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
MAKING YOUR OWN VARIABLE RESISTORS
Anything conductive, accessible, doesn't oxidise, able to be "wiped" with a wiper (resistor pickup) with adequate dying "too quickly".
As this is as much for fun as anything else "too quickly" may be able to be of lower duration in time or cycles than usually.
Resistance values that you generate may be lower than not, depending on material used.
Properly "potentiometer" mans a 3 terminal device with voltage across it and a sliding voltage tap but I'll take it to also just mean 'variable resistor.
Connection to start may be with "crocodile clips or pushed in "drawing pins" / "thumbtacks/other.
Pencil lead.
Select an old (or new) pencil.
Sharpen both ends.
Measure resistance to see what sort of pot resistance you are going to get.
CAREFULLY break open and remove the lead intact.
May need a few pencils to get it right.
Connect clips at either end.
Connect ohmmeter to one end.
Run other ohm-meter along length and note variation in resistance.
If you connect a voltage across length then you can use a slider to puck off variable voltage with position.
Resistance wire
In place of the pencil lead above you can us a length of new or used resistance wire.
Wire can be strethed tight between eg thumbtacks or nail in a piece of wood.
Note that wood becones part of the resistor.
New Nichome or Chomel wire canbe bought for modest cost.
Ohms per metre varies with thickness - thinnest possible is liable to be best.
Around 10 ohms/meter is common but higher R is possible.
Nichrome from old heater or toaster element works.
This may be somewhat oxidised with age and may be brittle.
You can sand surface carefully once stretched in place.
Butyl Rubber and friends
Black rubber used for roofing has carbon black in it.
Take meter and wander round sticking probes in rubber on sale and other material.
When you find a sheet of substance that has some resistance acquire a small sample by best permissible means and cut strips to make a pot.
Paper and salt water.
Lay out a strip of newspaper
Wet well but not until soggy with salt in water solution.
Test pot.
Note how result varies with salt concentration, degree of saturation of paper, passage of time, ...
Try copper-sulphate in place of salt.
Try "Epsom Salts"
Try ... ?
Copper wire
!!!
(1) Get thinnest possible bare copper wire.
Measure resistance.
Low but usable.
(2) Now for a very good trick.
Get thinnest (within reason) enamel or varnish or polyurethane insulated copper wire. Sortthat insulation can be "sanded" off with care.
Find a "former" that is an insulator and that you can wind your copper wire on.
Round or oval cross section is good.
Once you have built one you will get a better feel for shape that is needed.
Wind copper carefully and neatly in a long ish coil along formr. Many turns.
Not too too many the first time.
Wind neatly so turns stack against each other neatly.
Fasten carefully at both ends.
Now CAREFULLY use fine sand paper to sand along top surcae of coil so you expose copper on each turn BUT DO NOT TAKE OFF SO MUCH THAT COIL TURNS ALL GET SHORTED TOGETHER.
Run a wiper along the bared copper.
You have a wire-wound variable resistor.
"Plastic"
Use epoxy resin and silicone rubber.
Fill with various amounts of carbon black, or pencil graphite or powdered metal etc.
Make a track.
Let set.
Test.
Also silver filled compound used for PCB track repair.
Also ??? - look around you ... .
By now you should have a few other ideas.
Report back :-) !!!!
Best Answer
What are you going to do with the result? Things are mostly controlled digitally nowadays, which is why there is little point to non-linear pots anymore. Set up the pot to drive the A/D input of a microcontroller, then perform whatever non-linearities you want on the resulting linear reading.
Even in the unusual case where you really do want a non-linear analog voltage from the user setting, you can still use a cheap micro (under $.50) to read the pot, perform the non-linear function, produce PWM from than, then a R-C filter to make the average voltage level. The linear pot plus micro is often cheaper than the fancy low-volume non-linear pot.