Phase margin measured at lowest positive-gain point, rather than 0dB point?

Question

When discussing voltage regulators, one often hears that phase margin should be measured at the 0dB gain crossing. This makes intuitive sense in a simple system where the phase has only a negative slope with increasing frequency. However, what about in more complicated systems where the phase dips and then recovers before gain goes to zero? It would seem that system stability would depend on the lowest phase margin at any frequency with positive gain, rather than just the amount of phase margin when gain finally goes to zero.

An example bode plot is below:

Using the oft-cited method of measuring phase margin at 0dB (70kHz, in this case), one would report a phase margin of 87 degrees. However, if we look at the worst-case, we actually have 58 degrees of phase margin at 23kHz. Either way, there's probably enough margin to comfortably call this stable, but that's not always the case.

Often, one will add a pole to the feedback loop to boost phase margin towards the higher frequencies in an attempt to improve stability. However, I've seen little emphasis on maintaining margin in the middle of the frequency range.

Am I fundamentally misunderstanding stability, or is lots of regulator design literature glossing over this important consideration when looking at phase response?

Answer 1

Phase margin is defined at the zero dB crossing of the gain. A system that has a phase dip that touches -180 degrees with positive gain will still be stable if the phase margin as measured where the gain crosses 0 dB is sufficient.

This is called "conditional stability" because if the gain changes so that the zero crossing occurs where the phase is close to -180 then the system can be unstable.

This can be problematic because the gain may go through an unstable point during power up, or over temperature or component variation, etc.

That's why it's desirable to have good gain margin as well. No matter how intuitive it seems, phase margin is NOT measured at the lowest phase dip where there is gain > 0dB, and a phase dip with gain >0dB doesn't predict system stability.

See the "conditional stability" section in this paper.

Answer 2

Phase and gain margins on the Bode plot are not the best metrics for determining stability as noted in this paper and this discussion. I believe the Nyquist criterion would probably be the simplest graphical approach. There are a variety of metrics that may be used, the gap-metric being my favorite. There is a lot of information in the book "Multivariable Feedback Control: Analysis and Design" by Skogestad and Postlethwaite.

Answer 3

Even if the phase dips below 180 degrees before the gain reaches 0dB, the system could still be stable.

Given the block diagram of a feedback system below, we can show why the only point to consider for phase margin is when the gain is 0dB.

The closed loop transfer function is below. This just says that the input voltage is multiplied by this "transfer function" to get the output voltage.

The punch line here is that if the denominator (1+AB) ever equals 0 then the equation will be unbounded and go to infinity! This is instability, you can't divide by 0. All other values have a finite result and are valid.

So how does the denominator get to 0? If AB ever equals -1, then the denominator is 0. The only way for AB to equal -1 is if the loop gain (AB) is 1 (or 0dB) with a -180 degree phase shift, which will make it -1.

So the system is only unstable when the loop gain is 1 with a -180 degree phase shift. It does not matter if the phase crosses -180 deg at other points. Of course you don't want to be anywhere near 180 deg phase shift at a gain of 1 or the system can be easily pushed into that unstable zone.

AB is a very special value and has a specific name, Loop Gain. Stability (phase margin) is measured from a systems loop gain, not closed loop gain. The value of AB (Loop Gain) has many other implications that are not mentioned here.

There is much much more to this topic, but I think having this basic understanding of where that phase margin criteria came from is important and a good place to start.