source frequency on circuit

volt-ages and currents The result of this analysis is the frequency response of a circuit

We've seen in previous chapters that a circuit's response depends on the types of elements in the circuit, the way the ele-ments are connected, and the impedance of the eleele-ments Although varying the frequency of a sinusoidal source does not change the element types or their connections, it does alter the impedance of capacitors and inductors, because the impedance

of these elements is a function of frequency As we will see, the careful choice of circuit elements, their values, and their con-nections to other elements enables us to construct circuits that pass to the output only those input signals that reside in a desired range of frequencies Such circuits are called

frequency-selective circuits Many devices that communicate

via electric signals, such as telephones, radios, televisions, and satellites, employ frequency-selective circuits

Frequency-selective circuits are also called filters because of

their ability to filter out certain input signals on the basis of fre-quency Figure 14.1 on page 524 represents this ability in a sim-plistic way To be more accurate, we should note that no practical frequency-selective circuit can perfectly or completely filter out

selected frequencies Rather, filters attenuate—that is, weaken or

lessen the effect of—any input signals with frequencies outside frequencies outside a particular frequency band Your home stereo system may have a graphic equalizer, which is an excellent example of a collection of filter circuits Each band in the graphic equalizer is a filter that amplifies sounds (audible frequencies) in the frequency range of the band and attenuates frequencies out-side of that band Thus the graphic equalizer enables you to change the sound volume in each frequency band

Practical Perspective

Pushbutton Telephone Circuits

In this chapter, we examine circuits in which the source

fre-quency varies The behavior of these circuits varies as the

source frequency varies, because the impedance of the

reac-tive components is a function of the source frequency These

frequency-dependent circuits are called filters and are used

in many common electrical devices In radios, filters are used

to select one radio station's signal while rejecting the signals

from others transmitting at different frequencies In stereo

systems, filters are used to adjust the relative strengths of the

low- and high-frequency components of the audio signal

Filters are also used throughout telephone systems

A pushbutton telephone produces tones that you hear

when you press a button You may have wondered about these

tones How are they used to tell the telephone system which

button was pushed? Why are tones used at all? Why do the

tones sound musical? How does the phone system tell the

dif-ference between button tones and the normal sounds of

peo-ple talking or singing?

The telephone system was designed to handle audio signals—those with frequencies between 300 Hz and 3 kHz Thus, all signals from the system to the user have to be audible—including the dial tone and the busy signal Similarly, all signals from the user to the system have to be audible, including the signal that the user has pressed a button I t is important to distinguish button signals from the normal audio signal, so a dual-tone-multiple-frequency (DTMF) design is employed When a number button is pressed, a unique pair of sinusoidal tones with very precise frequencies is sent by the phone to the telephone system The DTMF frequency and timing specifications make it unlikely that a human voice could pro-duce the exact tone pairs, even if the person were trying In the central telephone facility, electric circuits monitor the audio signal, listening for the tone pairs that signal a number

In the Practical Perspective example at the end of the chapter,

we will examine the design of the DTMF filters used to deter-mine which button has been pushed

523

Input

signal Filter

Output signal

Figure 14.1 • The action of a filter on an input signal

results in an output signal

We begin this chapter by analyzing circuits from each of the four major categories of filters: low pass, high pass, band pass, and band reject The transfer function of a circuit is the starting point for the frequency response analysis Pay close attention to the similarities among the trans-fer functions of circuits that perform the same filtering function We will employ these similarities when designing filter circuits in Chapter 15

14.1 Some Preliminaries

Vi(s)

Figure 14.2 A A circuit with voltage input and output

Recall from Section 13.7 that the transfer function of a circuit provides an easy way to compute the steady-state response to a sinusoidal input There,

we considered only fixed-frequency sources To study the frequency response

of a circuit, we replace a fixed-frequency sinusoidal source with a varying-frequency sinusoidal source The transfer function is still an immensely useful tool because the magnitude and phase of the output signal depend only on

the magnitude and phase of the transfer function H{ja))

Note that the approach just outlined assumes that we can vary the fre-quency of a sinusoidal source without changing its magnitude or phase angle Therefore, the amplitude and phase of the output will vary only if those of the transfer function vary as the frequency of the sinusoidal source is changed

To further simplify this first look at frequency-selective circuits, we will also restrict our attention to cases where both the input and output signals are sinusoidal voltages, as illustrated in Fig 14.2 Thus, the transfer function

of interest to us will be the ratio of the Laplace transform of the output

volt-age to the Laplace transform of the input voltvolt-age, or H(s) — V 0 (s)/Vi(s)

We should keep in mind, however, that for a particular application, a current may be either the input signal or output signal of interest

The signals passed from the input to the output fall within a band of

frequencies called the passband Input voltages outside this band have

their magnitudes attenuated by the circuit and are thus effectively pre-vented from reaching the output terminals of the circuit Frequencies not

in a circuit's passband are in its stopband Frequency-selective circuits are

categorized by the location of the passband

One way of identifying the type of frequency-selective circuit is to examine a frequency response plot A frequency response plot shows how

a circuit's transfer function (both amplitude and phase) changes as the source frequency changes A frequency response plot has two parts One is

a graph of \H(jai)\ versus frequency w This part of the plot is called the

magnitude plot The other part is a graph of d(Jw) versus frequency w This

part is called the phase angle plot

The ideal frequency response plots for the four major categories of fil-ters are shown in Fig 14.3 Parts (a) and (b) illustrate the ideal plots for a low-pass and a high-pass filter, respectively Both filters have one

pass-band and one stoppass-band, which are defined by the cutoff frequency that

separates them The names low pass and high pass are derived from the

magnitude plots: a low-pass filter passes signals at frequencies lower than the cutoff frequency from the input to the output, and a high-pass filter

passes signals at frequencies higher than the cutoff frequency Thus the

terms low and high as used here do not refer to any absolute values of

fre-quency, but rather to relative values with respect to the cutoff frequency Note from the graphs for both these filters (as well as those for the bandpass and bandreject filters) that the phase angle plot for an ideal filter varies linearly in the passband It is of no interest outside the passband because there the magnitude is zero Linear phase variation is necessary to avoid phase distortion

I

em

0°

9{ju c )

-Passband

1 Stopband

d(jw c )

0°

-Stopband Passband

d(ja>)

0°

0(/w c2 )

h)\

Stopband

Pass-band Stopband

0 ) c l (O c2 0)

\

- \

m

i

0°

0(M:l)

M

Passband Stop

band Passband

<u c ] a> C2 (o

Figure 14.3 • Ideal frequency response plots of the four types of filter circuits,

(a) An ideal low-pass filter, (b) An ideal high-pass filter, (c) An ideal bandpass filter, (d) An ideal bandreject filter

The two remaining categories of filters each have two cutoff frequen-cies Figure 14.3(c) illustrates the ideal frequency response plot of a

bandpass filter, which passes a source voltage to the output only when the

source frequency is within the band defined by the two cutoff frequencies

Figure 14.3(d) shows the ideal plot of a bandreject filter, which passes a

source voltage to the output only when the source frequency is outside the band defined by the two cutoff frequencies The bandreject filter thus rejects, or stops, the source voltage from reaching the output when its fre-quency is within the band defined by the cutoff frequencies

In specifying a realizable filter using any of the circuits from this chap-ter, it is important to note that the magnitude and phase angle characteris-tics are not independent In other words, the characterischaracteris-tics of a circuit that result in a particular magnitude plot will also dictate the form of the phase angle plot and vice versa For example, once we select a desired form for the magnitude response of a circuit, the phase angle response is also determined Alternatively, if we select a desired form for the phase angle response, the magnitude response is also determined Although there are some frequency-selective circuits for which the magnitude and phase angle behavior can be independently specified, these circuits are not presented here

The next sections present examples of circuits from each of the four filter categories They are a few of the many circuits that act as filters You should focus your attention on trying to identify what properties of a cir-cuit determine its behavior as a filter Look closely at the form of the transfer function for circuits that perform the same filtering functions Identifying the form of a filter's transfer function will ultimately help you

in designing filtering circuits for particular applications

All of the filters we will consider in this chapter are passive filters, so

called because their filtering capabilities depend only on the passive