[sdiy] Re: linear FM

[jhaible at debitel.net]

> While negative frequencies are phase reversed equivalents of the 
> positive frequency. Frequency is phase velocity and the integral of 
> that gives the momentaneous phase. Folding the momentaneous phase back 
> to any 2pi wide interval and mapping it appropriately to amplitude 
> gives an oscillator. Triangle relaxation oscillators work like that, 
> but there is an irony involved: they already employ a negative 
> frequency for half of their period, essentially using a chopper to 
> reverse the sign of frequency at pi intervals. 

Hmm, reversing the direction of current flow thru a capacitor doesn't
mean reversing frequency. Actually, a triangle wave is a mix of many
partials, all with their own frequencies (if you're using a fourier
series). And while it's certainly convenient to speak of "phase" to
describe a fraction of a triangle cycle (just as you would for a sine 
wave), I'm not sure if this can be backed up mathematically at all.
I guess it's all just a matter of definitions. But does this equation
f = d_phi/d_t make sense for any other waveform than a sine? (Other 
than referring to the _fundamental_ sine if you speak of the "phase 
of a triangle wave"?)

To illustrate this, imagine a fictious _sine_ oscillator which 
either runs smoothly by itself (like a filter with feedback of exactly 
unity loop gain), or which can (alternatively) be forced to reverse
direction by some fictious comparator-controlled mechanism.
Now imagine we have it running by itself, and force it to reverse
every now and then.
If you apply your "reverse trigger" exactly when the the sine is
at its peak value anyway, you will not notice any switching to 
negative frequency at the output. (Note #1)
If you apply your "reverse trigger" at any other time, you will
get a typical switch between positive and negative frequency at the 
output signal.

Note #1: If you think of an oscillating filter, you'd probably have 
the sine *and* cosine signal in your circuit somewhere, so this would
not work without a rapid discharge on the "cos" location in addition 
to the unobtrusive change of direction on the "sin" location in
your circuit. So I know the example is lacking somehow.
BUT going back to our triangle oscillator, with that hypotethical
sine oscillator example in mind, *this* only needs one capacitor,
so you don't have these problems. Therefore IMO it makes sense to
assume positive frequency for a triangle oscillator that just
reverses the current flow thru its capacitor at fixed comparator 
levels, but speaking of switching to negative frequency when the
current flow is switched anywhere in between. This is consistent 
with practical applications, too.

Does this make sense?

JH.

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