How can two poles be combined?

Hi, Everyone

In the book of Razavi, He said in the cascode OPA , two poles can be combined to one pole. How about the principle?Of cource , use small signal model, we can get that. But anybody can find it with just first look?

what page did you read that ?

Maybe he was mean to move RHP zero to eliminate one non-dominant pole

No, Not zero-pole pair.
Page 359-361

if you do zero-value time constant for the circuit you see that the little tiny high frequency poles at the cascode are indeed swamped by the output pole.

i think you should ignore this statement, razavi is using a hand-waving shortcut of zvtca. zvtca only tells you the worst case contribution - try using a very tiny load capacitance, uh oh! the "absorbed" pole now shows itself!

think of it this way - if the output branch is loaded down by a huge cap, the current in the whole branch can only change at a low frequency, right? this frequency is way too low for that tiny cascode cap to make a pole, so it looks like it does not exist.

but.... if the load cap is smaller, the current in the output branch can change very quickly. Now that tiny C pole becomes an interesting pole. Get your output pole too close, and you are doomed.

I think razavi's explanation was written in that confusing manner on purpose. The cascode is a nice amp for low voltage, but it is TERRRIBLY slow due to all the high-impedance nodes. You start with a 150MHz diff pair, and add a 20MHz cascode stage? Ouch! Ha ha - the way to avoid all these high frequency poles is to make your amp so slow with a big output cap, that it seems to have a single-pole rolloff.

Not that I dislike the cascode - the folded cascode is my favorite amp, because it can be used so many ways. But treat it like a GM, not an op amp.. That means every high impedance node gives a pole, slowing down that branch.

Thank you for your reply,electronrancher

You mean the cascode pole can be ignored , not merged?

And can you show me how to do "hand-waving zvtca" ?

yes - ignored, not merged.

and the way you do zvtca is to find the biggest cap in the system. calculate rout at that node, and calculate that pole alone. next, find the next biggest cap, etc etc.

basically treat each time constant (pole, cap) as an individual, and don't worry about how they act together.

as we see from razavi's blunder, zvtca comes with the disclaimer that it does NOT prevent poles from interacting, it just gives an easy calculation for those who want to pretend that each pole is separate.

one last note - if the poles are spaced by 2 decades or more, (ex-1MEG and 100MEG) then the phase contribution of the first pole will be completely done by the time the next pole starts to give phase shift.

Really. you should try to calculate it out, even if you're just using fake numbers. say you make your mirrors 3u long, giving Rout of 500kOhm. R is 500kohm at the tiny cap, and the tiny cap is 20ff. That gives a pole at about 16MHz. Output resistance is gm*1MEG^2, so we will assume gm=50micro since pmos, and Rout at the output is 12.5MEG. Now with even a 1pf on the output, the first pole occurrs at 12kHz, meaning the gain is down 60dB by the time we need to worry about the next pole (12MEG). Obviously, this amp would have phase margin problems with gain >=60dB unless you reduce Rout (more bias current or shorter L) or drive a smaller cap.

Hope it helps!
-er

This is all ok as it boils down to a dominant pole situation where the high pole comes into play at a freq where it can do no harm .. so it is not a case of combining but ignoring a pole - why use that word then ?


However not having read Razavi ..

Recall that using a 'Miller' capacitor you can 'split' poles in such a way that the original pole disappears and becomes two - a dominant one and a harmless one.