Standing wave ratio

In radio engineering and telecommunications, standing wave ratio is a measure of impedance matching of loads to the characteristic impedance of a transmission line or waveguide. Impedance mismatches result in standing waves along the transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode to the amplitude at a node along the line.
Voltage standing wave ratio is the ratio of maximum to minimum voltage on a transmission line. For example, a VSWR of 1.2 means a peak voltage 1.2 times the minimum voltage along that line, if the line is at least one half wavelength long.
A SWR can be also defined as the ratio of the maximum amplitude to minimum amplitude of the transmission line's currents, electric field strength, or the magnetic field strength. Neglecting transmission line loss, these ratios are identical.
The power standing wave ratio is defined as the square of the VSWR, however, this deprecated term has no direct physical relation to power actually involved in transmission.
SWR is usually measured using a dedicated instrument called an SWR meter. Since SWR is a measure of the load impedance relative to the characteristic impedance of the transmission line in use, a given SWR meter can interpret the impedance it sees in terms of SWR only if it has been designed for the same particular characteristic impedance as the line. In practice most transmission lines used in these applications are coaxial cables with an impedance of either 50 or 75 ohms, so most SWR meters correspond to one of these.
Checking the SWR is a standard procedure in a radio station. Although the same information could be obtained by measuring the load's impedance with an impedance analyzer, the SWR meter is simpler and more robust for this purpose. By measuring the magnitude of the impedance mismatch at the transmitter output it reveals problems due to either the antenna or the transmission line.

Impedance matching

SWR is used as a measure of impedance matching of a load to the characteristic impedance of a transmission line carrying radio frequency signals. This especially applies to transmission lines connecting radio transmitters and receivers with their antennas, as well as similar uses of RF cables such as cable television connections to TV receivers and distribution amplifiers. Impedance matching is achieved when the source impedance is the complex conjugate of the load impedance. The easiest way of achieving this, and the way that minimizes losses along the transmission line, is for the imaginary part of the complex impedance of both the source and load to be zero, that is, pure resistances, equal to the characteristic impedance of the transmission line. When there is a mismatch between the load impedance and the transmission line, part of the forward wave sent toward the load is reflected back along the transmission line towards the source. The source then sees a different impedance than it expects which can lead to lesser power being supplied by it, the result being very sensitive to the electrical length of the transmission line.
Such a mismatch is usually undesired and results in standing waves along the transmission line which magnifies transmission line losses. The SWR is a measure of the depth of those standing waves and is, therefore, a measure of the matching of the load to the transmission line. A matched load would result in an SWR of 1:1 implying no reflected wave. An infinite SWR represents complete reflection by a load unable to absorb electrical power, with all the incident power reflected back towards the source.
It should be understood that the match of a load to the transmission line is different from the match of a source to the transmission line or the match of a source to the load seen through the transmission line. For instance, if there is a perfect match between the load impedance _load and the source impedance that perfect match will remain if the source and load are connected through a transmission line with an electrical length of one half wavelength using a transmission line of any characteristic impedance ₀. However the SWR will generally not be 1:1, depending only on _load and ₀. With a different length of transmission line, the source will see a different impedance than _load which may or may not be a good match to the source. Sometimes this is deliberate, as when a quarter-wave matching section is used to improve the match between an otherwise mismatched source and load.
However typical RF sources such as transmitters and signal generators are designed to look into a purely resistive load impedance such as 50Ω or 75Ω, corresponding to common transmission lines' characteristic impedances. In those cases, matching the load to the transmission line, _load ₀, always ensures that the source will see the same load impedance as if the transmission line weren't there. This is identical to a 1:1 SWR. This condition also means that the load seen by the source is independent of the transmission line's electrical length. Since the electrical length of a physical segment of transmission line depends on the signal frequency, violation of this condition means that the impedance seen by the source through the transmission line becomes a function of frequency, even if _load is frequency-independent. So in practice, a good SWR implies a transmitter's output seeing the exact impedance it expects for optimum and safe operation.

Relationship to the reflection coefficient

The voltage component of a standing wave in a uniform transmission line consists of the forward wave superimposed on the reflected wave.
A wave is partly reflected when a transmission line is terminated with an impedance unequal to its characteristic impedance. The reflection coefficient can be defined as:
or
is a complex number that describes both the magnitude and the phase shift of the reflection. The simplest cases with measured at the load are:

: complete negative reflection, when the line is short-circuited,
: no reflection, when the line is perfectly matched,
: complete positive reflection, when the line is open-circuited.

The SWR directly corresponds to the magnitude of.
At some points along the line the forward and reflected waves interfere constructively, exactly in phase, with the resulting amplitude given by the sum of those waves' amplitudes:
At other points, the waves interfere 180° out of phase with the amplitudes partially cancelling:
The voltage standing wave ratio is then
Since the magnitude of always falls in the range , the SWR is always greater than or equal to unity. Note that the phase of V_f and V_r vary along the transmission line in opposite directions to each other. Therefore, the complex-valued reflection coefficient varies as well, but only in phase. With the SWR dependent only on the complex magnitude of, it can be seen that the SWR measured at any point along the transmission line obtains an identical reading.
Since the power of the forward and reflected waves are proportional to the square of the voltage components due to each wave, SWR can be expressed in terms of forward and reflected power:
By sampling the complex voltage and current at the point of insertion, an SWR meter is able to compute the effective forward and reflected voltages on the transmission line for the characteristic impedance for which the SWR meter has been designed. Since the forward and reflected power is related to the square of the forward and reflected voltages, some SWR meters also display the forward and reflected power.
In the special case of a load _L, which is purely resistive but unequal to the characteristic impedance of the transmission line ₀, the SWR is given simply by their ratio:
with the ratio or its reciprocal is chosen to obtain a value greater than unity.

The standing wave pattern

Using complex notation for the voltage amplitudes, for a signal at frequency, the actual voltages V as a function of time are understood to relate to the complex voltages according to:
Thus taking the real part of the complex quantity inside the parenthesis, the actual voltage consists of a sine wave at frequency with a peak amplitude equal to the complex magnitude of, and with a phase given by the phase of the complex. Then with the position along a transmission line given by, with the line ending in a load located at, the complex amplitudes of the forward and reverse waves would be written as:
for some complex amplitude of Either convention obtains the same result for.
According to the superposition principle the net voltage present at any point on the transmission line is equal to the sum of the voltages due to the forward and reflected waves:
Since we are interested in the variations of the magnitude of along the line, we shall solve instead for the squared magnitude of that quantity, which simplifies the mathematics. To obtain the squared magnitude we multiply the above quantity by its complex conjugate:
Depending on the phase of the third term, the maximum and minimum values of are and respectively, for a standing wave ratio of:
as earlier asserted. Along the line, the above expression for is seen to oscillate sinusoidally between and with a period of . This is half of the guided wavelength for the frequency . That can be seen as due to interference between two waves of that frequency which are travelling in opposite directions.
For example, at a frequency in a transmission line whose velocity factor is 0.67 , the guided wavelength would be At instances when the forward wave at is at zero phase then at it would also be at zero phase, but at it would be at 180° phase. On the other hand, the magnitude of the voltage due to a standing wave produced by its addition to a reflected wave, would have a wavelength between peaks of only Depending on the location of the load and phase of reflection, there might be a peak in the magnitude of at Then there would be another peak found where at whereas it would find minima of the standing wave at 8.8 m, etc.