EE 541
USC Viterbi School of Engineering
J. Choma
U niversity of S outhern C alifornia
USC Viterbi School of Engineering
Ming Hsieh Department of Electrical Engineering
EE 541: Homework Assignment #05
Due: 10/25/2012
Fall, 2012
Choma
Problem #23:
Figure (P23) offers the schematic diagram of the so-called Tow-Thomas biquadratic
filter. Each of the three operational amplifiers utilized in this active filter can be viewed as ideal
in the senses of at least providing infinitely large input impedances and infinitely large open
loop gains.
R4
R3
Vout
C1
R1
Vin

op-amp

C2
R2

op-amp

R
R


op-amp
Figure (P23)
(a). For the indicated input signal, Vin, the output response shown as Vout delivers a bandpass
frequency response. Derive an expression for the transfer function, Av(s) = Vout /Vin.
(b). For the transfer function determined in (a), give expressions for the tuned center frequency, ωo, the circuit quality factor, Q, and the center frequency voltage gain, Av(ωo).
(c). What designable circuit parameters must be rendered large or small to achieve a sharply
defined bandpass frequency response without affecting the tuned center frequency?
Problem #24:
The network shown in Figure (P24) is to be designed so that its voltage transfer
function, Av(s), is given by
V
s 2
A (s)  o 
,
v
2
V
1  s    s  
s

a

b
where the time constants, τ, τa, and τb, are all positive. In the given network, the load resistance,
R, which terminates the output port of the circuit is the characteristic impedance of the filter.
Moreover, observe that the Thévenin resistance of the applied signal source is also R. DeterHomework #05
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Fall Semester, 2012
EE 541
USC Viterbi School of Engineering
J. Choma
mine the required impedances, Za and Zb, and show a final schematic diagram.
Zin(s)
Vi
Vo
Za
Zb
R

Vs
R
R

Figure (P24)
Problem #25:
Figure (P25) depicts the schematic diagram of the active Delyiannis-Friend bandpass
filter, where parameter k is a smaller than unity positive number. The amplifier is ideal in the
senses that its input impedance is infinitely large, its output impedance is zero, and its voltage
gain, A, is frequency invariant. However, gain A is a finite number.
C
R1
k
R2
C

A

Vi
R1
(1k)
Vo
Figure (P25)
(a). Derive an expression for the transfer function, H(s) = Vo/Vi, of the filter.
(b). Give expressions for the center frequency, ωc, the quality factor, Qc, and the I/O gain, Hc,
at the center frequency. Finally, give an expression for the 3-dB bandwidth, Bc.
(c). Design the circuit for a center frequency of 500 MHz, a quality factor of 10, and a center
frequency gain of 20 volts/volt. Assume a 50  signal source resistance. Confirm the propriety of your design by simulating it on HSPICE and plotting the frequency response of
the designed filter. To the latter end, the amplifier in the filter can be modeled by a simple
voltage-controlled voltage source.
Problem #26:
A modified version of the Delyiannis-Friend architecture addressed in the preceding
problem is given in Figure (P26). Parameter a in this diagram is a less than unity positive constant. Derive an expression for the voltage transfer function, H(s) = Vo/Vi, of the subject network and in the process, discuss the impact that the aR3 — (1 − a)R3 divider exerts on the
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Fall Semester, 2012
EE 541
USC Viterbi School of Engineering
J. Choma
original version of the bandpass network.
C
R1
k
C
R2

A

Vi
Vo
(1a)R3
R1
(1k)
aR3
Figure (P26)
Problem #27:
Figure (P27) depicts the schematic diagram of the active Sallen-Key lowpass filter.
The amplifier is ideal in the senses that its input impedance is infinitely large, its output impedance is zero, and its voltage gain, A, is very large and frequency invariant.
C1
R1
R2

A

Vi
Vo
R4
C2
R3
Figure (P27)
(a). Derive an expression for the transfer function, H(s) = Vo/Vi, of the filter.
(b). In terms of the zero frequency I/O envelope delay, Tdo, discuss how the circuit parameters
must be selected to realize a second order Bessel filter. Can a zero frequency gain whose
magnitude exceeds one be realized?
(c). Give an expression for the 3-dB bandwidth, say B, associated with the magnitude response
of the network.
(d). Design the circuit for a zero frequency envelope delay of 5 nSEC and arbitrary zero frequency gain. Assume a 50  signal source resistance. Confirm the propriety of your design by simulating it on HSPICE and plotting the envelope delay response of the designed
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Fall Semester, 2012
EE 541
USC Viterbi School of Engineering
J. Choma
filter. Also, use HSPICE to plot the magnitude response and the output response to a unit
step input excitation. To the latter ends, the amplifier in the filter can be modeled by a
simple voltage-controlled voltage source.
Problem #28:
The three transconductor units in the operational transconductor amplifier-capacitor
(OTA-C) filter of Figure (P28) are ideal in that they possess infinitely large input and output impedances.
 g
m1

Vi
C2
C1
 g
m2

 g
m3

Vo
Figure (P28)
(a). Derive the expression for the voltage transfer function, Vo/Vs, of the filter.
(b). What type of filter (lowpass, highpass, etc.) is realized by the structure in Figure (P28)?
(c). Give an expression for the quality factor, say Q, of the filter.
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Fall Semester, 2012