A fundamental problem of exponential VCOs?

[Rene Schmitz]

At 17:30 10.07.1997 +0100, you wrote:
>Hi!
>
>I spent some thoughts on beating between a pair of exponential
>(V/Octave) VCOs.
>...
>On an exponential VCO we usually have 2 trimmers (I omit the HFT here),
>detune
>and scale.
>If you have perfect matching of scales of the two VCOs, you can produce
>some beating
>with the Detune trimmer. Then you'll *always* have a certain beat rate
>at a certain note
>*and* the double beat rate one octave higher. And so on. So you can only
>choose between
>too much detuning in ithe upper range or almost non-existing beating in
>the lower range.

This is due to the fact that *every* harmonic content will be transposed
when the pitch goes up. Same with the beat frequency.
See when you have say a pitch of 100 Hz and a beat note of 1Hz
that means 101Hz on the other VCO and you transpose both up one 
octave you get 200 Hz and 202 Hz so the beat is 2 Hz.
(Of course, you had done the maths before but I include this here to
avoid confusion)
I think it can only be "compensated" by using dedicated linear VCO behaviour.

>Then I thought of compensating this by slightly changing the Scale
>trimmer of one VCO.

One could archive this by sort of linearizing the VCOs by using logarithmic
offset voltages. As the played pitch goes up the detuning becomes smaller.
But I think this will be to complicated since one would have to do 
complex analog computation. (someone included the equation in a previous reply)
(This would be like a linear FM input which just offsets a few Hz, that
 would be independant of the played pitch)
The beat frequency would be independant of the played pitch 
and this is what you want right?

>To me, it looks like this: Most of us use exponential VCOs to get rid of
>the offset voltage
>adjustments you need in linear (V/Hz) VCOs. But exactly this (offset
>misadjustment) would
>be the means to get a constant beat rate term over the whole keyboard !!
>
>To be more precise: The offset must be introduced *after* the
>exponential converter.
>We get *rid* of this by generating an exponential current which charges
>some capacitor
>directly. The only way to *introduce* an offset here would be an offset
>*current*. Maybe
>some leakage current has some (positive) effect, but it's really hard to
>control such small
>amounts of current. (If 1kHz corresponds to some uA, then a 0.2 Hz
>difference would
>correspond to some nA ... ) (Method A)

This may be not state-of-the-art but in my (homebrew) VCO design I use
an opamp integrator to which the expo. control current is fed.
By using a large resistor and an trimpot I could (although didn't)
trim a small offset current into the cap. 
The design looks like a linear VCO with the input resistor replaced by the 
current source. This trimpot would be the same as an 0V->0Hz adjustment pot
needed in a linear design.
The input offset current introduces an error here obviously. 
Unfortunately the VCO is a single child, so I have nothing to compare it to.;-( 

>Another way to introduce an offset is using an exponential
>voltage-to-voltage converter
>(just drive the collector current into a resistor instead of the VCO
>capacitor), and use
>this voltage to drive a linear VCO. Some Korg Synths did this (Polysix,
>Trident), and
>as far as I know the first Moog Modular VCOs as well. Problem here: You
>must take care
>that the offset voltage doesn't grow too *large*, which reduces the
>usable range to
>approx. 5 or 6 octaves. (To cover a larger range with linear VCOs, you
>have to switch
>resistors, or do some waveshaping/dividing tricks.) (Method B)

This is more or less the idea that I had when I made my VCO.
I intend to do a double VCO with a single expo. converter to 
have a pair that tracks very good. 

>So we have two possibillities to make controlled use of a constant term
>in beat rate:
>Either use method A and find a way to control very small amounts of
>current
>(a linear current sorce in parallel of your exponential one, adjustable
>from 0 to
>a few nanoamperes), or
>use Method B and extend the usable range with an opamp that has
>extremely low
>offset voltage drift.
>
>I encourage you to make experiments, if this sounds reasonable to you.
>If not, please tell me - comments invited (;->)

Maybe I should clone my VCO just to see!!

-René