[sdiy] VCO reset time

[Magnus Danielson]

From: "JH." <jhaible at debitel.net>
Subject: Re: [sdiy] VCO reset time
Date: Sun, 6 Jun 2004 14:14:12 +0200
Message-ID: <000401c44bbf$c95a1fa0$ae76b9d9 at jhsilent>

Jürgen,

> >    > 4. A Pulse-Width Modulated output based on triangle does not have
> >    > the phase- modulation that a sawtooth based variant has. This may
> >    > or may not be a good thing depending on what you are after.
> >
> > True.
> 
> Well, it has much _less_ than the saw version at least.
> I did the maths for this some years ago, and was surprised that even with
> triangle based PWM there are some sidebands produced.
> I only was partially surprised, because tri based PWM still _sounds_ like
> having some angle modulation components; it's just surprising when you look
> at the modulated waveform and see that its symmetry is is never changed.
> I would have to dig this up, but I'm sure Magnus you are faster developing
> the formula yourself than me finding the old calculations. If memory serves,
> the fundamental of a tri based pwm has no angle modulation component, but the
> higher harmonics still do.

Instead of developing my own formulas and the possible errors I would make
while doing that I instead do what I usually do, check my Beta Mathematics
Handbook (only a armstrech away as I sit at the computer) and then I see that
for a PWM waveform of peak-to-peak amplitude h, high-level h and low-level 0
with a length L and the high-to-low at +cL and low-to-high at -cL (thus
symmetric around the axis) then it's Fourier series formula of (this is where
people should set their email-handlers into monospace/non-proprotional fonts
for best viewing):

              oo
              ---
       1      \         
f(t) = - a  +  >  (a cos 2*Pi*f*n*t + b sin 2*Pi*f*n*t)
       2  0   /     n                  n
              ---
              n=1

has the following identities:

a  = 2*c*h
 0 
     2*h*sin(n*Pi*c)
a  = --------------- (n >= 1)
 n         n*Pi

b  = 0
 n

These formulas are streight from the mathematical formula handbook, and it is
easy enought to verify that they are correct.

The a0 component is the DC component, and I think most people should be able to
verify it's vality fairly easy.

It is interesting that all bn components are zero, this effectively means that
there is no sine components. Also, this means that there is no shift between
an and bn for some n and thus there is _no_ phase shift as we change the PWM
factor c (which was called the greek letter alpha, but I avoided that in this
text). So, we can actually conclude that Jürgen was incorrect in saying it was
phase modulation. He is however correct in that there is modulation, but that
is amplitude modulation of the cosines, since the pulse width factor c is part
of the formulation of the cosine amplitudes of an. So, each cosine (fundamental
and overtones) is amplitude modulated, and that sure enought creates sidebands
just like any modulation does.

For small modulations we are probably not as sensitive to which form of
modulation it is but more sensitive to the existence of a modulation. For
larger modulations I think the modulationform becomes really important, but
that is nothing more than pure speculation on my part.

Cheers,
Magnus - who now goes back into the garden to enjoy the sun