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Standing wave ratio

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In telecommunications, 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 [V_f]) superimposed on the reflected wave (with amplitude [V_r]).

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 [\Gamma] is defined thus:

[\Gamma = .]
[\Gamma] is a complex number that describes both the magnitude and the phase shift of the reflection. The simplest cases, when the imaginary part of [\Gamma] is zero, are: For the calculation of VSWR, only the magnitude of [\Gamma], denoted by ρ, is of interest.

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

[V_\max = V_f + V_r = V_f + \rho V_f = V_f (1 + \rho).\,]
At other points, the waves interfere destructively, and the resulting amplitude [V_\min] is the difference between their amplitudes:

[V_\min = V_f - V_r = V_f - \rho V_f = V_f ( 1 - \rho).\,]
The voltage standing wave ratio is then equal to:

[VSWR = = .]
As ρ, the magnitude of [\Gamma], 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. [E_\max/E_\min].

Contents

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:

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

[V_r(x,t) = \rho A \sin (\omega t + kx).\,]
The total voltage [V_t] on the transmission line is given by the principle of superposition, which is just a matter of adding the two waves:

[V_t(x,t) = A \sin (\omega t - kx) + \rho A \sin (\omega t + kx).\,]
Using standard trigonometric identities, this equation can be converted to the following form:

[V_t(x,t) = A \sqrt \cos(\omega t + \phi),\,]
where [=\cot(kx).]

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

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

Standing wave ratio for ρ = 0.5
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 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.

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.
  4. The higher voltages associated with a sufficiently high SWR could damage the transmitter. solid state radios which have a lower tolerance for high voltages may automatically reduce output power to prevent damage. Tube radios may arc. The high voltages may also cause transmission line dielectric to break down and/or burn.
  5. VSWR measurements may be taken to ensure that a waveguide is contiguous and has no leaks or sharp bends. If such bends or holes are present in the waveguide surface, they may diminish the performance of both TX and RX equipment strings. Arcing may occur if there is a hole, if transmitting at high power, usually 200 watts or more (Need reference for the power statement). Waveguide plumbing[link] is crucial for proper waveguide performance. Reflected power may occur and damage equipment as well. Another cause of bad VSWR in a waveguide is moisture build-up, which can typically be prevented with silica gel.

See also

References

 

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