# Example 1237 a three pole feedback system exhibits

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Example 12.37 A three-pole feedback system exhibits the frequency response depicted in Fig. 12.65. Does this system oscillate? Solution Yes, it does. The loop gain at is greater than unity, but we note from the analysis in Fig. 12.64 that indefinite signal growth still occurs, in fact more rapidly. After each trip around the loop, a sinusoid at experiences a gain of and returns with opposite phase to the subtractor. Exercise Suppose is halved in value. Does the system still oscillate?

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 657 (1) Sec. 12.8 Stability in Feedback Systems 657 ω 0 ω 0 ω 90 180 270 (log scale) (log scale) H 20log (c) ω ω ω p1 p2 p3 1 KH Figure 12.65 12.8.3 Stability Condition Our foregoing investigation indicates that if and , then the negative feedback system oscillates. Thus, to avoid instability, we must ensure that these two conditions do not occur at the same frequency. Figure 12.66 depicts two scenarios wherein the two conditions do not coincide. Are both of ω 0 ω 0 180 20log ω 1 ω 0 ω 0 180 ω 2 ω 1 (a) (b) KH 20log KH KH KH Figure 12.66 Systems with phase reaching (a) before and (b) after the loop gain reaches unity. these systems stable? In Fig. 12.66(a), the loop gain at exceeds unity (0 dB), still leading to oscillation. In Fig. 12.66(b), on the other hand, the system cannot oscillate at (due to insufficient phase shift) or (because of inadequate loop gain). The frequencies at which the loop gain falls to unity or the phase shift reaches play such a critical role as to deserve specific names. The former is called the “gain crossover fre- quency” ( ) and the latter, the “phase crossover frequency” ( ). In Fig. 12.66(b), for example, and . The key point emerging from the two above scenarios is that stability requires that (12.183)
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 658 (1) 658 Chap. 12 Feedback In summary, to guarantee stability in negative-feedback systems, we must ensure that the loop gain falls to unity before the phase shift reaches so that Barkhausen’s criteria do not hold at the same frequency. Example 12.38 We wish to apply negative feedback with around the three-stage amplifier shown in Fig. 12.67(a). Neglecting other capacitances and assuming identical stages, plot the frequency M 1 R D C 1 M R D C 1 M R D C 1 V DD out V in V 2 3 ω 0 ω 0 270 (log scale) (log scale) H H 20log R D C 1 1 60 dB/dec 135 ω 0 ω 0 (log scale) (log scale) H H 20log R D C 1 1 180 (a) (b) (c) ω PX H P P g m R D ( ) 3 Figure 12.67 response of the circuit and determine the condition for stability. Assume . Solution The circuit exhibits a low-frequency gain of and three coincident poles given by . Thus, as depicted in Fig. 12.67(b), begins to fall at a rate of 60 dB/dec at . The phase begins to change at one-tenth of this frequency, reaches at , and approaches at .

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