Solving an Op Amp Circuit with Mathematica

[engineer23]

Here's my problem:

Rs is a resistor whose value depends on temperature. Rs = R0 - 1.8T, where R0 = 5000 ohms and T is in Kelvin.

If the ideal op amp has +/- 15 V power rails and you want to maximize the circuit's sensitivity, what should Rf be on the op amp? What is your maximum sensitivity at this resistance?

A drawing of the circuit is attached as a file. R2 corresponds to Rs (the temperature-dependent resistor) and R7 = R8 correspond to Rf. Please note that the drawing shows the circuit solved for T = 72 degrees Fahrenheit and Rf = 300 kOhm.

What I have done:
I have solved the circuit (using Mathematica) to obtain an expression for Vout in terms of Rf and Rs. Circuit sensitivity is dVout/dT (first derivative of voltage with respect to temperature), so I also have an expression for this. Since the power rails on the op amp dictate the maximum Vout, I should have Vout = 15. Maximum sensitivity is the second derivative of Vout with respect to T set equal to zero, correct? So then I have two equations and two unknowns and can solve for Rf and Rs.

Is this approach correct? Or am I missing something easier?

 

[anorlunda]

ping @berkeman

 

[berkeman]

Hmm, I'd have to write the equations to be sure, but I would think that the sensitivity is monotonic with respect to the feedback resistor values. Increasing the feedback resistance should increase the gain of the bridge output circuit, so it seems more likely that the limiting equation would come from the opamp railing out. For that, we'd need to know the maximum temperature range that is desired to measure, and then that gives us the extremes of the Rs values, which would then constrain the max value of Rs.

And to get close to full rail-to-rail output on the opamp, it would need to be a fairly high quality FET model. (I know they are assuming an ideal opamp in this problem)