The math behind the Smith chart

All kinds of parameters can be described using complex numbers of the form real + j * imaginary. The numerical values of real and imaginary can vary over very wide ranges. Trying to graph them directly can be difficult or just plain impractical, but we did do that once in this blog on transmission line velocity factor.

The problem lends itself to the good work of Phillip H. Smith (1905-1987) and T. Mizuhashi (b. 1937). The Smith chart is a graph-based method of simplifying complex math. Instead of plotting the real and imaginary numbers directly on x-y coordinates, a new parameter called “gamma” is derived and the real and imaginary parts of that new gamma are themselves plotted on x-y coordinates instead. Why is it called gamma? William Shakespeare’s line seems appropriate: “A rose by any other name would smell as sweet.”

The governing equations are as follows. The algebra isn’t hard so please trace it through.

algebra equations for a Smith chart

This is the algebra of the Smith chart.

We graphically plot the real and imaginary parts of gamma. If we hold a constant value of R and allow X to vary, we get one set of curves that look like circles tangent to each other at the far right. If we hold a constant value of X and allow R to vary, we get a second set of curves above and below the horizontal axis which seem to emanate from that point of tangency we just spoke of.

Varying R or varying X causes both the real and the imaginary parts of gamma to vary. It all looks like the following.

smith chart results of varying R and X

These two annotated views of the Smith chart show the results of varying R or varying X.

The outermost circle shown here for R = 0 is not an absolute limit. We can extend this plot to negative values of R as well, but then the outermost circle diameter can get really big.

John Dunn is an electronics consultant, and a graduate of The Polytechnic Institute of Brooklyn (BSEE) and of New York University (MSEE).

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