Fundamentals-of-Microelectronics-Behzad-Razavi.pdf

We begin with an amplifier having a high but poorly

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We begin with an amplifier having a high, but poorly-controlled gain and apply negative feed- back around it so as to obtain a better-defined, but inevitably lower gain. This concept was also extensively employed in the op amp circuits described in Chapter 8. The gain desensitization property of negative feedback means that any factor that influences the open-loop gain has less effect on the closed-loop gain. Thus far, we have blamed only process and temperature variations, but many other phenomena change the gain as well. As the signal frequency rises, may fall, but remains relatively constant. We therefore expect that negative feedback increases the bandwidth (at the cost of gain). If the load resistance changes, may change; e.g., the gain of a CS stage depends on the load resistance. Negative feedback, on the other hand, makes the gain less sensitive to load variations. The signal amplitude affects because the forward amplifier suffers from nonlinearity. For example, the large-signal analysis of differential pairs in Chapter 10 reveals that the small- signal gain falls at large input amplitudes. With negative feedback, however, the variation of the open-loop gain due to nonlinearity manifests itself to a lesser extent in the closed- loop characteristics. That is, negative feedback improves the linearity. We now study these properties in greater detail. 12.2.2 Bandwidth Extension Let us consider a one-pole open-loop amplifier with a transfer function (12.16) Here, denotes the low-frequency gain and the -dB bandwidth. Noting from (12.2) that negative feedback lowers the low-frequency gain by a factor of , we wish to determine the resulting bandwidth improvement. The closed-loop transfer function is obtained by substitut- ing (12.16) for in (12.2): (12.17)
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 609 (1) Sec. 12.2 Properties of Negative Feedback 609 Multiplying the numerator and the denominator by gives (12.18) (12.19) In analogy with (12.16), we conclude that the closed-loop system now exhibits: (12.20) (12.21) In other words, the gain and bandwidth are scaled by the same factor but in opposite directions, displaying a constant product. Example 12.5 Plot the closed-loop frequency response given by (12.19) for , 0.1, and 0.5. Assume . Solution For , the feedback vanishes and reduces to as given by (12.16). For , we have , noting that the gain decreases and the bandwidth increases by the same factor. Similarly, for , , yielding a proportional reduction in gain and increase in bandwidth. The results are plotted in Fig. 12.8. ω A 0 K = 0.1 K = 0 K = 0.5 Figure 12.8 Exercise Repeat the above example for . Example 12.6 Prove that the unity-gain bandwidth of the above system remains independent of if and .
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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 610 (1) 610 Chap. 12 Feedback Solution The magnitude of (12.19) is equal to (12.22) Equating this result to unity and squaring both sides, we write (12.23) where denotes the unity-gain bandwidth. It follows that (12.24) (12.25) (12.26) which is equal to the gain-bandwidth product of the open-loop system. Figure 12.9 depicts the results.
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