# Standing wave ratio

In radio engineering and telecommunications, standing wave ratio (SWR) 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 (maximum) to the amplitude at a node (minimum) along the line.

The SWR is usually thought of in terms of the maximum and minimum AC voltages along the transmission line, thus called the voltage standing wave ratio or VSWR (sometimes pronounced "viswar"[1] [2]). For example, the VSWR value 1.2:1 denotes an AC voltage due to standing waves along the transmission line reaching a peak value 1.2 times that of the minimum AC voltage along that line. The SWR can as well be 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 (PSWR) is defined as the square of the VSWR,[3] however this terminology has no physical relation to actual powers involved in transmission.

The SWR can be measured with an instrument called an SWR meter. Since SWR is defined relative to the transmission line's characteristic impedance, the SWR meter must be constructed for that impedance; in practice most transmission lines used in these applications are coaxial cables with an impedance of either 50 or 75 ohms. Checking the SWR is a standard procedure in a radio station, for instance, to verify impedance matching of the antenna to the transmission line (and transmitter). Unlike connecting an impedance analyzer (or "impedance bridge") directly to the antenna (or other load), the SWR does not measure the actual impedance of the load, but quantifies the magnitude of the impedance mismatch just performing a measurement on the transmitter side of the transmission line.

## Impedance matching

{{#invoke:main|main}} SWR is used as a measure of impedance matching of a load to the characteristic impedance of a transmission line carrying radio frequency (RF) 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 both the source and load to be real, 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 (or in some cases, more) 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 (significant at higher frequencies and for longer cables). 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 Zload and the source impedance Zsource=Z*load, that perfect match will remain if the source and load are connected through a transmission line with an electrical length of one half wavelength (or a multiple of one half wavelengths) using a transmission line of any characteristic impedance Z0. However the SWR will generally not be 1:1, depending only on Zload and Z0. With a different length of transmission line, the source will see a different impedance than Zload 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, Zload=Z0, always insures 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 ( Zload=Z0) 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 (especially if the line is long), even if Zload is frequency-independent. So in practice, a good SWR (near 1:1) implies a transmitter's output seeing the exact impedance it expects for optimum and safe operation.

## Relationship to the reflection coefficient

Incident wave (blue) is fully reflected (red wave) out of phase at short-circuited end of transmission line creating a net voltage (black) standing wave. Γ=-1, SWR=∞.
Standing waves on transmission line, net voltage shown in different colors during one period of oscillation. Incoming wave from left (amplitude = 1) is partially reflected with (top to bottom) Γ= .6, -.333, and .8∠60°. Resulting SWR = 4, 2, 9.

The voltage component of a standing wave in a uniform transmission line consists of the forward wave (with complex amplitude ${\displaystyle V_{f}}$) superimposed on the reflected wave (with complex amplitude ${\displaystyle V_{r}}$).

A wave is partly reflected when a transmission line is terminated with other than a pure resistance equal to its characteristic impedance. The reflection coefficient ${\displaystyle \Gamma }$ is defined thus:

${\displaystyle \Gamma ={V_{r} \over V_{f}}.}$

${\displaystyle \Gamma }$ is a complex number that describes both the magnitude and the phase shift of the reflection. The simplest cases with ${\displaystyle \Gamma }$ measured at the load are:

The SWR directly corresponds to the magnitude of ${\displaystyle \Gamma }$.

At some points along the line the forward and reflected waves interfere constructively, exactly in phase, with the resulting amplitude ${\displaystyle V_{\max }}$ given by the sum of their those waves' amplitudes:

${\displaystyle |V_{\max }|=|V_{f}|+|V_{r}|=|V_{f}|+|\Gamma V_{f}|=|V_{f}|(1+|\Gamma |).\,}$

At other points, the waves interfere 180° out of phase with the amplitudes partially cancelling:

${\displaystyle |V_{\min }|=|V_{f}|-|V_{r}|=|V_{f}|-|\Gamma V_{f}|=|V_{f}|(1-|\Gamma |).\,}$

The voltage standing wave ratio is then equal to:

${\displaystyle VSWR={|V_{\max }| \over |V_{\min }|}={{1+|\Gamma |} \over {1-|\Gamma |}}.}$

Since the magnitude of ${\displaystyle \Gamma }$ always falls in the range [0,1], the SWR is always greater than or equal to unity. Note that the phase of Vf and Vr vary along the transmission line in opposite directions to each other. Therefore the complex valued reflection coefficient ${\displaystyle \Gamma }$ varies as well, but only in phase. With the SWR dependent only on the complex magnitude of ${\displaystyle \Gamma }$, it can be seen that the SWR measured at any point along the transmission line (neglecting transmission line losses) 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 as follows:

${\displaystyle SWR={\frac {1+{\sqrt {P_{r}/P_{f}}}}{1-{\sqrt {P_{r}/P_{f}}}}}}$

In fact, most SWR meters operate by measuring both the forward power and the reflected power. Normalizing the power readings according to the forward power, a reading of the reflected power is then directly read off a meter in terms of SWR.

In the special case of a load RL which is purely resistive but unequal to the characteristic impedance of the transmission line Z0, the SWR is given simply by their ratio:

${\displaystyle SWR=\left({\frac {R_{\text{L}}}{Z_{\text{0}}}}\right)^{\pm 1}}$

with the ±1 chosen to obtain a value greater than unity.

## The standing wave pattern

Using complex notation for the voltage amplitudes, for a signal at frequency ν, the actual (real) voltages Vactual as a function of time t are understood to relate to the complex voltages according to:

${\displaystyle V_{actual}=\Re (e^{i2\pi \nu t}V)}$ .

Thus taking the real part of the complex quantity inside the parenthesis, the actual voltage consists of a sine wave at frequency ν with a peak amplitude equal to the complex magnitude of V, and with a phase given by the phase of the complex V. Then with the position along a transmission line given by x, with the line ending in a load located at x0, the complex amplitudes of the forward and reverse waves would be written as:

${\displaystyle V_{f}(x)=e^{-ik(x-x_{0})}A}$
${\displaystyle V_{r}(x)=\Gamma e^{ik(x-x_{0})}A}$

for some complex amplitude A (corresponding to the forward wave at x0). Here k is the wavenumber due to the guided wavelength along the transmission line. Note that some treatments use phasors where the time dependence is according to ${\displaystyle e^{-i2\pi \nu t}}$ and spatial dependence (for a wave in the +x direction) of ${\displaystyle e^{+ik(x-x_{0})}}$. Either convention obtains the same result for Vactual.

According to the superposition principle the net voltage present at any point x on the transmission line is equal to the sum of the voltages due to the forward and reflected waves:

${\displaystyle V_{net}(x)=V_{f}(x)+V_{r}(x)}$
${\displaystyle =e^{-i(x-x_{0})}\left(1+\Gamma e^{i2k(x-x_{0})}\right)A}$

Since we are interested in the variations of the magnitude of Vnet along the line (as a function of x), 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:

${\displaystyle |V_{net}(x)|^{2}=V_{net}(x)V_{net}^{*}(x)}$
${\displaystyle =e^{-i(x-x_{0})}\left(1+\Gamma e^{i2k(x-x_{0})}\right)A\,e^{+i(x-x_{0})}\left(1+\Gamma ^{*}e^{-i2k(x-x_{0})}\right)A^{*}}$
${\displaystyle =\left(1+|\Gamma |^{2}+2\Re (\Gamma e^{i2k(x-x_{0})})\right)|A|^{2}}$

Depending on the phase of the third term, one can see that the maximum and minimum values of Vnet (the square root of the quantity in the equations) are (1+|Γ|)|A| and (1-|Γ|)|A| respectively, for a standing wave ratio of:

${\displaystyle SWR={\frac {|V_{max}|}{|V_{min}|}}={\frac {1+|\Gamma |}{1-|\Gamma |}}}$

as we had earlier asserted. Along the line, the above expression for ${\displaystyle |V_{net}(x)|^{2}}$ is seen to oscillate sinusoidally between ${\displaystyle |V_{min}|^{2}}$ and ${\displaystyle |V_{max}|^{2}}$ with a period of 2π/2k. This is half of the guided wavelength λ = 2π/k for the frequency ν. 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 ν=20 MHz (free space wavelength of 15m) in a transmission line whose velocity factor is 2/3, the guided wavelength (distance between voltage peaks of the forward wave alone) would be λ =10m. At instances when the forward wave at x=0 is at zero phase (peak voltage) then at x=10m it would also be at zero phase, but at x=5m it would be at 180° phase (peak negative voltage). 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 λ/2 =5m. Depending on the location of the load and phase of reflection, there might be a peak in the magnitude of Vnet at x=1.3m. Then there would be another peak found where |Vnet|=Vmax at x=6.3m, whereas it would find minima of the standing wave |Vnet|=Vmin at x=3.8m, 8.8m, etc.

## Practical implications of SWR

The most common case for measuring and examining SWR is when installing and tuning transmitting antennas. When a transmitter is connected to an antenna by a feed line, the driving point impedance of the antenna must be resistive and matching the characteristic impedance of the feed line in order for the transmitter to see the impedance it was designed for (the impedance of the feedline, usually 50 or 75 ohms).

The impedance of a particular antenna design can vary due to a number of factors that cannot always be clearly identified. This includes the transmitter frequency (as compared to the antenna's design or resonant frequency), the antenna's height above the ground and proximity to large metal structures, and variations in the exact size of the conductors used to construct the antenna.[4]

When an antenna and feedline do not have matching impedances, the transmitter sees an unexpected impedance, where it might not be able to produce its full power, and can even damage the transmitter in some cases.[5] The reflected power in the transmission line increases the average current and therefore losses in the transmission line compared to power actually delivered to the load.[6] It is the interaction of these reflected waves with forward waves which causes standing wave patterns,[5] with the negative repercussions we have noted.[7]

Matching the impedance of the antenna to the impedance of the feed line can sometimes be accomplished through adjusting the antenna itself, but otherwise is possible using an antenna tuner, an impedance matching device. Installing the tuner between the feed line and the antenna allows for the feedline to see a load close to its characteristic impedance, while sending most of the transmitter's power (a small amount may be dissipated within the tuner) to be radiated by the antenna despite its otherwise unacceptable feedpoint impedance. Insalling a tuner in between the transmitter and the feed line can also transform the impedance seen at the transmitter end of the feedline to one preferred by the transmitter. However in the latter case, the feedline still has a high SWR present, with the resulting increased feedline losses unmitigated.

The magnitude of those losses are dependent on the type of transmission line, and its length. They always increase with frequency. For example, a certain antenna used well away from its resonant frequency may have an SWR of 6:1. For a frequency of 3.5 MHz, with that antenna fed through 75 meters of RG-8A coax, the loss due to standing waves would be 2.2 dB. However the same 6:1 mismatch through 75 meters of RG-8A coax would incur 10.8 dB of loss at 146 MHz.[5] Thus, a better match of the antenna to the feedline, that is, a lower SWR, becomes increasingly important with increasing frequency, even if the transmitter is able to accommodate the impedance seen (or an antenna tuner is used between the transmitter and feedline).

## Power standing wave ratio

The term power standing wave ratio (PSWR) is sometimes referred to, and defined as the square of the voltage standing wave ratio. The term is widely cited as "misleading."[8] In the words of Gridley:[9]

The expression "power standing-wave ratio", which may sometimes be encountered is even more misleading, for the power distribution along a loss-free line is constant.....

—J. H. Gridley

In other words, there are no actual powers being compared. Patently a misnomer, the term "power standing wave ratio" is not the ratio of any two physical quantities.

However it does correspond to one type of measurement of SWR using what was formerly a standard measuring instrument at microwave frequencies. A slotted line involves a waveguide (or air-filled coaxial line) in which a small sensing antenna measures the electric field along the transmission line directly. The electric field strength is commonly measured using a crystal detector or Schottky barrier diode. These detectors have a square law output for low levels of input. Readings therefore corresponded to the square of the electric field along the slot, E2(x), with maximum and minumum readings of E2max and E2min found as the probe is moved along the slot. The ratio of these yields the PSWR directly, the square root of which is the VSWR.[10]

## Implications of SWR on medical applications

SWR can also have a detrimental impact upon the performance of microwave based medical applications. In microwave electrosurgery an antenna that is placed directly into tissue may not always have an optimal match with the feedline resulting in an SWR. The presence of SWR can affect monitoring components used to measure power levels impacting the reliability of such measurements.[11]

## References

1. {{#invoke:citation/CS1|citation |CitationClass=book }}
2. {{#invoke:citation/CS1|citation |CitationClass=book }}
3. Samuel Silver, Microwave Antenna Theory and Design, p. 28, IEE, 1984 (originally published 1949) ISBN 0863410170.
4. {{#invoke:citation/CS1|citation |CitationClass=book }}
5. {{#invoke:citation/CS1|citation |CitationClass=book }}
6. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
7. {{#invoke:citation/CS1|citation |CitationClass=book }}
8. Christian Wolff, "Standing Wave Ratio", radartutorial.eu
9. J. H. Gridley, Principles of Electrical Transmission Lines in Power and Communication, p. 265, Elsevier, 2014 ISBN 1483186032.
10. Bernard Vincent Rollin, An Introduction to Electronics, p. 209, Clarendon Press, 1964 Template:OCLC.
11. Template:Cite web