Okay, here’s a breakdown of the provided text, focusing on the core ideas and how they relate to building a Smith chart (though the author explicitly states they won’t explain how to use it, only how to make it). I’ll organize it into key concepts and their connections.
Overall Goal: The text explains the mathematical basis for why a Smith chart looks the way it does. It’s about understanding the geometric transformation performed by a specific Möbius transformation, and how that transformation maps elements of the complex plane (the z*-plane) onto the Smith chart itself (the *w-plane).
1.The Möbius Transformation (f(z))
* The core of the Smith chart’s construction is a specific Möbius transformation.The exact formula isn’t given in this excerpt, but it’s implied to be the transformation that maps the right half-plane to the interior of the unit circle.
* Key Property: Möbius transformations map generalized circles (circles and lines) to generalized circles. This is the fundamental theorem the author relies on.
2. Mapping the Imaginary Axis
* The imaginary axis in the z*-plane (where the real part is zero) is mapped to the *unit circle in the w*-plane.
* Proof: The author demonstrates this by:
* Recognizing the imaginary axis is a line, so its image must be a line or a circle.
* Choosing three points on the imaginary axis (0, *i, –i*).
* calculating their images under *f(z): f*(0) = -1, *f(i) = i, f*(-i) = -i.
* These three points (-1, *i, –i*) lie on the unit circle. Since three points uniquely define a circle, the image of the imaginary axis *is the unit circle.
3.mapping the Right Half-Plane
* The right half-plane (where the real part is positive) is mapped to the interior of the unit circle.
* Proof:
* The imaginary axis is the boundary of the right half-plane. Since it maps to the unit circle, the right half-plane must be either inside or outside the unit circle.
* The point z* = 1 (wich is in the right half-plane) maps to *w = 0 (which is inside the unit circle). Thus, the right half-plane maps to the interior.
4. Mapping Vertical Lines
* Vertical lines in the z*-plane (lines with constant positive real part) are mapped to *circles in the w*-plane.
* Key Properties of these circles:
* They are all inside the unit circle.
* They are all tangent to the unit circle at the point *w = 1.
* Reasoning:
* Vertical lines contain the point at infinity (∞).
* f*(∞) = 1.
* Therefore,the image of each vertical line is a circle passing through *w = 1.
5. Mapping horizontal Lines
* The text begins to discuss horizontal lines in the z*-plane (lines with constant imaginary part). It states that their image is either a line or a circle.The reasoning is that a line is mapped to a line if it contains infinity, otherwise it’s a circle. The text stops mid-sentence, so the conclusion about horizontal lines isn’t fully presented in this excerpt.
How this relates to the Smith Chart’s appearance:
* Unit Circle: The unit circle in the *w-plane is the outer boundary of the Smith chart.
* Circles Tangent to the unit Circle: The circles formed by the images of vertical lines are the circles within the Smith chart that are tangent to the unit circle. These represent constant resistance.
* Lines Intersecting the Unit Circle: The horizontal lines, when mapped, will become lines (or circles) that intersect the unit circle. These represent constant reactance.
in essence, the author is showing how the geometry of the Smith chart arises naturally from the properties of a specific Möbius transformation. The chart isn’t just an arbitrary collection of circles and lines; it’s a carefully constructed geometric representation of complex impedances, based on a solid mathematical foundation
