Electrical – Log-Taper Variable Resistor from Linear-Taper

It sounds like the built-in pot model you are using in your circuit simulator only lets you set the pot position once on the schematic, and then the position is constant during the simulation.

The Potentiometer Model at eCircuit shows how to build a model that acts like a linear pot that turns during the simulation. That's exactly what you need, right?

That model has a spice file that uses a piecewise linear source (PWL) that controls the position of the pot vs. time.

* WIPER POSITION: 0V=CCW, 1V=CW
VPOS    20  0   PWL(0MS 0V   1000MS 1V)

You could either use the "voltage" of VPOS as the X coordinate on your graph, representing pot position; or perhaps it's simpler to plot X as time and pick a PWL that linearly turns the pot proportional to time.

Then you run the simulation, and plot output voltage vs. time. Perhaps pipe in a square-wave at some audio frequency, and plot the output voltage vs time; then when viewing several seconds of simulation, you'll see a solid mass (the oscillations are too fast too see, more than 1 cycle per pixel width) that shows the envelope of the output waveform, and you can use either the top or the bottom as an estimate of the gain.

To simulate a non-linear pot, you could (a) edit the PWL line to turn the pot at a non-linear rate, but plot X as time, something like:

* nonlinear turn
VPOS    20  0   EXP(TIME)
VPOS    20  0   LOG10(TIME)

Or you could (b) build a model of a non-linear pot, and keep the PWL turning that pot at a linear rate, using something like

EPOS  21 0 TABLE{V(20,0)} = (0 0.7) (1 7.0) (2 700) (3 7k) (4 70k)

Both (a) and (b) give the same resistance-vs-time characteristics, right? Hopefully you can find some function or polynomial or a set of points to feed into PWL or TABLE that gives a close-enough approximation to the actual resistance of your real-world nonlinear pot.

I'm assuming you already have software tools that let you draw a circuit schematic and simulate it, that also accept SPICE models. If not, I'm pretty sure there is something suitable in the List of free electronics circuit simulators.

EDIT:

Or at the Chiphacker list of freeware SPICE simulators.

To plot AC signal gain as a function of pot position, first run a transient (time) simulation. Then plot the output (the voltage on the wire going to the speaker) vs. time. (Or you could plot it vs. the "turn signal", V(20) in the above code). You might have a pull-down menu option to do this; the old-school method is something like:

* WARNING: untested code
* ANALYSIS
.TRAN   5US  1000MS
*
* VIEW RESULTS
.PRINT  TRAN    V(1) V(2) V(20) V(77)
*
.PROBE
.END

The -3dB point is your cutoff frequency. It's just standard practice to define it that way. In order to find what your values should be, I'd go with equal element implementation (it's simpler, and you can correct for gain with a simple gain stage later if you need to). Choose R1=R2, C1=C2, and pick a value for either R or C. This yields the following formula for the cutoff frequency: $$f_0=\frac{1}{2\pi RC}$$ I generally choose a value of C initially, as it's easier to find or make a resistor with a strange value, whereas it's more difficult with capacitors. So, set your cutoff frequency equal to f0, and solve for R.

Here's an example: let's say I want a LPF with f0=250 Hz. I'll choose C to be 0.1 micro and solve for R. $$250=\frac{1}{2\pi RC} \rightarrow 250=\frac{1}{2\pi R(0.1x10^{-6})}\rightarrow R\approx6400\Omega.$$

From there, all you need to do is implement your circuit. Once you know what your value for R is supposed to be, you can use a dual-channel potentiometer that has the correct resistance within it's range in place of the two resistors (for the above example, something like a 10k ohm potentiometer would do the trick). This will allow you to change your cutoff frequency, since it's based upon both R and C.

Edit: As Matt Young suggested in the comments, adding a resistor in series with the potentiometer will set the maximum cutoff, and prevent shorts. It's an excellent addition to the circuit, and will keep some sanity when adding the potentiometers.