# Standing wave ratio

In telecommunication, standing wave ratio (SWR) is the ratio of the amplitude of a partial standing wave at an antinode (maximum) to the amplitude at an adjacent node (minimum).

The SWR is usually defined as a voltage ratio called the VSWR, for voltage standing wave ratio. It is also possible to define the SWR in terms of current, resulting in the ISWR, which has the same numerical value. The power standing wave ratio (PSWR) is defined as the square of the SWR.

The voltage component of a standing wave in a uniform transmission line consists of the forward wave (with amplitude [itex]V_f[itex]) superimposed on the reflected wave (with amplitude [itex]V_r[itex]).

Reflections occur as a result of discontinuities, such as an imperfection in an otherwise uniform transmission line, or when a transmission line is terminated with other than its characteristic impedance. The reflection coefficient Γ is defined thus:

[itex]\Gamma = {V_r \over V_f}[itex].

Γ is a complex number that describes both the magnitude and the phase shift of the reflection. The simplest cases, when the imaginary part of Γ is zero, are:

• [itex]\Gamma=-1[itex]: maximum negative reflection, when the line is short-circuited,
• [itex]\Gamma=0[itex]: no reflection, when the line is perfectly matched,
• [itex]\Gamma=+1[itex]: maximum positive reflection, when the line is open-circuited.

For the calculation of VSWR, only the magnitude of Γ, denoted by ρ, is of interest.

At some points along the line the two waves interfere constructively, and the resulting amplitude [itex]V_{max}[itex] is the sum of their amplitudes:

[itex]V_{max} = V_f + V_r = V_f + \rho V_f = V_f (1 + \rho)\,\![itex].

At other points, the waves interfere destructively, and the resulting amplitude [itex]V_{min}[itex] is the difference between their amplitudes:

[itex]V_{min} = V_f - V_r = V_f - \rho V_f = V_f ( 1 - \rho)\,\![itex].

The voltage standing wave ratio is then equal to:

[itex]VSWR = {V_{max} \over V_{min}} = {{1 + \rho} \over {1 - \rho}}[itex]

As ρ, the magnitude of Γ, is always >= 0, the VSWR is always >= +1.

The SWR can also be defined as the ratio of the maximum amplitude of the electric field strength to its minimum amplitude, i.e. [itex]E_{max}/E_{min}[itex].

## 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:

[itex]V_f(x,t) = A \sin (\omega t - kx),\,\![itex]

where A is the amplitude of the forward wave, ω is its angular frequency and k is a constant (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:

[itex]V_r(x,t) = \rho A \sin (\omega t + kx)\,\![itex].

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

[itex]V_t(x,t) = A \sin (\omega t - kx) + \rho A \sin (\omega t + kx)\,\![itex].

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

[itex]V_t(x,t) = A \sqrt {4\rho\cos^2 kx+(1-\rho)^2} \cos(\omega t + \phi)\,\![itex],

where [itex]{\tan \phi}={{(1+\rho)}\over{(1-\rho)}}\cot(kx)[itex].

This form of the equation shows, if we ignore some of the details, that the maximum voltage over time [itex]V_{mot}[itex] at a distance x from the transmitter is the periodic function

[itex]V_{mot} = A \sqrt {4\rho\cos^2 kx+(1-\rho)^2}[itex]

This varies with x from a minimum of [itex]A(1-\rho)[itex] to a maximum of [itex]A(1+\rho)[itex], as we saw in the earlier, simplified discussion. A graph of [itex]V_{mot}[itex] against x, in the case when ρ = 0.5, is shown below. [itex]V_{min}[itex] and [itex]V_{max}[itex] are the values used to calculate the SWR.

Missing image
Swr.png
Standing wave ratio for ρ = 0.5

It is important to note that this graph does not show the instantaneous voltage profile along the transmission line. It only shows the 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.

## Practical implications of SWR

SWR has a number of implications that are directly applicable to radio use.

1. SWR is an indicator of reflected waves bouncing back and forth within the transmission line, and as such, an increase in SWR corresponds to an increase in power in the line beyond the actual transmitted power. This increased power will increase RF losses, as increased voltage increases dielectric losses, and increased current increases resistive losses.
2. Matched impedances give ideal power transfer; mismatched impedances give high SWR and reduced power transfer.
3. Higher power in the transmission line also leaks back into the radio, which causes it to heat up. This is a big concern for solid state radios, but the higher voltages associated with a sufficiently high SWR could cause tube radios to arc or the transmission line dielectric to break down.

## References

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy