Audiocraze wrote:

How about looking at the gain and phase difference between the input and output to predict stability with resistor load?

Actually whereas this is a

*necessary* condition for stability, it is not a

*sufficient* condition for stability. What you really need to do to predict stability is look at the complex loop gain (i.e.

**Aß**) as a function of frequency. If this is plotted in a polar format, then the

*sufficient* condition for stability is to limit the response to those frequencies for which the curve is outside of the unit circle centered at (1, j0).

Let's see if I can explain in a less

*mathematical* and more

*conceptual* manner. Look at the two plots contained below.

Attachment:

Polar Feedback Diagrams.png

These are polar plots of the complex loop gain

**Aß** (magnitude and phase) as a function of frequency for an inverting amplifier chain (i.e. 180º phase shift at midband frequency). The plot on the left is for amplifier with purely resistive load (note that this is only theoretical because there is always some stay capacitance and inductance in real circuits, but its good approximation at audio frequencies) and the one on the right is for an amplifier with complex reactive loads. As you increase frequency beyond the midband you travel clockwise along these contours from the midband point, when you decrease frequency below the midband point you travel counterclockwise along these contours from the midband point. On the right hand diagram is illustrated the unit circle I mentioned above. At frequencies where the curve is inside this circle, the amplifier is conditionally unstable. If the closed response contour encloses the point (1,j0), then the amplifier is absolutely unstable. This is the classical Nyquist Stability Criteria. So let's talk about what this really means.

Instability happens because there is a frequency response in the feedback loop which falls in the unit circle centered at (1,j0). The instability can happen even if the amplifier is not driven at the frequency in question. If the amplifier is regenerative, random thermal noise will drive it into oscillation. So how do we prevent this? We should take our clues from these two stability diagrams.

Starting with the figure on the right, this amplifier is absolutely stable. As we increase from the midband point the curve starts to approach the instability circle (it's still there even if it's not drawn on that diagram). However, the magnitude of the loop gain decreases so that by the time the frequency is high enough to get to that region the loop gain is insignificant and there are no instability issues. Likewise, as the frequency decreases from midband the same phenomenon is exhibited.

Now moving to the figure on the left something different happens. In this case, as frequency increases, the loop gain doesn't shrink fast enough to keep the response out of the instability region. (

*In fact, it even increases at some higher frequency point. This represents a response peak at the top end of the pass band.*) At the frequencies above the point labeled 'P', the amplifier becomes conditional unstable and ringing or oscillation can occur.

So we see that the critical factor for insuring stability seems to be limiting the complex loop gain,

**Aß**, outside of the pass band. Now there are two ways to accomplish this. One is to limit the feedback

**ß** and let the amplifier run open loop. The problem with this approach is that the amplifier response suffers both in magnitude / phase linearity and distortion at either end of the pass band. The other way to accomplish this is to limit the forward gain

**A** of the amplifier outside of the pass band. This is a much better approach as the effects of feedback are not perturbed and the feedback relation

**A/(1 - Aß)** still works at either end of the pass band. But we also have to be careful of how fast we let phase change outside of the pass band. If the gain does not decrease fast enough and the phase changes bring the response into the right half plane, we risk stability issues.

So the proper approach for amplifier stability is to limit the magnitude of the forward gain outside of the pass band, and to not force too many poles into the rolloff function so we control the rate of phase change verses frequency.

This begins to explain why so many designers have difficulties with stability in feedback operation amplifier circuits. Modern operational amplifiers are very high gain, very linear, and very high bandwidth. If the designer does't take care to control the forward gain frequency response, then there is still significant gain at high frequencies to drive the overall response into the instability region. Add to this the problems with the complexities of a speaker load and things get even more difficult to control. In transformer coupled tube output stages, stability is usually much less of a problem because there are natural bandwidth limitations in the output transformer and the input Miller capacitance of the output stage. In this case, it's the double pole phase characteristic of the transformer high end roll off in conjunction with too much high end bandwidth in other places that usually leads to stability issues.

I hope this explanation helps to put general stability criteria into a more understandable form. If you have questions, just let me know.