The reader may wonder if an ambiguity exists with

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The reader may wonder if an ambiguity exists with respect to the direction of the signal flow in the loop gain test. For example, can we modify the topology of Fig. 12.4(b) as shown in Fig. 12.6? This would mean applying to the output of and expecting to observe a signal at its A 1 K X V test V N Figure 12.6 Incorrect method of applying test signal. input and eventually at . While possibly yielding a finite value, such a test does not represent the actual behavior of the circuit. In the feedback system, the signal flows from the input of to its output and from the input of the feedback network to its output. 12.2 Properties of Negative Feedback 12.2.1 Gain Desensitization Suppose in Fig. 12.1 is an amplifier whose gain is poorly controlled. For example, a CS stage provides a voltage gain of while both and vary with process and temperature; the

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BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 607 (1) Sec. 12.2 Properties of Negative Feedback 607 gain thus may vary by as much as . Also, suppose we require a voltage gain of 4.00. How can we achieve such precision? Equation (12.2) points to a potential solution: if , we have (12.10) a quantity independent of . From another perspective, Eq. (12.4) indicates that leads to a small error, forcing to be nearly equal to and hence nearly equal to . Thus, if can be defined precisely, then impacts negligibly and a high precision in the gain is attained. The circuit of Fig. 12.2 exemplifies this concept very well. If , then (12.11) (12.12) Why is more precisely defined than is? If and are made of the same ma- terial and constructed identically, then the variation of their value with process and temperature does not affect their ratio. As an example, for a closed-loop gain of 4.00, we choose and implement as the series combination of three “unit” resistors equal to . Illustrated in Fig. 12.7, the idea is to ensure that and “track” each other; if increases by , so does each unit in and hence the total value of , still yielding a gain of . R 2 R 1 Figure 12.7 Construction of resistors for good matching. Example 12.4 The circuit of Fig. 12.2 is designed for a nominal gain of 4. (a) Determine the actual gain if . (b) Determine the percentage change in the gain if drops to 500. Solution For a nominal gain of 4, Eq. (12.12) implies that . (a) The actual gain is given by (12.13) (12.14) Note that the loop gain . (b)If falls to 500, then (12.15) Some analog-to-digital converters (ADCs) require very precise voltage gains. For example, a 10-bit ADC may call for a gain of 2.000.
BR Wiley/Razavi/ Fundamentals of Microelectronics [Razavi.cls v. 2006] June 30, 2007 at 13:42 608 (1) 608 Chap. 12 Feedback Thus, the closed-loop gain changes by only if drops by factor of 2. Exercise Determine the percentage change in the gain if falls to 200. The above example reveals that the closed-loop gain of a feedback circuit becomes relatively independent of the open-loop gain so long as the loop gain, , remains sufficiently higher than unity. This property of negative feedback is called “gain desensitization.” We now see why we are willing to accept a reduction in the gain by a factor of .

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