Download Homework 9 - Engineering Class s - University of Southern California

U niversity of S outhern C alifornia
School Of Engineering
Department Of Electrical Engineering
EE 348:
Homework Assignment #09
(Due 04/25/2002)
Spring, 2002
Choma
Problem #35:
In the common source feedback amplifier depicted in Fig. (P35), the current
sources, IQi and IQo are ideal biasing currents in the sense that their terminal resistances can be
taken as infinitely large. In the interest of analytical simplification, the channel resistance, ro, of
the transistor can also be viewed as infinitely large. However, bulk-induced modulation of the
forward transconductance cannot be ignored.
Vdd
Rf
IQo
Rs
Vo
Rout

Vs
IQi
Rin

Rss
Rl
Fig. (P35)
(a). Use the low frequency small signal model of a MOSFET to show that the indicated driving
point input resistance, Rin, is given by
R R
f
l
Rin 
,
1  g me Rl
where gme represents the effective forward transconductance,
g
m
gme @
.
1 1 g R

b

m ss
(b). Use the low frequency small signal model of a MOSFET to show that the indicated driving
point output resistance, Rout, is given by
R R
f
s
Rout 
.
1  g me Rs
(c). A match-terminated amplifier, which is commonly exploited in broadband communication
EE 348
University of Southern California
J. Choma, Jr.
system applications, is designed to ensure that Rs = Rl = Rin = Rout  R. In terms of resistance R and effective forward transconductance gme, how must the feedback resistance, Rf,
be selected to realize match-terminated performance?
(d). Show that the small signal voltage gain, Av, of the match-terminated amplifier is
V
1 g R
me
A  o 
.
v
Vs
2
Problem #36:
The transistors in the voltage reference circuit of Fig. (P31) operate in their
saturation regimes. All transistors are identical except for the fact that while the gate aspect
ratio of transistor M4 is unity, the gate aspect ratios of the remaining three devices are each 4.
All transistor substrate terminals are grounded. Derive an expression for the indicated static
voltage, Vref.
Vdd
IQ
M3
M4
Vref
M1
M2
Rout
Fig. (P36)
Problem #37:
In the current reference circuit of Fig. (P37), transistors M1, M2, and M3
are identical, except for the fact that the gate aspect ration of M2 is k-times larger than that of
M1, and the gate aspect ratio of M3 is k-times larger than that of M2. Transistors M4 and M5 are
identical, inclusive of identical gate aspect ratios. All transistors operate in saturation, with the
substrates of M1, M2, and M3 returned to ground and the substrates of M4 and M5 returned to
the positive bus voltage. The drain of transistor M5 is connected to a load that is not shown in
the diagram. Derive an expression for the static reference current, Iout.
Homework #09
46
Spring Semester, 2002
EE 348
University of Southern California
J. Choma, Jr.
Vdd
M4
M5
Iout
Rout
R
M2
M3
M1
Fig. (P37)
Problem #38:
Revisit the circuits addressed in the preceding two problems.
(a). Derive an expression for the small signal output resistance, Rout, in the circuit of Fig. (P37).
(b). Derive an expression for the small signal output resistance, Rout, in the circuit of Fig. (P36).
Problem #39:
The NMOS transistors in the amplifier of Fig. (P39a) are identical except
for the fact that the gate aspect ratio of the driver transistor, MD, is k–times larger than the gate
aspect ratio of the load device, transistor ML. The drain-source channel resistances of both transistors are large, but they are not infinitely large. Transistor MD is saturated.
 Vdd
ML
Vo
MD
Rs
CL

Rth
Vs

Vos

CL
KthVs


Vgg

 Vss
(a).
(b).
Fig. (P39)
(a). Reduce the small signal equivalent model of the subject amplifier to the Thévenin equiva-
Homework #09
47
Spring Semester, 2002
EE 348
University of Southern California
J. Choma, Jr.
lent form abstracted in Fig. (P39b). Give “exact” and approximate expressions for the
Thévenin parameters, Kth and Rth.
(b). What is the time constant associated with the pole established by the load capacitance, CL?
Approximate your “exact” result.
(c). What is the overall voltage transfer function, Av(s) = Vos(s)/Vs(s)? Give an approximate
expression for the voltage gain at zero signal frequency.
(d). What are the 3–dB bandwidth and unity gain frequency of the amplifier? Approximate
your “exact” results.
Problem #40:
In the CMOS amplifier of Fig. (P40a), both transistors operate in their saturated domains and do not have identical small signal parameters. Moreover, their channel resistances are not infinitely large.
(a). Reduce the small signal equivalent model of the subject amplifier to the Norton equivalent
form abstracted in Fig. (P40b). Give “exact” and approximate expressions for the Norton
parameters, Gn and Rn.
(b). What is the time constant associated with the pole established by the load capacitance, CL?
Approximate your “exact” result.
(c). What is the overall voltage transfer function, Av(s) = Vos(s)/Vs(s)? Give an approximate
expression for the voltage gain at zero signal frequency.
(d). What are the 3–dB bandwidth and unity gain frequency of the amplifier? Approximate
your “exact” results.
 Vdd
MP
Vbb
Vo
MN
Rs
CL

Vs
Vos

GnVs

Rn
CL
Vgg

 Vss
(a).
(b).
Fig. (P40)
Problem #41:
The NMOS transistors in the buffer of Fig. (P41a) are identical except for
the fact that the gate aspect ratio of transistor M2 is k–times smaller than the gate aspect ratio of
Homework #09
48
Spring Semester, 2002
EE 348
University of Southern California
J. Choma, Jr.
the driver device, transistor M1. The drain-source channel resistances of both transistors are
large, but they are not infinitely large. All transistors are saturated.
 Vdd
Rs
Rin
M1
Rout

Vs

Rth
Vo
M2
CL
Vbb

Rout
Vos

CL
KthVs

Vgg

 Vss
(a).
(b).
Fig. (P41)
(a). Reduce the small signal equivalent model of the subject amplifier to the Thévenin equivalent form abstracted in Fig. (P41b). Give “exact” and approximate expressions for the
Thévenin parameters, Kth and Rth.
(b). What is the time constant associated with the pole established by the load capacitance, CL?
Approximate your “exact” result.
(c). What is the overall voltage transfer function, Av(s) = Vos(s)/Vs(s)? Give an approximate
expression for the voltage gain at zero signal frequency.
(d). What is the 3–dB bandwidth of the amplifier? Approximate your “exact” results. Why is
finding the unity gain frequency of the circuit inappropriate?
Problem #42:
The current source, Idd, in the common gate amplifier of Fig. (P42a) is ideal
in the sense that its small signal terminal resistance is infinitely large. On the other hand, Is is a
signal current source. The channel resistance of the transistor, which operates in saturated
mode, is large, but it is not infinitely large.
(a). Reduce the small signal equivalent model of the subject amplifier to the Norton equivalent
form abstracted in Fig. (P42b). Give “exact” and approximate expressions for the Norton
parameters, Gn and Rn.
(b). What is the time constant associated with the pole established by the load capacitance, CL?
Approximate your “exact” result.
(c). Give “exact” and approximate expressions for the low frequency input resistance, Rin.
(d). What is the overall transimpedance function, Zv(s) = Vos(s)/Is? Give an approximate
relationship for this transimpedance at zero signal frequency.
(e). What are the 3–dB bandwidth and unity transimpedance gain frequency of the amplifier?
Approximate your “exact” results.
Homework #09
49
Spring Semester, 2002
EE 348
University of Southern California
J. Choma, Jr.
 Vdd
Idd
Vo
MN
Rs
CL

Rin
Vs

Rout
Vos
Rs
Is

GnVs
Vgg

(a).
Rn
CL
(b).
Fig. (P39)
Problem #43:
In the current amplifier of Fig. (P43), transistors M1, M2, and M3 are identical, save for the fact that the gate aspect ratio of transistor M1 is k–times smaller than that of
M2 and M3. The signal is applied as the current, Is.
(a). In terms of k, determine the small signal, low frequency current gain, Aio = IL/Is.
(b). Give an expression for the 3–dB bandwidth, B, of the circuit. Assume that the capacitances intrinsic to all transistors are negligible.
 VDD
R
RL
C
M3
IL
M1
Is
M2
Fig. (P43)
Homework #09
50
Spring Semester, 2002
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 #09
(SOLUTIONS: Due 04/25/2002)
Spring, 2002
Choma
Problem #35:
Homework #09
51
Spring Semester, 2002