Mathematical Derivation of the Smith Chart: Understanding the Möbius Transformation Behind Electrical Engineering's Impedance Tool
By
tzury
Properly proved. Has structure, has flavour, has a point.
Summary
This article provides a mathematical derivation of the Smith chart, a graphical tool used in electrical engineering for impedance matching. The author explains that the Smith chart is the image of a Cartesian grid in the right half-plane under the Möbius transformation f(z) = (z - 1)/(z + 1). The post focuses on deriving the mathematical properties of this transformation, showing how lines in the z-plane correspond to circles in the w-plane, and specifically how the Smith chart represents constant resistance and reactance circles. The article is purely mathematical in nature, explaining how to construct a Smith chart rather than how to use it in engineering applications.
Key quotes
· 5 pulledThe 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.
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