By paying attention to mistakes, we invest more time and effort to correct them. The result is that you make the mistake work for you.
— Jason Moser
Yesterday, I had a question from a reader on how to develop mathematical formulas for different potentiometer tapers. Normally, I would simply answer the questioner without a separate post, but my solution for this particular question provided a nice illustration of basic coordinate transformations. Since I have not shown any coordinate transformation applications in this blog before, I thought it would be worthwhile to make a post of my response.
There many different names (e.g. "M", "W", "S") assigned to the common potentiometer tapers. To my knowledge, the taper names vary by vendor. For this post, I will use the taper names as stated in Figure 1 by State Electronics, which the questioner referred to. My work here will focus on the M and W tapers, which are closely related.
My analysis assumes that the potentiometer taper is an actual exponential curve. For ease of manufacture, many potentiometer suppliers approximate the exponential curve using a piecewise linear approximation. For example, Figure 1 shows a W taper that appears to be composed of three linear segments.
I should also mention that I add a constant term to my exponential function to allow my curve fit to pass through zero, which is what happens with real potentiometers – a true exponential curve, i.e. , would not pass through zero.
This discussion will focus on the common, three-terminal potentiometer. Figure 2(a) shows how State Electronics defines the terminals and Figure 2(b) shows what a potentiometer looks like inside.
|Figure 2(a): Terminal Definitions.||Figure 2(b): Physical Construction of a Three-Terminal Potentiometer (source).|
The W taper is sometimes referred to as the antilog taper because it related to the exponential function. Its specific functional form of the potentiometer resistance between terminals 1 and 2 is dictated by the following definition.
The “W” taper attains 20% resistance value at 50% of
clockwise rotation (left-hand).
I should mention that you rarely see the W taper described in terms of an actual function – it resistance versus wiper position is almost always shown as a graph. Remember that these are physical parts and they vary quite a bit from their nominal specifications.
The M taper has a sigmoid shape and its resistance between terminals 1 and 2 is defined in terms of the W taper as follow.
The “M” taper is such that a “W” taper is attained from
either the 1 or 3 terminal to the center of the element.
Figure 3 shows my approach to developing a W taper functional relationship.
Figure 4 shows my approach to developing an M taper functional relationship.
Figure 5 shows my combined plot of the M and W tapers. They are very similar to that shown in Figure 1.
This post demonstrated how to develop functional relationships for the resistance of two common types of potentiometer tapers. It does seem odd that these functions are never actually stated in the vendor documentation, but hopefully I have alleviated that shortcoming here.