Finding the gain expression for this CMOS operational transconductance amplifier

Question

So here's the circuit:

I think M2 is a common source amplifier and M4 a common gate amplifier, so they form a cascode amplifier where the gain is given by : Av = -gm2 * r0, r0 is the impedance at the node where the drain of the common gate amplifier is at (Vop).

r0 = ( [rds8 // (rds2a * ( gm4a / gds4a) )] * ( gm6 / gds6 ) ) // ( rds2 * (gm4 / gds4) )

I think the low frequency gain is this and there is no need to worry about the rest of the circuit because this circuit is symmetric.

So, is this how the gain is calculated or am i doing something wrong?

Also, i think transistors M5,M6,M7,M8 and the one where VDD is at are just for polarizing purposes but i dont know what's the purpose of transistors M1a,M2a,M3a and M4a. Why are they there?

Edit: So i tried computing the gain for the folded cascode.

I got these equations:

Are they right?

Answer 1

As the image mentions, the amplifier you've shown is a hybrid telescopic and folded cascode amplifier. Essentially, it is a telescopic cascode amplifier and a folded-cascode amplifier combined so that they output at the same node.

Your analysis of the telescopic cascode is basically correct. Your equation for the output impedance at Vop is difficult to parse (please typeset with LaTeX in the future) but seems to be on the right track.

M1a, M2a, M3a, and M4a form the input stage of the folded cascode. You should be able to analyze this part of the circuit separately by superposition and then combine with your result for the telescopic cascode to get overall gain.

I have no idea what you mean by "polarizing purposes". The top transistor is a current source for the input differential pair, and the others form a cascoded load network.

Both telescopic and folded cascodes are analyzed in detail in Razavi's Analog IC Design textbook.