Reflection coefficient

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In physics and electrical engineering the reflection coefficient is a parameter that describes how much of an electromagnetic 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. The reflection coefficient is closely related to the transmission coefficient . The reflectance of a system is also sometimes called a "reflection coefficient".

Physics Study of the fundamental properties of matter and energy

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Electrical engineering field of engineering that deals with electricity

Electrical engineering is a technical discipline concerned with the study, design and application of equipment, devices and systems which use electricity, electronics, and electromagnetism. It emerged as an identified activity in the latter half of the 19th century after commercialization of the electric telegraph, the telephone, and electrical power generation, distribution and use.

The amplitude of a periodic variable is a measure of its change over a single period. There are various definitions of amplitude, which are all functions of the magnitude of the difference between the variable's extreme values. In older texts the phase is sometimes called the amplitude.


A wave experiences partial transmittance and partial reflectance when the medium through which it travels suddenly changes. The reflection coefficient determines the ratio of the reflected wave amplitude to the incident wave amplitude. Partial transmittance.gif
A wave experiences partial transmittance and partial reflectance when the medium through which it travels suddenly changes. The reflection coefficient determines the ratio of the reflected wave amplitude to the incident wave amplitude.

Different specialties have different applications for the term.

Transmission lines

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 () to that of the incident wave (). This is typically represented with a (capital gamma) and can be written as:

Telecommunication Transmission of information between locations using electromagnetics

Telecommunication is the transmission of signs, signals, messages, words, writings, images and sounds or information of any nature by wire, radio, optical or other electromagnetic systems. Telecommunication occurs when the exchange of information between communication participants includes the use of technology. It is transmitted through a transmission media, such as over physical media, for example, over electrical cable, or via electromagnetic radiation through space such as radio or light. Such transmission paths are often divided into communication channels which afford the advantages of multiplexing. Since the Latin term communicatio is considered the social process of information exchange, the term telecommunications is often used in its plural form because it involves many different technologies.

Transmission line specialized cable or other structure designed to carry alternating current of radio frequency

In radio-frequency engineering, a transmission line is a specialized cable or other structure designed to conduct alternating current of radio frequency, that is, currents with a frequency high enough that their wave nature must be taken into account. Transmission lines are used for purposes such as connecting radio transmitters and receivers with their antennas, distributing cable television signals, trunklines routing calls between telephone switching centres, computer network connections and high speed computer data buses.

Ratio relationship between two numbers of the same kind

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six. Similarly, the ratio of lemons to oranges is 6:8 and the ratio of oranges to the total amount of fruit is 8:14.

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). Similarly in acoustics one uses the acoustic pressure and velocity respectively.

Electronic circuit electrical circuit with active components such as transistors, valves and integrated circuits; electrical network that contains active electronic components, generally nonlinear and require complex design and analysis tools

An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or traces through which electric current can flow. To be referred to as electronic, rather than electrical, generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another.

The impedance of free space, Z0, is a physical constant relating the magnitudes of the electric and magnetic fields of electromagnetic radiation travelling through free space. That is, Z0 = |E|/|H|, where |E| is the electric field strength and |H| is the magnetic field strength. Its presently accepted value is

Acoustics science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound

Acoustics is the branch of physics that deals with the study of all 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 while someone working in the field of acoustics technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of modern society with the most obvious being the audio and noise control industries.

Simple circuit configuration showing measurement location of reflection coefficient. Reflection Coefficient Circuit.svg
Simple circuit configuration showing measurement location of reflection coefficient.

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 . 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 thus reduced by .

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 corresponds directly to a specific impedance as seen at the point it is measured. A load impedance of (using a reference impedance ) corresponds to a reflection coefficient of


If that load, , were measured not directly but through a transmission line, then the magnitude of the reflection coefficient is identical (as are the powers in the forward and reflected waves). However its phase will have shifted according to

where is the electrical length (expressed as phase) of that length of transmission line at the frequency considered. 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 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 for this purpose.

Smith chart

The Smith chart, invented by Phillip H. Smith (1905–1987), is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assist in solving problems with transmission lines and matching circuits. The Smith chart can be used to simultaneously display multiple parameters including impedances, admittances, reflection coefficients, scattering parameters, noise figure circles, constant gain contours and regions for unconditional stability, including mechanical vibrations analysis. The Smith chart is most frequently used at or within the unity radius region. However, the remainder is still mathematically relevant, being used, for example, in oscillator design and stability analysis.


A nomogram, also called a nomograph, alignment chart or abaque, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Nomograms use a parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates.

Standing wave ratio

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 above calculation assumes that has been calculated using Z0 as the reference impedance. Since it uses only the magnitude of , 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 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 or antenna system. It is most often measured at the transmitter side of a transmission line, but having, as explained, the same value as would be measured at the antenna (load) itself.


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

Optics and microwaves

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.

See also

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