Fundamentals-of-Microelectronics-Behzad-Razavi.pdf

Solution for the sinusoids depicted in fig 1014b the

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Solution For the sinusoids depicted in Fig. 10.14(b), the circuit operates linearly because the maximum
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 481 (1) Sec. 10.2 Bipolar Differential Pair 481 V in1 V in2 0 I EE I I I EE 2 V in1 V in2 0 2 V CC V V V CC R C I EE V CC R C I EE out2 out1 V in1 V in2 0 R C I EE V V out1 out2 R C I EE + C2 C1 Figure 10.13 Variation of currents and voltages as a function of input. Q Q 1 2 R V I EE V in1 V in2 CC P C R C V out1 V out2 t V CM V V in1 in2 1 mV t V CM V V in1 in2 100 mV t 1 t V CM V V t V V t 1 g m R C 1 mV x out2 out1 out2 out1 (d) (c) (a) (b) (e) 2 V CC R C I EE V CC V CC R C I EE Figure 10.14 differential input is equal to mV. The outputs are therefore sinusoids having a peak amplitude of [Fig. 10.14(d)]. On the other hand, the sinusoids in Fig. 10.14(c) force a maximum input difference of mV, turning or off. For example, as approaches 50 mV above and reaches 50 mV below (at ), absorbs most of the tail current, thus producing (10.79) (10.80)
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 482 (1) 482 Chap. 10 Differential Amplifiers Thereafter, the outputs remain saturated until falls to less than 100 mV. The result is sketched in Fig. 10.14(e). We say the circuit operates as a “limiter” in this case, playing a role similar to the diode limiters studied in Chapter 3. Exercise What happens to the above results if the tail current is halved? 10.2.3 Small-Signal Analysis Our brief investigation of the differential pair in Fig. 10.11 revealed that, for small differential inputs, the tail node maintains a constant voltage (and hence is called a “virtual ground”). We also obtained a voltage gain equal to . We now study the small-signal behavior of the circuit in greater detail. As explained in previous chapters, the definition of “small signals” is somewhat arbitrary, but the requirement is that the input signals not influence the bias currents of and appreciably. In other words, the two transistors must exhibit approximately equal transconductances—the same condition required for node to appear as virtual ground. In prac- tice, an input difference of less than 10 mV is considered “small” for most applications. Assuming perfect symmetry, an ideal tail current source, and , we construct the small-signal model of the circuit as shown in Fig. 10.15(a). Here, and represent small changes in each input and must satisfy for differential operation. Note that the tail current source is replaced with an open circuit. As with the foregoing large-signal analysis, let us write a KVL around the input network and a KCL at node : (10.81) (10.82) With and , (10.82) yields (10.83) and since , (10.81) translates to (10.84) That is, (10.85) (10.86) Thus, the small-signal model confirms the prediction made by (10.32). In Problem 28, we prove that this property holds in the presence of Early effect as well.
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