U niversity of S outhern C alifornia
School Of Engineering
Department Of Electrical Engineering
EE 348:
Homework Assignment #02
(Due 02/01/2001)
Spring, 2001
Choma
Problem #05:
The amplifier module in Fig. (P5) has a finite voltage gain, A, a finite input
resistance, Rin , and a non- zero output resistance, Rout. A voltage source whose Thévenin resistance is Rs excites the amplifier, and the load is the indicated capacitance, C.
Rs
Rin
+
+
A
−
Vi
+
R out
−
Vo
R2
Vs
−
R1
C
Fig. (P5)
(a). Derive a generalized expression for the low frequency voltage gain, Av(0) ∆ Avo = Vo /Vs.
In the process of arriving at this relationship, you may wish to introduce the algebraic substitution,
R
+ R2
A + out
Rin
K ∆
,
Rout + R2
1 +
R1
(b).
(c).
(d).
(e).
which physically represents the ratio of the voltage developed across resistance R1 –to– the
voltage, Vi.
Derive a generalized expression for the low frequency voltage transfer function, Vi /Vs.
Using the expression deduced in Part (b), can you deduce a generalized relationship for the
input resistance, say Ri, seen by the signal source?
Derive an expression for the Thévenin resistance, say Rthc, seen by the capacitor, C.
Assume here that the amplifier input resistance, Rin , is sufficiently large to enable its
approximation by an infinitely large resistance.
Give an expression, in terms of Rthc and C, for the 3-dB bandwidth, B, of the entire circuit.
EE 348
University of Southern California
J. Choma, Jr.
(f).
If the amplifier module is an idealized operational amplifier, Rin and A are both infinitely
large, and Rout is zero. To what do the results of Parts (a), (b), (c), (d), and (e) reduce
under this idealized operating constraint? What can specifically be asserted with respect to
the input voltage, Vi, and the current supplied by the signal source?
(g). Can you make use of the observations in Part (f) to find the voltage gain, Avo, and the input
resistance, Rin , virtually by inspection? What happens to the bandwidth, B? Can you use
this bandwidth observation to state a general strategy for designing wide bandwidth circuits that must drive capacitive loads?
(h). Take Rs = 300 Ω, Rin = 4 MegΩ, Rout = 100 Ω, R1 = 1 KΩ, R2 = 9 KΩ, C = 20 pF, and A
= 100. Numerically calculate the results of Parts (a) through (e). Verify the accuracy of
these calculations by simulating the circuit on SPICE. Submit SPICE results in the form
of a gain magnitude frequency response and a frequency response of the input impedance
seen by the signal source.
Problem #06:
The amplifier module in Fig. (P6) has a finite voltage gain, A, an infinitely
large input resistance, Rin , and zero output resistance, Rout. A voltage source whose Thévenin
resistance is Rs excites the amplifier, and the load is the indicated resistance, Rl.
Cf
Rf
Rs
+
Vs
−
Rout
+
−
A
+
Vi
−
R in
Vo
R1
Fig. (P6)
(a). Derive a generalized expression for the low frequency voltage gain, Av(0) ∆ Avo = Vo /Vs.
(b). Derive a generalized expression for the low frequency voltage transfer function, Vi /Vs.
(c). Using the expression deduced in Part (b), deduce a generalized relationship for the input
resistance, Ri, seen by the signal source.
(d). Derive an expression for the Thévenin resistance, say Rthc, seen by the capacitor, Cf.
(e). Give an expression, in terms of Rthc and C, for the 3-dB bandwidth, B, of the entire circuit.
(f). If the amplifier module is an idealized operational amplifier, to what do the results of Parts
(a), (b), (c), (d), and (e) reduce? What can be asserted specifically with respect to the input
voltage, Vi, and the current supplied by the signal source?
Homework #02
15
Spring Semester, 2001
EE 348
University of Southern California
J. Choma, Jr.
(g). Can you make use of the observations in Part (f) to find the voltage gain, Avo, and the input
resistance, Rin , virtually by inspection? What happens to the bandwidth, B? Can you use
this bandwidth observation to state a general strategy for designing wide bandwidth circuits having the architecture in the subject schematic diagram?
(h). Take Rs = 300 Ω, Rin = 4 MegΩ, Rout = 10 Ω, Rl = 1 KΩ, Rf = 12 KΩ, Cf = 10 pF, and A =
98 dB. Numerically calculate the results of Parts (a) through (e). Verify the accuracy of
these calculations (in light of the idealized input and output resistance approximations) by
simulating the circuit on SPICE. Submit SPICE results in the form of a gain magnitude
frequency response and a frequency response of the input impedance seen by the signal
source.
Problem #07:
The circuit in Fig. (P7) is an example of a lowpass filter realized with an
operational amplifier (op-amp) and several passive elements.
C
R
R
+
Op-Amp
−
+
Vo
R2
Vs
−
C
R1
Fig. (P7)
(a). Assuming that the op-amp is ideal in the senses of infinitely large gain, infinitely large
input resistance, and zero output resistance, show that the voltage transfer function, Av(s) =
Vo (s)/V i(s), is of the form,
Av( 0 )
Av( s ) =
;
2
 s   s 
1+
+

 Qω   ω 

o   o 
specifically, derive expressions for the zero frequency gain, Av(0), the circuit quality factor, Q, and the natural, or undamped resonant, frequency, ? o .
(b). Must a limit be imposed on the resistor ratio, R2 /R1 ? What is the basis of this limiting
value? What is the maximum resistance ratio value and what is he corresponding maximum possible low frequency gain?
(c). With R = R1 = 500 Ω, design the circuit for a 10 MHz bandwidth and a maximally flat frequency response.
Homework #02
16
Spring Semester, 2001
EE 348
University of Southern California
J. Choma, Jr.
(d). Verify the accuracy of your design by simulating the circuit on SPICE. For this simulation, consider the op-amp to have a gain of 98 dB, an input resistance of 10 MegΩ, and an
output resistance of 10 Ω. Submit your simulation results as a frequency response that
clearly shows the value of the 3-dB bandwidth.
Problem #08:
Bandpass filters, such as the unit drawn in Fig. (P8), are used to provide
amplification in a narrow range, or band, of frequencies, while providing attenuation outside of
this band. In other words, the bandpass filter provides frequency selective amplification. As
such, these filters are useful in radio receiver and other communication system applications.
C1
R1
R2
C2
−
Op-Amp
+
+
R3
Vs
Vo
−
Fig. (P8)
(a). Assuming an ideal operational amplifier, show that the transfer function of the filter in Fig.
(P8) is expressible as,
Av ( jω ) =
Vo ( jω )
Vs ( jω )
=
Av ( jω o )
;
ω
ωo 
1 + jQ 
−

ω

ω
 o

specifically, give expressions for the center frequency, ? o , the center frequency gain,
Av(j? o ), and the circuit quality factor, Q.
(b). Using EXCEL or suitable other software, plot the normalized gain magnitude,
|Av(j?)/A v(j? o )| -versus- the normalized signal frequency, ?/? o , for Q = 1, 3, 5, and 10. If
you were to use this filter in a radio receiver application, would you want large or small Q?
Explain your answer.
(c). Study the plots deduced in Part (b). Note in particular that a gain that is 3-dB below the
center frequency gain is achieved at two signal frequencies. One such frequency, say ? low,
is smaller than the center frequency, ? o , while the other frequency, say ? high, is larger than
? o . Derive expressions for ? low and ? high.
(d). The 3-dB bandwidth of a bandpass filter is defined as the difference between the aforementioned frequencies, ? low and ? high. Show that B ∆ ? high – ? low = ? o /Q.
Homework #02
17
Spring Semester, 2001
EE 348
University of Southern California
J. Choma, Jr.
U niversity of S outhern C alifornia
School Of Engineering
Department Of Electrical Engineering
EE 348:
Homework Assignment #02
(SOLUTIONS: Due 02/01/2001)
Spring, 2001
Choma
Problem #05:
Homework #02
18
Spring Semester, 2001