The Smith chart from electrical engineering is the image of a Cartesian grid under the function
f(z) = (z − 1)/(z + 1).
More specifically, it’s the image of a grid in the right half-plane.
This post will derive the basic mathematical properties of this graph but will not go into the applications. Said another way, I’ll explain how to make a Smith chart, not how to use one.
We will use z to denote points in the right half-plane and w to denote the image of these points under f. We will speak of lines in the z plane and the circles they correspond to in the w plane.
Möbius transformations
Our function f is a special case of a Möbius transformation. There is a theorem that says Möbius transformation map generalized circles to generalized circles. Here a generalized circle means a circle or a line; you can think of a line as a circle with infinite radius. We’re going to get a lot of mileage out of that theorem.
Image of the imaginary axis
The function f maps the imaginary axis in the z plane to the unit circle in the w plane. We can prove this using the theorem above. The imaginary axis is a line, so it’s image is either a line or a circle. We can take three points on the imaginary axis in the z plane and see where they go.
When we pick z equal to 0, i, and −i from the imaginary axis we get w values of −1, i, and −i. These three w values do not line on a line, so the image of the imaginary axis must be a circle. Furthermore, three points uniquely determine a circle, so the image of the imaginary axis is the circle containing −1, i, and −i, i.e. the unit circle.
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