In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
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
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
of the reflected wave to the incident wave, with each expressed as
phasor
In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
s. For example, it is used in
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
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
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
to calculate how much of the
electromagnetic wave
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visib ...
is reflected by an impedance discontinuity. The reflection coefficient is closely related to the ''
transmission coefficient
The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A transmission coefficient describes the amplitude, intensity, or total power of a transmitte ...
''. The
reflectance
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
of a system is also sometimes called a "reflection coefficient".
Different specialties have different applications for the term.
Transmission lines
In
telecommunication
Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
s and
transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
theory, the reflection coefficient is the
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
of the
complex amplitude
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
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 ''Z
0''. The reference impedance used is typically the
characteristic impedance
The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in ...
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
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
ratio of the voltage of the reflected wave (
) to that of the incident wave (
). This is typically represented with a
(capital
gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
) 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 ''Z''
0 (equal to the
impedance of free space The impedance of free space, , is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, , where is the electric field strength and is the magnetic fiel ...
in a vacuum). Similarly in
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
one uses the acoustic pressure and velocity respectively.
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 Thévenin is a surname. Notable people with the surname include:
*Charles Thévenin (1764–1838), neoclassical French painter
* Denis Thévenin, birth name of French author Georges Duhamel
*Léon Charles Thévenin (1857–1926), French engineer
* ...
, driving the load
. For a real (resistive) source impedance
, if we define
using the reference impedance
then the source's
maximum power is delivered to a load
, in which case
implying no reflected power. More generally, the squared-magnitude of the reflection coefficient
denotes the proportion of that power that is "reflected" and absorbed by the source, with the power actually delivered to the load being
.
Anywhere along an intervening (lossless) transmission line of characteristic impedance
, the magnitude of the reflection coefficient
will remain the same (the powers of the forward and reflected waves stay the same) but with a different phase. In the case of a short circuited load (
), one finds
at the load. 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).
Relation to load impedance
The reflection coefficient is determined by the load impedance at the end of the transmission line, as well as the
characteristic impedance
The characteristic impedance or surge impedance (usually written Z0) of a uniform transmission line is the ratio of the amplitudes of voltage and current of a single wave propagating along the line; that is, a wave travelling in one direction in ...
of the line. A load impedance of
terminating a line with a characteristic impedance of
will have a reflection coefficient of
:
This is the coefficient at the load. 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 from the load. If the coefficient is measured at a point
meters from the load, so the
electrical distance from the load is
radians, the coefficient
at that point will be
:
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,
, corresponds to an impedance which is generally dissimilar to
present at the far side of the transmission line.
The complex reflection coefficient (in the region
, corresponding to passive loads) may be displayed graphically using a
Smith chart
The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist ...
. The Smith chart is a polar plot of
, therefore the magnitude of
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
around the chart's center. Using the scales on a Smith chart, the resulting impedance (normalized to
) can directly be read. Before the advent of modern electronic computers, the Smith chart was of particular use as a sort of
analog computer
An analog computer or analogue computer is a type of computer that uses the continuous variation aspect of physical phenomena such as electrical, mechanical, or hydraulic quantities (''analog signals'') to model the problem being solved. In c ...
for this purpose.
Standing wave ratio
The
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 transmi ...
(SWR) is determined solely by the ''magnitude'' of the reflection coefficient:
:
Along a lossless transmission line of characteristic impedance ''Z''
0, 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
has been calculated using ''Z''
0 as the reference impedance. Since it uses only the ''magnitude'' of
, the SWR intentionally ignores the specific value of the load impedance ''Z
L'' responsible for it, but only the magnitude of the resulting
impedance mismatch
In electronics, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize signal ...
. 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
only changes the phase, not magnitude of
. 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
In radio engineering, an antenna or aerial is the interface between radio waves propagating through space and electric currents moving in metal conductors, used with a transmitter or receiver. In transmission, a radio transmitter supplies an ...
or antenna system. It is most often
measured
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared t ...
at the transmitter side of a transmission line, but having, as explained, the same value as would be measured at the antenna (load) itself.
Seismology
Reflection coefficient is used in feeder testing for reliability of medium.
Optics and microwaves
In
optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
and electromagnetics in general, "reflection coefficient" can refer to either the amplitude reflection coefficient described here, or the
reflectance
The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...
, 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
The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
in
classical optics
Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
.
Acoustics
Acousticians use reflection coefficients to understand the effect of different materials on their acoustic environments.
See also
*
Microwave
Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
*
Mismatch loss Mismatch loss in transmission line theory is the amount of power expressed in decibels that will not be available on the output due to impedance mismatches and signal reflections. A transmission line that is properly terminated, that is, terminate ...
*
Reflections of signals on conducting lines
A signal travelling along an electrical transmission line will be partly, or wholly, reflected back in the opposite direction when the travelling signal encounters a discontinuity in the characteristic impedance of the line, or if the far end ...
*
Scattering parameters
Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals.
The parameters are useful f ...
*
Transmission coefficient
The transmission coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A transmission coefficient describes the amplitude, intensity, or total power of a transmitte ...
*
Target strength
The target strength or '' acoustic size'' is a measure of the area of a sonar target. This is usually quantified as a number of decibels. For fish such as salmon, the target size varies with the length of the fish and a 5 cm fish could have a t ...
*
Hagen–Rubens relation
*
Reflection phase change A phase change sometimes occurs when a wave is reflected, specifically from a medium with faster wave speed to the boundary of a medium with slower wave speed. Such reflections occur for many types of wave, including light waves, sound waves, and w ...
References
*
* Figure 8-2 and Eqn. 8-1 Pg. 279
External links
Flash tutorial for understanding reflectionA flash program that shows how a reflected wave is generated, the reflection coefficient and VSWR
Application for drawing standing wave diagrams including the reflection coefficient, input impedance, SWR, etc.
{{Authority control
Geometrical optics
Electronic engineering
Physical optics
Seismology measurement
Telecommunication theory
Dimensionless numbers