[sdiy] VCO reset time

[Magnus Danielson]

From: Don Tillman <don at till.com>
Subject: Re: [sdiy] VCO reset time
Date: 06 Jun 2004 09:25:42 -0700
Message-ID: <m2y8n07iwp.fsf at till.com>

Hi Don,

>    > From: "JH." <jhaible at debitel.net>
>    > Date: Sun, 6 Jun 2004 14:14:12 +0200
>    > 
>    > 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.
> 
> Email posts are not the best medium for this, so we'll just have to
> pretend we're all sitting around a table with a large pad of paper, a
> few pens, and a pitcher of "Fat Tire Amber Ale".

I didn't have the friends and a pitcher of Ale around, but I did have a pad of
paper, a few pens, a formula book and the lovely summery nature around the
table.

Let's do the friends and pitcher around a table at some later date, OK? ;O)

> The topic is PWM from a triangle vs. PWM from a sawtooth.

Indeed.

> Magnus just reached over and pulled out the equations for triangle-
> based PWM, and, strangely enough, the fundamental and all the
> harmonics are all in phase.

I actually took half-a-page to just reherse myself and then one page of
derivations to find the Fourier series for both forms. They are:

Tri-based:

     2*h
a  = ---- sin(n*Pi*alpha)
 n   n*Pi

b  = 0
 n

Saw-based:

     2*h
a  = ---- sin(n*Pi*alpha)*cos(-n*Pi + n*Pi*alpha)
 n   n*Pi

     2*h
b  = ---- sin(n*Pi*alpha)*sin(-n*Pi + n*Pi*alpha)
 n   n*Pi

Notice how these represent the same energy, but that the saw based have the
additional cos and sin terms due to the phase-modulation due to changing alpha.

Also, I would actually like to object to the "strangely enought" comment. It's
not as strange since when you look at this waveform it is symmetric around t=0
and thus is an even function. It comes as no major supprise that the overtones
are also even.

> Triangle-based PWM sounds great.  It's like the harmonics are all
> phasing around.  Yet remarkably, the phases of the harmonics are not
> actually changing.

Indeed.

> What's happening is a comb filter effect.  At this point I take the
> pad of paper and draw out the harmonic spectrum of a pulse stream, and
> it shows that the strength of the harmonics take the overall shape of
> a decaying full-wave-rectified sine wave.  [scribble-scribble...]

Exactly.

> As the pulse width narrows, the FWR sine shape spreads out.  And for
> the theoretical case of an infinitely narrow pulse, the spectrum is
> flat and the harmonics all have the same level.  [scribble...]

Exactly.

> As the width of the pulses increase, going toward a square wave, the
> FWR sine shape compresses.  And for the case of a symmetrical square
> wave there's a null for each of the even harmonics and we get the
> classic all odd harmonics sound.  [scribble...]
> 
> It's the moving shape of the spectrum that we hear as the
> triangle-based PMW sound.

Indeed. This is just what the formulas actually say. Look at the sin term,
this is having exactly this effect. You can view the PWM-ration alpha in the
above formulas as being the relative comb-filter frequency.

sin(n*Pi*alpha) will have a first null at 180 degrees, that is n*Pi*alpha = Pi
thus giving n * alpha = 1 => n = 1/alpha. Thus, the higher alpha the lower n
for the first null. 

Don't say Fourier-series is booring, it can be very exciting when in hands of
creative people - been more creative! ;O)

> Now...
> 
> Triangle-based PWM and sawtooth-based PWM are essentially the same
> thing.  The difference is that with sawtooth-based PWM, everything is
> moving back and forth in time with the modulating signal compared to
> the triangle version.  So the phase of the fundamental is changing,
> and the phase of each harmonic is changing by multiples of that.

Right.

> For an audio VCO, triangle-based PWM is more "correct".  The duty
> cycle of the waveform is changing and nothing else.  An audio VCO with
> sawtooth-based PWM has all these other phases and timings changing;
> that may sound better for some applications, and it makes a great
> optional feature, but it's not what was asked for.

Indeed.

Now this makes it relevant to ask the question on how you acheive the variable
amount of such phase-modulation. One way is naturally to use it in combination
with my little sawtooth phase modulator. You could use that one in combination
with a PWM curcuit to do a tri-based PWM equalent, but then on top of that
provide the modulation. There is naturally a more direct fashion to acheive it.

> For an LFO, folks care less about phase, it's more about where the
> waveform starts.  So for an LFO it may be more reasonable to have
> falling-sawtooth-based PWM because the effective start of the
> waveform, the rising point, stays in place as the modulating
> voltage changes.

Again, it all depends on the application.

Cheers,
Magnus