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. It is equal to the ratio of the amplitude of the reflected wave to the incident wave, with each expressed as phasors. 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. The reflection coefficient is closely related to the transmission coefficient. The reflectance of a system is also sometimes called a reflection coefficient.
Different specialties have different applications for the term.
See also: Reflections of signals on conducting lines and Signal reflection.
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. In terms of the forward and reflected waves determined by the voltage and current, the reflection coefficient is defined as the complex ratio of the voltage of the reflected wave (
V-
V+
\Gamma
\Gamma=
V- | |
V+ |
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:
\Gamma=-
I- | |
I+ |
=
V- | |
V+ |
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). Similarly in acoustics one uses the acoustic pressure and velocity respectively.
In the accompanying figure, a signal source with internal impedance
ZS
ZS
ZL
ZS
\Gamma
Z0=ZS
ZL=Z0
\Gamma=0
|\Gamma|2
1-|\Gamma|2
Anywhere along an intervening (lossless) transmission line of characteristic impedance
Z0
|\Gamma|
ZL=0
\Gamma=-1
The reflection coefficient is determined by the load impedance at the end of the transmission line, as well as the characteristic impedance of the line. A load impedance of
ZL
Z0
\Gamma={ZL-Z0\overZL+Z0}.
\phi
L
\phi=2\piL/λ
\Gamma'
\Gamma'=\Gammae-i
Note that the phase of the reflection coefficient is changed by twice the phase length of the attached transmission line. 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. The reflection coefficient so measured,
\Gamma'
ZL
The complex reflection coefficient (in the region
|\Gamma|\le1
\Gamma
\Gamma
|\Gamma|=1
2\phi
Z0
The standing wave ratio (SWR) is determined solely by the magnitude of the reflection coefficient:
SWR={1+|\Gamma|\over1-|\Gamma|}.
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 above calculation assumes that
\Gamma
\Gamma
ZL
\Gamma
See main article: Reflection seismology. Reflection coefficient is used in feeder testing for reliability of medium.
See main article: Fresnel equations. In optics and electromagnetics in general, reflection coefficient can refer to either the amplitude reflection coefficient described here, or the reflectance, depending on context. 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.