# Transfer function of a difference amplifier

#### AetherWerx

Joined Dec 30, 2011
16
Hello,

First post here.

I am trying to figure out how to derive the transfer function of a difference amplifier when differential gain, common mode gain, and feedforward are used in the model.

I have attached two PDF's one with my sloppy work trying to figure out how it was done in "On the Measurement of CMRR."

The equation in question is equation 12 (just below figure 8) on pg. 4. It is referencing figure 7, a standard difference amplifier configuration, on pg. 3.

I am totally lost as to where these terms are coming from. It seems like Superposition, but I do not understand how each term is derived - especially the term that is multiplied by A(d).

This is not homework and I do not need a full solution; I am looking for insight on where to begin.

This is for a test bench that will eventually be put into an Excel or LabView program.

I could simply copy the equation, but what fun is in that?

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#### AetherWerx

Joined Dec 30, 2011
16
No one?

#### steveb

Joined Jul 3, 2008
2,436
I can't speak for anyone else, but I didn't respond to this because I couldn't figure out what you are asking and what you are trying to do. I was hoping that someone else was smarter and might figure it out, but perhaps others were confused too? I'm not sure, but it would help if you could clarify some things to increase the chance someone can help you.

One thing that is confusing is that you are asking how to derive something, but you are referencing a measurement paper.

Another confusing thing is that you are asking how to derive the transfer function of a difference amplifier, but your circuit looks like an inverting amplifier.

Also, as you said, your work is sloppy, which makes it harder to figure out what you are doing.

One suggestion is that if you are firstly trying to figure out how to derive something, then provide a clear schematic of the circuit (which should be a difference amplifier according to your question) and specify exactly what you are trying to derive. Presumably you might be interested in how to define the common mode and the differential mode signals - then how to derive the common mode and differential gain, and further how to define and derive the CMRR etc. Or, perhaps I have it all wrong and you are more interested in the measurement side of this. Again, please clarify.

#### steveb

Joined Jul 3, 2008
2,436
I ran across this document which may help get you started.

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#### PaulEE

Joined Dec 23, 2011
423
I think I know what you're after but you didn't ask it correctly...or maybe you did...in either case, I derived (picture below) the expression for a non-inverting opamp circuit with negative feedback for someone else inquiring about a unity-gain buffer in another forum post. This might give you a kick on whatever you're trying to duke out on your papers.

Please elaborate if this wasn't what you were after.

Happy new year!

#### AetherWerx

Joined Dec 30, 2011
16
One thing that is confusing is that you are asking how to derive something, but you are referencing a measurement paper.
Indeed it is a measurement paper. It is a measurement paper that discusses the various different circuits and their pros/cons used for measuring CMRR.

My question is about the difference amplifier configuration and the inclusion of the "feedforward" path.

The paper states the following about the difference amplifier configuration:

"The second problem with this technique is the omission of output impedance of the operational amplifier. Resistors R1a, R2a, and output impedance Rout, consist of a forward path, take this forward path into account, there are three paths from input to output: differential gain, common-mode gain, and feedforward."

So I am trying to derive the transfer function shown in equation 12 in the paper. It is all algebra from equation 12 to equation 15 which is easy, but it is understanding how equation 12 is arrived at.

I am assuming that I did not fully include the R1b and R2b with Vmid in my analysis and in doing so perhaps terms cancel out and one arrives at equation 12.

I know how to derive the signal gain of a difference amplifier; however, in this case the terms shown do not look familiar. This is the first time I have seen mention of differential gain, common mode gain, and a feedforward path for this circuit configuration.

I have attached the circuit and equations by themselves for clarity.

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