Question 9.7: Objective: Calculate the common-mode rejection ratio of a di...

From Equation (9.50(b)), we have

v_{O} = \left(1 + \frac{R_{2}}{R_{1}} \right) \left\lgroup \frac{ \frac{R_{4}}{R_{3}}}{1  + \frac{R_{4}}{R_{3}}} \right\rgroup v_{I 2}  −  \left(\frac{R_{2}}{R_{1}} \right) v_{I 1}                    (9.50(b))

v_{O} = (1 + 10) \left(\frac{11}{1  +  11} \right) v_{I 2}  −  (10) v_{I 1}
or
v_{O} = 10.0833 v_{I 2}  −  10 v_{I 1}                   (9.60)
The differential-mode input voltage is defined as
v_{d} = v_{I 2}  −  v_{I 1}

and the common-mode input voltage is defined as
v_{cm} = (v_{I 1} + v_{I 2})/2
Combining these two equations produces
v_{I 1} = v_{cm}  −  \frac{v_{d}}{2}                  (9.61(a))
and
v_{I 2} = v_{cm} + \frac{v_{d}}{2}                     (9.61(b))
If we substitute Equations (9.61(a)) and (9.61(b)) in Equation (9.60), we obtain
v_{O} = (10.0833) \left(v_{cm} + \frac{v_{d}}{2} \right)  −  (10) \left(v_{cm}  −  \frac{v_{d}}{2} \right)
or
v_{O} = 10.042 v_{d} + 0.0833 v_{cm}                   (9.62)
The output voltage is also
v_{O} = A_{d}  v_{d} + A_{cm}  v_{cm}                   (9.63)
If we compare Equations (9.62) and (9.63), we see that
A_{d} = 10.042                  and              A_{cm} = 0.0833
Therefore, from Equation (9.59), the common-mode rejection ratio, is

CMRR(dB) = 20 \log_{10} \left|\frac{A_{d}}{A_{cm}} \right|                   (9.59)
CMRR(dB) = 20 \log_{10} \left( \frac{10.042}{0.0833} \right) = 41.6  dB
Comment: For good differential amplifiers, typical CMRR values are in the range of 80–100 dB. This example shows how close the ratios R_{2}/R_{1} and R_{4}/R_{3} must be in order to achieve a CMRR value in that range.
Computer Verification: A PSpice analysis was performed on the differential amplifier in this example with a µA-741 op-amp. For input voltages of v_{I 1} = −50  mV and v_{I 2} = +50  mV, the output voltage is v_{O} = 1.0043  V, which gives a differential voltage gain of 10.043. For input voltages of v_{I 1} = v_{I 2} = 5  V, the output voltage is v_{O}= 0.4153  V, which gives a common-mode voltage gain of A_{cm} = 0.4153/5 = 0.0831. The common-mode rejection ratio is then CMRR = 10.043/0.0831 = 120.9 ⇒ 41.6 dB, which agrees with the hand analysis. This result demonstrates that at this point, the nonideal characteristics of the µA-741 op-amp do not affect these results