Electric Circuits, 9th Edition P58 potx

Bandreject filters are characterized by the same parameters as pass filters: the two cutoff frequencies, the center frequency, the band-width, and the quality factor.. This series RLC ba

546 Introduction to Frequency Selective Circuits

R

C

(c)

Figure 14.28 • (a) A series RLC bandreject filter

(b) The equivalent circuit for 00 = 0 (c) The equivalent

circuit for co = 00

\ff(h*)\

90°

-Figure 14.29 • The frequency response plot for the

series RLC bandreject filter circuit in Fig 14.28(a)

before they reach the output at frequencies between the two cutoff fre-quencies (the stopband) Bandpass filters and bandreject filters thus per-form complementary functions in the frequency domain

Bandreject filters are characterized by the same parameters as pass filters: the two cutoff frequencies, the center frequency, the band-width, and the quality factor Again, only two of these five parameters can

be specified independently

In the next sections, we examine two circuits that function as band-reject filters and then compute equations that relate the circuit compo-nent values to the characteristic parameters for each circuit

The Series RLC Circuit—Qualitative Analysis

Figure 14.28(a) shows a series RLC circuit Although the circuit components and connections are identical to those in the series RLC bandpass filter in

Fig 14.19(a), the circuit in Fig 14.28(a) has an important difference: the out-put voltage is now defined across the inductor-capacitor pair As we saw in the case of low- and high-pass filters, the same circuit may perform two dif-ferent filtering functions, depending on the definition of the output voltage

We have already noted that at to = 0, the inductor behaves like a

short circuit and the capacitor behaves like an open circuit, but at

a) = 00, these roles switch Figure 14.28(b) presents the equivalent

cir-cuit for a) = 0; Fig 14.28(c) presents the equivalent circir-cuit for to = 00 In

both equivalent circuits, the output voltage is defined over an effective open circuit, and thus the output and input voltages have the same

mag-nitude This series RLC bandreject filter circuit then has two

pass-bands—one below a lower cutoff frequency, and the other above an upper cutoff frequency

Between these two passbands, both the inductor and the capacitor have finite impedances of opposite signs As the frequency is increased from zero, the impedance of the inductor increases and that of the capac-itor decreases Therefore the phase shift between the input and the

out-put approaches —90° as toL approaches 1/OJC A S soon as coL exceeds 1/wC, the phase shift jumps to +90° and then approaches zero as to

con-tinues to increase

At some frequency between the two passbands, the impedances of the inductor and capacitor are equal but of opposite sign At this frequency, the series combination of the inductor and capacitor is that of a short cir-cuit, so the magnitude of the output voltage must be zero This is the

cen-ter frequency of this series RLC bandreject filcen-ter

Figure 14.29 presents a sketch of the frequency response of the series

RLC bandreject filter from Fig 14.28(a) Note that the magnitude plot is

overlaid with that of the ideal bandreject filter from Fig 14.3(d) Our qual-itative analysis has confirmed the shape of the magnitude and phase angle plots We now turn to a quantitative analysis of the circuit to confirm this frequency response and to compute values for the parameters that charac-terize this response

^(•v)

Figure 14.30 • The s-domain equivalent of the circuit

in Fig 14.28(a)

The Series RLC Circuit—Quantitative Analysis

After transforming to the s-domain, as shown in Fig 14.30, we use voltage division to construct an equation for the transfer function:

R + sL + —

sC

s 2 + 1

LC

•> R l

(14.43)

[H(jm)\ = LC

LC + ?)"

(14.44)

6(ja>) = — tan -1

wR

L

(14.45)

LC

Note that Eqs 14.44 and 14.45 confirm the frequency response shape pictured in Fig 14.29, which we developed based on the qualita-tive analysis

Wc use the circuit in Fig 14.30 to calculate the center frequency For the bandreject filter, the center frequency is still defined as the frequency for which the sum of the impedances of the capacitor and inductor is zero

In the bandpass filter, the magnitude at the center frequency was a maxi-mum, but in the bandreject filter, this magnitude is a minimum This is because in the bandreject filter, the center frequency is not in the pass-band; rather, it is in the stopband It is easy to show that the center fre-quency is given by

Substituting Eq 14.46 into Eq 14.44 shows that \H(jco 0 )\ = 0

The cutoff frequencies, the bandwidth, and the quality factor are defined for the bandreject filter in exactly the way they were for the bandpass filters Compute the cutoff frequencies by substituting the constant ( l / V 2 ) / /m a x for the left-hand side of Eq 14.44 and then

solv-ing for co cl and wc2- Note that for the bandreject filter,

#max = 1^(/0)1 = \H(j oo)\, and for the series RLC bandreject filter in Fig 14.28(a), H BUX = l.Thus,

0>c\ = -A _, fJLY

2L + 2LJ LC (14.47)

R

^- = 2L +

R_

Use the cutoff frequencies to generate an expression for the band-width, jS:

Finally, the center frequency and the bandwidth produce an equation for

the quality factor, Q:

Q =

548 Introduction to Frequency Selective Circuits

Again, we can represent the expressions for the two cutoff frequencies

in terms of the bandwidth and center frequency, as we did for the band-pass filter:

to* - - + + cof (14.52)

Alternative forms for these equations express the cutoff frequencies in terms of the quality factor and the center frequency:

(0 ct

w c 2

= t0 o '

.-¾ + V 1 + (¾)

i

(14.53)

(14.54)

Example 14.8 presents the design of a series RLC bandreject filter

Using the series RLC circuit in Fig 14.28(a),

com-pute the component values that yield a bandreject

filter with a bandwidth of 250 Hz and a center

fre-quency of 750 Hz Use a 100 nF capacitor Compute

values for R, L, a) ch a>c.2, and Q

Solution

We begin by using the definition of quality factor to

compute its value for this filter:

Q = <oJ(3 = 3

Use Eq 14.46 to compute L, remembering to

con-vert io a to radians per second:

colC

[2TT(750)] 2 (100 X 10" 9 )

= 450 mH

Use Eq 14.49 to calculate R:

R = (3L

= 277(250)(450 x 10"3)

= 707 a The values for the center frequency and band-width can be used in Eqs 14.51 and 14.52 to com-pute the two cutoff frequencies:

co c] = 0

3992.0 rad/s,

0) c2

P + PV + uf

= 5562.8 rad/s

The cutoff frequencies are at 635.3 Hz and 885.3 Hz Their difference is 885.3 - 635.3 = 250 Hz, con-firming the specified bandwidth The geometric mean is V(635.3)(885.3) = 750 Hz, confirming the specified center frequency

As you might suspect by now, another configuration that produces a

bandreject filter is a parallel RLC circuit Whereas the analysis details of

the parallel RLC circuit are left to Problem 14.34, the results are

summa-rized in Fig 14.31, along with the series RLC bandreject filter As we did

for other categories of filters, we can state a general form for the transfer

functions of bandreject filters, replacing the constant terms with /3 and OJ ( ;

s z + jSs + <a% 2' (14.55) A Transfer function for RLC bandreject filter

Equation 14.55 is useful in filter design, because any circuit with a transfer

function in this form can be used as a bandreject filter

H(s)=~2 s 2 + l/LC

s 1 + (R/L)s + l/LC

> 0 = VT/LC P = R/L

"•©

sL

sC

+

H(s) = s 2 + l/LC

s 2 + s/RC + l/LC

co 0 = VljLC p = l/RC

Figure 14.31 • Two RLC bandreject filters, together

with equations for the transfer function, center

frequency, and bandwidth of each

/ A S S E S S M E N T PROBLEMS

Objective 4—Know the RLC circuit configurations that act as bandreject filters

14.10 Design the component values for the series

RLC bandreject filter shown in Fig 14.28(a) so

that the center frequency is 4 kHz and the

quality factor is 5 Use a 500 nF capacitor

Answer: L = 3.17 mH,

R = 15.92 H

NOTE: Also try Chapter Problems 14.35 and 14.36

14.11 Recompute the component values for Assessment Problem 14.10 to achieve a band-reject filter with a center frequency of 20 kHz The filter has a 100 fi resistor The quality fac-tor remains at 5

Answer: L = 3.98 mH,

C = 15.92 nF

550 Introduction to Frequency Selective Circuits

Practical Perspective

69? * *

770 #

%

8521» 1

941 H*

O '

Figure 14.32 • Tones generated by the rows and

columns of telephone pushbuttons

Pushbutton Telephone Circuits

In the Practical Perspective at the start of this chapter, we described the dual-tone-multiple-frequency (DTMF) system used to signal that a button has been pushed on a pushbutton telephone A key element of the DTMF system is the DTMF receiver—a circuit that decodes the tones produced by pushing a button and determines which button was pushed

In order to design a DTMF reciever, we need a better understanding of the DTMF system As you can see from Fig 14.32, the buttons on the tele-phone are organized into rows and columns The pair of tones generated by pushing a button depends on the button's row and column The button's row determines its low-frequency tone, and the button's column determines its high-frequency tone 1 For example, pressing the "6" button produces sinu-soidal tones with the frequencies 770 Hz and 1477 Hz

At the telephone switching facility, bandpass filters in the DTMF receiver first detect whether tones from both the low-frequency and high-frequency groups are simultaneously present This test rejects many extraneous audio signals that are not DTMF If tones are present in both bands, other filters are used to select among the possible tones in each band so that the frequencies can be decoded into a unique button signal Additional tests are performed to prevent false button detection For example, only one tone per frequency band is allowed; the high- and low-band frequencies must start and stop within a few milliseconds of one another to be considered valid; and the highl-and low-bhighl-and signal amplitudes must be sufficiently close to each other

You may wonder why bandpass filters are used instead of a high-pass filter for the high-frequency group of DTMF tones and a low-pass filter for the low-frequency group of DTMF tones The reason is that the telephone system uses frequencies outside of the 300-3 kHz band for other signal-ing purposes, such as rsignal-ingsignal-ing the phone's bell Bandpass filters prevent the DTMF receiver from erroneously detecting these other signals

NOTE: Assess your understanding of this Practical Perspective by trying Chapter Problems 14.46-14.48

1 A fourth high-frequency tone is reserved at 1633 Hz This tone is used infrequently and is not produced by a standard 12-button telephone

Summary

A frequency selective circuit, or filter, enables signals at

certain frequencies to reach the output, and it

attenu-ates signals at other frequencies to prevent them from

reaching the output The passband contains the

fre-quencies of those signals that are passed; the stopband

contains the frequencies of those signals that are

atten-uated (See page 524.)

The cutoff frequency, coc, identifies the location on the

frequency axis that separates the stopband from the

passband At the cutoff frequency, the magnitude of the transfer function equals (1/V2)//raax (See page 527.)

A low-pass filter passes voltages at frequencies below

(a c and attenuates frequencies above co c Any circuit

with the transfer function

/.>

H{s)

S + (o c

functions as a low-pass filter (See page 531.)

A high-pass filter passes voltages at frequencies above

and attenuates voltages at frequencies below Any

cir-cuit with the transfer function

H{s) =

-functions as a high-pass filter (See page 536.)

Bandpass filters and bandreject filters each have two

cut-off frequencies, a) c] and a;c2 These filters are further

char-acterized by their center frequency («„), bandwidth (/3),

and quality factor (Q) These quantities are defined as

w 0 = Vcoti • (o c2 , (3 = (o c2 - o)d ,

Q = "o/P

(See pages 539-540.)

A bandpass filter passes voltages at frequencies within

the passband, which is between a>cl and u) c2 - It attenuates

frequencies outside of the passband Any circuit with the transfer function

His) Ps

r + Pi

functions as a bandpass filter (See page 544.)

A bandreject filter attenuates voltages at frequencies

within the stopband, which is between co c] and wc2 It passes frequencies outside of the stopband Any circuit with the transfer function

S 2 + p s + <4

functions as a bandreject filter (See page 549.) Adding a load to the output of a passive filter changes its filtering properties by altering the location and mag-nitude of the passband Replacing an ideal voltage source with one whose source resistance is nonzero also changes the filtering properties of the rest of the circuit, again by altering the location and magnitude of the passband (See page 542.)

Problems

Section 14.2

14.1 a) Find the cutoff frequency in hertz for the RL

fil-ter shown in Fig P14.1

b) Calculate H(Jm) at G>C , 0.2W C , and 5w c

c) If Vj = lOcosatf V, write the steady-state

expression for v a when co = w(., co = 0.2wr, and

a) = 5oo c

Figure P14.1

10 mH

m rv^r*r\ « m

127 a

14.2 Use a 1 mH inductor to design a low-pass, RL,

pas-OESIGN sive filter with a cutoff frequency of 5 kHz

PROBLEM " J

PSPICE a) Specify the value of the resistor

MULTISIM

b) A load having a resistance of 68 fi is connected

across the output terminals of the filter What is

the corner, or cutoff, frequency of the loaded

fil-ter in hertz?

c) If you must use a single resistor from Appendix H

for part (a), what resistor should you use? What is

the resulting cutoff frequency of the filter?

14.3 A resistor, denoted as R/, is added in series with the

inductor in the circuit in Fig 14.4(a) The new low-pass filter circuit is shown in Fig P14.3

a) Derive the expression for His) where

His) = VJV b

b) At what frequency will the magnitude of H{J<D)

be maximum?

c) What is the maximum value of the magnitude

of » ( / » ) ?

d) At what frequency will the magnitude of Hija))

equal its maximum value divided by V2?

e) Assume a resistance of 75 O is added in series with the 10 mH inductor in the circuit in

Fig P14.1 Find ta e , # ( / 0 ) , ff(/»e)f / / ( / 0 3 0 ,

and Hij3a) c )

Figure P14.3

Ri

+•

•

L

4

: R

»

+

• 14.4 a) Find the cutoff frequency (in hertz) of the

low-pass filter shown in Fig PI4.4

b) Calculate H{joo) at w., 0.1w., and 10w

552 Introduction to Frequency Selective Circuits

DESIGN

PROBLEM

PSPICE

MULT1SIM

c) If Vj = 200 cos col mV, write the steady-state

expression for v a when to = co c , 0.1coc, and 10co c

Figure P14.4

o VW

100 nF

14.5 Use a 500 nF capacitor to design a low-pass passive

filter with a cutoff frequency of 50 krad/s

a) Specify the cutoff frequency in hertz

b) Specify the value of the filter resistor

c) Assume the cutoff frequency cannot increase by

more than 5% What is the smallest value of

load resistance that can be connected across the

output terminals of the filter?

d) If the resistor found in (c) is connected across

the output terminals, what is the magnitude of

H(joi) when co = 0?

14.6 Design a passive RC low pass filter (see Fig 14.7)

with a cutoff frequency of 100 Hz using a 4.7/AF

capacitor

a) What is the cutoff frequency in rad/s?

b) What is the value of the resistor?

c) Draw your circuit, labeling the component

val-ues and output voltage

d) What is the transfer function of the filter in

part (c)?

e) If the filter in part (c) is loaded with a resistor

whose value is the same as the resistor part (b),

what is the transfer function of this loaded filter?

f) What is the cutoff frequency of the loaded filter

from part (e)?

g) What is the gain in the pass band of the loaded

filter from part (e)?

14.7 A resistor denoted as R L is connected in parallel

with the capacitor in the circuit in Fig 14.7 The

loaded low-pass filter circuit is shown in Fig P14.7

a) Derive the expression for the voltage transfer

function V 0 /V r

b) At what frequency will the magnitude of H(joo)

be maximum?

c) What is the maximum value of the magnitude

d) At what frequency will the magnitude of H(joo)

equal its maximum value divided by V2?

e) Assume a resistance of 10 kH is added in

paral-lel with the 100 nF capacitor in the circuit in

Fig P14.4 Find a> c , H(jO), H(Jm c ), H(j0.1w c ),

and H(jl0co c )

Figure P14.7 +

m—

R

O 4

:R L

+

• 14.8 Study the circuit shown in Fig PI 4.8 (without the load resistor)

a) As co —> 0, the inductor behaves like what circuit

component? What value will the output voltage

v 0 have?

b) As co —>• oo, the inductor behaves like what

cir-cuit component? What value will the output

voltage v 0 have?

c) Based on parts (a) and (b), what type of filtering does this circuit exhibit?

d) What is the transfer function of the unloaded filter?

e) If R = 330 O and L = 10 mH, what is the cutoff

frequency of the filter in rad/s?

Figure P14.8

His) VM

V,<s)

14.9 Suppose we wish to add a load resistor in parallel with the resistor in the circuit shown in Fig PI 4.8 a) What is the transfer function of the loaded filter? b) Compare the transfer function of the unloaded filter (part (d) of Problem 14.8) and the trans-fer function of the loaded filter (part (a) of Problem 14.9) Are the cutoff frequencies differ-ent? Are the passband gains differdiffer-ent?

c) What is the smallest value of load resistance that can be used with the filter from Problem 14.8(e) such that the cutoff frequency of the resulting filter is no more than 5% different from the unloaded filter?

Section 14.3 14.10 a) Find the cutoff frequency (in hertz) for the

high-pass filter shown in Fig P14.10

b) Find H(joo) at oo c , 0.2co c , and 5co c

c) If Vj = 500 cos cot mV, write the steady-state

expression for v a when co = co c , to = 0.2co ct and

&) = 5co r

Figure P14.10

5nF

1(-50 kn

14.11 A resistor, denoted as R c , is connected in series

with the capacitor in the circuit in Fig 14.10(a) The

new high-pass filter circuit is shown in Fig P14.ll

a) Derive the expression for H(s) where

H(s) = VJV,

b) At what frequency will the magnitude of H(joo)

be maximum?

c) What is the maximum value of the magnitude

of H(joo)?

d) At what frequency will the magnitude of H(jco)

equal its maximum value divided by V2?

e) Assume a resistance of 12.5 kfl is connected in

series with the 5 nF capacitor in the circuit in

Fig P14.10 Calculate wc, H(jto c ), H(jQ.2a> c ),

and H(j5(i) c )

Figure P14.ll

Re

•

-vw-c

If

A'

14.12 Design a passive RC high pass filter (see Fig 14.10[a])

with a cutoff frequency of 500 Hz using a 220 pF

capacitor

a) What is the cutoff frequency in rad/s?

b) What is the value of the resistor?

c) Draw your circuit, labeling the component

val-ues and output voltage

d) What is the transfer function of the filter in

part (c)?

e) If the filter in part (c) is loaded with a resistor

whose value is the same as the resistor in (b),

what is the transfer function of this loaded filter?

f) What is the cutoff frequency of the loaded filter

from part (e)?

g) What is the gain in the pass band of the loaded

filter from part (e)?

14.13 Using a 100 nF capacitor, design a high-pass passive

filter with a cutoff frequency of 300 Hz

a) Specify the value of R in kilohms

b) A 47 kfl resistor is connected across the output terminals of the filter What is the cutoff fre-quency, in hertz, of the loaded filter?

DFSIGN

PROBLEM PSPICE MULTISIM

DESIGN

PROBLEM PSPICE

M U L T I S I M

14.14 Using a 100 juH inductor, design a high-pass, RL,

passive filter with a cutoff frequency of 1500 krad/s a) Specify the value of the resistance, selecting from the components in Appendix H

b) Assume the filter is connected to a pure resistive load The cutoff frequency is not to drop below

1200 krad/s What is the smallest load resistor from Appendix H that can be connected across the output terminals of the filter?

14.15 Consider the circuit shown in Fig P14.15

a) With the input and output voltages shown in the figure, this circuit behaves like what type of filter?

b) What is the transfer function, H(s) = 1/,,(^)/^:(^),

of this filter?

c) What is the cutoff frequency of this filter?

d) What is the magnitude of the filter's transfer

function at s — jcojl

Figure P14.15

150 a

A W

14.16 Suppose a 150 ft load resistor is attached to the

fil-ter in Fig P14.15

a) What is the transfer function, H(s) = V ( ,(s)/Vi(s),

of this filter?

b) What is the cutoff frequency of this filter?

c) How does the cutoff frequency of the loaded fil-ter compare with the cutoff frequency of the unloaded filter in Fig P14.15?

d) What else is different for these two filters? Section 14.4

14.17 Show that the alternative forms for the cutoff

fre-quencies of a bandpass filter, given in Eqs 14.36 and 14.37, can be derived from Eqs 14.34 and 14.35

14.18 Calculate the center frequency, the bandwidth, and

the quality factor of a bandpass filter that has an upper cutoff frequency of 121 krad/s and a lower cutoff frequency of 100 krad/s

554 Introduction to Frequency Selective Circuits

14.19 A bandpass filter has a center, or resonant, frequency

of 50 krad/s and a quality factor of 4 Find the

band-width, the upper cutoff frequency, and the lower

cut-off frequency Express all answers in kilohertz

14.20 Use a 5 nF capacitor to design a series RLC

band-PMBLEM Pa s s f i l t e r'a s s n o w n a t t n e t 0P o f Fig-14.27 The c e n"

PSPFCE ter frequency of the filter is 8 kHz, and the quality

MumsiM factor is 2

a) Specify the values of R and L

b) What is the lower cutoff frequency in kilohertz?

c) What is the upper cutoff frequency in kilohertz?

d) What is the bandwidth of the filter in kilohertz?

14.21 Design a series RLC bandpass filter using only

three components from Appendix Ff that comes

closest to meeting the filter specifications in

Problem 14.20

a) Draw your filter, labeling all component values

and the input and output voltages

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

center frequency and quality factor when

com-pared to the values specified in Problem 14.20

14.22 For the bandpass filter shown in Fig P14.22, find

"SPICE (a) co a , (b) f m (c) Q, (d) » At (e) f ch (f) coc2, (g) /c 2,

Figure P14.22

+

•—

8kfi

T~

llOmH ^

-A i

~10nF

i

+

•

DESIGN

PROBLEM

PSPICE

M U L T I S I M

14.23 Using a 50 nF capacitor in the bandpass circuit

shown in Fig 14.22, design a filter with a quality

fac-tor of 5 and a center frequency of 20 krad/s

a) Specify the numerical values of R and L

b) Calculate the upper and lower cutoff

frequen-cies in kilohertz

c) Calculate the bandwidth in hertz

14.24 Design a series RLC bandpass filter using only

three components from Appendix H that comes

closest to meeting the filter specifications in

Problem 14.23

a) Draw your filter, labeling all component values

and the input and output voltages

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

center frequency and quality factor when

com-pared to the values specified in Problem 14.23

14.25 For the bandpass filter shown in Fig P14.25, calculate

-SPICE the following: (a) f 0; (b) Q; (c) /c l; (d) f c2; and (e) 0 MULTISIM

Figure P14.25

20 H 40 mH 40 "F

- A W ^nrv> 1 ^

180 n v»

14.26 The input voltage in the circuit in Fig PI 4.25 is

500 cos cot mV Calculate the output voltage when (a) co = co 0 ; (b) co = co c i; and (c) co = co c2

14.27 Design a series RLC bandpass filter (see Fig 14.19[aJ)

with a quality of 8 and a center frequency of

50 krad/s, using a 0.01 /xF capacitor

a) Draw your circuit, labeling the component val-ues and output voltage

b) For the filter in part (a), calculate the bandwidth and the values of the two cutoff frequencies

14.28 The input to the series RLC bandpass filter designed

in Problem 14.27 is 50costttf mV Find the voltage

drop across the resistor when (a) co = eo () ; (b) eo= (o cl ; (c) co = o) c2 \ (d) co = 0.2wo; (e) co = 5co ()

14,29 The input to the series RLC bandpass filter designed

in Problem 14.27 is 50coswt mV Find the voltage drop across the series combination of the inductor

and capacitor when (a) eo = co a ; (b) to = o><;1;

(c) co = co c2 \ (d) co = 0.2oo o ; (e) oo = 5co ()

14.30 A block diagram of a system consisting of a

sinu-soidal voltage source, an RLC series bandpass

fil-ter, and a load is shown in Fig P14.30 The internal impedance of the sinusoidal source is

80 + ;0 fl, and the impedance of the load is

480 + /0 11

The RLC series bandpass filter has a 20 nF

capacitor, a center frequency of 50 krad/s, and a quality factor of 6.25

a) Draw a circuit diagram of the system

b) Specify the numerical values of L and R for the

filter section of the system

c) What is the quality factor of the interconnected system?

d) What is the bandwidth (in hertz) of the inter-connected system?

Figure P14.30

Sourc

* > Filt er

Load

Figure P14.32 lOOkft

-AAV

400 kH

14.31 The purpose of this problem is to investigate how a

resistive load connected across the output

termi-nals of the bandpass filter shown in Fig 14.19

affects the quality factor and hence the bandwidth

of the filtering system The loaded filter circuit is

shown in Fig PI 4.31

a) Calculate the transfer function VJV-, for the

cir-cuit shown in Fig P14.31

b) What is the expression for the bandwidth of

the system?

c) What is the expression for the loaded

width (/3{J as a function of the unloaded

band-width (ft/)?

d) What is the expression for the quality factor of

the system?

e) What is the expression for the loaded quality

factor (Qi) as a function of the unloaded quality

factor ( 0 j ) ?

f) What are the expressions for the cutoff

frequen-cies a) cl and ft>c2?

Figure PI4.31

14.33 The parameters in the circuit in Fig P14.31 are

R = 2.4 kO, C = 50 pF, and L = 2 fiH The quality

factor of the circuit is not to drop below 7.5 What is the smallest permissible value of the load resistor /?L?

Section 14.5 14.34 a) Show (via a qualitative analysis) that the circuit

in Fig P14.34 is a bandreject filter

b) Support the qualitative analysis of (a) by finding the voltage transfer function of the filter

c) Derive the expression for the center frequency

of the filter

d) Derive the expressions for the cutoff frequen-cies avi a nd o> C

2-e) What is the expression for the bandwidth of the filter?

f) What is the expression for the quality factor of the circuit?

Figure P14.34

C

It- R

14.32 Consider the circuit shown in Fig P14.32

a) Find co (>

PSPICE

MULTISIM

b) Find (3

c) FmdQ

d) Find the steady-state expression for v„ when

Vi = 250 cos a) 0 t mV

e) Show that if R L is expressed in kilohms the Q of

the circuit in Fig PI4.32 is

Q = 20

1 + 100//?L

f) Plot Q versus R L for 20 kfl < / ?L< 2 M H

14.35 For the bandreject filter in Fig PI4.35, calculate

PSPICE (a) (o a ; (b) f a ; (c) Q; (d) m cl; (e) / c l; (f) a> c2; (g) / , 2 ;

HULTC,M and (h) j8 in kilohertz

Figure P14.35

50 fiH

20 nF

750 a