# Determining the gain of a circuit with multiple op-amps

say I would need to find the gain (Vo/Vi) for a given circuit like this one with multiple op-amps.

I know how to find the gain for a simple circuit with a single op-amp, like an inverting circuit circuit would be -(R2/R1)Vin

But I can find the gain in those situations only because I know the gain formulas for those specific op-amps circuits. How can I determine the gain for unique circuits like this one?

I know for a start that the first op-amp is a voltage follower, so its output equals Vi, but that is about it... I am lost afterwards. I don't recognize any op-amp circuit on the right stage.

I am basically looking for a universal approach to calculating gain of op amps circuits • Find the two gains separately plug the first gain's output into the input of the second (VO1 = VI2). Feb 11, 2021 at 14:32
• Did you understand the answer to this earlier question you wrote? Feb 11, 2021 at 14:34
• I am basically looking for a universal approach to calculating gain of op amps circuits then you need to study "Operational Amplifiers". Basic properties of an opamp lets you calculate the gain and bandwidth, easily. Feb 11, 2021 at 14:35
• "I don't recognize any op-amp circuit on the right stage." Then you need to study more. The right-hand circuit is a standard Opamp circuit. Feb 11, 2021 at 14:40
• Stage by stage. You can easily split this into 3 stages, calculate the gain for each, and take the product of those 3 gains. Feb 11, 2021 at 15:07

Well, (using ideal op-amps) we get that the output of the first op-amp is equal to $$\\text{V}_\text{i}\$$. For the RC circuit, we get:

$$\frac{\text{V}_\text{RC}}{\text{V}_\text{i}}=\frac{\frac{1}{\text{sC}}}{\frac{1}{\text{sC}}+\text{R}}\tag1$$

And the second op-amp has the following transfer function:

$$\frac{\text{V}_\text{o}}{\text{V}_\text{RC}}=1+\frac{\text{R}\text{||}\frac{1}{\text{sC}}}{\text{R}}\tag2$$

So, in the end we get:

$$\frac{\text{V}_\text{o}}{\text{V}_\text{i}}=\frac{\frac{1}{\text{sC}}}{\frac{1}{\text{sC}}+\text{R}}\cdot\left(1+\frac{\text{R}\text{||}\frac{1}{\text{sC}}}{\text{R}}\right)=\frac{\frac{1}{\text{sC}}}{\frac{1}{\text{sC}}+\text{R}}\cdot\left(1+\frac{1}{\text{R}}\cdot\frac{\frac{1}{\text{sC}}\cdot\text{R}}{\frac{1}{\text{sC}}+\text{R}}\right)=$$ $$\frac{1}{1+\text{sRC}}\cdot\left(1+\frac{1}{1+\text{sRC}}\right)\tag3$$

• Thanks! I managed to find out it was a non inverting op amp, so as you mentioned, its transfer fct is Vout = Vin*(1 + R/Rgnd ). Thank you for the extra help with the RC circuit
– JCSB
Feb 12, 2021 at 14:12
• @JCSB You're welcome. Feb 12, 2021 at 14:16

Assuming the opamps are connected such that they have (virtually) zero output impedance (i.e. they are 'proper' opamp circuits), you simply multiply the gains of all the stages to get the theoretical total gain. (Practical issues may change this, like power supply rails, output impedances that really aren't zero, etc.)

So, if the stages only have resistors then it's pretty easy.

However, if one or more stages have inductors and capacitors, you will still be multiplying the gains, but it's done in a complex frequency domain. That is, you'd be multiplying complex spectra. You can use a brute force method where you pick several important frequencies and multiply the complex gain for each to produce the overall complex gain for that frequency. Then interpolate other frequencies to estimate their gains. The more inductors and capacitors, the higher the 'order' of the amplifier chain, and therefore the more frequencies must be chosen.