Electric Circuits, 9th Edition P67 potx

16.20 Derive the Fourier series for the periodic function described in Problem 16.12, using the form of Eq.. 16.21 Derive the Fourier series for the periodic function constructed in Pr

Figure P16.3

-150

v(t)V

8 0

-40

16.8 Show that if / ( / ) = - / ( / - T/2), the Fourier

coef-ficients bfr are given by the expressions

•50 50

-40 -80

(a)

«(t)V

150 250

t(fXS)

T/2

100

-45 -35

r(/xs)

•5 5 (b)

35 45

16.4 Derive the expressions for a v , a k , and b k for the

periodic voltage shown in Fig PI6.4 if V m = 6QTT V

Figure P16.4

v(t)

vm/z

T/4 T/2 37/4 T 5T/4

16.5 a) Verify Eqs 16.6 and 16.7

b) Verify Eq 16.8 Hint: Use the trigonometric

iden-tity cos a sin /3 = \ sin(a + /3) - \ sin(a - /3)

c) Verify Eq 16.9 Hint: Use the trigonometric

iden-tity sin a sin /3 — \ cos(a - / 3 ) - 2 c o s(a + £)

d) Verify Eq 16.10 Hint: Use the trigonometric

iden-tity cos a cos /3 = | c o s ( a — /3) 4- | c o s ( a + /3)

16.6 Derive Eq 16.5

Section 16.3

16.7 Derive the expressions for the Fourier coefficients of

an odd periodic function Hint: Use the same

tech-nique as used in the text in deriving Eqs 16.14-16.16

bk = Tf, / f(t)smka)jdt, for A: odd

Hint: Use the same technique as used in the text to

derive Eqs 16.28 and 16.29

16.9 Derive Eqs 16.36 Hint: Start with Eq 16.29 and

divide the interval of integration into 0 to T/4 and T/4 to T/2 Note that because of evenness and quarter-wave symmetry, f(t) = —f(T/2 — t) in the interval T/4 < t < T/2 Let x = T/2 - t in the

second interval and combine the resulting integral

with the integration between 0 and T/4

16.10 Derive Eqs 16.37 Follow the hint given in

Problem 16.9 except that because of oddness and

quarter-wave symmetry, f(t) = /(7//2 - t) in the interval T/4 < t < T/2

16.11 It is given that v(t) = 20 sin 7r\t\ V over the interval

- 1 < f < 1 s The function then repeats itself a) What is the fundamental frequency in radians per second?

b) Is the function even?

c) Is the function odd?

d) Does the function have half-wave symmetry?

16.12 One period of a periodic function is described by

the following equations:

/(/) = 5/ A, - 2 ms < t < 2 ms;

/(f) = 10 mA, 2 ms < f < 6 ms;

/(/) = 0.04 - 5/ A, 6 ms < / < 10 ms;

/(f) = - 1 0 mA, 10 ms < / < 14 ms

a) What is the fundamental frequency in hertz? b) Is the function even?

c) Is the function odd?

d) Does the function have half-wave symmetry? e) Does the function have quarter-wave symmetry?

f) Give the numerical expressions for a v , a k , and b k

16.13 Find the Fourier series of each periodic function

shown in Fig PI6.13

Figure P16.13

v(t)

V

-T •T/2

-v

16.17 a) Derive the Fourier series for the periodic

cur-rent shown in Fig P16.17

b) Repeat (a) if the vertical reference axis is shifted

T/2 units to the right

Figure P16.17

7/2 T

(a)

(b) 16.14 The periodic function shown in Fig P16.14 is odd

and has both half-wave and quarter-wave symmetry

a) Sketch one full cycle of the function over the

interval -7//4 < t < 37/4

b) Derive the expression for the Fourier

coeffi-cients a v , cik, and b k

c) Write the first three nonzero terms in the

Fourier expansion o f / ( / )

d) Use the first three nonzero terms to estimate

/(774)

Figure P16.14

/(0

16.18 It is sometimes possible to use symmetry to find the Fourier coefficients, even though the original function

is not symmetrical! With this thought in mind,

con-sider the function in Fig P16.4 Observe that v(t) can

be divided into the two functions illustrated in Fig P16.18(a) and (b) Furthermore, we can make

v 2 (t) an even function by shifting it 7 / 8 units to the

left This is illustrated in Fig P16.18(c) At this point

we note that v(t) = Vj(t) + v 2 (t) and that the

Fourier series of V\ (f) is a single-term series consisting

of V m /2 To find the Fourier series of v 2 (t), we first

find the Fourier series of v 2 (t + T/8) and then shift

this series T/8 units to the right Use the technique

just outlined to verify the Fourier coefficients found in Problem 16.4

Figure P16.18

vtit) V.J2

T

4

v 2 (t)

V,„/2

7

4

T 3T

(a)

T 57

4

16.15 It is given that /(f) = 4f2 over the interval

- 2 < f < 2 s

a) Construct a periodic function that satisfies this

/(f) between - 2 and +2 s, has a period of 8 s,

and has half-wave symmetry

b) Is the function even or odd?

c) Does the function have quarter-wave symmetry?

d) Derive the Fourier series for /(f)

e) Write the Fourier series for /(f) if /(f) is shifted

2 s to the right

16.16 Repeat Problem 16.15 given that /(f) = t 3 over the

interval - 2 < f < 2 s

-T 0

4

T T 37/

4 2 4 (b)

57

4

v 2 (t + 7/8)

V,„/2

T

4

7 37

2 4 (c)

7 57

4

Section 16.4

16.19 For each of the periodic functions in Fig PI6.3,

derive the Fourier series for v(t) using the form of

Eq 16.38

16.20 Derive the Fourier series for the periodic function

described in Problem 16.12, using the form of

Eq 16.38

16.21 Derive the Fourier series for the periodic function

constructed in Problem 16.15, using the form of

Eq 16.38

16.22 a) Derive the Fourier series for the periodic

func-tion shown in Fig PI 6.22 when /,„ = 5TT 2 A

Write the series in the form of Eq 16.38

b) Use the first five nonzero terms to estimate

/(7/4)

16.26 a) Show that for large values of C, Eq 16.67 can be

approximated by the expression

vM) -V T V Y m' r in

ARC ~RC '

Note that this expression is the equation

of the triangular wave for 0 < f < 7/2

Hints: (1) Let e~ t/,iC « 1 - (t/RC) and

e -T/2RC ^ j _ (T/2RC); (2) put the resulting

expression over the common denominator

2 - (T/2RC); (3) simplify the numerator; and (4) for large C, assume that T/2RC is much

less than 2

b) Substitute the peak value of the triangular wave into the solution for Problem 16.13 (sec Fig P16.13(b)) and show that the result is

Eq 16.59

Figure PI6.22

ii

* - 1

Section 16.5

16.23 Derive Eqs 16.69 and 16.70

16.24 a) Derive Eq 16.71 Hint: Note that b k =

4V„,/7rk + kto a RCa k Use this expression for b k

to find a\ + b\ in terms of a k Then use the

expression for a k to derive Eq 16.71

b) Derive Eq 16.72

16.25 Show that when we combine Eqs 16.71 and 16.72

with Eqs 16.38 and 16.39, the result is Eq 16.58

Hint: Note from the definition of fi k that

a k

— = - t a n p k ,

bk

and from the definition of 0 k that

t<m9 k = -cot/3*

Now use the trigonometric identity

tan x - cot(90 - x)

to show that 0 k = 90 + p k

16.27 The periodic square-wave voltage shown in

PSPICE Fig P16.13(a) with V m = 105ir V and T = ir ms is

applied to the circuit shown in Fig PI6.27

a) Derive the first three nonzero terms in the Fourier series that represents the steady-state

voltage v a

b) Which frequency component in the input volt-age is eliminated from the output voltvolt-age? Explain why

16.28

Figure P16.27

100 mH

The periodic square-wave voltage seen in Fig PI6.28(a) is applied to the circuit shown in Fig P16.28(b) Derive the first three nonzero terms

in the Fourier series that represents the steady-state

voltage v 0 if V m = 157r V and the period of the

input voltage is 4TT ms

Figure P16.28

ion

• -WW

+

Vm

0

I/

v in

,

1

T/2

1

+

10 mH u v

16.29 The full-wave rectified sine-wave voltage shown in

Fig PI6.29(a) is applied to the circuit shown in

Fig P16.29(b)

a) Find the first five nonzero terms in the Fourier

series representation of /„

b) Does your solution for i (> make sense? Explain

Figure P16.29

v g (V)

340

16.32 The periodic current described below is used to energize the circuit shown in Fig PI6.32 Write the time-domain expression for the third-harmonic

component in the expression for v ()

i R = 500f,

= 1 A

= 5 - 500f,

= - 1 A

- 2 ms s t < 2 ms;

2 ms < t < 8 ms;

8 ms < t <; 12 ms;

12 ms < t < 18 ms

Figure P16.32

1/120 1/60 1/40

(a)

16 H

312.5 nF

l k t t

(b) 16.30 The square-wave voltage shown in Fig PI6.30(a) is

PSPICE applied to the circuit shown in Fig PI6.30(b)

MULTISIM

a) Find the Fourier series representation of the

steady-state current i

b) Find the steady-state expression for /' by

straight-forward circuit analysis

Figure P16.30

Section 16.6 16.33 The periodic current shown in Fig PI6.33 is applied

to a 1 kft resistor

a) Use the first three nonzero terms in the Fourier series representation of /(f) to estimate the aver-age power dissipated in the 1 kft resistor

b) Calculate the exact value of the average power dissipated in the 1 kft resistor

c) What is the percentage of error in the estimated value of the average power?

Figure P16.33

i (mA)

2 4 0

-7/2 7 3-7/2

(a)

(b) 16.31 A periodic voltage having a period of 10-7T/XS is

given by the following Fourier series:

/Z7T

v g = 150 2 — sin — cos naj V

/1=1,3,5 " 2

This periodic voltage is applied to the circuit shown

in Fig PI6.31 Find the amplitude and phase angle

of the components of v a that have frequencies of

3 and 5 Mrad/s

Figure P16.31

250 kfi

vw

16.34 The periodic voltage across a 10 ft resistor is shown

in Fig PI6.34

a) Use the first three nonzero terms in the Fourier

series representation of v(t) to estimate the

average power dissipated in the 10 ft resistor b) Calculate the exact value of the average power dissipated in the 10 ft resistor

c) What is the percentage error in the estimated value of the average power dissipated?

Figure P16.34 w(V)

37 4"

16.35 The triangular-wave voltage source is applied to the

circuit in Fig P16.35(a) The triangular-wave

volt-age is shown in Fig P16.35(b) Estimate the avervolt-age

power delivered to the 5 0 v T O resistor when the

circuit is in steady-state operation

Figure P16.35

100 mH

lOjiiF 50V2ft

(a)

M V )

t (ms)

16.36 a) Find the rms value of the voltage shown in

Fig PI6.36 for V m = 100 V Note that the

Fourier series for this periodic voltage was

found in Assessment Problem 16.3

b) Estimate the rms value of the voltage, using the

first three nonzero terms in the Fourier series

representation of v g (t)

Figure P16.36

M0

16.37 The voltage and current at the terminals of a

network are

v = 15 + 400 cos 500/ + 100 sin 1500/ V,

i = 2 + 5 sin (500/ + 60°) + 3 cos (1500/ - 15°) A

The current is in the direction of the voltage drop

across the terminals

a) What is the average power at the terminals?

b) What is the rms value of the voltage?

c) What is the rms value of the current?

16.38 a) Estimate the rms value of the full-wave rectified

sinusoidal voltage shown in Fig PI6.38(a) by

using the first three nonzero terms in the

Fourier series representation of v(t)

b) Calculate the percentage of error in the

estimation

c) Repeat (a) and (b) if the full-wave rectified sinu-soidal voltage is replaced by the half-wave recti-fied sinusoidal voltage shown in Fig PI 6.38(b)

Figure P16.38

v(V)

*- t (ms)

v(V)

*- r(ms)

16.39 a) Estimate the rms value of the periodic

square-wave voltage shown in Fig PI 6.39(a) by using the first five nonzero terms in the Fourier series

representation of v(t)

b) Calculate the percentage of error in the estimation

c) Repeat parts (a) and (b) if the periodic square-wave voltage is replaced by the periodic triangu-lar voltage shown in Fig PI6.39(b)

Figure P16.39

v(V)

120

0

-120

5 10 -»- t (ms)

(a)

t (ms)

120

-16.40 a) Use the first four nonzero terms in the Fourier

series approximation of the periodic voltage shown in Fig PI6.40 to estimate its rms value b) Calculate the true rms value of the voltage c) Calculate the percentage of error in the esti-mated value

Figure P16.40

I O T T

-

2.577-

-2.577-1077

V p /V3 Verify this observation by finding the

rms value of the three waveforms depicted in Fig.P16.43(b)-(d)

Figure P16.43

J L J I L

T/A T/2 3T/A

16.41 a) Derive the expressions for the Fourier coefficients

for the periodic current shown in Fig PI6.41

b) Write the first four nonzero terms of the series

using the alternative trigonometric form given

by Eq 16.39

c) Use the first four nonzero terms of the expression

derived in (b) to estimate the rms value of L

d) Find the exact rms value of i g

e) Calculate the percentage of error in the

esti-mated rms value

Figure P16.41

o(

i

10

0

- 1 0

v.)

,

0 4

_ 1/

(c) Section 16.8

V » / M

/ 1 0

v{

10

0

- 1 0

v.)

1

^ \ 1 1 1

-(d)

I

i

i

L

0

- / , „

<?

i

T/A T/2^ \3T/A

/

T

16.42 Assume the periodic function described in

Problem 16.14 is a voltage v R with a peak

ampli-tude of 20 V

a) Find the rms value of the voltage

b) If this voltage exists in a 15 fl resistor, what is

the average power dissipated in the resistor?

c) If v g is approximated by using just the

fundamen-tal frequency term of its Fourier series, what is the

average power delivered to the 15 H resistor?

d) What is the percentage of error in the estimation

of the power dissipated?

16.43 The rms value of any periodic triangular wave

hav-ing the form depicted in Fig P16.43(a) is

independ-ent of t u and ?/, Note that for the function to be

single valued, L ^ t The rms value is equal to

/ ( s )

16.44 Use the exponential form of the Fourier series to write an expression for the voltage shown in Fig PI 6.44

Figure PI6.44

v(t)

T/A 0 T/A T/2 3T/A T Sir/4

16.45 Derive the expression for the complex Fourier coefficients for the periodic voltage shown in Fig PI 6.45

Figure P16.45

16.46 a) The periodic voltage in Problem 16.45 is applied

to a 10 a resistor If V m = 120 V what is the

average power delivered to the resistor?

b) Assume v(t) is approximated by a truncated

exponential form of the Fourier series consisting

of the first eight nonzero terms, that is,

n = 0 , 1 , 2, 3,4,5, 6 and 7 What is the rms value

of the voltage, using this approximation?

c) If the approximation in part (b) is used to

repre-sent v what is the percentage of error in the

cal-culated power?

16.47 The periodic voltage source in the circuit shown

in Fig P16.47(a) has the waveform shown in

Fig PI 6.47(b)

a) Derive the expression for C„

b) Find the values of the complex coefficients

C () , C^,C], C_2, C2, C_3, C3, C_4, and C4 for the

input voltage v g if V m = 54 V and T = IOTT /AS

c) Repeat (b) for v a

d) Use the complex coefficients found in (c) to

estimate the average power delivered to the

250 k l i resistor

Figure P16.47

62.5 11

-AW

(a)

V,

v,„

(b)

16.48 a) Find the rms value of the periodic voltage in

Fig.P16.47(b)

b) Use the complex coefficients derived in

Problem 16.47(b) to estimate the rms value of v*

c) What is the percentage of error in the estimated

rms value of v„?

Section 16.9 16.49 a) Make an amplitude and phase plot, based on

Eq 16.38, for the periodic voltage in Example 16.3

Assume V m is 40 V Plot both amplitude and phase

versus na) ( „ where n — 0 , 1 , 2 , 3 ,

b) Repeat (a), but base the plots on Eq 16.82

16.50 a) Make an amplitude and phase plot, based on

Eq 16.38, for the periodic voltage in Problem 16.33 Plot both amplitude and phase

versus no) a where n = 0 , 1 , 2 ,

b) Repeat (a), but base the plots on Eq 16.82

16.51 A periodic function is represented by a Fourier

series that has a finite number of terms The ampli-tude and phase spectra are shown in Fig P16.51(a) and (b), respectively

a) Write the expression for the periodic current using the form given by Eq 16.38

b) Is the current an even or odd function of t'!

c) Does the current have half-wave symmetry? d) Calculate the rms value of the current in milliamperes

e) Write the exponential form of the Fourier series f) Make the amplitude and phase spectra plots on the basis of the exponential series

Figure P16.51

A n (fiA)

(11,025)

(1225)

y

(441) (225)

180°

in 30 50 70 krad/s

16.52 A periodic voltage is represented by a truncated

Fourier series The amplitude and phase spectra are shown in Fig P16.52(a) and (b), respectively

a) Write an expression for the periodic voltage using the form given by Eq 16.38

b) Is the voltage an even or odd function of f? c) Does the voltage have half-wave symmetry? d) Does the voltage have quarter-wave symmetry?

Figure P16.52

t A,

A,

(a)

A-,

-I »• nco,

t)

90°

0

on°

n

,

<*><>

1

3c

5o) 0

I

t) 0

1 1 1

l0) o

r

(b)

the Fourier series that represents the steady-state output voltage of the filter

16.56 The transfer function {VJV g ) for the narrowband

bandpass filter circuit in Fig P16.56(a) is

s- + /35 + c4

a) Find K 0 , /3, and u> 20 as functions of the circuit parameters JRj, /?2> ^3- Q» a nd C2

b) Write the first three terms in the Fourier series

that represents v a if v H is the periodic voltage in Fig PI 6.56(b)

c) Predict the value of the quality factor for the fil-ter by examining the result in part (b)

d) Calculate the quality factor for the filter using /3

and o) a and compare the value to your predic-tion in part (c)

16.53 The input signal to a unity-gain third-order low-pass

Butterworth filter is a half-wave rectified sinusoidal

voltage The corner frequency of the filter is

100 rad/s The amplitude of the sinusoidal voltage is

54-77 V and its period is 57r ms Write the first three

terms of the Fourier series that represents the

steady-state output voltage of the filter

Sections 16.1-16.9

16.54 The input signal to a unity-gain second-order

low-pass Butterworth filter is the periodic

triangular-wave voltage shown in Fig PI 6.54 The corner

frequency of the filter is 2 krad/s Write the first

three terms of the Fourier series that represents the

steady-state output voltage of the filter

Figure P16.54

v g (V)

Figure P16.56

Vi ( m V )

2.257T,

\ ' Y

\ - 0 1 T T /

V-2.257T2

I '

\ O.ITT / Y ' ^

0.27T

f(T

(b)

16.55 The input signal to a unity-gain second-order

low-pass Butterworth filter is a full-wave rectified sine

wave with an amplitude of 2.5TT V and a

fundamen-tal frequency of 5000 rad/s The corner frequency

of the filter is 1 krad/s Write the first two terms in

16.57 a) Find the values for K, /3, and col f °r t r i e

band-pass filter shown in Fig 16.20(b)

b) Find the first three terms in the Fourier series

for V[) in Fig 16.20(b) if the input to the filter is

the waveform shown in Fig 16.20(a)

CHAPTER CONTENTS

17.1 The Derivation of the Fourier

Transform p 646

17.2 The Convergence of the Fourier

Integral p 648

17.3 Using Laplace Transforms to Find Fourier

Transforms p 650

17A Fourier Transforms in the Limit p 653

17.5 Some Mathematical Properties p 655

17.6 Operational Transforms p 657

17.7 Circuit Applications p 661

17.8 Parseval's Theorem p 664

/ C H A P T E R O B J E C T I V E S

1 Be able to calculate the Fourier transform of a

function using any or all of the following:

• The definition of the Fourier transform;

• Laplace transforms;

• Mathematical properties of the Fourier

transform;

• Operational transforms

2 Know how to use the Fourier transform to find

the response of a circuit

3 Understand Parseval's theorem and be able to

use i t to answer questions about the energy

contained within specific frequency bands

The Fourier Transform

In Chapter 16, we discussed the representation of a periodic

function by means of a Fourier series This series representation enables us to describe the periodic function in terms of the frequency-domain attributes of amplitude and phase The Fourier transform extends this frequency-domain description to functions that are not periodic Through the Laplace transform, we already introduced the idea of transforming an aperiodic function from the time domain to the frequency domain You may wonder, then, why yet another type of transformation is necessary Strictly speaking, the Fourier transform is not a new transform It is a spe-cial case of the bilateral Laplace transform, with the real part of the complex frequency set equal to zero However, in terms of physical interpretation, the Fourier transform is better viewed as

a limiting case of a Fourier series We present this point of view in Section 17.1, where we derive the Fourier transform equations The Fourier transform is more useful than the Laplace trans-form in certain communications theory and signal-processing sit-uations Although we cannot pursue the Fourier transform in depth, its introduction here seems appropriate while the ideas underlying the Laplace transform and the Fourier series are still fresh in your mind

644

^u

Practical Perspective

Filtering Digital Signals

It is common to use telephone lines to communicate

informa-tion from one computer to another As you may know,

com-puters represent all information as collections of 1's and O's

Usually the value 1 is represented as a voltage, usually 5 V,

and 0 is represented as 0 V, as shown below

The telephone line has a frequency response characteristic that is similar to a low pass filter We can use Fourier trans-forms to understand the effect of transmitting a digital value using a telephone line that behaves like a filter

0111010010

645