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I i ( jv) + Vi ( jv) C R
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Z o = R1 + f1 ( ) = f2 ( ) =
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2 R1 LC R1 R2 [R1 R2 C + L] 2 LC 1 CR2
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Figure P610
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c Determine the four cutoff frequencies at which f1 ( ) = +1 or 1 and f2 ( ) = +1 or 1 d Determine the resonant frequency of the circuit
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e Plot the magnitude of the impedance (in dB) as a function of the log of the frequency, ie, a Bode plot
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Determine: a How the voltage transfer function: HV (j ) = V0 (j ) Vi (j )
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behaves at extremes of high and low frequencies b An expression for the voltage transfer function and show that it can be manipulated into the form: Hv (j ) = where Ho = R2 R1 + R 2 f ( ) = R1 R2 C R1 + R 2 Ho 1 + jf ( )
Figure P612
613 Determine an expression for the circuit of Figure
P613(a) for the equivalent impedance in standard form Choose the Bode plot from Figure P613(b) that best describes the behavior of the impedance as a function of frequency and describe how (a simple one-line statement with no analysis is suf cient) you would obtain the resonant and cutoff frequencies and the magnitude of the impedance where it is constant over some frequency range Label the Bode plot to indicate which feature you are discussing
c The cutoff frequency at which f ( ) = 1 and the value of Ho in dB d The value of the voltage transfer function at the cutoff frequency and at = 25 rad/s, 250 rad/s, 25 krads, and 250 krad/s e How the magnitude (in dB) and the angle of the transfer function behave at low frequencies, the cutoff frequency, and high frequencies
I i ( jv) R1 RC C L Z eq (a) + Vi ( jv) C R2
I o( jv) + V ( jv) o
Figure P614
|Zeq|db
|Zeq|db
615 The circuit shown in Figure P615 is not a lter but
illustrates the undesirable effects of capacitances (and sometimes inductances) in a circuit The circuit is a simple model of an ampli er state Capacitors are often necessary for the proper operation of such circuits, or they may be unwanted but inherent in one of the circuit components At high or low frequencies these capacitors adversely affect the proper operation of the circuit The input impedance is used to demonstrate Determine: a An expression, in the form: Zi (j ) = Vi (j ) = Zo Ii (j ) 1 + jf1 ( ) 1 + jf2 ( )
|Zeq|db |Zeq|db
(b)
Figure P613
for the input impedance Note the output current = 0 b The cutoff frequencies at which f1 ( ) = 1 and f2 ( ) = 1 if: R1 = 13 k C = 05 F R2 = 56 k gm = 35 mS
614 In the circuit of Figure P614:
R2 = 19 k R1 = 13 k C = 05182 F
c The limiting value of Zi as increases toward in nity As decreases toward zero
6
Frequency Response and System Concepts
d The Bode plot for the input impedance
Ii(v) + Vi ( jv) R1 Vbe gmVbe
frequency where: Vi = 707 V R1 = 22 k 8 R2 = 38 k Xc = 5 k XL = 125 k c The output voltage if the frequency of the input voltage doubles so that: XC = 25 k XL = 25 k
+ C R2 V ( jv) o
d The output voltage if the frequency of the input voltage again doubles so that: XC = 125 k XL = 5 k
Figure P615
e The possible type of lter this might be, considering how the output voltage changes with frequency
616 The circuit shown in Figure P616 is a very
simpli ed model of a transistor ampli er stage The capacitance C is an internal effect of the transistor It causes the transfer function: Vo (j ) Hv (j ) = Vi (j ) where R = 100 k C = 0125 nF R = 750 gm = 75 mS
Figure P617
+ R2 C R2 L + V ( jv) o
Vi ( jv)
to decrease at high frequencies as shown in the Bode plot Determine: a The two cutoff frequencies b The magnitude of the transfer function at very low and very high frequencies
618 Are the lters shown in Figure P618 low-pass,
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