Reflection_coefficient

# Reflection coefficient The **reflection coefficient** is a dimensionless, generally complex number equal to the ratio of a reflected wave's amplitude to the amplitude of the wave incident on a boundary where the medium or circuit a wave is travelling through changes abruptly. Written Γ (gamma) and evaluated as a ratio of phasors at a single frequency, its magnitude reports what fraction of the incoming wave's amplitude returns from the boundary, and its phase reports how the returning wave is shifted relative to the wave that arrived. A three.js sketch further down this page follows Γ for a transmission line referenced to 50 ohms as its termination is changed. Every field that carries energy as a travelling wave meets the same problem at a boundary: part of the wave keeps going forward and part comes back the way it arrived, and the reflection coefficient is the standard way of keeping score of the returning part. On a [[Transmission_line|transmission line]] or in an [[Electrical_engineering|electrical]] network it is set by a mismatch between two impedances; in seismology and [[Acoustics|acoustics]] by a mismatch in acoustic impedance between two media; in optics and [[Microwave|microwave]] engineering by a mismatch in the wave impedance the two sides present. The formula keeps the same shape in every case; only which impedance is being compared changes. ## Transmission lines ### Relation to load impedance On a [[Transmission_line]] of characteristic impedance Z0 ending in a load Z_L, the reflection coefficient the load presents is `Gamma = (Z_L - Z0)/(Z_L + Z0)`. A load equal to Z0 is matched and reflects nothing, Γ = 0; an open circuit reflects the full voltage in phase, Γ = +1; a short circuit reflects it inverted, Γ = −1. For any passive load with non-negative resistance, |Γ| never exceeds 1, since a load cannot hand back more energy than the line delivered to it. Moving the measurement point back along a lossless line by an electrical length θ does not change |Γ|; it turns the phase, `Gamma_in = Gamma_L * exp(-j*2*theta)`, so the same reflection coefficient reappears every half wavelength back toward the source. Plotting Γ this way, as a point inside a unit circle, is exactly what a [[Smith_chart]] does, and its rim, |Γ| = 1, is the locus of every purely reactive, lossless termination.[^smith1939] ### Standing wave ratio A mismatched line carries both a forward wave and the reflected wave it produces, and the two add and cancel at fixed points along the line, building a standing pattern of voltage maxima and minima that does not move. The ratio of a maximum to the neighbouring minimum is the [[Standing_wave_ratio|standing wave ratio]], related to the reflection coefficient's magnitude by `VSWR = (1 + |Gamma|)/(1 - |Gamma|)`. VSWR is never below 1, reaches 1 only at a perfect match, and grows without bound as |Γ| approaches 1, which is what makes it a convenient quantity to read directly off a slotted line or a directional-coupler meter without ever measuring a phase. ## Electrical networks The same ratio applies anywhere a [[Signal|signal]] source of one impedance drives a load of another, whether or not either component sits on a distributed transmission line. An amplifier's input, a filter's port or an [[Antenna_(radio)|antenna]]'s feed point each has a reflection coefficient defined against whatever reference impedance the surrounding system uses, commonly 50 ohms in radio-frequency work. Network engineers usually report the same number logarithmically, as return loss in [[Decibel|decibels]], `RL = -20*log10(|Gamma|)`, so that a well-matched port with a small Γ shows a large, favourable return loss and a badly mismatched one shows a small or even negative figure. A return loss of 20 dB, for example, corresponds to |Γ| = 0.1: only one part in a hundred of the incident power returns, and the remaining fraction actually absorbed by the load is `1 - |Gamma|^2`, essentially all of it. In a multi-port network the reflection coefficient looking into one port, with every other port correctly terminated, is one entry of the network's scattering matrix, and the same mismatch that wastes forward power also determines how much of a neighbouring port's own signal can leak backward through the network rather than reach the port it was meant for.[^cn-sparam] A connector or a length of [[Coaxial_cable|coaxial cable]] built to the wrong characteristic impedance for the system around it adds a reflection of its own at each end, on top of whatever mismatch the components it joins already contribute, which is why a system's reference impedance is chosen once and held to for every part in the chain rather than fixed at only the source and the load. ## Seismology A downward-travelling seismic pulse meets a reflection coefficient at every boundary where the rock's acoustic impedance changes, `Gamma = (Z2 - Z1)/(Z2 + Z1)`, with each layer's impedance the product of its density and the seismic wave's speed in it. A sharp increase in density or stiffness, such as the boundary between ordinary sedimentary rock and a buried salt dome or a hard limestone bed, reflects a much larger share of the pulse's energy than a gradual change does, and the sign of Γ records whether the deeper layer is the stiffer one or the softer one. Reflection seismology builds a picture of what lies underground by recording the pattern of returning pulses at the surface and working backward from their timing and relative strength to the sequence of boundaries that produced them, an exploration technique that plays the same role for a geologist that pulsed ranging plays for [[Radar|radar]], with a column of rock standing in for open air and a seismic source standing in for a transmitter.[^cn-seismic] The simple formula above is exact only for a wave arriving straight on; a real seismic survey usually records reflections that arrive at an angle, where some of the energy also converts from a compressional wave into a shear wave at the same boundary and has to be tracked separately. Even so, the normal-incidence reflection coefficient is the number that tells a surveyor how strong a boundary's echo ought to be before the angle-dependent corrections are added, and a strong, sharp reflection in the recorded data is still read first as a sign of a sharp jump in acoustic impedance underground. ## Optics and microwaves At a boundary between two transparent media of refractive index n1 and n2, light meeting the interface head-on has an amplitude reflection coefficient `r = (n2 - n1)/(n2 + n1)`, and the fraction of power reflected is its square. For ordinary window glass in air, where the indices are about 1.5 and 1.0, this gives r ≈ 0.2 and a power reflectance of about 4 percent, close to what a clean glass surface is actually measured to return at normal incidence, which is why a stack of several glass plates dims a beam of light noticeably even though each surface reflects only a small share of it.[^cn-fresnel] The identical relation, with wave impedance standing in for refractive index, governs how much of a [[Microwave|microwave]] signal reflects at a connector, a waveguide transition or the boundary between two dielectric materials in a printed circuit board, which is why microwave design spends so much of its effort on keeping every such transition's two impedances as close together as the transmission-line case above. ## Acoustics Sound meeting a boundary between two media reflects according to the same form, with acoustic impedance, the product of a medium's density and its sound speed, standing in for the electrical or optical impedance used elsewhere on this page. Because a gas differs enormously in that product from a liquid or a solid, sound meeting an air-water or an air-tissue boundary reflects nearly all of its energy rather than crossing it, which is the reason a diver underwater hears almost nothing of a conversation in the air above, and the reason medical ultrasound equipment applies a coupling gel to remove the air gap that would otherwise reflect an ultrasound pulse straight back before it ever reached the body.[^cn-retloss] The same idea, at a much smaller impedance contrast, governs how a room sounds. A hard, rigid, smooth wall presents an acoustic impedance close enough to that of the air in the room that most of the [[Sound|sound]] energy reaching it reflects rather than being absorbed or transmitted through it, which is what makes a bare room with hard walls reverberant; a soft, porous material such as an acoustic panel is built to present a poorer impedance match to the air, so more of the same sound is absorbed on each reflection instead of bouncing back. Every echo, from a shout returning off a distant cliff to the reverberant tail in a concert hall, is this same partial reflection repeated at whatever surfaces the sound wave happens to meet. ## Microsims A three.js companion elsewhere on this page renders this same idea for a matched 50-ohm transmission line, following Γ as its termination departs from a perfect match. The fullest worked version of the underlying idea already lives on the [[Transmission_line]] page, whose primary sketch drives a lossless line terminated in an adjustable complex load and plots the resulting Γ as a moving point on a Smith chart while the line's electrical length is dragged. *Try:* in that sketch, set the load to a short or an open circuit and watch Γ jump out to the rim of the chart, then drag the line length toward a quarter wavelength and watch the input impedance the readout computes flip to its inverse. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Reflection_coefficient) : [Wikitube](https://en.wikitube.io/wiki/Reflection_coefficient) Skeleton mirrored at revision 1375113310. Prose, emphasis and the microsims are Wikitube's own. ## See also - [[Standing_wave_ratio]] - [[Smith_chart]] - [[Transmission_line]] - [[Coaxial_cable]] - [[Radar]] ## References [^smith1939]: Smith, P. H. "Transmission Line Calculator." *Electronics*, January 1939 (Bell Telephone Laboratories). Scan of the original: https://www.rfcafe.com/references/electronics-mag/transmission-line-calculator-phillip-h-smith-electronics-magazine-january-1939.htm . [^cn-sparam]: Citation needed: a primary source for the earliest formal treatment of a two-port network's input reflection coefficient as the S11 element of a scattering matrix. [^cn-seismic]: Citation needed: a primary source and date for the earliest practical use of acoustic-impedance reflection coefficients in reflection-seismology exploration. [^cn-fresnel]: Citation needed: a primary citation (memoir, journal and exact year) for Augustin-Jean Fresnel's derivation, in the 1820s, of the amplitude reflection and transmission coefficients at a dielectric interface. [^cn-retloss]: Citation needed: a primary source for when "return loss" became the standard decibel expression of |Γ| in radio-frequency engineering practice. ## External links - Smith, P. H. "Transmission Line Calculator." *Electronics*, January 1939: https://www.rfcafe.com/references/electronics-mag/transmission-line-calculator-phillip-h-smith-electronics-magazine-january-1939.htm