Reflection coefficient

This article is about reflections of waves. For the use of the term with capillary membrames, see Starling equation § Reflection coefficient. For intensity ratios, see Reflectance.

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.

A wave is partially transmitted and partially reflected 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 disciplines have different applications for the term.

Transmission lines

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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^-} ) to that of the incident wave (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V^+} ). This is typically represented with a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} (capital gamma) and can be written as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma = \frac{V^-}{V^+} }

It can also 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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. With electromagnetic plane waves, one uses the ratio of the electric fields of the reflected to that of the incident wave (or magnetic fields, again with a minus sign); the ratio of each wave's electric field E to its magnetic field H is the medium's characteristic impedance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0} , (equal to the impedance of free space if the medium is a vacuum).

Error creating thumbnail:

Simple circuit configuration showing measurement location of reflection coefficient

In the accompanying figure, a signal source with internal impedance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_S} possibly followed by a transmission line of characteristic impedance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_S} is represented by its Thévenin equivalent, driving the load Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_L} . For a real (resistive) source impedance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_S} , if we define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} using the reference impedance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0 = Z_S} then the source's maximum power is delivered to a load Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_L = Z_0} , in which case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma=0} implying no reflected power. More generally, the squared-magnitude of the reflection coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Gamma|^2} denotes the proportion of that power that is reflected back to the source, with the power actually delivered toward the load being Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-|\Gamma|^2} .

Anywhere along an intervening (lossless) transmission line of characteristic impedance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0} , the magnitude of the reflection coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_L=0} ), one finds Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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).

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 of the line. A load impedance of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_L} terminating a line with a characteristic impedance of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_0\,} will have a reflection coefficient of

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma ={Z_L-Z_0 \over Z_L+Z_0} .}

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi} from the load. If the coefficient is measured at a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} meters from the load, so the electrical distance from the load is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi = 2\pi L/\lambda} radians, the coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma'} at that point will be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma' =\Gamma e^{-i \, 2 \phi} }

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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma'} , corresponds to an impedance which is generally dissimilar to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_L} present at the far side of the transmission line.

The complex reflection coefficient (in the region Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \Gamma| \le 1} , corresponding to passive loads) may be displayed graphically using a Smith chart. The Smith chart is a polar plot of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , therefore the magnitude of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} is given directly by the distance of a point to the center (with the edge of the Smith chart corresponding to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Gamma|=1} ). Its evolution along a transmission line is likewise described by a rotation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\phi} around the chart's center. Using the scales on a Smith chart, the resulting impedance (normalized to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 reflected power in terms of the reflection coefficient is:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{reflected} = P_{incident} |\Gamma|^2} .

Standing wave ratio

Main article: Standing wave ratio

The standing wave ratio (SWR) is determined solely by the magnitude of the reflection coefficient:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} has been calculated using Z₀ as the reference impedance. Since it uses only the magnitude of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} , 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. 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_L} only changes the phase, not magnitude of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.

Electrical networks

A transmission line is an example of a 2-port electrical network, but reflection coefficients are useful in the analysis of any electrical networks. A reflection coefficient for each port in the same way as for the boundary of a transmission line. It will, however, also depend on the properties of connections at other ports and so is not a property intrinsic to the network itself. For a 2-port network with the 2x2 scattering matrix S, and with a source and load connected to its input and output, where the reflections off the source back into the input are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_S} and the reflections off the load back into the output are Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_L} , then the reflection coefficients at the input and output are given by:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Gamma_\mathrm{in}| = \left|S_{11} + \frac{S_{12}S_{21}\Gamma_L}{1-S_{22}\Gamma_L}\right|} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Gamma_\mathrm{out}| = \left|S_{22} + \frac{S_{12}S_{21}\Gamma_S}{1-S_{11}\Gamma_S} \right|}

Seismology

Main article: Reflection seismology

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

Optics and microwaves

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.

Acoustics

Main article: Acoustic wave § Reflection

Acousticians use reflection coefficients to understand the effect of different materials on their acoustic environments. The field properties used to define the reflection coefficient are typically the acoustic pressure and velocity in the incident and reflected acoustic waves.

See also

• Microwave
• Mismatch loss
• Reflections of signals on conducting lines
• Scattering parameters
• Transmission coefficient
• Target strength
• Hagen–Rubens relation
• Reflection phase change

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
• Pozar, David M. (2012). Microwave Electronics (Fourth ed.). John Wiley & Sons Inc. ISBN 9781118213636.

External links

• Flash tutorial for understanding reflection A 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. Archived 2020-11-25 at the Wayback Machine