Electric Circuits, 9th Edition P63 ppt

15.7 a Using only three components from Appendix H, design a low-pass filter with a cutoff frequency and passband gain as close as possible to the specifications in Problem 15.6a.. 15.8

15.7 a) Using only three components from Appendix H,

design a low-pass filter with a cutoff frequency

and passband gain as close as possible to the

specifications in Problem 15.6(a) Draw the

cir-cuit diagram and label all component values

b) Calculate the percent error in this new filter's

cutoff frequency and passband gain when

com-pared to the values specified in Problem 15.6(a)

15.8 Design an op amp-based high-pass filter with a

cut-off frequency of 4 kHz and a passband gain of 8

using a 250 nF capacitor

a) Draw your circuit, labeling the component

val-ues and the output voltage

b) If the value of the feedback resistor in the filter

is changed but the value of the resistor in the

forward path is unchanged, what characteristic

of the filter is changed?

15.9 The input to the high-pass filter designed in

Problem 15.8 is 250 cos ot mV

a) Suppose the power supplies are ±Vcc What is

the smallest value of Vcc that will still cause the

op amp to operate in its linear region?

b) Find the output voltage when a) = (o c

c) Find the output voltage when u> = 0.2&><)

d) Find the output voltage when w = 5ft>0

15.10 a) Use the circuit in Fig 15.4 to design a high-pass

filter with a cutoff frequency of 8 kHz and a

passband gain of 14 dB Use a 3.9 nF capacitor in

the design

b) Draw the circuit diagram of the filter and label

all the components

15.11 Using only three components from Appendix H,

design a high-pass filter with a cutoff frequency and

passband gain as close as possible to the

specifica-tions in Problem 15.10

a) Draw the circuit diagram and label all

compo-nent values

b) Calculate the percent error in this new filter's

cut-off frequency and passband gain when compared

to the values specified in Problem 15.10(a)

Section 15.2

15.12 The voltage transfer function for either high-pass

prototype filter shown in Fig P15.12 is

Figure P15.12

C = I F

If

R = l a v,

(a)

R = \a

L = 1 H v

(b)

15.13 The voltage transfer function of either low-pass

prototype filter shown in Fig P15.13 is

s + 1

Show that if either circuit is scaled in both magni-tude and frequency, the scaled transfer function is

(s/k f ) + 1

+

C = 1 F v„

(a)

L= 1H

R = 1 fl v

(b)

H(s) =

s + 1

15.14 The voltage transfer function of the prototype

bandpass filter shown in Fig P15.14 is Show that if either circuit is scaled in both

magni-tude and frequency, the scaled transfer function is

H'(s) = {s/kf)

(s/k f ) + 1

H(s)

s 2 +

Q s + 1

Show that if the circuit is scaled in both magnitude

and frequency, the scaled transfer function is

!)(±

H>(s)

Figure P15.14

<h + 1

C = 1 F

L = 1H

R = ^ v 0

15.15 a) Specify the component values for the prototype

passive bandpass filter described in Problem 15.14

if the quality factor of the filter is 20

b) Specify the component values for the scaled

bandpass filter described in Problem 15.14 if

the quality factor is 20; the center, or resonant,

frequency is 40 krad/s; and the impedance at

resonance is 5 kft

c) Draw a circuit diagram of the scaled filter and

label all the components

15.16 A n alternative to the prototype bandpass filter

illustrated in Fig P15.14 is to make m 0 = 1 rad/s,

R = 1 ft, and L = Q henrys

a) What is the value of C in the prototype

filter circuit?

b) What is the transfer function of the

prototype filter?

c) Use the alternative prototype circuit just described

to design a passive bandpass filter that has a

qual-ity factor of 16, a center frequency of 25 krad/s,

and an impedance of 10 kft at resonance

d) Draw a diagram of the scaled filter and label all

the components

e) Use the results obtained in Problem 15.14 to

write the transfer function of the scaled circuit

DESIGN

PROBLEM

15.17 The passive bandpass filter illustrated in Fig 14.22

has two prototype circuits In the first prototype

circuit, co 0 = 1 rad/s, C = 1 F, L = 1 H, and

R = Q ohms In the second prototype circuit,

co 0 = 1 rad/s, R = 1 ft, C = Q farads, and

L = (1/(2) henrys

a) Use one of these prototype circuits (your

choice) to design a passive bandpass filter that

has a quality factor of 25 and a center frequency

of 50 krad/s The resistor R is 40 kft

b) Draw a circuit diagram of the scaled filter and

label all components

15.18 The transfer function for the bandreject filter

shown in Fig 14.28(a) is

s 2 + H(s)

1

LC

'.I l$ LC

Show that if the circuit is scaled in both magnitude and frequency, the transfer function of the scaled circuit is equal to the transfer function of the

unsealed circuit with s replaced by (s/kf), where kf

is the frequency scale factor

15.19 Show that the observation made in Problem 15.18

with respect to the transfer function for the circuit

in Fig 14.28(a) also applies to the bandreject filter circuit (lower one) in Fig 14.31

15.20 The passive bandreject filter illustrated in

Fig 14.28(a) has the two prototype circuits shown

in Fig P15.20

a) Show that for both circuits, the transfer function is

H(s) = s 2 + l

s 2 + ( - )s + 1

b) Write the transfer function for a bandreject fil-ter that has a cenfil-ter frequency of 50 krad/s and

a quality factor of 5

Figure PI5.20

-pvQ;

(b)

15.21 The two prototype versions of the passive

band-reject filter shown in Fig 14.31 (lower circuit) are shown in Fig P15.21(a) and (b)

Show that the transfer function for either ver-sion is

H(s) s 2 + 1

s 2 + s + 1

Figure P15.21

(a) (b)

15.22 The circuit in Fig P13.22 is scaled so that the 1 Q

resistors are replaced by 1 k l l resistors and the

1 F capacitor is replaced by a 200 nF capacitor

a) What is the scaled value of L?

b) What is the expression for i(> in the scaled circuit?

15.23 Scale the circuit in Problem 13.31 so that the 50 0

resistor is increased to 5 kH and the frequency

of the voltage response is increased by a factor of

5000

Find/„(0-15.24 Scale the bandpass filter in Problem 14.22 so that

the center frequency is 200 kHz and the quality

fac-tor is still 8, using a 2.5 nF capacifac-tor Determine the

values of the resistor, the inductor, and the two

cut-off frequencies of the scaled filter

15.25 Scale the bandreject filter in Problem 14.35 to get a

center frequency of 50 krad/s, using a 200 /u,H

inductor Determine the values of the resistor, the

capacitor, and the bandwidth of the scaled filter

15.26 a) Show that if the low-pass filter circuit illustrated

in Fig 15.1 is scaled in both magnitude and

fre-quency, the transfer function of the scaled circuit

is the same as Eq 15.1 with s replaced by s/kf,

where kf is the frequency scale factor

b) In the prototype version of the low-pass filter

circuit in Fig 15.1, toc = 1 rad/s, C = 1 F,

R 2 = 1 O, and R x = \/K ohms What is the

transfer function of the prototype circuit?

c) Using the result obtained in (a), derive the

transfer function of the scaled filter

15.27 a) Show that if the high-pass filter illustrated in

Fig 15.4 is scaled in both magnitude and

fre-quency, the transfer function is the same as

Eq 15.4 with s replaced by s/kf, where kf is the

frequency scale factor

b) In the prototype version of the high-pass filter

circuit in Fig 15.4, o) c = 1 rad/s, R\ = 1 il,

C = 1 F, and R 2 = K ohms What is the transfer

function of the prototype circuit?

c) Using the result in (a), derive the transfer func-tion of the scaled filter

Section 15.3

15.28 a) Using 0.1 fx¥ capacitors, design an active

broad-OESIGN band first-order bandpass filter that has a lower

PROBLEM r

PSPICE cutoff frequency of 1000 Hz, an upper cutoff

fre-MULTISIM quency of 5000 Hz, and a passband gain of 0 dB

Use prototype versions of the low-pass and high-pass filters in the design process (see Problems 15.26 and 15.27)

b) Write the transfer function for the scaled filter c) Use the transfer function derived in part (b) to

find H(ja) v ), where o) () is the center frequency of the filter

d) What is the passband gain (in decibels) of the

fil-ter at o)a ?

e) Using a computer program of your choice, make

a Bode magnitude plot of the filter

15.29 a) Using 10 nF capacitors, design an active

broad-DESIGN band first-order bandreject filter with a lower

PSPICE cutoff frequency of 400 Hz, an upper cutoff

MULTISIM frequency of 4000 Hz, and a passband gain of

0 dB Use the prototype filter circuits intro-duced in Problems 15.26 and 15.27 in the design process

b) Draw the circuit diagram of the filter and label all the components

c) What is the transfer function of the scaled filter? d) Evaluate the transfer function derived in (c) at the center frequency of the filter

e) What is the gain (in decibels) at the center frequency?

f) Using a computer program of your choice, make a Bode magnitude plot of the filter trans-fer function

15.30 Design a unity-gain bandpass filter, using a cascade

connection, to give a center frequency of 200 Hz

and a bandwidth of 1000 Hz Use 5 fx¥ capacitors Specify fch f cZ , R L , and R H

15.31 Design a parallel bandreject filter with a center

fre-quency of 1000 rad/s, a bandwidth of 4000 rad/s, and a passband gain of 6 Use 0.2 JJL¥ capacitors, and specify all resistor values

15.32 Show that the circuit in Fig P15.32 behaves as a

bandpass filter {Hint—find the transfer function

for this circuit and show that it has the same form as

the transfer function for a bandpass filter Use the

result from Problem 15.1.)

a) Find the center frequency, bandwidth and gain

for this bandpass filter

b) Find the cutoff frequencies and the quality for

this bandpass filter

Section 15.4 15.34 The circuit in Fig 15.21 has the transfer function

given by Eq 15.34 Show that if the circuit in Fig 15.21 is scaled in both magnitude and fre-quency, the transfer function of the scaled circuit is

1 Figure P15.32

400 O

+

»i

•

50 (xF

\( i

\\ *

1

10 JJLF

\(

\\

5kO

r P ^ ,

1

+

%

o

2 C,C l<-2

RCMf) + R 2 CtC 2

* f

15.33 For circuits consisting of resistors, capacitors,

induc-tors, and op amps, \H(jco) I2 involves only even

pow-ers of a) To illustrate this, compute \H(ja))\ 2 for the

three circuits in Fig PI5.33 when

Figure P15.33

™-v,-R

dB/dec

+

•

w <

<

>

1

~sC

i

—•

+

V,

— •

(a)

Ri

-AA/V-R 2

-vw-1

(b)

15.35 The purpose of this problem is to illustrate the

advantage of an «th-order low-pass Butterworth

fil-ter over the cascade of n identical low-pass sections

by calculating the slope (in decibels per decade) of

each magnitude plot at the corner frequency o) c To

facilitate the calculation, let y represent the

magni-tude of the plot (in decibels), and let x = log10&>

Then calculate dy/dx at a>c for each plot

a) Show that at the corner frequency

(w c = 1 rad/s) of an wth-order low-pass

proto-type Butterworth filter,

dy -T = - 1 0 « dB/dec

dx b) Show that for a cascade of n identical low-pass prototype sections, the slope at (o c is

di _ -20n(2 ]/ " - 1)

c) Compute dy/dx for each type of filter for

n = 1,2, 3, 4, and oo

d) Discuss the significance of the results obtained

in part (c)

15.36 a) Determine the order of a low-pass Butterworth

filter that has a cutoff frequency of 2000 Hz and

a gain of no more than - 3 0 dB at 7000 Hz

b) What is the actual gain, in decibels, at 7000 Hz?

15.37 a) Write the transfer function for the prototype

low-pass Butterworth filter obtained in Problem 15.36(a)

b) Write the transfer function for the scaled filter

in (a) (see Problem 15.34)

c) Check the expression derived in part (b) by using it to calculate the gain (in decibels) at

7000 Hz Compare your result with that found in Problem 15.36(b)

* — • V ,

DESIGN PROBLEM

(c)

15.38 a) Using 1 kO resistors and ideal op amps, design a

circuit that will implement the low-pass Butterworth filter specified in Problem 15.36 The gain in the passband is one

b) Construct the circuit diagram and label all com-ponent values

15.39 a) Using 10 nF capacitors and ideal op amps,

PROBLEM design a high-pass unity-gain Butterworth filter

with a cutoff frequency of 2 kHz and a gain of no

more than - 4 8 dB at 500 Hz

b) Draw a circuit diagram of the filter and label all

component values

15.40 Verify the entries in Table 15.1 for n = 5 and n = 6

15.45 Show that if co ( , = 1 rad/s and C = 1 F in the cir-cuit in Fig 15.26, the prototype values of Rh R 2 , and R 3 are

R, =

R, =

R,

Q

Q 2Q 2 2Q

K

15.41 The circuit in Fig 15.25 has the transfer function

given by Eq 15.47 Show that if the circuit is scaled

in both magnitude and frequency, the transfer

func-tion of the scaled circuit is

H'(s) =

ft

+ • \ k f l H " D D ^ 2 R { R 2 a

DESIGN

PROBLEM

Hence the transfer function of a scaled circuit is

obtained from the transfer function of an unsealed

circuit by simply replacing s in the unsealed

trans-fer function by s/kf, where kf is the frequency

scal-ing factor

15.42 a) Using 1 kf! resistors and ideal op amps, design a

low-pass unity-gain Butterworth filter that has a

cutoff frequency of 8 kHz and is down at least

48 dB at 32 kHz

b) Draw a circuit diagram of the filter and label all

the components

15.43 The high-pass filter designed in Problem 15.39 is

cascaded with the low-pass filter designed in

Problem 15.42

a) Describe the type of filter formed by this

interconnection

b) Specify the cutoff frequencies, the

mid-frequency, and the quality factor of the filter

c) Use the results of Problems 15.36 and 15.40 to

derive the scaled transfer function of the filter

d) Check the derivation of (c) by using it to calculate

H(Ja) 0 ), where 0) o is the midfrequency of the filter

15.44 a) Use 20 nF capacitors in the circuit in Fig 15.26

to design a bandpass filter with a quality factor

of 16, a center frequency of 6.4 kHz, and a

pass-band gain of 20 dB

b) Draw the circuit diagram of the filter and label

all the components

DESIGN

PROBLEM

15.46

DESIGN

PROBLEM

15.47

15.48

15.49

15.50

DESIGN PROBLEM

a) Design a broadband Butterworth bandpass fil-ter with a lower cutoff frequency of 500 Hz and

an upper cutoff frequency of 4500 Hz The pass-band gain of the filter is 20 dB The gain should

be down at least 20 dB at 200 Hz and 11.25 kHz Use 15 nF capacitors in the high-pass circuit and

10 kH resistors in the low-pass circuit

b) Draw a circuit diagram of the filter and label all the components

a) Derive the expression for the scaled transfer function for the filter designed in Problem 15.46 b) Using the expression derived in (a), find the gain (in decibels) at 200 Hz and 1500 Hz

c) Do the values obtained in part (b) satisfy the fil-tering specifications given in Problem 15.46? Derive the prototype transfer function for a sixth-order high-pass Butterworth filter by first writing the transfer function for a sixth-order prototype

low-pass Butterworth filter and then replacing s by

\/s in the low-pass expression

The sixth-order Butterworth filter in Problem 15.48

is used in a system where the cutoff frequency is

25 krad/s

a) What is the scaled transfer function for the filter? b) Test your expression by finding the gain (in deci-bels) at the cutoff frequency

The purpose of this problem is to guide you through the analysis necessary to establish a design procedure for determining the circuit components

in a filter circuit The circuit to be analyzed is shown

in Fig P15.50

a) Analyze the circuit qualitatively and convince yourself that the circuit is a low-pass filter with a

passband gain of Rj/Rh

b) Support your qualitative analysis by deriving the

transfer function V0 fV- v {Hint: In deriving the

transfer function, represent the resistors with their

equivalent conductances, that is, Gx = l/i?i, and

so forth.) To make the transfer function useful in terms of the entries in Table 15.1, put it in the form

H(s) = -Kb,

s 2 + bis + b 0

c) Now observe that we have five circuit

compo-nents—/?!, R2 , i?3, Cb and C2 —and three

trans-fer function constraints—#, b h and b 0 At first

glance, it appears we have two free choices

among the five components However, when we

investigate the relationships between the circuit

components and the transfer function constraints,

we see that if C2 is chosen, there is an upper limit

on C\ in order for R 2 {G 2 ) to be realizable With

this in mind, show that if C 2 = 1 F, the three

con-ductances are given by the expressions

G\ — KG 2 \

[ G 2](

G 2 = h } ± Vbj - 4b () (l + K)C {

2(1 + K) For G2 to be realizable,

C,

46,,(1 +K)'

d) Based on the results obtained in (c), outline the

design procedure for selecting the circuit

com-ponents once K, b w and b\ are known

Figure P15.50

» i

| 1

f&2 *

i

" C !

«3

- Q

i '

+

v„

DESIGN

PROBLEM

15.51 Assume the circuit analyzed in Problem 15.50 is

part of a third-order low-pass Butter worth filter

having a passband gain of 4 (Hint: implement the

gain of 4 in the second-order section of the filter.)

a) If C 2 = 1 F in the prototype second-order

sec-tion, what is the upper limit on Ci?

b) If the limiting value of C] is chosen, what are the

prototype values of R\, R2 , and /?3?

c) If the corner frequency of the filter is 2.5 kHz

and C 2 is chosen to be 10 nF, calculate the scaled

values of Cb R h R 2 , and /?3

d) Specify the scaled values of the resistors and the

capacitor in the first-order section of the filter

e) Construct a circuit diagram of the filter and

label all the component values on the diagram

DESIGN

PROBLEM

15.52 Interchange the Rs and Cs in the circuit in Fig P15.50; that is, replace Rx with Ch R 2 with C2, i?3 with C3, C\ with R\, and C2 with R2

a) Describe the type of filter implemented as a result of the interchange

b) Confirm the filter type described in (a) by

deriv-ing the transfer function V a /Vj Write the

trans-fer function in a form that makes it compatible with Table 15.1

c) Set C2 = C3 = 1 F and derive the expressions

for Q , /?i, and R 2 in terms of K, b h and b 0 (See Problem 15.50 for the definition of b\ and b(r )

d) Assume the filter described in (a) is used in the same type of third-order Butterworth filter that has a passband gain of 8 With C2 = C3 = 1 F,

calculate the prototype values of Ch R h and R2

in the second-order section of the filter

DESIGN PROBLEM

15.53 a) Use the circuits analyzed in Problems 15.50 and

15.52 to implement a broadband bandreject fil-ter having a passband gain of 0 dB, a lower cor-ner frequency of 400 Hz, an upper corcor-ner frequency of 6400 Hz, and an attenuation of at least 30 dB at both 1000 Hz and 2560 kHz Use

10 nF capacitors whenever possible

b) Draw a circuit diagram of the filter and label all the components

15.54 a) Derive the transfer function for the bandreject

filter described in Problem 15.53

b) Use the transfer function derived in part (a) to find the attenuation (in decibels) at the center frequency of the filter

DESIGN PROBLEM

15.55 The purpose of this problem is to develop the

design equations for the circuit in Fig PI5.55 (See Problem 15.50 for suggestions on the development

of design equations.) a) Based on a qualitative analysis, describe the type

of filter implemented by the circuit

b) Verify the conclusion reached in (a) by deriving

the transfer function V 0 /Vi Write the transfer

function in a form that makes it compatible with the entries in Table 15.1

c) How many free choices are there in the selec-tion of the circuit components?

d) Derive the expressions for the conductances

G\ = l/Ri and G2 = l/R 2 in terms of C h C2,

and the coefficients b0 and b% (See Problem 15.50 for the definition of b0 and b\.)

e) Are there any restrictions on C\ or C2? f) Assume the circuit in Fig P15.55 is used to design a fourth-order low-pass unity-gain Butterworth filter Specify the prototype values

of Rx and R2 in each second-order section if 1 F

capacitors are used in the prototype circuit

Figure PI5.55 Section 15.5

15.56 The fourth-order low-pass unity-gain Buttcrworth

PROBLEM ^t e r *n Problem 15.55 is used in a system where the

cutoff frequency is 3 kHz The filter has 4.7 nF

capacitors

a) Specify the numerical values of R{ and R2 in

each section of the filter

b) Draw a circuit diagram of the filter and label all

the components

15.57 Interchange the Rs and Cs in the circuit in

DESIGN fig P15.55, that is, replace i?, with Ci, Ri with C2,

PROBLEM ° * r ' l •

-and vice versa

a) Analyze the circuit qualitatively and predict the

type of filter implemented by the circuit

b) Verify the conclusion reached in (a) by deriving

the transfer function V Q /Vf Write the transfer

function in a form that makes it compatible with

the entries in Table 15.1

c) How many free choices are there in the

selec-tion of the circuit components?

d) Find R\ and R 2 as functions of b m b\, C h and C 2

e) Are there any restrictions on C} and C2 1

f) Assume the circuit is used in a third-order

Butterworth filter of the type found in (a) Specify

the prototype values of R\ and R 2 in the

second-order section of the filter if C\ = C2 = 1 F

DESIGN

PROBLEM

15.58 a) The circuit in Problem 15.57 is used in a

third-order high-pass unity-gain Butterworth filter

that has a cutoff frequency of 5 kHz Specify the

values of Rl and R2 if 75 nF capacitors are

avail-able to construct the filter

b) Specify the values of resistance and capacitance

in the first-order section of the filter

c) Draw the circuit diagram and label all the

components

d) Give the numerical expression for the scaled

transfer function of the filter

e) Use the scaled transfer function derived in (d)

to find the gain in dB at the cutoff frequency

15.59 a) Show that the transfer function for a prototype

narrow band bandreject filter is

H{s) s l + 1

s 2 + (1/Q)s + 1

DESIGN

PROBLEM

b) Use the result found in (a) to find the transfer function of the filter designed in Example 15.13

15.60 a) Using the circuit shown in Fig 15.29, design a

narrow-band bandreject filter having a center frequency of 1 kHz and a quality factor of 20 Base the design on C = 15 nF

b) Draw the circuit diagram of the filter and label all component values on the diagram

c) What is the scaled transfer function of the filter?

Sections 15.1-15.5 15.61 Using the circuit in Fig 15.32(a) design a volume

J£2S™ control circuit to give a maximum gain of 20 dB and

a gain of 17 dB at a frequency of 40 Hz Use an 11.1 kfi resistor and a 100 kfl potentiometer Test your design by calculating the maximum gain at

o) = 0 and the gain at &> = X/R^Cy using the selected values of R[} R 7 , and C v

PERSPECTIVE DESIGN PROBLEM

15.62 Use the circuit in Fig 15.32(a) to design a bass

vol-PERSPECTIVE u m e control circuit that has a maximum gain of

DESIGN 13.98 dB that drops off 3 dB at 50 Hz

PROBLEM

15.63 Plot the maximum gain in decibels versus a when

<w = 0 for the circuit designed in Probh

a vary from 0 to 1 in increments of 0.1

PRACTICAL w = 0 for the circuit designed in Problem 15.61 Let

PERSPECTIVE

PRACTICAL PERSPECTIVE

15.64 a) Show that the circuits in Fig PI 5.64(a) and (b)

are equivalent

b) Show that the points labeled x and y in Fig P15.64(b) are always at the same potential c) Using the information in (a) and (b), show that the circuit in Fig 15.33 can be drawn as shown in

Fig P15.64(c)

d) Show that the circuit in Fig PI5.64(c) is in the form of the circuit in Fig 15.2, where

/?! + (! - a)R2 + RiR 2 C x s

Zi

z / =

1 + R2 C]S

R } + aR 2 + R^C^

1 + R2 C t s

Figure P15.64

lfsCi

HM-K

(l-a)R 2 otR 2

(a)

(l-a)R 2 y cxR 2

(b)

R 4 + 2/?3

(c)

15.65 An engineering project manager has received a

PRACTICAL proposal from a subordinate who claims the circuit

PERSPECTIVE r r

shown in Fig PI 5.65 could be used as a treble

vol-ume control circuit if R4 » R^ + R$ + 2R2 , The

subordinate further claims that the voltage transfer

function for the circuit is

*«-£

-{(2J?3 + R 4 ) + [(1 - /3)i?4 + Rg](fiR4 + R 3 )C 2 s}

{(27?3 + Hi) + [(1 - Aft* + ^ 1 ( / ^ 4 + #e>)C2*}

where i?(, = /?t + 7?3 + 2R 2 Fortunately the project

engineer has an electrical engineering undergraduate

student as an intern and therefore asks the student to check the subordinate's claim

The student is asked to check the behavior of the

transfer function as co—>0; as w—»oo; and the behavior when co = oo and /3 varies between 0 and 1

Based on your testing of the transfer function do you think the circuit could be used as a treble volume control? Explain

Figure P15.65

15.66 In the circuit of Fig P15.65 the component values

PRACTICAL are R x = R 2 = 20 k l l , R^ = 5.9 kO, R 4 = 500 kft,

and C2 = 2.7 nF

a) Calculate the maximum boost in decibels

b) Calculate the maximum cut in decibels

c) Is R4 significantly greater than Ra 7 d) When p = 1, what is the boost in decibels when

co = 1/R 3 C 2 7

e) When /3 = 0, what is the cut in decibels when

co = l/i?3C2? f) Based on the results obtained in (d) and (e),

what is the significance of the frequency \/R3 C 2 when R4 » i?0?

15.67 Using the component values given in

PRACTICAL Problem 15.66, plot the maximum gain in decibels

PERSPECTIVE ' r °

versus /3 when co is inifinite Let /3 vary from 0 to 1

in increments of 0.1

CHAPTER

M

Fourier Series

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

16.1 Fourier Series Analysis:

An Overview p 607

16.2 The Fourier Coefficients p 608

16.3 The Effect of Symmetry on the Fourier

Coefficients p 611

16.4 An Alternative Trigonometric Form of the

Fourier Series p 617

16.5 An Application p 529

16.6 Average-Power Calculations with Periodic

Functions p 623

16.7 The rms Value of a Periodic

Function p 6£6"

16.8 The Exponential Form of the Fourier

Series p 627

16.9 Amplitude and Phase Spectra p 530

1 Be able to calculate the trigonometric form of

the Fourier coefficients for a periodic waveform

using the definition of the coefficients and the

simplifications possible if the waveform exhibits

one or more types of symmetry

2 Know how to analyze a circuit's response to a

periodic waveform using Fourier coefficients

and superposition

3 Be able to estimate the average power

delivered to a resistor using a small number

of Fourier coefficients

4 Be able to calculate the exponential form of the

Fourier coefficients for a periodic waveform and

use them to generate magnitude and phase

spectrum plots for that waveform

604

In the preceding chapters, we devoted a considerable amount

of discussion to steady-state sinusoidal analysis One reason for this interest in the sinusoidal excitation function is that it allows

us to find the steady-state response to nonsinusoidal, but

peri-odic, excitations A periodic function is a function that repeats

itself every T seconds For example, the triangular wave

illus-trated in Fig 16.1 on page 606 is a nonsinusoidal, but periodic, break waveform

A periodic function is one that satisfies the relationship

fit) = f(t ± nT), (16.1)

where n is an integer (1,2, 3, ) and T is the period The

func-tion shown in Fig 16.1 is periodic because

f(to) = f(to ~T) = f(tQ + T) = f(t0 + 2T) =

-for any arbitrarily chosen value of t0 Note that T is the smallest

time interval that a periodic function may be shifted (in either direction) to produce a function that is identical to itself

Why the interest in periodic functions? One reason is that many electrical sources of practical value generate periodic waveforms For example, nonfiltered electronic rectifiers driven from a sinusoidal source produces rectified sine waves that are nonsinusoidal, but periodic Figures 16.2(a) and (b) on page 606 show the waveforms of the full-wave and half-wave sinusoidal rectifiers, respectively

The sweep generator used to control the electron beam of a cathode-ray oscilloscope produces a periodic triangular wave like the one shown in Fig 16.3 on page 606

Electronic oscillators, which are useful in laboratory testing of equipment, are designed to produce nonsinusoidal periodic waveforms Function generators, which are capable of producing square-wave, triangular-wave, and rectangular-pulse waveforms, are found in most testing laboratories Figure 16.4 on page 606 illustrates typical waveforms

m

Practical Perspective

Active High-Q Filters

In Chapters 14 and 15, we discovered that an important

char-acteristic of bandpass and bandreject filters is the quality

fac-tor, Q The quality factor provides a measure of how selective

the filter is at its center frequency For example, a bandpass

fil-ter with a large value of Q will amplify signals at or near its

center frequency and will attentuate signals at all other

fre-quencies On the other hand, a bandreject filter with a small

value of Q will not effectively distinguish between signals at

the center frequency and signals at frequencies quite different

from the center frequency

In this chapter, we learn that any periodic signal can be

represented as a sum of sinusoids, where the frequencies of the

sinusoids in the sum are comprised of the frequency of the periodic signal and integer multiples of that frequency We can use a periodic signal like a square wave to test the quality fac-tor of a bandpass or bandreject filter To do this, we choose a square wave whose frequency is the same as the center fre-quency of a bandpass filter, for example I f the bandpass filter has a high quality factor, its output will be nearly sinusoidal, thereby transforming the input square wave into an output sinusoid If the filter has a low quality factor, its output will still look like a square wave, as the filter is not able to select from among the sinusoids that make up the input square wave

We present an example at the end of this chapter

h A

VIMy IAAAAA/ 1

uvuVuu

High-(?

Bandpass

\J V

605