Transfer function for CMOS opamp design

Question

I am currently working on my bachelor thesis about readout interface for resistive gas sensors. In the whole schematic a use an opamp which I am currently trying to design. At first, I want to have the circuit described properly in my work so I can learn something in the process.

Operational amplifier consists of two blocks - differential block and amplifying block. I want to do a small-signal analysis on this circuit so I can better understand what's going on.

Here's the schematic of the circuit:

I have applied the small-signal model which looks like this (I neglected gate-source capacitances):

Then, to solve this circuit to get the transfer function, I used the nodal analysis which gave me system of two equations below:

Calculating this I got this final transfer function:

Now my questions are:

• Is this the correct approach I used?
• Is there any simplier way to get poles from the equation using some approximation so I don't have to work my ass off?
• What will be the frequency at which we have unity gain?
• I want to have a phase margin at least 60 degrees at the unity-gain frequency. Are there any steps to get there?

Hopefully the final transfer function is correctly calculated because I don't want to mislead you.

PS: I realise that those questions should be aimed at mathematicians, but if anyone can help me and dealt with this before, I would be glad.

Thank you for your help.

Answer 1

Your general analysis looks correct. The design and analysis of a two stage op-amp is a well-known and well-treaded problem.

questions:

Is this the correct approach I used?

It looks correct. You can check various papers on small signal analysis to double check your work.

Is there any simplier way to get poles from the equation using some approximation so I don't have to work my ass off?

Yes, as I mentioned in the comments, you can

1. Use general factor decompositions, to factor the two poles and zero.
2. Assume 2nd pole and zero location are much greater than unity gain frequency, and therefore those factors will be negligible, and you will have a much simpler equation to work with. See Allen Hollberg slides page 120-11 to see the algebraic steps more clearly.

What will be the frequency at which we have unity gain?

As I previously mentioned, you can simplify the general gain equation by decomposing into pole/zero factors then set the resulting 1st order gain equation to 1 and solve for the frequency (|s|=|wu|). Solving for this gives a very simple unity gain frequency approximation. You can check this paper to see an example in section II.

I want to have a phase margin at least 60 degrees at the unity-gain frequency. Are there any steps to get there?*

Pushing out the 2nd non-dominant pole and zero locations much further than the unity gain frequency should make this a simple task. A simple estimate of the phase margin is $$PM \approx 90 - atan{(\frac{GBW}{p_2}}) - atan{(\frac{GBW}{z}})$$ you should see that making the 2nd pole and zero much greater than the GBW will diminish those terms.

Notice that all of these steps/questions help to understand why pole splitting is often used in introductory two stage op amps. In order to make the assumptions valid, you must design around and verify the assumptions (namely non-dominant pole and zero are pushed far out beyond the unity gain bandwidth).