The Secret Life of Potentiometer Equations

[Single Coil]

I tried following the the math on The Secret Life of Pots page and I got a little lost. I want to make a graph to see how the 500K reverse audio taper potentiometer would change as it's turned from its min to max setting. What does the equation look like for the Reverse Log series resistor with actual values on the two axis? Then, I wanted to make a graph of the most commonly used pots. It just sounds like fun and I might learn something. Mostly, I just got confused with applying the equation to the reverse log plot.

Thanks for any help you can provide.

[R.G.]

If you have at least an algebra background, this is pretty simple.

(1) if you have two resistors in series, put a voltage across the series combination, and measure the voltage at the middle, you find that the voltage at the middle is always

Vmiddle = Vtop * (Rbottom/(Rtop+Rbottom))
Which is just the voltage divider equation, derived from Ohm's law.

(2) Let's change to the voltage ratio by dividing by Vtop on both sides; we get
Vratio = R2/(R1+R2) where I let R1 be the top resistor and R2 be the top resistor.

(3) For a pot, R1 and R2 are not independent. At fully counterclockwise R2=0 and R1 = Rpot. At fully clockwise rotation, R2 = Rpot and R1 = 0.
For any intermediate rotation Y where Y = 0.1 to 1.0, the fraction of the full travel that the pot is rotated, R1 = Rpot*(1-Y) and R2= Rpot*Y. So at Y= 0, R2 =0 and R1 = Rpot, and the votlage ratio is Vratio = 0/(0+Rpot) = 0. At y = 0.25 (1/4 of rotation), Vratio = 0.25*Rpot/(0.25Rpot + 0.75Rpot) =0.25/1.0 = 0.25.

(4) If you parallel either R1 or R2 with a fixed resistor, then they are no longer just a fraction of Rpot, but are the parallel combination of a fixed resistor and a fraction of Rpot.  So if you put a resistor across R2 to cause a log/audio taper, the bottom resistor is no longer Y*Rpot, but is Y*Rpot in parallel with Rt, the tapering resistor.

So instead of R2 = Y*Rpot, R2 in the voltage divider equation becomes R2 = (Rpot*Y*Rt)/(Rpot*Y+Rt)

Now the voltage ratio becomes
Vratio = (Rpot*Y*Rt)/(Rpot*Y +Rt)/ ((Rpot*Y*Rt)/(Rpot*Y +Rt)+(1-Y)*Rpot)

If you do this same procedure for paralleling R1 with Rt, you get the reverse log tapering equation

Vratio = (Rpot*Y)/(Rpot*Y+ (Rpot*(1-Y)*Rt)/((Rpot*(1-Y)+Rt)))
Which is correct if I got all my parentheses to balance.

Plotting this is a pain if you do all the calculations. I cheat - I put them into a spreadsheet. I make a column filled with, for instance, 0.0, 0.1, ... 1.0, that being the rotation.
I make one labeled cell be Rpot and one labeled cell be Rt. Then I type the equation into the first cell to the right of Y = 0.0 and get that working right. Finally I copy the equation in the cells down through 1.0. If you get 0.0 for a result for 0.0 and 1.0 for a result at 1.0, you did it right. Now you can substitute in values for Rpot and Rt to your heart's content, and let the spreadsheet do the math, as well as making the graphs up for you.

There is one more fillip that you will want to do.

Express Rt as a fraction of Rpot by replacing Rt in the equations by K*Rpot. This should result in Rpot dropping out, and you should get equations in only Y and K to plot. So do these new equations by setting K to some value between 0.0 and, say, five, and plot a set of points of Vratio for Y = 0 to 1. Then change K and plot that new set of points. This set of parametric equations in K defines all possible curves for voltage ratios you can get by putting a fixed resistor in parallel with the top of a pot. It also defines it in terms of  resistor that is a fixed fraction of the pot. So you get the same curve for a 1K pot and a 100 ohm tapering resistor as you get for a 10K pot and a 1K tapering resistor. All that changes is the scaling factor. The curve is the same.

Again, spreadsheets can do the math and the plotting for you very quickly.