Computing Impedance Values





I recently wanted to calculate some impedance values using some of the equations given in this web site. I was embarrassed to find out how long it took me to figure it out!.  Since you might have the same problems, here are the tricks I finally remembered. If you need a refresher on complex numbers, see "Simple Stuff about Complex Numbers"

The one equation that is key to translating the equations given on this web site and elsewhere is

e^jx = cos(x) + j sin(x)

It is called Euler's Equation and can be proved by expanding the sine, cosine, and exponential functions as Taylor Series. The two sides can be shown to be identical. The proof is left to the reader.

Not only does this allow us to exponentiate an imaginary number, but it is also the clue to other arithmetic operations. For example, if we substitute x=pi / 2 (or 90°), we get


Although this looks like the hard way to write j, taking the square root of both sides gives

sqrt(j)=exp(j pi/4)

We need the square root of j to calculate the values of diffusional impedances, W, O, and T.

A similar trick helps us to calculate the impedance of a Constant Phase Element.


Some other relationships may be needed as well. I found the identities involving tanh( jx ) and coth( jx ) in the CRC Handbook. More tips are in Numerical Recipes, along with techniques to make things more "computable," i.e.,  how to compute values without crashing your program! See Sec 5.4.




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