University of Southern C alifornia School Of Engineering Department Of Electrical Engineering

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1 University of Southern C alifornia School Of Engineering Department Of Electrical Engineering EE 348: Homework Assignment #05&6 Spring, 2004 (Due 03/09/2004) Choma Problem #18: In the common source feedback amplifier depicted in Fig.(P35), the current sources, I Qi and I Qo 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, r o, of the transistor can also be viewed as infinitely large. However, bulk-induced modulation of the forward transconductance cannot be ignored. V dd R f I Qo R in I Qi s R l Fig. (P35) Use the low frequency small signal model of a MOSFET to show that the indicated driving point input resistance, R in, is given by R R f l R =, in 1 g R me l where g me represents the effective forward transconductance, g m g. me 1 ( 1 λ b) g R m ss Use the low frequency small signal model of a MOSFET to show that the indicated driving point output resistance,, is given by R R f s R =. out 1 g R me s (c). A match-terminated amplifier, which is commonly exploited in broadband communication

2 system applications, is designed to ensure that = R l = R in = R. In terms of resistance R and effective forward transconductance g me, how must the feedback resistance, R f, be selected to realize match-terminated performance? (d). Show that the small signal voltage gain, A v, of the match-terminated amplifier is V 1 g R o me A = =. v V 2 Problem #19: s 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, V ref. V dd I Q M3 M4 V ref Problem #20: Fig. (P36) In the current reference circuit of Fig. (P37), transistors,, and M3 are identical, except for the fact that the gate aspect ration of is k-times larger than that of, and the gate aspect ratio of M3 is k-times larger than that of. Transistors M4 and M5 are identical, inclusive of identical gate aspect ratios. All transistors operate in saturation, with the substrates of,, 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, I out. Homework #09 46 Spring Semester, 2002

3 V dd M4 M5 I out R M3 Fig. (P37) Problem #21: Revisit the circuits addressed in the preceding two problems. Derive an expression for the small signal output resistance,, in the circuit of Fig. (P37). Derive an expression for the small signal output resistance,, in the circuit of Fig. (P36). Problem #22: 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. V dd ML MD KV th s R th s s Fig. (P39) Reduce the small signal equivalent model of the subject amplifier to the Thévenin equiva- Homework #09 47 Spring Semester, 2002

4 lent form abstracted in Fig. (P39b). Give exact and approximate expressions for the Thévenin parameters, K th and R th. What is the time constant associated with the pole established by the load capacitance,? (c). What is the overall voltage transfer function, A v (s) = s (s)/ (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 #23: 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. 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, G n and R n. What is the time constant associated with the pole established by the load capacitance,? (c). What is the overall voltage transfer function, A v (s) = s (s)/ (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. V dd MP V bb MN s GV n s R n s Fig. (P40) Problem #24: The NMOS transistors in the buffer of Fig. (P41a) are identical except for the fact that the gate aspect ratio of transistor is k times smaller than the gate aspect ratio of Homework #09 48 Spring Semester, 2002

5 the driver device, transistor. The drain-source channel resistances of both transistors are large, but they are not infinitely large. All transistors are saturated. V dd R in V bb KV th s R th s s Fig. (P41) 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, K th and R th. What is the time constant associated with the pole established by the load capacitance,? (c). What is the overall voltage transfer function, A v (s) = s (s)/ (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 #25: The current source, I dd, 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, I s is a signal current source. The channel resistance of the transistor, which operates in saturated mode, is large, but it is not infinitely large. 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, G n and R n. What is the time constant associated with the pole established by the load capacitance,? (c). Give exact and approximate expressions for the low frequency input resistance, R in. (d). What is the overall transimpedance function, Z v (s) = s (s)/i s? 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

6 V dd I dd MN R in I s GV n s R n s Fig. (P39) Problem #26: In the current amplifier of Fig. (P43), transistors,, and M3 are identical, save for the fact that the gate aspect ratio of transistor is k times smaller than that of and M3. The signal is applied as the current, I s. In terms of k, determine the small signal, low frequency current gain, A io = I L /I s. Give an expression for the 3 db bandwidth, B, of the circuit. Assume that the capacitances intrinsic to all transistors are negligible. V DD R C M3 I L R L I s Fig. (P43) Homework #09 50 Spring Semester, 2002

7 Homework #09 51 Spring Semester, 2002