Speckle pattern

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), in an electrical transmission line.

The SWR is usually defined as a voltage ratio called the VSWR, (sometimes pronounced "viswar" ), for voltage standing wave ratio. For example, the VSWR value 1.2:1 denotes a maximum standing wave amplitude that is 1.2 times greater than the minimum standing wave value. 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 VSWR. To avoid confusion, wherever SWR is used without modification in this article, assume it is referring to the VSWR.

SWR is used as an efficiency measure for transmission lines, electrical cables that conduct radio frequency signals, used for purposes such as connecting radio transmitters and receivers with their antennas, and distributing cable television signals. A problem with transmission lines is that impedance mismatches in the cable tend to reflect the radio waves back toward the source end of the cable, preventing all the power from reaching the destination end. SWR measures the relative size of these reflections. An ideal transmission line would have an SWR of 1:1, with all the power reaching the destination and no reflected power. An infinite SWR represents complete reflection, with all the power reflected back down the cable. The SWR of a transmission line can be measured with an instrument called an SWR meter, and checking the SWR is a standard part of installing and maintaining transmission lines.

Relationship to the reflection coefficient

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:

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

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

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

For the calculation of SWR, only the magnitude of $\Gamma$, denoted by $\rho$, is of interest. Therefore, we define

$\rho =|\Gamma |$.

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:

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

As $\rho$, the magnitude of $\Gamma$, always falls in the range [0,1], the SWR 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, $E_{\max }/E_{\min }$.

Since power is proportional to V², VSWR can be expressed in terms of forward and reflected power as follows:

${\frac {P_{r}}{P_{f}}}=\left({\frac {VSWR-1}{VSWR+1}}\right)^{2}$

Further analysis

Template:Accuracy-section If the standing waves are only the result of a mismatch between the characteristic impedance and load impedance of the line, the VSWR can be expressed as one of the two equations below (pick the one that gives a value greater than 1):

$VSWR={\frac {Z_{\text{0}}}{R_{\text{L}}}}$ Template:SpaceTemplate:SpaceTemplate:SpaceTemplate:SpaceTemplate:SpaceTemplate:Spaceor Template:SpaceTemplate:SpaceTemplate:SpaceTemplate:SpaceTemplate:SpaceTemplate:Space$VSWR={\frac {R_{\text{L}}}{Z_{\text{0}}}}$

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 the wave number (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 superposition principle, 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:

$V_{t}(x,t)=A{\sqrt {4\rho \cos ^{2}kx+(1-\rho )^{2}}}\cos(\omega t+\phi ),\,$

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

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

$V_{\mathrm {mot} }=A{\sqrt {4\rho \cos ^{2}kx+(1-\rho )^{2}}}.$

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 $V_{\mathrm {mot} }$ against kx, for a range of ρ, is shown below. The maximum and minimum of V_mot in a period are $V_{\min }$ and $V_{\max }$ and are the values used to calculate the SWR.

Standing wave ratio for a range of ρ. In this graph, A and k are set to unity.

Note that this graph does not show the instantaneous voltage profile, V_t(x,t), along the transmission line. It only shows V_t(x) or the voltage amplitude as a function of space at a single point in time. 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

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 impedance of the antenna and feed line must match exactly for maximum energy transfer from the feed line to the antenna to be possible. The impedance of the antenna varies based on many factors including: the antenna's natural resonance at the frequency being transmitted, the antenna's height above the ground, and the size of the conductors used to construct the antenna.

When an antenna and feedline do not have matching impedances, some of the electrical energy cannot be transferred from the feedline to the antenna. Energy not transferred to the antenna is reflected back towards the transmitter. It is the interaction of these reflected waves with forward waves which causes standing wave patterns. Reflected power has three main implications in radio transmitters: Radio Frequency (RF) energy losses increase, distortion on transmitter due to reflected power from load and damage to the transmitter can occur.

Matching the impedance of the antenna to the impedance of the feed line is typically done using an antenna tuner. The tuner can be installed between the transmitter and the feed line, or between the feed line and the antenna. Both installation methods will allow the transmitter to operate at a low SWR, however if the tuner is installed at the transmitter, the feed line between the tuner and the antenna will still operate with a high SWR, causing additional RF energy to be lost through the feedline.

Many amateur radio operators consider any impedance mismatch a serious matter. Power loss will increase as the SWR increases. For example, a dipole antenna tuned to operate at 3.75 MHz—the center of the 80 meter amateur radio band—will exhibit an SWR of about 6:1 at the edges of the band. However, if the antenna is fed with 250 feet of RG-8A coax, the loss due to standing waves is 2.2dB, which may seem like a small loss, but is on a logarithmic scale. If running a typical 100W transmitter on the HF band, 2.2dB of loss would reduce the output power to 60W. That is a 40% reduction in power. Feed line loss typically increases with frequency, so VHF and above antennas must be matched closely to the feedline. The same 6:1 mismatch to 250 feet of RG-8A coax would incur 10.8dB of loss at 146 MHz. However, a length of 250 feet would not likely be used for 2m VHF radios. Antennas for the 80m band frequently involve large or complex designs typically mounted on a tall tower with great distances needed between buildings and thus the transmitter. VHF requires a much smaller antenna, and unless being used on a high powered repeater, does not have a very tall tower. The most common usage of 2m band is mobile single or dual band VHF or VHF/UHF mobiles. Also in part due to the typical output power of a VHF band is 50W, due to the FCC requirement of RF exposure evaluations needing to be conducted on power greater than 50W in the 2m band. This 50W with the 250 feet of cable would be reduced to 5W with 10dB of loss. On the less common occasions where a long transmission line is needed for 146 MHz, a higher quality low-loss transmission line would be used instead of the relatively cheap RG-8A.

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

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• Template:FS1037C MS188

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