The Variation of Complex Reflection Coefficient With Position Along The Line
The complex voltage reflection coefficient is defined as the ratio of the reflected wave to the incident (or forward) wave. Therefore
where C is also a constant.
For a uniform transmission line (in which is constant), the complex reflection coefficient of a standing wave varies according to the position on the line. If the line is lossy ( is non-zero) this is represented on the Smith chart by a spiral path. In most Smith chart problems however, losses can be assumed negligible and the task of solving them is greatly simplified. For the loss free case therefore, the expression for complex reflection coefficient becomes
The phase constant may also be written as
where is the wavelength within the transmission line at the test frequency.
Therefore
This equation shows that, for a standing wave, the complex reflection coefficient and impedance repeats every half wavelength along the transmission line. The complex reflection coefficient is generally simply referred to as reflection coefficient. The outer circumferential scale of the Smith chart represents the distance from the generator to the load scaled in wavelengths and is therefore scaled from zero to 0.50.
Read more about this topic: Smith Chart, Mathematical Basis, The Normalised Impedance Smith Chart
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