Variable ramp again

[Martin Czech]

:::Well, in fact it IS really working on my desktop :-)

Ahh. What did you do against the pole @ 0 of the 1/s integrator.
Some kind of feedback? You know p=0 means no energy dissipation, ie. any
disturbation will be preserved eternally, or simply the integrator will
drift, this cicuit is not stable, as you mentioned before.

I thought about a feedback loop (also as exercise):

integrator             : g/s (some integrator gain g)
lowpass                : f/(s+a) (some dc gain f/a , single real pole (a positive))
integrator input summer: integrator in = integrator signal - k* loop signal

we don't really need k, but maybe there in actual circuit.

Transfer function of this integrator with feedback loop is:

              s+a
H(s)= g*---------------
         (s**2)+a*s+f*g*k

This means a zero at -a

and two poles at -a/2 +- sqrt((a**2)/4-f*g*k)

a > 0 is required for stability, otherwise real part=0 (double integrator)
or even >0 (sine oscillator, or underdamped sine oscillator)

if f*g*k < 0 the sqrt gets too large and one pole is positive, ie. instable

if f*g*k = 0 we get one pole in 0, expected because of no loop

if 0 < f*g*k <= (a**2)/4 we have stable real poles

if f*g*k > (a**2)/4 we get stable complex poles

Our first design goal was stability, now we know how to dimension the
circuit in oder to have that.

Our second goal is of course that it works like an integrator.  Now,
we have a negative real zero and two poles, that means the phase shift
asymptote will be -90deg, which is what we want. We will only get near
to these -90deg if the lowpass is allready in stopband, ie. frequency > a.
How good the phase will meet -90deg arround frequency a is determined
by the product f*g*k, the lower, the better.
OTOH a very small product means very high dc gain.

H(0)= a/(f*k).

So, at least f*k shouldn't be choosen too small.

Btw., larger f*g*k will flaten the lower frequency integrator
bode curve, for f*g*k>(a**2)/4 we get the usuall reonance hump,
this will certainly destroy our wanted integrator performance.

So this could be a starting point in order to stabilize the Bartkowian
PWM-Integrator.

m.c.