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Fig. 9-3 Noninverting ampli er Example 9.3. For the noninverting ampli er of Fig. 9-3, assume that the current into the inverting terminal of the op amp is zero, so that vd % 0 and v1 % v2 . Derive an expression for the voltage gain vo =v2 . With zero input current to the basic op amp, the currents through R2 and R1 must be identical; thus, vo v1 v 1 R2 R1 and Av vo vo R % 1 2 v2 v1 R1 9:7

Thus, we have proved:

COMMON-MODE REJECTION RATIO The common-mode gain is de ned (see Fig. 9-1) as Acm vo v2 9:8

Partnership pro ts and losses may be divided in any manner the partners may agree upon. In general, a partner may be expected to share in pro portion to the amount of capital and/or services he or she contributes. In

CHAP. 9]

Theorem 12.1: Dx (Ax + B) = A By letting A = 0 in Theorem 12.1, we obtain: Corollary 12.2: Dx (B) = 0; that is, the derivative of a constant function is 0. Letting A = 1 and B = 0 in Theorem 12.1, we obtain: Corollary 12.3: Dx (x) = 1 By the computations in Problems 11.1, 11.2, and 11.3(a), we have: Theorem 12.4: (i) Dx (x 2 ) = 2x (ii) Dx (x 3 ) = 3x 2 (iii) Dx 1 x = 1 x2

where v1 v2 by explicit connection. Usually, Acm is much less than unity Acm 0:01 being typical). Common-mode gain sensitivity is frequently quantized via the common-mode rejection ratio (CMRR), de ned as CMRR and expressed in decibels as CMRRdb 20 log AOL 20 log CMRR Acm 9:10 AOL Acm 9:9

Typical values for the CMRR range from 100 to 10,000, with corresponding CMRRdb values of from 40 to 80 db.

We shall need to know how to differentiate functions built up by arithmetic operations on simpler functions. For this purpose, several rules of differentiation will be proved. RULE 1. (i) Dx ( f (x) + g(x)) = Dx f (x) + Dx g(x) The derivative of a sum is the sum of the derivatives. (ii) Dx ( f (x) g(x)) = Dx f (x) Dx g(x) The derivative of a difference is the difference of the derivatives. For proofs of (i) and (ii), see Problem 12.1(a). EXAMPLES

Example 9.4. Find the voltage-gain ratio Av of the noninverting ampli er of Fig. 9-3 in terms of its CMRR. Assume v1 v2 insofar as the common-mode gain is concerned. The ampli er output voltage is the sum of two components. The rst results from ampli cation of the di erence voltage vd as given by (9.1). The second, de ned by (9.8), is a direct consequence of the commonmode gain. The total output voltage is, then, vo AOL vd Acm v2 Voltage division (with iin 0) gives vd v1 v2 R1 v v2 R1 R2 o 9:12 9:11

and substituting (9.12) into (9.11) and rearranging give A R vo 1 OL 1 AOL Acm v2 R1 R2 Av vo AOL Acm AOL AOL =CMRR v2 1 AOL R1 = R1 R2 1 AOL R1 = R1 R2 1 AOL R1 = R1 R2

(a) Dx (x 3 + x 2 ) = Dx (x 3 ) + Dx (x 2 ) = 3x 2 + 2x (b) Dx x 2 1 x = Dx (x 2 ) Dx 1 x 1 = 2x 2 x 1 = 2x + 2 x