# Reflection factor

In physics, the term reflection factor (also reflection coefficient ) is the amplitude ratio between reflected and incident waves at the transition into another propagation medium.

The amplitude relates to the scalar or vector field size , for example the electrical voltage on a line , the pressure in sound or the electrical field strength in electromagnetic waves . The reflection factor is generally a complex quantity . Its amount indicates the proportion by which the reflected wave is weaker than the incident and its argument indicates which phase the reflected wave has with respect to the incident wave. The reflection factor depends on the angle of incidence. If a wave falls on an optically or acoustically denser medium, total reflection occurs for flat angles of incidence and the reflection factor is 1. In addition to the angle dependence, the reflection factor depends on the wave type: It is therefore different for longitudinal waves and transverse waves in acoustics and optics depending on the polarization of the wave. The latter is described by the Fresnel equations .

The amplitude ratio of the transmitted and incident waves is called the transmission factor . In order to calculate the energy transfer of the individual waves (incident, reflected, transmitted), the degree of reflection must be considered, which relates to the power or intensity of the wave. This is often given for an entire component instead of a single transition and can be strongly dependent on the wavelength due to interference .

## Reflection factor in line theory

When an electromagnetic wave of any shape is propagated along a (linear, homogeneous) line , a reflection occurs if the wave impedance of the line changes at a joint or if an interference point (e.g. a transverse resistance) is present on the line. In the case of a linear behavior of the point of impact or interference, a dimensionless reflection factor describes how the reflected voltage and current wave is generated from the incoming wave. The reflection factor is often represented in the literature by the symbols or . In contrast, the transmission factor (transmission coefficient, transition or refraction factor ) describes the proportion of the transmitted (passed) wave (provided the line does not end). Both factors generally depend on the direction in which the shaft traverses a joint. ${\ displaystyle \ Gamma}$${\ displaystyle r}$${\ displaystyle T}$

### Real reflection factor

If two lines with a real characteristic impedance (i.e. distortion-free or lossless lines) collide or if one ends and is terminated with an ohmic resistance, the reflection factor is calculated as the ratio of reflected voltage to incoming voltage using the following equation: ${\ displaystyle U _ {\ mathrm {r}}}$${\ displaystyle U _ {\ mathrm {h}}}$

${\ displaystyle \ Gamma = {\ frac {U _ {\ mathrm {r}}} {U _ {\ mathrm {h}}}} = - {\ frac {I _ {\ mathrm {r}}} {I _ {\ mathrm {h}}}} = {\ frac {Z _ {\ mathrm {w2}} -Z _ {\ mathrm {w1}}} {Z _ {\ mathrm {w2}} + Z _ {\ mathrm {w1}}}}, \ quad | \ Gamma | \ leq 1}$

It is the characteristic impedance before the jump point and the wave impedance to the jump point or the size of an ohmic resistance at the final line. The following borderline cases arise: ${\ displaystyle Z _ {\ mathrm {w1}}}$${\ displaystyle Z _ {\ mathrm {w2}}}$

${\ displaystyle Z _ {\ mathrm {w2}}}$ Reflection factor ${\ displaystyle \ Gamma}$ meaning
0 −1 Total reflection at the short-circuited end of a line
${\ displaystyle Z _ {\ mathrm {w1}}}$ 0 No reflection through adaptation
${\ displaystyle \ infty}$ 1 Total reflection at the open end of a line

The transmission factor can be calculated directly from the reflection factor: ${\ displaystyle T}$

${\ displaystyle T = {\ frac {2 \ cdot Z _ {\ mathrm {w2}}} {Z _ {\ mathrm {w2}} + Z _ {\ mathrm {w1}}}} = 1+ \ Gamma}$

### Complex reflection factor

If a line is operated with sinusoidal voltage, it is analyzed using the complex alternating current calculation . In this case, the reflection factor is defined as the ratio of the complex amplitudes of the reflected and incoming voltage wave and is calculated from the now generally complex wave or terminating resistances according to the same formula as the real reflection factor. In this case, however, it is itself a complex phasor that is dependent on the frequency and never becomes greater than 1 in terms of magnitude . His argument determines the phase jump of the reflected wave at the point of impact. If it is not 0, the so-called standing waves are created by the interference of the waves traveling back and forth . When a line is terminated with a pure reactance , the amount of the reflection factor is also equal to 1 and total reflection occurs in this case as well . In order to "present" its frequency dependence, the reflection factor can be represented as a locus curve .

If, for example, a line with the real characteristic impedance is terminated with a capacitance , the reflection factor is obtained ${\ displaystyle Z}$${\ displaystyle C}$

${\ displaystyle {\ underline {\ Gamma}} (j \ omega) = {\ frac {{\ frac {1} {j \ omega C}} - Z} {{\ frac {1} {j \ omega C} } + Z}} = {\ frac {1-j \ omega CZ} {1 + j \ omega CZ}} = e ^ {- 2j \ cdot \ arctan (\ omega CZ)}}$

### Generalized complex reflection factor

While the reflection factor is generally only precisely defined at the point of impact or interference, in the case of sinusoidal waves, its definition is generalized to the entire line as the ratio of the phasors of the returning and moving voltage wave at any point. One speaks of a transformation of the reflection factor . For this generalized reflection factor, the following applies to the distance from the joint ${\ displaystyle y}$

${\ displaystyle {\ underline {\ Gamma}} (y) = {\ underline {\ Gamma}} (0) \ cdot e ^ {\ pm 2y \ cdot \ gamma} = {\ underline {\ Gamma}} (0 ) \ cdot e ^ {\ pm 2y \ cdot \ alpha} \ cdot e ^ {\ pm 4 \ pi j {\ frac {y} {\ lambda}}}}$

It is the complex propagation constant of the line and not with the reflection factor to be confused. is the reflection factor at the junction, which is rotated in phase by the factor ( is the wavelength) with increasing distance and attenuated by the factor in the case of a lossy line ( depending on the direction of the wave). ${\ displaystyle \ gamma = \ alpha + j \ beta}$${\ displaystyle \ Gamma}$${\ displaystyle \ Gamma (0)}$${\ displaystyle e ^ {\ pm 4 \ pi j {\ frac {y} {\ lambda}}}}$${\ displaystyle \ lambda}$${\ displaystyle e ^ {\ pm 2y \ cdot \ alpha}}$${\ displaystyle \ pm}$

### The reflection factor as an operator

In general, the wave resistances and / or the element of the fault point have a reactance component. Then a non-sinusoidal reflected wave at the jump or interference point is not only changed in size compared to the incoming wave, but also " linearly distorted " in shape . Although the same calculation formulas also apply in this case, the reflection and transmission factors are complicated operators in the sense of an operator calculation and the calculations can generally only be carried out using numerical methods .

The above example of a line with the real characteristic impedance and a termination with the capacitance then gives the operator of the reflection factor with the complex frequency${\ displaystyle Z}$${\ displaystyle C}$ ${\ displaystyle s}$

${\ displaystyle {\ underline {\ Gamma}} (s) = {\ frac {{\ frac {1} {sC}} - Z} {{\ frac {1} {sC}} + Z}} = {\ frac {1-sCZ} {1 + sCZ}}}$

After multiplication with the image function of the incoming wave, one obtains the image function of the reflected wave, which ultimately has to be transformed back into a time function .

### Return loss

The term return loss is often used, especially when describing line properties . The return loss factor describes the ratio of transmitted power to reflected power. Since the power is proportional to the square of the magnitude of the field size like the voltage, the return loss factor can be expressed by the reflection factor: ${\ displaystyle R}$

${\ displaystyle R = {\ frac {P _ {\ mathrm {h}}} {P _ {\ mathrm {r}}}} = \ left | {\ frac {U _ {\ mathrm {h}}} {U _ {\ mathrm {r}}}} \ right | ^ {2} = {\ frac {1} {\ left | \ Gamma \ right | ^ {2}}}}$

If you take the logarithm of the return loss factor , you get the return loss measure , which is usually given in the pseudo unit decibel (dB): ${\ displaystyle a}$

{\ displaystyle {\ begin {aligned} a & = 10 \, \ mathrm {dB} \ cdot \ lg R \\ & = - 20 \, \ mathrm {dB} \ cdot \ lg \ left | \ Gamma \ right | \ \ & = - 20 \, \ mathrm {dB} \ cdot \ lg \ left | {\ frac {Z _ {\ mathrm {w2}} -Z _ {\ mathrm {w1}}} {Z _ {\ mathrm {w2}} + Z _ {\ mathrm {w1}}}} \ right | \ end {aligned}}}

### Standing wave ratio

In the case of sinusoidal waves on lossless lines, the relationship between the complex reflection factor and the standing wave ratio is given by ${\ displaystyle \ Gamma}$ ${\ displaystyle SWR}$

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

## Water waves

Reflection coefficient C ( f )
Reflection coefficient C ( x )

In the case of monochromatic water waves, the reflection coefficient is defined as the quotient of the height of the reflected wave and the height of the incoming wave . ${\ displaystyle H _ {\ mathrm {r}}}$${\ displaystyle H _ {\ mathrm {i}}}$

${\ displaystyle C _ {\ mathrm {r}} = H _ {\ mathrm {r}} / H _ {\ mathrm {i}} <1}$

It can be determined experimentally from the resulting water level deflections of the wave partially standing on a structure.

${\ displaystyle C _ {\ mathrm {r}} = {\ frac {H _ {\ mathrm {r}}} {H _ {\ mathrm {i}}}} = {\ frac {H _ {\ max} -H _ {\ min}} {H _ {\ max} + H _ {\ min}}}}$

Therein mean:

• ${\ displaystyle H _ {\ max} = H _ {\ mathrm {i}} + H _ {\ mathrm {r}}}$
• ${\ displaystyle H _ {\ min} = H _ {\ mathrm {i}} -H _ {\ mathrm {r}}}$

For the analysis of the frequency-dependent reflection of wave spectra offshore of a building, the extreme values ​​of the integrated energy density and can be used for defined frequency bands instead of the superimposed vertical water level deflections . ${\ displaystyle i}$ ${\ displaystyle E _ {\ max, i}}$${\ displaystyle E _ {\ min, i}}$

${\ displaystyle C _ {\ mathrm {r}, i} = {\ frac {{\ sqrt {E _ {\ max, i}}} - {\ sqrt {E _ {\ min, i}}}} {{\ sqrt {E _ {\ max, i}}} + {\ sqrt {E _ {\ min, i}}}}}}$

With

• ${\ displaystyle E _ {\ max, i}}$ = Amount of the energy maximum of the frequency components contributing to the partial wave at the antinode and
• ${\ displaystyle E _ {\ min, i}}$ = Amount of the energy minimum of the frequency components contributing to the partial wave at the oscillation node.

Taking into account the phase jump (phase difference between the incident and the reflected wave ) that occurs with partial reflection on inclined walls (embankments) , a complex reflection coefficient can be defined in such a way that it contains the phase shift in addition to the wave height ratio : ${\ displaystyle \ Delta \ varphi}$${\ displaystyle C _ {\ mathrm {r}} = H _ {\ mathrm {r}} / H _ {\ mathrm {i}}}$${\ displaystyle \ Delta \ varphi}$

${\ displaystyle \ Gamma = C _ {\ mathrm {r}} \ cdot e ^ {i \ Delta \ varphi}}$