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Conditional stability

Conditional stability, I like to think about it this way:

The ultimate test of stability is knowing whether the poles of the closed loop system are in the LHP. If so, it is stable.

We get at the poles of the system by looking at the characteristic equation, 1+T(s). Unfortunately, we don't have the math available (except in classroom exercises) we have an empirical system that may or may not be reduced to a mathematical model. For power supplies, even if they can be reduced to a model, it is approximate and just about always has significant deviations from the hardware. That is why measurements persist in this industry.

Nyquist came up with a criterion for making sure that the poles are in the LHP by drawing his diagram. When you plot the vector diagram of T(s) is must not encircle the -1 point.

Bode realized that the Nyquist diagram was not good for high gain since it plotted a linear scale of the magnitude, so he came up with his Bode plot which is what everyone uses. The Bode criteria only says that the phase must be above -180 degrees when it crosses over 0 dB. There is nothing that says it can't do that before 0 dB.

If you draw the Nyquist diagram of a conditionally stable system, you'll see it doesn't surround the -1 point.

If you like, I can put some figures together. Or maybe a video would be a good topic.

All this is great of course, but it's still puzzling to think of how a sine wave can chase itself around the loop, get amplified and inverted, phase shifted another 180 degrees, and not be unstable!

Having said all this about Nyquist, it is not something I plot in the lab. I just use it as an educational tool. In the lab, in courses, or consulting for clients, the Bode plot of gain and phase is what we use.

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