Solving the Multiple Op-Amps Problem for Optimal Gain: A Simplified Approach

[psycovic23]

Hey,

I've been working on this problem for a while now to no avail. The question is to choose R1 and R2 such that Vo = 10IsRx.

I've gotten as far as solving for each input voltage for the op amps and their accompanying output voltages, but I end up with a huge expression in terms of Is and Rx that I can't really simplify.

Written out, this is what I have:

$$V_o = \frac {20K}{R_2}V_3 + V_2(\frac{20K}{R_2} + 1) = 10I_s R_x$$

where V1 is the input to the first op-amp, or:
$$V_1 = 100I_s$$

V3 is output of opamp 1, or:
$$(\frac{v_1}{R_1})20K + V_1$$

V2 is input to op-amp 2:
$$V_2 = (100+R_x)I_s$$

Looking at V_o, I can see that the gain for the second opamp should be $$(\frac{20K}{R_2} + 1)$$ and the first would be $$(\frac{20K}{R_1}+1)$$ but I don't know what to put for those individual gains. Any advice? Thanks!

 

[LightHero]

The best way to approach this problem is by doing a bit of simplification. We can start by breaking down the equation for Vo and rearranging it: Vo = 10I_sR_x = \frac {20K}{R_2}V_3 + V_2(\frac{20K}{R_2} + 1) = \frac {20K}{R_2}(V_1/R_1 + V_1) + (100+R_x)I_s(\frac{20K}{R_2} + 1)= \frac {20K}{R_2}V_1 + 20KV_1 + (100+R_x)I_s(\frac{20K}{R_2} + 1)= \frac {20K R_1 + 20K R_2 + (100 + R_x)(20K + R_2)}{R_2} I_sNow, we can solve for the resistors: R_2 = \frac {10I_sR_xR_2 - 20K R_1 - 20K R_2 - 100 R_2 - 10I_sR_x}{20K + R_2}R_1 = \frac {10I_sR_xR_2 - 10I_sR_x - 20K R_2 - 100 R_2 - 20K R_1}{20K}From here, you should be able to solve for your resistors using any numerical solver or graphing software. Hope this helps!

 

[piercas]

Hi there,

Thank you for sharing your progress on this problem. It seems like you have made some good progress in solving for the input and output voltages of the op-amps. However, as you mentioned, the expression for Vo is quite complex and difficult to simplify.

In situations like this, it can be helpful to take a step back and look at the problem from a different perspective. Have you considered using a different approach or method to solve for R1 and R2? Sometimes, a simplified approach can lead to a more manageable solution.

Additionally, have you checked your calculations and equations to make sure they are correct? It's always a good idea to double check your work to ensure accuracy.

I hope this helps and good luck with solving the multiple op-amps problem for optimal gain!