Re: Feedback loops bug me

CC wrote:
Kevin Aylward wrote:
Consider yourself challanged!

I can think of an example where this is true, but its a cheat.

Consider a response where the gain falls down past unity, hitting
unity at 90 degrees. Then have the gain increase back up again to
some value above unity. Then have the gain fall off at a >= second
order rate back through a final unity gain. This will be unstable.
However, the 1st x-ing is a red herring. Stability only cares about
the final x-ing, not intermediate ones.

Isn't this almost a case of effectively separate circuits? I mean
that say in a transistor stage, if there is a unity gain point at some
frequency where the crossing phase is <180, and that is the intended
region of operation, then that circuit can be functional even though
in a higher frequency range the gain again pops up over unity, and
crosses again with oscillatory conditions. Hence a parasitic
oscillation in a system that otherwise appears to be working
normally. Some situations can tolerate this without it even
degrading circuit performance in the region of interest, while others
display peculiar anomalies such as unexplained drifts, etc.

Other than that, there is no realistsic way for the system to be
unstable if gain < 1 at phase < 180. The converse is not true. Its
quite easy to construct systems with gain > 1 at 180 deg (net
positive fb) that are quite stable.

What exactly are the criteria for stability? In my not-formally
derived understanding (a lot of which came from AoE's PLL and opamp
discussions) negative FB at DC and <180 phase at f0 are the only
essentials.
Is this correct?

Not exactly. This will guarantee stability, but it is not a necessary
condition for stability.

The fundermental idea to stability is is there a pole in the right hand
plane. That is for:

Av(s) = ()()().../()()()...


where () are 1+k.S terms, k different for each pole. e.g. (1+RC.S) terms

All small signal stability analysis is an attempt to find out where the
demominater goes to zero for a particular value of S. If it does, the
system is unstable. The basic isue is that finding the roots equatons is
difficult. The demominator of a system is usually express as a power
series, not in factored form.

Nyquist plots and bode plotes are just methods to graphically try and
determine where the roots of the denominator of the loop gain is.

For Nyquist plots, "it can be shown" that the number of net
encirclements of the -1 point tells you whether or not there is a pole.
The summary is that stability effectivly only depends on the gain and
phase at the last zero x-ing point. This means that intermediate
frequencies can have gain and positive feedback and still be stable. If
the phase does go to net 0 deg (180 in NF systems) then the system is
"conditionally" stable. That is, it is conditioned on the gain. If the
gain were to fall, the system could go unstable, e.g. vacuum tubes when
warming up. Note that in RF, conditionally stable might well mean stable
independent of load and source impedances, which is a different meaning.


However, I suppose a more general statement has to include
considerations of matters such as right plane poles and zeros, which
if present might complicate matters. I have some interest in
eventually dealing with this since I am aware of the fact that SMPS
topologies can have right plane poles/zeros, and so I suppose
frequency domain analysis isn't quite enough for these.

Yes and no. RHZ cause extra phase shift, but as far as the analysis is
concerned, its all in the wash. You need to to transient sims because
real systems are always non-linear.

Kevin Aylward B.Sc.
431infoEXTRACT@xxxxxxxxxxxxx
http://www.anasoft.co.uk
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