Voltage Standing Wave Ratio Derivation
A transmission line introduces two primary factors that account for the application specific behavior of the transmission line: attenuation and phase delay. These factors dominate what differentiate a transmission line from normal lumped circuit analysis. While they are typically present in a lumped circuit, their effect is so minimal that it can be ignored. However, once the interconnections become longer than approximately 1/10 wavelength, the transmission line effects will become measurable and therefore should be factored into the analysis.

The attenuation constant and phase constant of a transmission line are expressed as α and β respectively. Both of these constants are frequency dependent. Combined, they form the complex propagation constant γ given as γ=α+jβ. Both attenuation and phase delay increase as the length of the transmission line increases. Skipping the lengthy proof, we can say that this effect is expressed as the complex term eγl where l is the length of the line. Given the voltage or current of the signal generator, it is now possible to predict the amplitude and phase of the voltage and current at any length l of a transmission line terminated in its characteristic impedance ZO or more generally, the amplitude and phase of the voltage and current of the incident (generator inducted) wave.

When a transmission line is not terminated in its characteristic impedance, a reflection of part or all of the incident energy occurs. The characteristics of the reflection include an amplitude and phase component. This complex term is given as Γ.

The reflected energy must also travel along the length of the line back toward the generator. As a result, the reflected wave once again encounters the eγl effect of the complex propagation constant but traveling in the reverse direction. But recall that what caused the reflection may also have introduced some attenuation (the returned energy compared to the incident energy) and phase change. The combined effect is therefore Γeγl. But this expresses the combined effect at the junction of the transmission line and the terminating impedance. To allow this effect to be determined at any point z along the line, as measured from the terminating impedance, we must account for the effect of the complex propagation constant as the incident energy travels from point z toward the terminating impedance and again as the reflected energy returns to point z. Since actual distance traveled is 2z, we arrive at the formula Γeγ2z to express the magnitude of the reflected wave at point z. By convention, reflected current is expressed as a negative term to indicate its direction of travel.

Therefore to determine the voltage or current at any point z along the transmission line, one must simply add the incident and reflected terms at position z along the line to arrive at the desired result. This is also defined as the amplitude of the standing wave at position z. It is also possible then to calculate the impedance at point z since the relative or absolute amplitude and phase of the voltage and current are easily calculated.