Help with potentiometer math equation

Question

It's the first two equations of the section called "Theory of operation. How is that they go from the equation that shows the resistors in parallel to the equation right under that one. All they say is if you expand it fully it becomes this but they don't show how they expanded it. I have been trying everything but I cant get it in that form.

http://en.wikipedia.org/wiki/Potentiomet...

Answer 1

You are right, it is not obvious at all how they did it. Maybe by doing a star-delta transform and simplifying that?

Don't worry though, I have been designing with potentiometers for 28 years and it is the very first time I have seen that equation. It works, but how the h**l are you supposed to remember that thing?!

I would normally do it the intuitive way which is to calculate the resistance of R2 and RL in parallel, then plug that value into the normal potentiometer equation.

Answer 2

Click on the link for paralleled resistors right there. It shows:

R2||RL = 1/(1/R2 + 1/RL)

Substitute for both instances of R2||RL.

Now, you have nasty denominators in a fraction on both the top and the bottom. To get rid of them, multiply the whole thing by (1/R2 + 1/RL)/(1/R2 + 1/RL). Since this is equal to 1, there is no change in value. This gets rid of the denominators in both the top and the bottom and leaves:

R = 1/[(1/R2 + 1/RL)*R1 + 1]

Multiply again by R2*RL / R2*RL, also equal to 1 so it doesn't change the value, to yield the second result.