[sdiy] analog through zero FM dummy

[René Schmitz]

Hi Martin, 

comments inline.

>There was a thread started by Rene about TZFM. It all started with the
>fact that switching triangle polarity due to modulator and switching due
>to level can interact in a messy way and then came several attempts to
>solve the conflict. So far, so good, but now I don't understand the whole
>concept of this kind of modulation any more, I mean: is this really FM?
>Let me see if I can get it together:
>
>we start with a tri wave:
>
>tri(p(t)) 
>
>where p(t) is our phase function. Phase relates to charge
>in this kind of oscillators.
>
>We want to write down everthing in terms of frequency, because that
>would relate to current, makes things easier.
>We exploit 
>
>p(t)=INT{f(t)}dt +p0
>
>simply stating that phase is integrated frequency information,
>or OTOH charge is integrated current. See if it works for 
>f(t)=wc=const:
>
>tri(wc*t)
>
>ok, that seems to be wellknown.
>
>Now some modulation:
>
>p(t)=INT{wc+M*sin(wm*t)}dt +p0
>
>This would be the right formula for sinuoidal frequency modulation
>with some modulation frequency wm arround some carrier wc.  I hope you
>all aggree. We can put that plain into our tri wave, forgetting about
>some signs:
>
>tri(wc*t+M/wm*cos(wm*t)+p0)
>
>I hope you're still with me. The point is: the modulation term simply
>ADDS to the running carrier phase.

So far I agree, but we want a carrier of 0Hz. 
Your maths is just too general. ;-)

That in mind I'd say:
tri(M/wm*cos(wm*t)+p0)

>OTOH, what do the known FM circuits do? They should ADD some current
>into the timing cap, this would mean another current source side by
>side to the normal expo current source. But all the circuits I know
>seem to influence the REFERENCE current of the expo source instead.
>This gives a current like:
>
>i(t)=(i0+M*sin(wm*t))*exp{C*U/Ut}

Here is where I start to disagree:
i(t)=i0*M*sin(wm*t)*exp{C*U/Ut} 

I.e. in these TZ circuits the reference current is multiplied (not added as
it might look) along with the modulation signal, which is here assumed to
be sinusodual.

That multiplication takes place in the OTA in my circuit, and it takes
place by making the reference current of the expo proportional to the
modulating input in the EN circuit. (Iout = Iref * exp {...}, substitute
Iref with i0*M*sin(wm*t) and you're there. This works only for one quadrant
hence the rectification and sign restoration.) 
Lets look at the TZVCOs, for 0V at the linear input: there is no current
fed into the timing cap! In my circuit OTA is then off. And there is no
reference current fed into the expo in the EN circuit. Without modulation
signal the output signal is 0Hz (or DC)!


>or:
>
>i(t)=i0*exp{C*U/Ut}+M*sin(wm*t)*exp{C*U/Ut}

Thus:
i(t)=i0*M*sin(wm*t)*exp{C*U/Ut}

If we integrate that we get the same result that you gave above except for
some extra scaling constants (i0*exp{...})

>this is corresponding to our frequency function.  My point is now that
>the modulation index M is also scaled with exp{C*U/Ut}, ie. if the 
>carrier frequnency gets higher, so does effective M. This seems not to be the
>same as in the theoretical case above which should have translated
>into:
>
>i(t)=i0*exp{C*U/Ut}+M*sin(wm*t)

With the restriction of 0Hz with a input signal of 0V, applied to your
theoretical case, and going back to it from the formula for the TZVCO:

i(t)=i0*M*sin(wm*t)*exp{C*U/Ut}
u(t) ~ tri(INT{i0*M*sin(wm*t)*exp{C*U/Ut}}dt+p0) 
-> tri(i0*M*exp(C*U/Ut)*1/wm*cos(wm*t)+p0), calling i0*M*exp(...) =: M'
-> tri(M'/wm*cos(wm*t)+p0)

This is where you started, just that M' is a function of the voltage on the 
expo input now, and the carrier is 0Hz.

>Or speaking electronic wise: 
>
>not the reference current should be modulated, instead a linear
>current should be additionally injected into the timing cap.
>This would mean another ota in most cases. Not that it would help much
>with our problems we have with the usual tri oscillator due to polarity
>switching...
>btw. Rene's tricky circuit follows the same path in this concern.
>The multiplication happens in the ota...
>
>So I claim: analog TZFM is not the same as "theoretical FM".

Depends on the definition of "theoretical FM" i guess ;-)

Some remarks here: one must track the modulation frequency along with the
carrier frequency, only then you get constant peak freq deviation, i.e.
constant timbre. (To modulate the timbre dynamically you can use the expo
input of the TZFM-VCO, varying the PFD.)

And, there is one other thing one should notice, when applying waveshaping
to the generated tzfm-triangle (to get sinusodial output) one gets a
different result than with a generic tzfm-sinusodial oscillator.
I still wonder if a real sinewave oscillator can be made running back and
forth?

Bye,
 René

-- 
uzs159 at uni-bonn.de
http://www.uni-bonn.de/~uzs159