# Thus q 1 in v r e a r out g m π v π v π r p r o x

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Thus, Q 1 in V R E (a) R out g m π v π v π r P r O X v X i R E X i (b) Figure 5.46 (a) Output impedance of degenerated stage, (b) equivalent circuit. (5.194) where the negative sign arises because the positive side of is at ground. We also recognize that carries a current of and hence sustains a voltage of . Adding this voltage to that across ( ) and equating the result to , we obtain (5.195) (5.196) It follows that (5.197) (5.198) Recall from (5.146) that the intrinsic gain of the transistor, , and hence (5.199) (5.200) Interestingly, emitter degeneration raises the output impedance from to the above value, i.e., by a factor of . The reader may wonder if the increase in the output resistance is desirable or undesirable. The “boosting” of output resistance as a result of degeneration proves extremely useful in circuit design, conferring amplifiers with a higher gain as well as creating more ideal current sources. These concepts are studied in Chapter 9. It is instructive to examine (5.200) for two special cases and . For , we have and (5.201) (5.202)

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 218 (1) 218 Chap. 5 Bipolar Amplifiers because . Thus, the maximum resistance seen at the collector of a bipolar transistor is equal to —if the degeneration impedance becomes much larger than . For , we have and (5.203) Thus, the output resistance is boosted by a factor of . In the analysis of circuits, we sometimes draw the transistor output resistance explicitly to emphasize its significance (Fig. 5.47). This representation, of course, assumes itself does not contain another . Q 1 in V R E R out r O Figure 5.47 Stage with explicit depiction of . Example 5.28 We wish to design a current source having a value of 1 mA and an output resistance of 20 k . The available bipolar transistor exhibits and V. Determine the minimum required value of emitter degeneration resistance. Solution Since , degeneration must raise the output resistance by a factor of two. We postulate that the condition holds and write (5.204) That is, (5.205) (5.206) Note that indeed . Exercise What is the output impedance if is doubled? Example 5.29 Calculate the output resistance of the circuit shown in Fig. 5.48(a) if is very large. Solution Replacing and with an ac ground and with an open circuit, we arrive at the simplified
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 219 (1) Sec. 5.3 Bipolar Amplifier Topologies 219 Q 1 I 1 R 1 R 2 C 1 Q 1 R 1 R 2 (a) (b) R out R out Q 1 R 2 R (c) out1 V b Figure 5.48 (a) CE stage example, (b) simplified circuit, (c) resistance seen at the collector. model in Fig. 5.48(b). Since appears in parallel with the resistance seen looking into the collector of , we ignore for the moment, reducing the circuit to that in Fig. 5.48(c). In analogy with Fig. 5.40, we rewrite Eq. (5.200) as (5.207) Returning to Fig. 5.48(b), we have (5.208) (5.209) Exercise What is the output resistance if a very large capacitor is tied between the emitter of and ground?

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