# Minimum CMRR design requirement

I've just started studying op amps recently and I am having trouble with the following problem from my textbook:

Design the difference amplifier circuit to realize a differential gain of 1000, a differential input resistance of $$\ 2 kΩ \$$ , and a minimum CMRR of 88 dB. Assume the op amp to be ideal. Specify both the resistor values and their required tolerance (e.g., better than x%). Attempted solution:

Solving for differential output:

$$\ V_{id} = V_{i2} \cdot (1 + {{R_{2}}\over{R_{1}}}) \cdot {{R_{4}}\over{R_{3}+R_{4}}} - V_{i1} \cdot {{R_{2}} \over {R_{1}}} \$$

Since the differential gain of 1000 is needed, pick $$\{{R_{2}}\over{R_{1}}} = {{R_{4}}\over{R_{3}}}\$$.

The differential gain becomes:

$$\A_{d} = {{R_{2}}\over{R_{1}}}\$$

(I hope I can use this approximation; otherwise everything becomes even more complicated).

Next, solving for common node gain:

$$\ A_{cm} = {{R_{4}}\over{R_{3}+R_{4}}} \cdot (1 - {{R_{2}}\over{R_{1}}}{{R_{3}}\over{R_{4}}})\$$

Since, $$\CMRR = 20 log {|A_{o}| \over |A_{cm}|} \$$ and a minimum CMRR of 88dB is required, this is the expression that I have that takes this constraint in consideration:

$$\ {|A_{d}| \over |A_{cm}|} = {|{{R_{2}}\over{R_{1}}}| \over |{{R_{4}}\over{R_{3}+R_{4}}} \cdot (1 - {{R_{2}}\over{R_{1}}}{{R_{3}}\over{R_{4}}})|} = 10^{{88} \over {20}}\$$

Another constraint is $$\ R_{id} = 2kΩ\$$, which means that $$\R_{1} + R_{3} = 2kΩ\$$

I don't really know how to proceed from here. Of course, I can try plugging different resistor values and testing them assuming different tolerances (keeping the constraints in mind), but that doesn't seem like an efficient way to solve this problem. How would you approach a problem like this?

• "Since the differential gain of 1000 is needed, pick R2R1=R4R3R2R1=R4R3." True, but "(I hope I can use this approximation; otherwise everything becomes even more complicated" Say hello to complication. For perfectly matched resistors the CMRR is infinite. You need to find the smallest mismatch which gives your desired CMRR, and specify this in terms of resistor tolerance. Jan 22 '17 at 22:56