Yesterday in 2-op-amp instrumentation amp, I worked through the analysis of the differential gain and common-mode gain for an instrumentation amp made from two op amps. There is a classic design of an instrumentation amp made from 3 op amps, and I was wondering how it compared in performance. Both instrumentation amp designs are available as fairly low cost integrated circuits, but the ones with better common-mode rejection seem to use the 3-op-amp design. Is there are reason for that?

3-op-amp instrumentation amp. Schematic drawn with SchemeIt (using PNG export rather than a screenshot, since I have no special characters).

The 3-op-amp instrumentation amplifier consists of two separately analyzable parts. The first stage consists of two op amps that provide high-impedance inputs and amplify the differential signal without changing the common-mode signal. The second stage is a differential amplifier that zeros out the common-mode signal and provides unity gain on the differential signal.

We can analyze the first stage by setting the currents through the resistors equal:

.

If we solve this for Va and Vb, we get

If we define and , we can rewrite the output of the first stage as

The differential output of the first stage depends only on the differential input:

It may seem strange that I defined the common voltage so that I have an extra Vref floating around in the formula, but that simplifies out in the analysis of the second stage (which would not be the case if I had defined Vcomm as the just the average of Vp and Vm).

The differential amplifier that is the second stage can be described in terms of the two voltage dividers:

.

Solving this for Vout gives us

If we substitute in the values of Va and Vb (and use Maple plus some hand algebra), we can simplify to

.

At R3=R4 and R5=R6, we can simplify to

,

which is . (Note: this result does not require that R1=R2 nor that R4=R5, though instrumentation amps are usually designed with those additional constraints, perhaps to maximize the range of acceptable input values, which this ideal-op-amp analysis ignores.)

The worst-case common-mode gain with 1% errors in the resistors would be with R4 and R6 high, and R3 and R5 low, giving a gain of 0.02 (or vice versa, for a common-mode gain of –0.02). Comparing this to the 2-op-amp design’s common-mode gain of 0.04, we see that we gain a factor of 2 (6dB) in common-mode rejection by using an extra op amp.

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