# Calculating gain of ICF amplifier

I'm designing a circuit with an ICF (indirect current feedback) amplifier, and am trying to calculate the gain of the resultant circuit.

Using the AD8237 I followed the example here when designing with one modification. Instead of using a voltage follower, I used a resistor divider as the datasheet said I could (the ICF amplifier does not require a low impedance Vref like other 3 amp instrumentation amplifiers). Doing this does affect the gain, as can be seen from the snapshot from the datasheet below:

The gain when Vref = 0 is G = (1 + R2/R1). I tried dusting off some Thevenin equivalent circuit knowledge (it has been quite a while) and calculated the resistance with R1 removed and the voltage sources and calculated it as (R2 + R3||R4). Replacing R2 in the gain equation with this gives G = (1 + (R2 + R3||R4)/R1) which is the same as what is in the datasheet. So far so good!

The challenge comes when adding the next part of the circuit - an offset voltage Vdac and resistor Ra as seen in the circuit below.

Doing the same as before, I calculated the resistance with R1 removed and all voltage sources shorted. From this I calculated it as (R3||R4 + R2||Ra) which when replacing R2 in the gain equation results in a gain of G = (1 + (R3||R4 + R2||Ra)/R1).

Is this the correct way of calculating this? Is the resultant gain I've calculated correct?

Edit: I don't think my math/understanding here is correct. I tried to derive equation (5) from the example as a check on my understanding and didn't have any luck. Can someone help me understand how that gain equation is being derived?

Many thanks!

In my opinion you might be making things too complicated. You have R3 and R4 that set the voltage at the reference pin so, instead of trying to drive your DAC offset control voltage to the FB pin via $R_A$, just drive the reference pin with your DAC output. Use a small value DAC output resistor (maybe 100 ohms) followed by a 10 uF decoupler to stop DAC clock noise then feed the ref pin.