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Infinity fits inside a circle
The problem with infinity
An impedance can take any value between a short circuit, where it is zero, and an open circuit, where it is infinite. A chart of raw impedances would have to be infinite too. Bell Labs engineer Philip Smith, with two colleagues who specialized in mathematics, Ferrell and McRae, used a property of complex numbers to get around that.
How a conformal map works
Multiplying a complex number by a real constant scales it, and multiplying by the imaginary number j rotates it by 90 degrees. A function such as 1/z applied to the whole plane warps it, and the lines curve, but zoom in far enough and every tiny patch still looks square. A map like that preserves shapes and angles and is called a conformal map. Values that used to stretch off to infinity now fit inside the chart.
Why everything fits
Smith's choice was the reflection coefficient, which compares the reflected wave with the forward wave. On a lossless line its magnitude stays constant along the line and only its phase changes, and the reflected wave can at most equal the forward wave, so the coefficient can never grow past one. Plot a line of constant resistance through that transformation and it becomes a circle. Do it for every positive resistance and every reactance, and you get two families of circles in which each point holds an impedance and a reflection coefficient at the same time.
Why does a conformal map keep angles intact?
A conformal map comes from a complex function that is smooth enough to be differentiable. Near any single point, such a function acts like a rotation combined with a uniform scaling. Because that treatment is the same in every direction, tiny shapes keep their form and crossing curves keep their angles. The Mercator projection of the globe is a familiar conformal map, which is why it preserves local shapes while inflating areas near the poles.
Why does a straight line become a circle?
The function that carries impedances onto the chart is a Möbius transformation, which sends every circle or straight line to another circle or straight line. A line of constant resistance is straight on the impedance plane. After the transformation it becomes a circle, and every one of those circles passes through the same point on the right edge of the chart. That shared point is where infinite impedance lands, which is how the transformation squeezes the endless ends of those lines into one location.
Why can a reflection never exceed the forward wave?
A passive load such as an antenna or a resistor has no source of energy, so it cannot send back more power than it receives. Power in a wave depends on the square of its amplitude, so the reflected share of the power is the reflection coefficient squared. An open or short circuit absorbs nothing and reflects everything, which puts it at a coefficient of exactly one. The share of power a load actually absorbs is one minus that squared value.
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