Reflection coefficient

In physics and electrical engineering the reflection coefficient is a parameter that describes how much of a wave is reflected by an impedance discontinuity in the transmission medium.

For example, it is used in optics to calculate the amount of light that is reflected from a surface with a different index of refraction, such as a glass surface, or in an electrical transmission line to calculate how much of the electromagnetic wave is reflected by an impedance discontinuity.

In telecommunications and transmission line theory, the reflection coefficient is the ratio of the complex amplitude of the reflected wave to that of the incident wave.

The voltage and current at any point along a transmission line can always be resolved into forward and reflected traveling waves given a specified reference impedance Z0.

The reference impedance used is typically the characteristic impedance of a transmission line that's involved, but one can speak of reflection coefficient without any actual transmission line being present.

(capital gamma) and can be written as: It can as well be defined using the currents associated with the reflected and forward waves, but introducing a minus sign to account for the opposite orientations of the two currents: The reflection coefficient may also be established using other field or circuit pairs of quantities whose product defines power resolvable into a forward and reverse wave.

For instance, with electromagnetic plane waves, one uses the ratio of the electric fields of the reflected to that of the forward wave (or magnetic fields, again with a minus sign); the ratio of each wave's electric field E to its magnetic field H is again an impedance Z0 (equal to the impedance of free space in a vacuum).

In the accompanying figure, a signal source with internal impedance

possibly followed by a transmission line of characteristic impedance

is represented by its Thévenin equivalent, driving the load

then the source's maximum power is delivered to a load

Anywhere along an intervening (lossless) transmission line of characteristic impedance

will remain the same (the powers of the forward and reflected waves stay the same) but with a different phase.

This implies the reflected wave having a 180° phase shift (phase reversal) with the voltages of the two waves being opposite at that point and adding to zero (as a short circuit demands).

The reflection coefficient can also be measured at other points on the line.

The magnitude of the reflection coefficient in a lossless transmission line is constant along the line (as are the powers in the forward and reflected waves).

However its phase will be shifted by an amount dependent on the electrical distance

That is to take into account not only the phase delay of the reflected wave, but the phase shift that had first been applied to the forward wave, with the reflection coefficient being the quotient of these.

, corresponding to passive loads) may be displayed graphically using a Smith chart.

is given directly by the distance of a point to the center (with the edge of the Smith chart corresponding to

Its evolution along a transmission line is likewise described by a rotation of

Using the scales on a Smith chart, the resulting impedance (normalized to

The standing wave ratio (SWR) is determined solely by the magnitude of the reflection coefficient: Along a lossless transmission line of characteristic impedance Z0, the SWR signifies the ratio of the voltage (or current) maxima to minima (or what it would be if the transmission line were long enough to produce them).

, the SWR intentionally ignores the specific value of the load impedance ZL responsible for it, but only the magnitude of the resulting impedance mismatch.

That SWR remains the same wherever measured along a transmission line (looking towards the load) since the addition of a transmission line length to a load

While having a one-to-one correspondence with reflection coefficient, SWR is the most commonly used figure of merit in describing the mismatch affecting a radio antenna or antenna system.

Reflection coefficient is used in feeder testing for reliability of medium.

Typically, the reflectance is represented by a capital R, while the amplitude reflection coefficient is represented by a lower-case r. These related concepts are covered by Fresnel equations in classical optics.

Acousticians use reflection coefficients to understand the effect of different materials on their acoustic environments.