[sdiy] 2164 Expo correction

[David G. Dixon]

(Apologies: long posting)

So, further to the 2164 Expo tracking saga, I bought a Fluke 1900A frequency
counter off some dude on Craigslist today ($50!) and used it in tandem with
my DVM to play with the 2164 Expo Tricore VCO some more.

First of all, as an aside, let me say that the frequency counter in my
cheapo Victor VC921A auto-range pocket multimeter gives exactly the same
reading as the Fluke, and has 4 digit precision at any frequency (unlike the
Fluke, which can only go down to 1/10 of a Hz).  Plus, at the 1/10 Hz
setting, the Fluke is dreadfully slow to respond, which makes using it a
real chore.  I'm going to go buy another Victor so I can dedicate one to
voltage readings and the other to frequency readings.  I love this little
unit, and it only cost 25 bucks!

http://www.victorelectronics.com/specifications/vc921_specs.htm

Anyway, once the frequency exceeded 100Hz, I used the Fluke and the Victor
in tandem to make the measurements.

When I was measuring the lowest frequency (16 Hz), I noticed that the VC
voltage kept drifting upward.  And then it hit me: I had to let the VCO warm
up before making these measurements!  So I left it turned on, came in the
house for 10 or 15 minutes, and then went back and made the measurements.

Result: the tracking was much better than yesterday!  The response was
essentially flat at the lower frequencies, and only started to show
significant tracking error above about 256 Hz.  This is a little lower than
my original simulations led me to expect.  However, the shape of the
response suggested that a BJT-based HF correction scheme should correct the
tracking nicely.

I did two sets of measurements: one where I fixed the frequencies to powers
of 2 (from 16 to 8192 Hz) and measured the VC voltage, and another where I
set the VC voltage at 200 mV increments (from 1400 to -400 mV) and measured
the frequency.  It turns out that the effective 2164 gain factor (at low
frequencies, where the response is flat) is just about 188 mV per octave --
a little higher than I indicated yesterday, but still well lower than 200.

Next, I did an iterative spreadsheet calculation where I fit the base-2 log
of frequency as a sixth-order polynomial function of the 2164 VC voltage
(and vice versa), estimated the new VCs which would give perfect tracking,
calculated the correction currents which must be added to the CV summing
node to achieve these new VC voltages, related this to the V_BE of the
2N3904 transistor with a best-fit Ebers-Moll model, figured out the bias
(intercept) and gain (slope) of the HF correction opamp inverter to give
these V_BE values as a linear function of the VC voltage, calculated the new
frequencies, and then found the CV summer gain factor which minimized the
sum of square tracking errors over the audio range.  This was all done in an
iterative feedback loop which converged around the entire expo circuit to a
specific set of values.

Results: At the optimum CV summer gain factor of 186.1 mV per decade, the
average tracking error over about 8.6 octaves (from 12.3 Hz to 4.77 kHz) was
0.00% (which means that the 9th octave would be in perfect tune with the
first octave), the average absolute tracking error was 0.05%, and the
maximum tracking error was 0.13% (at about the seventh octave, between 1 and
2 kHz).  The tracking error will be about 1% at 10 kHz, but I don't really
care!

Finally, assuming a 2164 gain factor of 188 mV per decade, the optimum
tempco voltage is 271.2 mV.  This should give maximum tempcos of about
+/-200 ppm per degree C between 20 and 30 degrees C, with typical values
less than 50 ppm per degree C.  In other words, perfect tempco.

Needless to say, I'm going to stick with my original design.