[sdiy] CA3046 VCA

[Tim Stinchcombe]

Hi Ryan,
	First off, let's not beat about the bush - what you are trying to do
is hard, and most people would baulk at the large amounts of algebra
involved. Having said that, it must be do-able, the trick will be in
spotting some nice substitution to make to simplify it all (one tends to end
up with a sea of expressions, with many unknowns, and deciding on a strategy
to eliminate them is what mostly causes the difficulty!). I made two
attempts at it from different directions: one sort of produced an answer,
but was messy; the other stopped dead in its tracks, for want of that
'killer' substitution.

Secondly, a couple of comments on your method. You can safely forget about
the '-1' in the Ebers-Moll type Ic=Is*e^(whatever) expressions: I think
almost without exception, none of the books I've seen that tackle this stuff
ever bother with it, the effects you're after are wrapped up in the
exponential, and whether it is possible to make a linear approx to it
somewhere down the line. As for trying to solve for things like 'Vb' in
these non-linear expressions, generally there is no need to even try, unless
you are particularly interested in some numerical results. Normally what you
aim for is to either eliminate them via expressions obtained from elsewhere
in the circuit (which is what most of these 'linearizing' circuit
arrangements are all about), or at some suitable point, assume things are
'small' and make some linear approximation or other.

So, to the circuit itself. It is a fairly 'standard' set-up: Q3/Q4 a
differential pair, whose output currents are inherently tanh (hyperbolic
tangent) of the input voltage; to get around the non-linearity of the tanh,
Q2/Q5 'predistort' the input voltage using a tanh^-1 relationship; Q1 just
'programs' the current in Q2/Q5 (the arrangement is similar to the
linearizing diodes one sees in the LM13600 OTA for example). The thing (for
me) that makes this circuit tricky is the means of controlling the gain: Q3
being in the feedback loop of IC1 means that Q3 collector current is
'programmed' by 'V control', this gives the circuit an awkward
'one-sidedness', which I found caused me difficulties in analysing it
(normally the gain would be controlled through Iee, here it is through one
'arm' of it!). The single-ended output arrangement also compounds things
somewhat.

Attempt No. 1: comfortable starting point Q3/Q4:
Ic3,4=Iee/2[1+/-tanh((Vb3-Vb4)/(2Vt))], '+' for 3, '-' for 4 (Vt=kT/q). (If
you don't know how to get this, I have coincidently last night finished a
note on how to derive the transfer function of the Moog ladder filter, which
requires manipulating very similar relationships, and includes how to derive
this tanh expression - it's in pdf format, so I could send you a copy if you
like.) I then turned my attention to Q2/Q5: added another input resistor at
Q4 base to make symmetric (made it easier to see for me!); assume Q2, Q5
collector currents are I+deltaI, I-deltaI; write down Ebers-Moll for each;
divide, then note tanh^-1(x)=1/2 log(1+x)/(1-x), giving the input, Vb3-Vb4,
to Q3/Q4 as inverse tanh of deltaI/I. I is constant (it's where Q1 comes
in), then some straightforward analysis of the input resistors gives deltaI
in terms of the input voltages (but I did need to make an approximation
here!). The tanh^-1 then cancels tanh in Ic4 expression; then Vout in terms
of Ic4 and 'Vcontrol' is easy; Ic3=Vcontrol/47k and Iee=Ic3+Ic4 gives means
to rid Iee, and you finally get an expression for Vout in terms of Vcontrol
and the input, but its quite a mess! (I don't get a straight
Vout=f(Vcontrol*Vin), I get Vin in the numerator *and* denominator, so may
have made a mistake/missed a simplification...)

Attempt No. 2: treat IC1/Q3/Q4 as a standard exponential converter. This
quickly gives Ic4=Ic3*e^((Vb4-Vb3)/Vt)=Vcontrol/47k*e^(ditto), but now
you've lost the nice cancelling tanh/tanh^-1 aspect, and are left trying to
rid the exponential term by another suitably derived from Q2/Q5, which I
couldn't spot, and so that is where I gave up!

If anyone else has read this far *and* has sent an analysis to Ryan
off-list, I'd also love to see it (please)!

Books I have covering this stuff: Wilmshurst, Analog Circuit Techniques,
Newnes 2001, has a small section on Gilbert cell etc., but its an MSc
conversion course book, so its very terse, and you need to supply much
working yourself. I also have an old Gray & Meyer, Analog Integrated
Circuits, 2nd ed, Wiley 1984 - it has quite a big section on multipliers,
Gilbert cell too, but again, much detail is left to the reader to supply.
I've yet to see a book with a decent treatment of all this stuff in it - I
suspect it's rather too specialised for most.

Hope this helps (!), and if it hasn't put you off completely and you need
more pointers on the above, let me know.

Good luck,
	Tim
__________________________________________________________
Tim Stinchcombe 

Cheltenham, Glos, UK
email: tim102 at tstinchcombe.freeserve.co.uk

> hi,
> 
> Hope this isn't too much, electronics and not enough synth for you... 
> I'm trying to learn about some different VCA designs. I'm 
> looking at the 
> CA3046 VCA that was in Modulus E-Zine 5. here: 
> http://www.modulus.synth.net/modular/modulus5.> pdf
> 
> It 
> consists of a CA3046, and 2 Opamps, 1 diode, and 
> some resistors. This 
> circuit was also used in the Oakley ADSR/VCA.  I'm trying to 
> figure out 
> how it works but am having some trouble. I wonder if anyone 
> can give me 
> a little hint as to how to go about analyzing it.  They keep 
> teaching us 
> in school about small signal analysis, but I want to keep 
> track of the 
> nonlinear effects of each transistor so thats no good.
> 
> my idea was to break the circuit apart, and first just try to 
> find out 
> how the transistors which are not part of the differential 
> pair affect 
> the base voltages of the differential pair.  I tried solving for the 
> voltages at the bases of these 3 transistors (which are all 
> connected), 
> and ended up with equations that look like this:
> 
> Ic=(15V-Vb)/100K, emitter is grounded, base & collector are 
> shorted for 
> this one.
> and
> Ic=Is*(e^(Vb*q/kT)-1)
> (I know it's not completely accurate, but I'll start simple)
> 
> There's my problem, the only way I know to solve these for 'Vb' are 
> graphically or with newton's method, and I'm trying to leave, 
> 'Is', and 
> 'q/kT' in the equations as letters so that I can check how well the 
> circuit works with temperature change (I think it probably 
> cancels out, 
> atleast 'Is' in the pairs?).  I'm thinking there must be a 
> better way to 
> go about doing this?
> 
> I've spent a few days trying to figure it out. I don't think 
> I can use 
> this VCA for my synth, unless I can understand how it works 
> completely.
> 
> also, If anyone knows of any books that deal with this type 
> of circuit 
> in detail. Please let me know.
> 
> Thanks,
> Ryan
>