# butterworth configuration of multi-stage amplifier

When designing a multi-stage operational amplifier, people usually use butterworth polynomial to set closed loop pole location of the op amp in feedback configuration, and then back calculate to open loop pole locations.

For one example, in this particular compensation scheme, http://repository.ust.hk/ir/bitstream/1783.1-2351/1/200409TCASI_AFFC.pdf, the author states that,

The stability of the AFFC amplifier is achieved by following Butterworth frequency response to arrange the location of poles.

Requires p3=2p2=4ωun for stability (Butterworth response)

, or in countless other literatures. It seems it is a common sense that butterworth placement is the assumption, no matter what compensation you are doing.

In butterworth's original paper, he certainly didn't envision people in the future that is going to generalize his idea into multi-stage op amp design.

My question is why use butterworth? how is this generalization done? How is this filter design theory generalize to multi-stage op amp stability design?

For most people that is familiar with two stages design, we do the placement like: based on how much phase margin and how much unity-gain bandwidth we want, we will place the open loop non-dominant pole a cetain factor larger than the unitiy-gain frequency.

For three stages or more stage op-amp, for an amateur like me, I can imagine I will place the poles/zeros like I do in two stage amp recursively, by treating the already-compensated amp as another single-stage, although the phase margin for the intermediate steps may be somehow done in an ad-hoc way.

So I cannot understand why butterworth pole placement strategy is preferred. It would be nice if this can be explained in comparison to other compensation strategies in terms of power, bandwidth, transient overshoot/phase margin or other metrics of interests analytically.

• have you checked the wikipedia article on the butterworth filter? If you really want to understand why the Butterworth is the flattest filter of them all, you might want to tead the original paper by Butterworth. Jul 9, 2020 at 10:25
• Re: stability: where are the poles? Is such a filter stable? Re: phase margin: what's the phase response of that filter? Jul 9, 2020 at 10:27
• Your opening sentence doesn't make sense - poles that are open-loop vs poles that are closed loop???? Jul 9, 2020 at 11:07
• @MarcusMüller Yeah. I checked. What I was asking is a generalization of filter theory into stability of multi-stage amp. Jul 10, 2020 at 4:44
• @Eagle Shou---I do not agree that "people usually use Butterworth polynominals" for setting opamp closed-loop poles. This was not done in the past. In the referenced article a specific amplifier topology with ACTIVE feedback is analyzed. As you know, people writing articles always want to desribe something new - and that is OK. However, it is another question if the new circuit will find a large field of application.
– LvW
Jul 10, 2020 at 11:05

Butterworth configuration is often used in three stage amplifier design because it is good compromise with transient behavior and power consumption.

Say if you design an two-stage amplifier configured in Bessel, which damps more than butterworth, so the non-dominant pole needs to be at higher frequency than its butterworth counterpart -- more bias current.

Another thing to look at it is , people trying to publish a paper talking about a new amp architecture, often wish to compare to previous work in some standard, that their circuits use less resources to perform better. Butterworth configuration is a good standard. The corresponding open loop transfer function of a butterworth configured closed loop amp has the property of independent phase margin regardless what order it is, 60. So some people think cross comparison of power efficiency between 3-stage/2-stage/4-stage is easier when we always have the phase margin fixed. But indeed I think this is sort of cheating, as gain margin does vary with order.

60 degree phase margin is usually enough to combat component variation.

Maximally flat property is also desired to avoid frequency domain peaking. Good for both switch-cap or audio applications.

Last, the corresponding third order open-loop poles of a butterworth configured unity-gain closed-loop amp can be viewed as a butterworth configured second-order closed loop amp adding a dominant pole, thus it is easy for people to design a nesting structure. And butterworth pole placement lies on a semicircle, which is also easy to calculate without too much messing with some polynomial coefficients, people are lazy in general.

The paper says it's attempting active feedback compensation:

This paper presents a low-power stability strategy to significantly reduce the power consumption of a three-stage amplifier using active-feedback frequency compensation (AFFC). The bandwidth of the amplifier can also be enhanced. (pg.1, right in the beginning)

and it does so by placing the poles according to the Butterworth criterion:

AFFC amplifier follows Butterworth frequency response for both stabilization and bandwidth maximization (pg.3)

not that Butterworth is needed in order to have stability. It could have been Chebyshev, Cauer, Pascal, Legendre, whatever else, but Butterworth had been chosen for its flatness. This flatness means that all the derivatives up to N-1 are smooth and monotonic. The same is true for the passband of the inverse Chebyshev, for example, but the zeroes in the stopband may not be needed here.

The paper also shows that there are two considered frequency responses, one given by Butterworth and one optimized for power consumption.