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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%). op amp schematic

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?

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  • \$\begingroup\$ "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. \$\endgroup\$ Jan 22 '17 at 22:56
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Your calculation of the gain (assuming an ideal op-amp) is not an approximation- it is exact.

The problem is underconstrained. Try setting R1 = R3, which, of course, means that R2 = R4, from your second line above.

The CMRR will only enter into the tolerance calculation. You can make the tolerance of each resistor worst-case in the correct direction that it changes the gain either up or down (pick one).

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  • \$\begingroup\$ OK, that was what I was going to do from the beginning. I just thought that maybe there is a more systematic approach to this problem, instead of just trying different tolerances for resistors in CMRR equation. I picked R1=R3=1kΩ with really low (zero) tolerance and R2=R4=1MΩ with 2% tolerance. In the worst case, the CMRR is 88.14 dB. \$\endgroup\$ Jan 25 '17 at 21:51
  • \$\begingroup\$ I though I described a systematic approach. The right answer is likely 1% tolerance for all resistors (the 88dB being picked by the person that set the question in order to yield that answer). \$\endgroup\$ Jan 25 '17 at 22:07

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