# Stability test for Feedback Amplifiers

• eyeweyew

#### eyeweyew

Homework Statement
Does the following simple test deduced from Nyquist criterion also apply to positive feedback or it needs to be modified?
Relevant Equations
If magnitude of loop gain i.e. |T (jω)| > 1 at the frequency where phase(T (jω))= −180◦, then the amplifier is unstable.
Does the following simple test deduced from Nyquist criterion also apply to positive feedback or it needs to be modified?

If magnitude of loop gain i.e. |T (jω)| > 1 at the frequency where phase(T (jω))= −180◦, then the amplifier is unstable.

Reference:

eyeweyew said:
If magnitude of loop gain i.e. |T (jω)| > 1 at the frequency where phase(T (jω))= −180◦, then the amplifier is unstable.
The statement means that you have less than zero gain margin, which is bad/unstable. You want positive gain and phase margins in order to help ensure stability.

I'm not sure what you mean about positive feedback though. Positive feedback is pretty much guaranteed to be unstable, no? Unless you are introducing extra phase shifts in that "positive" feedback to make it negative...

A few places such as wikipedia mentioned that if loop gain > 1, the system will be unstable. I am not sure if loop gain less than unity imply stable system or not?

Reference: https://en.wikipedia.org/wiki/Positive_feedback

First of all, feedback nearly always has a phase shift that varies with frequency. The two cases of positive and negative feedback are only literally applicable when the phase shift is at or near 0o or 180o. This is often a shorthand for the behavior at low frequencies (or maybe passbands) where stability isn't an issue. In the discussion of feedback systems, you'll just have feedback with a gain and phase. You'll do well to get away from general terms like positive and negative feedback, they are too general and imprecise to be useful.

Now for the Nyquist stability criterion...
[Disclaimer: I hate Nyquist plots and have very successfully avoided using them for my entire multi decade career in analog electronics, control systems, and power supply design. They are only mentioned in passing in control theory classes, mostly for historical value. What is actually used is bode plots for convenience, or pole/zero locations in more modern control theory.]

If you read the text book you linked carefully you'll see that Nyquist's stability criterion states that the plot can not ENCLOSE the point (-1,0). This is kind of a pedantic point but if you're going to invoke Nyquist, this is what he actually said, and sometimes it matters.

The authors then state 'From the above criterion for stability, a simpler test can be derived that is useful in most common cases. “If |T (jω)| > 1 at the frequency where ph T (jω) = −180◦, then the amplifier is unstable.”' (my emphasis added). There are unusual cases where you need to look more carefully. These are cases where there are multiple frequencies where the phase is 180o.

There are plots that can have |T|>1 at 180o phase that are stable. There is an example here in the discussion about "counting encirclements".

PS: Aside from examples in graduate controls courses, having worked with lots and lots of feedback systems, I've never personally seen a real, practical, system that exhibits stability in spite of |T|>1 @ ∠T=180o. They do exist, for real, but they are uncommon.

Also, in my experience, there is a huge disconnect from the academic study of stability and real world requirements. We typically aren't paid to design "stable" systems, we are paid to design "well behaved" systems, which also have to meet performance requirements like settling time and such. There are many control systems instructors out there that haven't spent enough time working for real companies. Or if they have, they won't tell their students that real world performance is much less well defined, in a general sense, than "stability". Stability is an easily defined mathematical condition that is more amenable to textbooks than what people actually want from their machines. If you design a control system with a phase margin of 5o, you might get fired. -- Sorry, \end of diatribe\

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difalcojr and berkeman
Thanks for all the answers! I understand there are gaps between the academic and reality.

In academic/mathematical perspective, assume this simple test for negative feedback system deduced from Nyquist criterion for stability is correct(at least in most common cases):

eyeweyew said:
If magnitude of loop gain i.e. |T (jω)| > 1 at the frequency where phase(T (jω))= −180◦, then the amplifier is unstable.

And one could proceed using Nyquist stability criterion for positive feedback system just by multiplying the gain factor H by -1(see reference).

So assume the loop gain for a positive feedback system is T(s) = G(s)H, does the test for positive feedback becomes

"If magnitude of loop gain i.e. |T(s)| > 1 at the frequency where phase(T(s))= 0◦, then the amplifier is unstable."?

since it is equivalent to:

"If magnitude of loop gain i.e. |-T(s)| > 1 at the frequency where phase(-T(s))= −180◦, then the amplifier is unstable."

Reference:
https://engineering.stackexchange.c...ist-stability-criterion-for-positive-feedback

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In this context, there is no positive feedback, there is no negative feedback. There is just feedback with a gain and phase that varies with frequency. You can't think this way when discussing feedback stability.

OK, but... sure ∠(-T) = 180o is the same as ∠(T) = 0o, and |T| = |-T|.

berkeman
DaveE said:
In this context, there is no positive feedback, there is no negative feedback. There is just feedback with a gain and phase that varies with frequency. You can't think this way when discussing feedback stability.

OK, but... sure ∠(-T) = 180o is the same as ∠(T) = 0o, and |T| = |-T|.
Yes - I fully agree to the above statements.
Because:
* A 180 deg phase shift within the feedback loop at a certain frequency can be called "negative feedback";
* A 360 deg phase shift within the loop at a certain frequency can be called "positive feedback"
* Question: What about the feedback status at all the other frequencies? Positive/negative?

One exception: Surely, when we have feedback for DC with no phase shift within the feedback loop we have really positive feedback which will not allow a fixed and stable operating point.
All other cases are covered by the mentioned stability criterion (which does not need any decision between positive/negative)

DaveE and berkeman