# 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 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 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.

## Contents

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 (${\displaystyle V^{-}}$) to that of the incident wave (${\displaystyle V^{+}}$). This is typically represented with a ${\displaystyle \Gamma }$ (capital gamma) and can be written as:

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.

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.

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.

${\displaystyle \Gamma ={\frac {V^{-}}{V^{+}}}}$

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:

${\displaystyle \Gamma =-{\frac {I^{-}}{I^{+}}}={\frac {V^{-}}{V^{+}}}}$

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.

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 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.

In the accompanying figure, a signal source with internal impedance ${\displaystyle Z_{S}\,}$ possibly followed by a transmission line of characteristic impedance ${\displaystyle Z_{S}\,}$ is represented by its Thévenin equivalent, driving the load ${\displaystyle Z_{L}}$. For a real (resistive) source impedance ${\displaystyle Z_{S}}$, if we define ${\displaystyle \Gamma }$ using the reference impedance ${\displaystyle Z_{0}}$=${\displaystyle Z_{S}\,}$ then the source's maximum power is delivered to a load ${\displaystyle Z_{L}}$=${\displaystyle Z_{0}}$, in which case ${\displaystyle \Gamma =0}$ implying no reflected power. More generally, the squared-magnitude of the reflection coefficient ${\displaystyle |\Gamma |^{2}}$ 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 ${\displaystyle 1-|\Gamma |^{2}}$.

Anywhere along an intervening (lossless) transmission line of characteristic impedance ${\displaystyle Z_{0}}$, the magnitude of the reflection coefficient ${\displaystyle |\Gamma |}$ 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 (${\displaystyle Z_{L}=0}$), one finds ${\displaystyle \Gamma =-1}$ 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).

The reflection coefficient corresponds directly to a specific impedance as seen at the point it is measured. A load impedance of ${\displaystyle Z_{L}}$ (using a reference impedance ${\displaystyle Z_{0}\,}$) corresponds to a reflection coefficient of

${\displaystyle \Gamma ={Z_{L}-Z_{0} \over Z_{L}+Z_{0}}}$ .

If that load, ${\displaystyle Z_{L}}$, 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

${\displaystyle \Gamma '=\Gamma e^{-i\,2\phi }}$

where ${\displaystyle \phi }$ 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, ${\displaystyle \Gamma '}$, corresponds to an impedance which is generally dissimilar to ${\displaystyle Z_{L}}$ present at the far side of the transmission line.

The complex reflection coefficient (in the region ${\displaystyle |\Gamma |\leq 1}$, corresponding to passive loads) may be displayed graphically using a Smith chart. The Smith chart is a polar plot of ${\displaystyle \Gamma }$, therefore the magnitude of ${\displaystyle \Gamma }$ is given directly by the distance of a point to the center (with the edge of the Smith chart corresponding to ${\displaystyle |\Gamma |=1}$). Its evolution along a transmission line is likewise described by a rotation of ${\displaystyle 2\phi }$ around the chart's center. Using the scales on a Smith chart, the resulting impedance (normalized to ${\displaystyle Z_{0}}$) 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.

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:

${\displaystyle SWR={1+|\Gamma | \over 1-|\Gamma |}}$ .

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 ${\displaystyle \Gamma }$ has been calculated using Z0 as the reference impedance. Since it uses only the magnitude of ${\displaystyle \Gamma }$, 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 ${\displaystyle Z_{L}}$ only changes the phase, not magnitude of ${\displaystyle \Gamma }$. 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.

## Seismology

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.

## Acoustics

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

## Related Research Articles

The Fresnel equations describe the reflection and transmission of light when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresnel who was the first to understand that light is a transverse wave, even though no one realized that the "vibrations" of the wave were electric and magnetic fields. For the first time, polarization could be understood quantitatively, as Fresnel's equations correctly predicted the differing behaviour of waves of the s and p polarizations incident upon a material interface.

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 the absence of reflections in the other direction. Alternatively and equivalently it can be defined as the input impedance of a transmission line when its length is infinite. Characteristic impedance is determined by the geometry and materials of the transmission line and, for a uniform line, is not dependent on its length. The SI unit of characteristic impedance is the ohm.

The propagation constant of a sinusoidal electromagnetic wave is a measure of the change undergone by the amplitude and phase of the wave as it propagates in a given direction. The quantity being measured can be the voltage, the current in a circuit, or a field vector such as electric field strength or flux density. The propagation constant itself measures the change per unit length, but it is otherwise dimensionless. In the context of two-port networks and their cascades, propagation constant measures the change undergone by the source quantity as it propagates from one port to the next.

In telecommunications, return loss is the loss of power in the signal returned/reflected by a discontinuity in a transmission line or optical fiber. This discontinuity can be a mismatch with the terminating load or with a device inserted in the line. It is usually expressed as a ratio in decibels (dB);

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 transmission line, and SWR is defined as the ratio of the partial standing wave's amplitude at an antinode (maximum) to the amplitude at a node (minimum) along the line.

A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting expansion to one dimension or two. There is a similar effect in water waves constrained within a canal, or guns that have barrels which restrict hot gas expansion to maximize energy transfer to their bullets. Without the physical constraint of a waveguide, wave amplitudes decrease according to the inverse square law as they expand into three dimensional space.

Electrical impedance is the measure of the opposition that a circuit presents to a current when a voltage is applied. The term complex impedance may be used interchangeably.

In electronics, impedance matching is the practice of designing the input impedance of an electrical load or the output impedance of its corresponding signal source to maximize the power transfer or minimize signal reflection from the load.

The input impedance of an electrical network is the measure of the opposition to current (impedance), both static (resistance) and dynamic (reactance), into the load network that is external to the electrical source. The input admittance (1/impedance) is a measure of the load's propensity to draw current. The source network is the portion of the network that transmits power, and the load network is the portion of the network that consumes power.

Scattering parameters or S-parameters describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals.

The SWR meter or VSWR meter measures the standing wave ratio in a transmission line. The meter can be used to indicate the degree of mismatch between a transmission line and its load, or evaluate the effectiveness of impedance matching efforts.

The telegrapher's equations are a pair of coupled, linear partial differential equations that describe the voltage and current on an electrical transmission line with distance and time. The equations come from Oliver Heaviside who in the 1880s developed the transmission line model. The model demonstrates that the electromagnetic waves can be reflected on the wire, and that wave patterns can appear along the line. The theory applies to transmission lines of all frequencies including high-frequency transmission lines, audio frequency, low frequency and direct current.

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 transmitted wave relative to an incident wave.

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, terminated with the same impedance as that of the characteristic impedance of the transmission line, will have no reflections and therefore no mismatch loss. Mismatch loss represents the amount of power wasted in the system. It can also be thought of as the amount of power gained if the system was perfectly matched. Impedance matching is an important part of RF system design; however, in practice there will likely be some degree of mismatch loss. In real systems, relatively little loss is due to mismatch loss and is often on the order of 1dB.

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 of the line is not terminated in its characteristic impedance. This can happen, for instance, if two lengths of dissimilar transmission lines are joined together.

Metal-mesh optical filters are optical filters made from stacks of metal meshes and dielectric. They are used as part of an optical path to filter the incoming light to allow frequencies of interest to pass while reflecting other frequencies of light.

Slotted lines are used for microwave measurements and consist of a movable probe inserted into a slot in a transmission line. They are used in conjunction with a microwave power source and usually, in keeping with their low-cost application, a low cost Schottky diode detector and VSWR meter rather than an expensive microwave power meter.

Electric power transmission is the bulk movement of electrical energy from a generating site, such as a power plant, to an electrical substation. The interconnected lines which facilitate this movement are known as a transmission network. This is distinct from the local wiring between high-voltage substations and customers, which is typically referred to as electric power distribution. The combined transmission and distribution network is known as the "power grid" in North America, or just "the grid". In the United Kingdom, India, Tanzania, Myanmar, Malaysia and New Zealand, the network is known as the "National Grid".

## References

•  This article incorporates  public domain material from the General Services Administration document "Federal Standard 1037C" (in support of MIL-STD-188 ).
• Bogatin, Eric (2004). Signal Integrity - Simplified. Upper Saddle River, New Jersey: Pearson Education, Inc. ISBN   0-13-066946-6. Figure 8-2 and Eqn. 8-1 Pg. 279