Standing Wave Ratio - Further Analysis

Further Analysis

To understand the standing wave ratio in detail, we need to calculate the voltage (or, equivalently, the electrical field strength) at any point along the transmission line at any moment in time. We can begin with the forward wave, whose voltage as a function of time t and of distance x along the transmission line is:

where A is the amplitude of the forward wave, ω is its angular frequency and k is the wave number (equal to ω divided by the speed of the wave). The voltage of the reflected wave is a similar function, but spatially reversed (the sign of x is inverted) and attenuated by the reflection coefficient ρ:

The total voltage on the transmission line is given by the superposition principle, which is just a matter of adding the two waves:

Using standard trigonometric identities, this equation can be converted to the following form:

where

This form of the equation shows, if we ignore some of the details, that the maximum voltage over time V_mot at a distance x from the transmitter is the periodic function

This varies with x from a minimum of to a maximum of, as we saw in the earlier, simplified discussion. A graph of against x, in the case when ρ = 0.5, is shown below. The maximum and minimum V_mot in a periods are and and are the values used to calculate the SWR.

It is important to note that this graph does not show the instantaneous voltage profile along the transmission line. It only shows the maximum amplitude of the oscillation at each point. The instantaneous voltage is a function of both time and distance, so could only be shown fully by a three-dimensional or animated graph.