Standing wave ratio

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 a node (minimum), on an electrical transmission line.

The SWR is usually defined as the ratio of maximum and minimum AC voltages along the transmission line, thus called the VSWR, (sometimes pronounced "viswar" ), for voltage standing wave ratio. 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. It is likewise possible to define the SWR in terms of AC current along the line, whose value is identical to the VSWR.

The power standing wave ratio (PSWR) is defined as the square of the VSWR, however this terminology has no physical relation to actual powers involved in transmission.

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. This 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. An ideal transmission line would have 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.

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.

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 $V_{f}$) superimposed on the reflected wave (with complex amplitude $V_{r}$).

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 $\Gamma$ is defined thus:

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

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

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

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

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

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

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

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

The voltage standing wave ratio is then equal to:

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

Since the magnitude of $\Gamma$ always falls in the range [0,1], 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 $\Gamma$ varies as well, but only in phase. With the SWR dependent only on the complex magnitude of $\Gamma$, it can be seen that the SWR measured at any point along the transmission line (neglecting transmission line losses) obtains an identical reading.

The SWR can as well be defined as the ratio of the maximum amplitude to minimum amplitude of the transmission line's currents, the electric field strength around the conductors of transmission line, or the magnetic field strength in those regions. Neglecting transmission line loss, these ratios are identical.

Since the power of the forward and reflected waves are proportional to the square of the voltage components due to each wave, VSWR can be expressed in terms of forward and reflected power as follows:

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

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 R_L which is purely resistive but unequal to the characteristic impedance of the transmission line Z₀, the SWR is given simply by their ratio:

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

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 V_actual as a function of time t are understood to relate to the complex voltages according to:

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

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:

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

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

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 $e^{-i2\pi \nu t}$ and spatial dependence (for a wave in the +x direction) of $e^{+ik(x-x_{0})}$. Either convention obtains the same result for V_actual.

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:

$V_{net}(x)=V_{f}(x)+V_{r}(x)$

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

Since we are interested in the variations of the magnitude of V_net 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:

$|V_{net}(x)|^{2}=V_{net}(x)V_{net}^{*}(x)$

$=e^{-i(x-x_{0})}\left(1+\Gamma e^{i2k(x-x_{0})}\right)A\,e^{+i(x-x_{0})}\left(1+\Gamma ^{*}e^{-i2k(x-x_{0})}\right)A^{*}$

$=\left(1+|\Gamma |^{2}+2\Re (\Gamma e^{i2k(x-x_{0})})\right)|A|^{2}$

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

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

as we had earlier asserted. Along the line, the above expression for $|V_{net}(x)|^{2}$ is seen to oscillate sinusoidally between $|V_{min}|^{2}$ and $|V_{max}|^{2}$ 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=20MHz (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 V_net at x=1.3m. Then there would be another peak found where |V_net|=V_max at x=6.3m, whereas it would find minima of the standing wave |V_net|=V_min 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.

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. 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. It is the interaction of these reflected waves with forward waves which causes standing wave patterns, with the negative repercussions we have noted.

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 3.5MHz, with that antenna fed through 75 meters of RG-8A coax, the loss due to standing waves would be 2.2dB. However the same 6:1 mismatch through 75 meters of RG-8A coax would incur 10.8dB of loss at 146 MHz. 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 is sometimes refered to, and defined as the square of the standing wave ratio. The term is widely cited as "misleading." In the words of Gridley :

"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....." In other words, there are no actual powers being compared. The term is patently a misnomer, since "power standing wave ratio" is not the ratio of any two physical quantities whether directly measurable or inferred.

— J. H. Gridley

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.

See also

• Return loss
• Time-domain reflectometer
• SWR meter
• Impedance
• Mismatch loss

References

•  This article incorporates public domain material from Federal Standard 1037C. General Services Administration. Archived from the original on 2022-01-22. (in support of MIL-STD-188).

Further reading

• Understanding the Fundamental Principles of Vector Network Analysis, Hewlett Packard Application note 1287-1, 1997

External links

• Reflection and VSWR A flash demonstration of transmission line reflection and SWR
• VSWR—An online conversion tool between SWR, return loss and reflection coefficient
• Online VSWR Calculator