Cascading Op-Amps: Where to set the gain?

[LoveLaika]

I'm working on a 3-stage circuit involving cascading op-amps. I was wondering if there was a general consensus on where the gain should be?

My three stages involve:
Stage 1 - Input : Basic Passive Filtering, no op-amps here
Stage 2 - Sum/Difference here
Stage 3 - Sum/Difference here, output

Stages 2 and 3 use op-amps, and I want my output to have a certain gain involving the inputs. Question is, do I place the gain at Stage 2 or Stage 3, or is it better to split the gain?

Stage 1 filters the inputs to a certain cutoff frequency. Working with op-amps, the max bandwidth for a closed loop circuit is the GBW, provided that gain = 1. As gain increases, the bandwidth decreases linearly. Assuming an inverting op-amp configuration, you can put a small capacitor in parallel with the feedback network to further filter signals. Keeping that in mind, I see some pros and cons to both approaches.

If I put the gain in stage 3, I can use the max bandwidth in stage 2, keeping with a gain of 1 at that stage. Furthermore, with a feedback capacitor, that would also filter out the signals even further, ensuring that stage 3 sees signals that can fall within the op-amp bandwidth at a set gain. So, assuming that all signals are within the bandwidth of the op-amp, stage 3 should have no problems with the signals.

On the other hand, if I put the gain in stage 2, it will also filter out certain frequencies even without the feedback capacitor (as the closed loop gain is only good up to the corner frequency found using the GBW and gain). However, here, I have to make sure that the reduced bandwidth as a result of having a gain here will be fine with the frequencies filtered from stage 1. After that, stage 3 can utilize its max bandwidth since it doesn't need to have gain here.

Both methods seem valid, but I was wondering what more experienced engineers think?

[Kleinstein]

It really depends on the application and signal levels. If noise or offset drift is important, it helps to have gain already in first active stage. Noise and the offset of later stages is less important than. If a high BW is important it can help to split the gain between stages, so that all have a similar BW.
For stability of the gain with temperature an time it helps to have only one stage with gain, as than less stable resistors are needed - difference stages need resistors anyway so this would not apply there.

With sum / diffferences in both stages, it makes a difference for the signal level where the gain is, so it really depends. It may be possible to just skip one stage and do the sum / difference with weight factors in 1 amplification stage.

[David Hess]

For maximum bandwidth, the gain should be distributed between stages according to their gain-bandwidth product.  The exception is that the first gain stage should have sufficient gain to overwhelm the noise of the following gain stage if input noise must be minimized.

[rstofer]

Dave did a video that talks about this in the context of increasing overall bandwidth

[bdunham7]

You're asking a very general question that needs specifics to have a meaningful answer.  You need to state:

The desired input impedance, or the impedance of the source

The desired output impedance

The overall gain needed

The types of op-amps you are using

The actual signal levels involved.

For some simple examples, if you wanted 100X gain in two stages and you want the best bandwidth possible and no extraordinary noise or impedance issues, you might use 10X gain at each stage with the same model op-amp.

If you wanted 10X gain, but needed low output impedance, you might use 10X gain with a decompensated (not unity-gain stable) op-amp followed by a 1X output with a different op-amp (unity-gain stable, of course).
 
etc, etc.

[TimFox]

Another important consideration, depending on amplifier choice, power supply levels, etc. is to make sure that no earlier stage before a gain adjustment (like the traditional volume control on an audio system) saturates under the maximum input signal condition, since you do not want to put the passive gain adjustment before a low-noise input stage.  E.g., don't use x100 gain on the input stage and then reduce the gain by 1/20 in normal operation unless you check the signal levels.