Transmission Lines: Reflection Coefficient and VSWR

Previous: Tx Line Models and Z0
Tx Line Table of Contents
Next: Input Impedance of a Transmission Line
Related Topics
Antennas (Home)

We are now aware of the characteristic impedance of a transmission line, and that the tx line gives rise to forward and backward travelling voltage and current waves. We will use this information to determine the voltage reflection coefficient, which relates the amplitude of the forward travelling wave to the amplitude of the backward travelling wave.

To begin, consider the transmission line with characteristic impedance Z0 attached to a load with impedance ZL:

model for determining gamma, reflection coefficient

At the terminals where the transmission line is connected to the load, the overall voltage must be given by:

ratio of total voltage to total current is determined by load impedance       [1]

Recall the expressions for the voltage and current on the line (derived on the previous page):

voltage and current waves

If we plug this into equation [1] (note that z is fixed, because we are evaluating this at a specific point, the end of the transmission line), we obtain:

solving for reflection coefficient

The ratio of the reflected voltage amplitude to that of the forward voltage amplitude is the voltage reflection coefficient. This can be solved for via the above equation:

voltage reflection coefficient

The reflection coefficient is usually denoted by the symbol gamma. Note that the magnitude of the reflection coefficient does not depend on the length of the line, only the load impedance and the impedance of the transmission line. Also, note that if ZL=Z0, then the line is "matched". In this case, there is no mismatch loss and all power is transferred to the load. At this point, you should begin to understand the importance of impedance matching: grossly mismatched impedances will lead to most of the power reflected away from the load.

Note that the reflection coefficient can be a real or a complex number.

Standing Waves

We'll now look at standing waves on the transmission line. Assuming the propagation constant is purely imaginary (lossless line), We can re-write the voltage and current waves as:

voltage and current waves with reflection coefficient

If we plot the voltage along the transmission line, we observe a series of peaks and minimums, which repeat a full cycle every half-wavelength. If gamma equals 0.5 (purely real), then the magnitude of the voltage would appear as:

standing waves along transmission line

Similarly, if gamma equals zero (no mismatch loss) the magnitude of the voltage would appear as:

impedance matched transmission line

Finally, if gamma has a magnitude of 1 (this occurs, for instance, if the load is entirely reactive while the transmission line has a Z0 that is real), then the magnitude of the voltage would appear as:

reflection coefficient with magnitude 1 produces standing waves

One thing that becomes obvious is that the ratio of Vmax to Vmin becomes larger as the reflection coefficient increases. That is, if the ratio of Vmax to Vmin is one, then there are no standing waves, and the impedance of the line is perfectly matched to the load. If the ratio of Vmax to Vmin is infinite, then the magnitude of the reflection coefficient is 1, so that all power is reflected. Hence, this ratio, known as the Voltage Standing Wave Ratio (VSWR) or standing wave ratio is a measure of how well matched a transmission line is to a load. It is defined as:

voltage standing wave ratio

This parameter is commonly quoted in antenna spec sheets. It is typically given over a bandwidth, so that you have an idea of how much power is reflected by the antenna over a frequency range (or alternatively, how much power the antenna radiates).


Next: Input Impedance of a Transmission Line

Previous: Tx Line Models and Z0

Tx Line Table of Contents

Antenna Tutorial (Home)