Nyquist stability. Vector addition. Bad diagram?

  • Thread starter schlunk
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This isn't exactly coursework, since I am no longer in school, but I think this may be the most appropriate forum to post in. This link is for a TI/Unitrode app note, "Control Loop Cookbook", which is about control loops for switching converters.
http://focus.ti.com/lit/ml/slup113a/slup113a.pdf [Broken]

I have been trying to read it, but I cannot understand a concept which he starts talking about on page 5-3. He seems to be saying that at frequencies below the crossover frequency (point where open loop gain =1), the system is always stable, even with 180 of phase lag (i.e. in-phase) an greater than unity gain. He admits that it seems counter-intuitive, and then points to a figure which explains it using vector addition (fig 3).

I think the figure must simply be a bad figure, because it doesn't seem to follow any rules of vector addition that I'm aware of. In 3a and 3b he has the two vectors meeting head to head. In figure 3b it seems to show a right triangle with a hypotenuse shorter than a leg.

Is this a bad graphic, or am I not getting it? Could someone please explain to me what he is trying to say?

Thank you for your help!
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Tbh I've studied some control theory, and I don't have a clue what he's trying to show with the vectors. Somehow he's using the Nyquist stability criterion without a Nyquist plot, hmmmm.

I'd find a better textbook to explain it, if you can. Or at least take a look here: http://www.facstaff.bucknell.edu/mastascu/eControlHTML/Freq/Nyquist3.html

I've actually got some good lecture notes in pdf form on basic control theory, so if you like then PM me with your email address and I'll send you them.