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Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
electrical network An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage sources, c ...
s when undergoing various
steady state In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
stimuli by electrical signals. The parameters are useful for several branches of
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, including
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
,
communication systems A communications system or communication system is a collection of individual telecommunications networks, transmission systems, relay stations, tributary stations, and terminal equipment usually capable of interconnection and interoperatio ...
design, and especially for
microwave engineering Microwave engineering pertains to the study and design of microwave circuits, components, and systems. Fundamental principles are applied to analysis, design and measurement techniques in this field. The short wavelengths involved distinguish this ...
. The S-parameters are members of a family of similar parameters, other examples being:
Y-parameters Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the el ...
,
Z-parameters Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electr ...
,
H-parameters A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
, T-parameters or
ABCD-parameters A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
. They differ from these, in the sense that ''S-parameters'' do not use open or short circuit conditions to characterize a linear electrical network; instead, matched loads are used. These
termination Termination may refer to: Science *Termination (geomorphology), the period of time of relatively rapid change from cold, glacial conditions to warm interglacial condition *Termination factor, in genetics, part of the process of transcribing RNA ...
s are much easier to use at high signal frequencies than open-circuit and short-circuit terminations. Contrary to popular belief, the quantities are not measured in terms of power (except in now-obsolete six-port network analyzers). Modern vector network analyzers measure amplitude and phase of voltage traveling wave
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
s using essentially the same circuit as that used for the demodulation of digitally modulated wireless signals. Many electrical properties of networks of components (
inductors An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a ...
,
capacitors A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a ...
,
resistors A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
) may be expressed using S-parameters, such as
gain Gain or GAIN may refer to: Science and technology * Gain (electronics), an electronics and signal processing term * Antenna gain * Gain (laser), the amplification involved in laser emission * Gain (projection screens) * Information gain in de ...
,
return loss In telecommunications, return loss is a measure in relative terms of the power of the signal reflected by a discontinuity in a transmission line or optical fiber. This discontinuity can be caused by a mismatch between the termination or load conne ...
,
voltage standing wave ratio 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 trans ...
(VSWR),
reflection coefficient 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 wa ...
and
amplifier An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It may increase the power significantly, or its main effect may be to boost the v ...
stability. The term 'scattering' is more common to
optical engineering Optical engineering is the field of science and engineering encompassing the physical phenomena and technologies associated with the generation, transmission, manipulation, detection, and utilization of light. Optical engineers use optics to solve ...
than RF engineering, referring to the effect observed when a plane electromagnetic wave is incident on an obstruction or passes across dissimilar
dielectric In electromagnetism, a dielectric (or dielectric medium) is an electrical insulator that can be polarised by an applied electric field. When a dielectric material is placed in an electric field, electric charges do not flow through the mate ...
media. In the context of S-parameters, scattering refers to the way in which the traveling
currents Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
and
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defi ...
ages in a
transmission line In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmis ...
are affected when they meet a discontinuity caused by the insertion of a network into the transmission line. This is equivalent to the wave meeting an impedance differing from the line's
characteristic impedance 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 ...
. Although applicable at any
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, S-parameters are mostly used for networks operating at
radio frequency Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upp ...
(RF) and
microwave Microwave is a form of electromagnetic radiation with wavelengths ranging from about one meter to one millimeter corresponding to frequencies between 300 MHz and 300 GHz respectively. Different sources define different frequency ran ...
frequencies where signal power and energy considerations are more easily quantified than currents and voltages. S-parameters change with the measurement frequency, so frequency must be specified for any S-parameter measurements stated, in addition to the
characteristic impedance 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 ...
or system impedance. S-parameters are readily represented in
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
form and obey the rules of matrix algebra.


Background

The first published description of S-parameters was in the thesis of
Vitold Belevitch Vitold Belevitch (2 March 1921 – 26 December 1999) was a Belgian mathematician and electrical engineer of Russian origin who produced some important work in the field of electrical network theory. Born to parents fleeing the Bolsheviks, he ...
in 1945. The name used by Belevitch was ''repartition matrix'' and limited consideration to lumped-element networks. The term ''scattering matrix'' was used by physicist and engineer
Robert Henry Dicke Robert Henry Dicke (; May 6, 1916 – March 4, 1997) was an American astronomer and physicist who made important contributions to the fields of astrophysics, atomic physics, physical cosmology, cosmology and gravity. He was the Albert Einstein ...
in 1947 who independently developed the idea during wartime work on radar. In these S-parameters and scattering matrices, the scattered waves are the so-called traveling waves. A different kind of S-parameters was introduced in the 1960s. The latter was popularized by Kaneyuki Kurokawa, who referred to the new scattered waves as 'power waves.' The two types of S-parameters have very different properties and must not be mixed up. In his seminal paper, Kurokawa clearly distinguishes the power-wave S-parameters and the conventional, traveling-wave S-parameters. A variant of the latter is the pseudo-traveling-wave S-parameters. In the S-parameter approach, an electrical network is regarded as a '
black box In science, computing, and engineering, a black box is a system which can be viewed in terms of its inputs and outputs (or transfer characteristics), without any knowledge of its internal workings. Its implementation is "opaque" (black). The te ...
' containing various interconnected basic electrical circuit components or
lumped elements The lumped-element model (also called lumped-parameter model, or lumped-component model) simplifies the description of the behaviour of spatially distributed physical systems, such as electrical circuits, into a topology consisting of discrete e ...
such as resistors, capacitors, inductors and transistors, which interacts with other circuits through ''ports''. The network is characterized by a square
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s called its S-parameter matrix, which can be used to calculate its response to signals applied to the ports. For the S-parameter definition, it is understood that a network may contain any components provided that the entire network behaves
linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
ly with incident small signals. It may also include many typical communication system components or 'blocks' such as
amplifier An amplifier, electronic amplifier or (informally) amp is an electronic device that can increase the magnitude of a signal (a time-varying voltage or current). It may increase the power significantly, or its main effect may be to boost the v ...
s, attenuators,
filters Filter, filtering or filters may refer to: Science and technology Computing * Filter (higher-order function), in functional programming * Filter (software), a computer program to process a data stream * Filter (video), a software component tha ...
,
couplers Coupler may refer to: Engineering Mechanical * Railway coupler, a mechanism for connecting rolling stock in a train device ** Janney coupler ** SA3 coupler ** Scharfenberg coupler for multiple unit passenger cars * Quick coupler, used in constru ...
and equalizers provided they are also operating under linear and defined conditions. An electrical network to be described by S-parameters may have any number of ports. Ports are the points at which electrical signals either enter or exit the network. Ports are usually pairs of terminals with the requirement that the
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
into one terminal is equal to the current leaving the other. S-parameters are used at frequencies where the ports are often
coaxial In geometry, coaxial means that several three-dimensional linear or planar forms share a common axis. The two-dimensional analog is ''concentric''. Common examples: A coaxial cable is a three-dimensional linear structure. It has a wire conduc ...
or
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
connections. The S-parameter
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
describing an ''N''-port network will be square of dimension ''N'' and will therefore contain N^2\, elements. At the test frequency each element or S-parameter is represented by a unitless
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
that represents
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
and
angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...
, i.e.
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
and
phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...
. The complex number may either be expressed in
rectangular In Euclidean geometry, Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a par ...
form or, more commonly, in
polar Polar may refer to: Geography Polar may refer to: * Geographical pole, either of two fixed points on the surface of a rotating body or planet, at 90 degrees from the equator, based on the axis around which a body rotates * Polar climate, the c ...
form. The S-parameter magnitude may be expressed in linear form or
logarithmic form In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne. Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' ...
. When expressed in logarithmic form, magnitude has the "
dimensionless unit A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
" of
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
s. The S-parameter angle is most frequently expressed in degrees but occasionally in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s. Any S-parameter may be displayed graphically on a polar diagram by a dot for one frequency or a
locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...
for a range of frequencies. If it applies to one port only (being of the form S_\,), it may be displayed on an impedance or admittance
Smith Chart The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assis ...
normalised to the system impedance. The Smith Chart allows simple conversion between the S_\, parameter, equivalent to the voltage reflection coefficient and the associated (normalised) impedance (or admittance) 'seen' at that port. The following information must be defined when specifying a set of S-parameters: #The frequency #The nominal characteristic impedance (often 50 Ω) #The allocation of port numbers #Conditions which may affect the network, such as temperature, control voltage, and bias current, where applicable.


The power-wave S-parameter matrix


A definition

For a generic multi-port network, the ports are numbered from 1 to ''N'', where ''N'' is the total number of ports. For port ''i'', the associated S-parameter definition is in terms of incident and reflected 'power waves', a_i\, and b_i\, respectively. Kurokawa defines the incident power wave for each port as :a_i = \frac\, k_i (V_i + Z_ I_i)\, and the reflected wave for each port is defined as :b_i = \frac\, k_i (V_i - Z_^ I_i)\, where Z_i\, is the impedance for port ''i'', Z_i^\, is the complex conjugate of Z_i\,, V_i\, and I_i\, are respectively the complex amplitudes of the voltage and current at port ''i'', and :k_i = \left(\sqrt\right)^\, Sometimes it is useful to assume that the reference impedance is the same for all ports in which case the definitions of the incident and reflected waves may be simplified to :a_i = \frac\, \frac\, and :b_i = \frac\, \frac\, Note that as was pointed out by Kurokawa himself, the above definitions of a_i and b_i are not unique. The relation between the vectors a and b, whose ''i''-th components are the power waves a_i and b_i respectively, can be expressed using the S-parameter matrix S: :\mathbf = \mathbf \mathbf\, Or using explicit components: : \begin b_1 \\ \vdots \\ b_n \end = \begin S_ & \dots &S_ \\ \vdots &\ddots &\vdots \\ S_ & \dots &S_ \end \begin a_1 \\ \vdots \\ a_n \end


Reciprocity

A network will be
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
if it is
passive Passive may refer to: * Passive voice, a grammatical voice common in many languages, see also Pseudopassive * Passive language, a language from which an interpreter works * Passivity (behavior), the condition of submitting to the influence of on ...
and it contains only reciprocal materials that influence the transmitted signal. For example, attenuators, cables, splitters and combiners are all reciprocal networks and S_ = S_\, in each case, or the S-parameter matrix will be equal to its
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...
. Networks which include non-reciprocal materials in the transmission medium such as those containing magnetically biased ferrite components will be non-reciprocal. An amplifier is another example of a non-reciprocal network. A property of 3-port networks, however, is that they cannot be simultaneously reciprocal, loss-free, and perfectly matched.


Lossless networks

A lossless network is one which does not dissipate any power, or: \Sigma\left, a_n\^2 = \Sigma\left, b_n\^2\,. The sum of the incident powers at all ports is equal to the sum of the reflected powers at all ports. This implies that the S-parameter matrix is
unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigrou ...
, that is (S)^H (S) = (I)\,, where (S)^H\, is the
conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex con ...
of (S)\, and (I)\, is the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial o ...
.


Lossy networks

A
lossy In information technology, lossy compression or irreversible compression is the class of data compression methods that uses inexact approximations and partial data discarding to represent the content. These techniques are used to reduce data size ...
passive network is one in which the sum of the incident powers at all ports is greater than the sum of the reflected powers at all ports. It therefore dissipates power: \Sigma\left, a_n\^2 \ne \Sigma\left, b_n\^2\,. Thus \Sigma\left, a_n\^2 > \Sigma\left, b_n\^2\,, and (I) - (S)^H (S)\, is
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: * Positive-definite bilinear form * Positive-definite f ...
.


Two-port S-parameters

The S-parameter matrix for the 2-port network is probably the most commonly used and serves as the basic building block for generating the higher order matrices for larger networks. In this case the relationship between the reflected, incident power waves and the S-parameter matrix is given by: :\beginb_1 \\ b_2 \end = \begin S_ & S_ \\ S_ & S_ \end\begin a_1 \\ a_2 \end\,. Expanding the matrices into equations gives: :b_1 = S_a_1 +S_a_2\, and :b_2 = S_a_1 +S_a_2\,. Each equation gives the relationship between the reflected and incident power waves at each of the network ports, 1 and 2, in terms of the network's individual S-parameters, S_\,, S_\,, S_\, and S_\,. If one considers an incident power wave at port 1 (a_1\,) there may result from it waves exiting from either port 1 itself (b_1\,) or port 2 (b_2\,). However, if, according to the definition of S-parameters, port 2 is terminated in a load identical to the system impedance (Z_0\,) then, by the
maximum power transfer theorem In electrical engineering, the maximum power transfer theorem states that, to obtain ''maximum'' external power from a power source with internal resistance, the resistance of the load must equal the resistance of the source as viewed from its out ...
, b_2\, will be totally absorbed making a_2\, equal to zero. Therefore, defining the incident voltage waves as a_1 = V_1^+ and a_2 = V_2^+ with the reflected waves being b_1 = V_1^- and b_2 = V_2^-, :S_ = \frac = \frac and S_ = \frac = \frac\,. Similarly, if port 1 is terminated in the system impedance then a_1\, becomes zero, giving :S_ = \frac = \frac\, and S_ = \frac = \frac\, The 2-port S-parameters have the following generic descriptions: :S_\, is the input port voltage reflection coefficient :S_\, is the reverse voltage gain :S_\, is the forward voltage gain :S_\, is the output port voltage reflection coefficient. If, instead of defining the voltage wave direction relative to each port, they are defined by their absolute direction as forward V^+ and reverse V^- waves then b_2 = V_2^+ and a_1 = V_1^+. The S-parameters then take on a more intuitive meaning such as the forward voltage gain being defined by the ratio of the forward voltages S_ = V_2^+/V_1^+. Using this, the above matrix may be expanded in a more practical way :V_1^-= S_V_1^+ +S_V_2^-\, :V_2^+ = S_V_1^+ +S_V_2^-\,


S-parameter properties of 2-port networks

An amplifier operating under linear (small signal) conditions is a good example of a non-reciprocal network and a matched attenuator is an example of a reciprocal network. In the following cases we will assume that the input and output connections are to ports 1 and 2 respectively which is the most common convention. The nominal system impedance, frequency and any other factors which may influence the device, such as temperature, must also be specified.


Complex linear gain

The complex linear gain G is given by :G = S_ = \frac\,. That is the linear ratio of the output reflected power wave divided by the input incident power wave, all values expressed as complex quantities. For lossy networks it is sub-unitary, for active networks , G, > 1 . It will be equal with the voltage gain only when the device has equal input and output impedances.


Scalar linear gain

The scalar linear gain (or linear gain magnitude) is given by :\left, G\ = \left, S_\\,. This represents the gain magnitude (absolute value), the ratio of the output power-wave to the input power-wave, and it equals the square-root of the power gain. This is a real-value (or scalar) quantity, the phase information being dropped.


Scalar logarithmic gain

The scalar logarithmic (decibel or dB) expression for gain (g) is: :g = 20\log_\left, S_\\, dB. This is more commonly used than scalar linear gain and a positive quantity is normally understood as simply a "gain", while a negative quantity is a "negative gain" (a "loss"), equivalent to its magnitude in dB. For example, at 100 MHz, a 10 m length of cable may have a gain of −1 dB, equal to a loss of 1 dB.


Insertion loss

In case the two measurement ports use the same reference impedance, the insertion loss () is the reciprocal of the magnitude of the transmission coefficient expressed in decibels. It is thus given by: IL = -20\log_\left, S_\\, dB. It is the extra loss produced by the introduction of the
device under test A device under test (DUT), also known as equipment under test (EUT) and unit under test (UUT), is a manufactured product undergoing testing, either at first manufacture or later during its life cycle as part of ongoing functional testing and calibra ...
(DUT) between the 2 reference planes of the measurement. The extra loss may be due to intrinsic loss in the DUT and/or mismatch. In case of extra loss the insertion loss is defined to be positive. The negative of insertion loss expressed in decibels is defined as insertion gain and is equal to the scalar logarithmic gain (see: definition above).


Input return loss

Input
return loss In telecommunications, return loss is a measure in relative terms of the power of the signal reflected by a discontinuity in a transmission line or optical fiber. This discontinuity can be caused by a mismatch between the termination or load conne ...
() can be thought of as a measure of how close the actual input impedance of the network is to the nominal system impedance value. Input return loss expressed in
decibel The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals whose levels differ by one decibel have a po ...
s is given by :RL_\mathrm = 10\log_\left, \frac \ = - 20\log_ \left, S_\\, dB. Note that for passive two-port networks in which , it follows that return loss is a non-negative quantity: . Also note that somewhat confusingly,
return loss In telecommunications, return loss is a measure in relative terms of the power of the signal reflected by a discontinuity in a transmission line or optical fiber. This discontinuity can be caused by a mismatch between the termination or load conne ...
is sometimes used as the negative of the quantity defined above, but this usage is, strictly speaking, incorrect based on the definition of loss.Trevor S. Bird
"Definition and Misuse of Return Loss"
''IEEE Antennas & Propagation Magazine'', vol.51, iss.2, pp.166–167, April 2009.


Output return loss

The output return loss () has a similar definition to the input return loss but applies to the output port (port 2) instead of the input port. It is given by :RL_\mathrm = - 20\log_\left, S_\\, dB.


Reverse gain and reverse isolation

The scalar logarithmic (decibel or dB) expression for reverse gain (g_\mathrm\,) is: :g_\mathrm = 20\log_\left, S_\\, dB. Often this will be expressed as reverse isolation (I_\mathrm\,) in which case it becomes a positive quantity equal to the magnitude of g_\mathrm\, and the expression becomes: :I_\mathrm = \left, g_\mathrm\ = \left, 20\log_\left, S_\\\, dB.


Reflection coefficient

The reflection coefficient at the input port (\Gamma_\mathrm\,) or at the output port (\Gamma_\mathrm\,) are equivalent to S_\, and S_\, respectively, so :\Gamma_\mathrm = S_\, and \Gamma_\mathrm = S_\,. As S_\, and S_\, are complex quantities, so are \Gamma_\mathrm\, and \Gamma_\mathrm\,. The reflection coefficients are complex quantities and may be graphically represented on polar diagrams or Smith Charts See also the
Reflection Coefficient 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 wa ...
article.


Voltage standing wave ratio

The voltage standing wave ratio (VSWR) at a port, represented by the lower case 's', is a similar measure of port match to return loss but is a scalar linear quantity, the ratio of the standing wave maximum voltage to the standing wave minimum voltage. It therefore relates to the magnitude of the voltage reflection coefficient and hence to the magnitude of either S_\, for the input port or S_\, for the output port. At the input port, the VSWR (s_\mathrm\,) is given by :s_\mathrm = \frac\, At the output port, the VSWR (s_\mathrm\,) is given by :s_\mathrm = \frac\, This is correct for reflection coefficients with a magnitude no greater than unity, which is usually the case. A reflection coefficient with a magnitude greater than unity, such as in a tunnel diode amplifier, will result in a negative value for this expression. VSWR, however, from its definition, is always positive. A more correct expression for port ''k'' of a multiport is; :s_k = \frac\,


4-port S-parameters

4 Port S Parameters are used to characterize 4 port networks. They include information regarding the reflected and incident power waves between the 4 ports of the network. :\beginS_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ \\ S_ & S_ & S_ & S_ \end They are commonly used to analyze a pair of coupled transmission lines to determine the amount of cross-talk between them, if they are driven by two separate single ended signals, or the reflected and incident power of a differential signal driven across them. Many specifications of high speed differential signals define a communication channel in terms of the 4-Port S-Parameters, for example the 10-Gigabit Attachment Unit Interface (XAUI), SATA, PCI-X, and InfiniBand systems.


4-port mixed-mode S-parameters

4-port mixed-mode S-parameters characterize a 4-port network in terms of the response of the network to common mode and differential stimulus signals. The following table displays the 4-port mixed-mode S-parameters. Note the format of the parameter notation SXYab, where "S" stands for scattering parameter or S-parameter, "X" is the response mode (differential or common), "Y" is the stimulus mode (differential or common), "a" is the response (output) port and b is the stimulus (input) port. This is the typical nomenclature for scattering parameters. The first quadrant is defined as the upper left 4 parameters describing the differential stimulus and differential response characteristics of the device under test. This is the actual mode of operation for most high-speed differential interconnects and is the quadrant that receives the most attention. It includes input differential return loss (SDD11), input differential insertion loss (SDD21), output differential return loss (SDD22) and output differential insertion loss (SDD12). Some benefits of differential signal processing are; * reduced electromagnetic interference susceptibility * reduction in electromagnetic radiation from balanced differential circuit * even order differential distortion products transformed to common mode signals * factor of two increase in voltage level relative to single-ended * rejection to common mode supply and ground noise encoding onto differential signal The second and third quadrants are the upper right and lower left 4 parameters respectively. These are also referred to as the cross-mode quadrants. This is because they fully characterize any mode conversion occurring in the device under test, whether it is common-to-differential SDCab conversion (EMI susceptibility for an intended differential signal SDD transmission application) or differential-to-common SCDab conversion (EMI radiation for a differential application). Understanding mode conversion is very helpful when trying to optimize the design of interconnects for gigabit data throughput. The fourth quadrant is the lower right 4 parameters and describes the performance characteristics of the common-mode signal SCCab propagating through the device under test. For a properly designed SDDab differential device there should be minimal common-mode output SCCab. However, the fourth quadrant common-mode response data is a measure of common-mode transmission response and used in a ratio with the differential transmission response to determine the network common-mode rejection. This common mode rejection is an important benefit of differential signal processing and can be reduced to one in some differential circuit implementations.


S-parameters in amplifier design

The reverse isolation parameter S_\, determines the level of feedback from the output of an amplifier to the input and therefore influences its stability (its tendency to refrain from oscillation) together with the forward gain S_\,. An amplifier with input and output ports perfectly isolated from each other would have infinite scalar log magnitude isolation or the linear magnitude of S_\, would be zero. Such an amplifier is said to be unilateral. Most practical amplifiers though will have some finite isolation allowing the reflection coefficient 'seen' at the input to be influenced to some extent by the load connected on the output. An amplifier which is deliberately designed to have the smallest possible value of \left, S_\\, is often called a
buffer amplifier A buffer amplifier (sometimes simply called a buffer) is one that provides electrical impedance transformation from one circuit to another, with the aim of preventing the signal source from being affected by whatever currents (or voltages, for a cu ...
. Suppose the output port of a real (non-unilateral or bilateral) amplifier is connected to an arbitrary load with a reflection coefficient of \rho_\,. The actual reflection coefficient 'seen' at the input port \rho_\mathrm\, will be given by :\rho_\mathrm = S_ + \frac\,. If the amplifier is unilateral then S_ = 0\, and \rho_\mathrm = S_\, or, to put it another way, the output loading has no effect on the input. A similar property exists in the opposite direction, in this case if \rho_\mathrm\, is the reflection coefficient seen at the output port and \rho_\, is the reflection coefficient of the source connected to the input port. :\rho_ = S_ + \frac\,


Port loading conditions for an amplifier to be unconditionally stable

An amplifier is unconditionally stable if a load or source of ''any'' reflection coefficient can be connected without causing instability. This condition occurs if the magnitudes of the reflection coefficients at the source, load and the amplifier's input and output ports are simultaneously less than unity. An important requirement that is often overlooked is that the amplifier be a linear network with no poles in the right half plane. Instability can cause severe distortion of the amplifier's gain frequency response or, in the extreme, oscillation. To be unconditionally stable at the frequency of interest, an amplifier must satisfy the following 4 equations simultaneously: :\left, \rho_s\ < 1\, :\left, \rho_L\ < 1\, :\left, \rho_\mathrm\ < 1\, :\left, \rho_\mathrm\ < 1\, The boundary condition for when each of these values is equal to unity may be represented by a circle drawn on the polar diagram representing the (complex) reflection coefficient, one for the input port and the other for the output port. Often these will be scaled as Smith Charts. In each case coordinates of the circle centre and the associated radius are given by the following equations:


\rho_ values for , \rho_\text, = 1 (output stability circle)

Radius r_L = \left, \frac \. Center c_L = \frac.


\rho_ values for , \rho_\text, = 1 (input stability circle)

Radius r_s = \left, \frac \. Center c_s = \frac. In both cases : \Delta = S_ S_ - S_ S_, and the superscript star (*) indicates a
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
. The circles are in complex units of reflection coefficient so may be drawn on impedance or admittance based Smith charts normalised to the system impedance. This serves to readily show the regions of normalised impedance (or admittance) for predicted unconditional stability. Another way of demonstrating unconditional stability is by means of the Rollett stability factor (K), defined as : K = \frac. The condition of unconditional stability is achieved when K > 1 and , \Delta, < 1.


Scattering transfer parameters

The Scattering transfer parameters or T-parameters of a 2-port network are expressed by the T-parameter matrix and are closely related to the corresponding S-parameter matrix. However, unlike S parameters, there is no simple physical means to measure the T parameters in a system, sometimes referred to as Youla waves. The T-parameter matrix is related to the incident and reflected normalised waves at each of the ports as follows: :\beginb_1 \\ a_1 \end = \begin T_ & T_ \\ T_ & T_ \end\begin a_2 \\ b_2 \end\, However, they could be defined differently, as follows : :\begina_1 \\ b_1 \end = \begin T_ & T_ \\ T_ & T_ \end\begin b_2 \\ a_2 \end\, The RF Toolbox add-on to MATLAB and several books (for example "Network scattering parameters") use this last definition, so caution is necessary. The "From S to T" and "From T to S" paragraphs in this article are based on the first definition. Adaptation to the second definition is trivial (interchanging T11 for T22, and T12 for T21). The advantage of T-parameters compared to S-parameters is that providing reference impedances are purely, real or complex conjugate, they may be used to readily determine the effect of cascading 2 or more 2-port networks by simply multiplying the associated individual T-parameter matrices. If the T-parameters of say three different 2-port networks 1, 2 and 3 are \beginT_1\end\,, \beginT_2\end\, and \beginT_3\end\, respectively then the T-parameter matrix for the cascade of all three networks (\beginT_T\end\,) in serial order is given by: :\beginT_T\end = \beginT_1\end\beginT_2\end\beginT_3\end\, Note that matrix multiplication is not commutative, so the order is important. As with S-parameters, T-parameters are complex values and there is a direct conversion between the two types. Although the cascaded T-parameters is a simple matrix multiplication of the individual T-parameters, the conversion for each network's S-parameters to the corresponding T-parameters and the conversion of the cascaded T-parameters back to the equivalent cascaded S-parameters, which are usually required, is not trivial. However once the operation is completed, the complex full wave interactions between all ports in both directions will be taken into account. The following equations will provide conversion between S and T parameters for 2-port networks.''S-Parameter Design''; Application Note AN 154; Agilent Technologies; p 14
/ref> From S to T: :T_ = \frac\, :T_ = \frac\, :T_ = \frac\, :T_ = \frac\, Where \det \beginS\end\, indicates the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the matrix \beginS\end, :\det \beginS\end\ = S_ \cdot S_ - S_ \cdot S_. From T to S :S_ = \frac\, :S_ = \frac\, :S_ = \frac\, :S_ = \frac\, Where \det \beginT\end\, indicates the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ...
of the matrix \beginT\end\,. :\det \beginT\end\ = T_.T_-T_.T_,


1-port S-parameters

The S-parameter for a 1-port network is given by a simple 1 × 1 matrix of the form (s_)\, where n is the allocated port number. To comply with the S-parameter definition of linearity, this would normally be a passive load of some type. An
antenna Antenna ( antennas or antennae) may refer to: Science and engineering * Antenna (radio), also known as an aerial, a transducer designed to transmit or receive electromagnetic (e.g., TV or radio) waves * Antennae Galaxies, the name of two collid ...
is a common one-port network for which small values of s_ indicate that the antenna will either radiate or dissipate/store power.


Higher-order S-parameter matrices

Higher order S-parameters for pairs of dissimilar ports (S_\,), where m \ne \; n\, may be deduced similarly to those for 2-port networks by considering pairs of ports in turn, in each case ensuring that all of the remaining (unused) ports are loaded with an impedance identical to the system impedance. In this way the incident power wave for each of the unused ports becomes zero yielding similar expressions to those obtained for the 2-port case. S-parameters relating to single ports only (S_\,) require all of the remaining ports to be loaded with an impedance identical to the system impedance therefore making all of the incident power waves zero except that for the port under consideration. In general therefore we have: :S_ = \frac\, and :S_ = \frac\, For example, a 3-port network such as a 2-way splitter would have the following S-parameter definitions :S_ = \frac = \frac\, :S_ = \frac = \frac\, :S_ = \frac = \frac\, :S_ = \frac = \frac\,


Measurement of S-parameters

S-parameters are most commonly measured with a
vector network analyzer A network analyzer is an instrument that measures the network parameters of electrical networks. Today, network analyzers commonly measure s–parameters because reflection and transmission of electrical networks are easy to measure at high ...
(VNA).


Output format of measured and corrected S-parameter data

The S-parameter test data may be provided in many alternative formats, for example: list, graphical (
Smith chart The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assis ...
or polar diagram).


List format

In list format the measured and corrected S-parameters are tabulated against frequency. The most common list format is known as Touchstone or SNP, where N is the number of ports. Commonly text files containing this information would have the filename extension '.s2p'. An example of a Touchstone file listing for the full 2-port S-parameter data obtained for a device is shown below: ! Created Fri 21 July, 14:28:50 2005 # MHZ S DB R 50 ! SP1.SP 50 -15.4 100.2 10.2 173.5 -30.1 9.6 -13.4 57.2 51 -15.8 103.2 10.7 177.4 -33.1 9.6 -12.4 63.4 52 -15.9 105.5 11.2 179.1 -35.7 9.6 -14.4 66.9 53 -16.4 107.0 10.5 183.1 -36.6 9.6 -14.7 70.3 54 -16.6 109.3 10.6 187.8 -38.1 9.6 -15.3 71.4 Rows beginning with an exclamation mark contains only comments. The row beginning with the hash symbol indicates that in this case frequencies are in megahertz (MHZ), S-parameters are listed (S), magnitudes are in dB log magnitude (DB) and the system impedance is 50 Ohm (R 50). There are 9 columns of data. Column 1 is the test frequency in megahertz in this case. Columns 2, 4, 6 and 8 are the magnitudes of S_\,, S_\,, S_\, and S_\, respectively in dB. Columns 3, 5, 7 and 9 are the angles of S_\,, S_\,, S_\, and S_\, respectively in degrees.


Graphical (Smith chart)

Any 2-port S-parameter may be displayed on a
Smith chart The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assis ...
using polar co-ordinates, but the most meaningful would be S_\, and S_\, since either of these may be converted directly into an equivalent normalized impedance (or admittance) using the characteristic Smith Chart impedance (or admittance) scaling appropriate to the system impedance.


Graphical (polar diagram)

Any 2-port S-parameter may be displayed on a polar diagram using polar co-ordinates. In either graphical format each S-parameter at a particular test frequency is displayed as a dot. If the measurement is a sweep across several frequencies a dot will appear for each.


Measuring S-parameters of a one-port network

The S-parameter matrix for a network with just one port will have just one element represented in the form S_\,, where n is the number allocated to the port. Most VNAs provide a simple one-port calibration capability for one port measurement to save time if that is all that is required.


Measuring S-parameters of networks with more than 2 ports

VNAs designed for the simultaneous measurement of the S-parameters of networks with more than two ports are feasible but quickly become prohibitively complex and expensive. Usually their purchase is not justified since the required measurements can be obtained using a standard 2-port calibrated VNA with extra measurements followed by the correct interpretation of the results obtained. The required S-parameter matrix can be assembled from successive two port measurements in stages, two ports at a time, on each occasion with the unused ports being terminated in high quality loads equal to the system impedance. One risk of this approach is that the return loss or VSWR of the loads themselves must be suitably specified to be as close as possible to a perfect 50 Ohms, or whatever the nominal system impedance is. For a network with many ports there may be a temptation, on grounds of cost, to inadequately specify the VSWRs of the loads. Some analysis will be necessary to determine what the worst acceptable VSWR of the loads will be. Assuming that the extra loads are specified adequately, if necessary, two or more of the S-parameter subscripts are modified from those relating to the VNA (1 and 2 in the case considered above) to those relating to the network under test (1 to N, if N is the total number of DUT ports). For example, if the DUT has 5 ports and a two port VNA is connected with VNA port 1 to DUT port 3 and VNA port 2 to DUT port 5, the measured VNA results (S_\,, S_\,, S_\, and S_\,) would be equivalent to S_\,, S_\,, S_\, and S_\, respectively, assuming that DUT ports 1, 2 and 4 were terminated in adequate 50 Ohm loads . This would provide 4 of the necessary 25 S-parameters.


See also

*
Admittance parameters Admittance parameters or Y-parameters (the elements of an admittance matrix or Y-matrix) are properties used in many areas of electrical engineering, such as power, electronics, and telecommunications. These parameters are used to describe the ele ...
*
Impedance parameters Impedance parameters or Z-parameters (the elements of an impedance matrix or Z-matrix) are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear ele ...
*
Two-port network A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them sati ...
*
X-parameters X-parameters are a generalization of Scattering parameters, S-parameters and are used for characterizing the amplitudes and relative phase of harmonics generated by nonlinear components under large input power levels. X-parameters are also refe ...
, a non-linear superset of S-parameters *
Belevitch's theorem Belevitch's theorem is a theorem in electrical network analysis due to the Russo-Belgian mathematician Vitold Belevitch (1921–1999). The theorem provides a test for a given scattering parameters, S-matrix to determine whether or not it can be con ...


References


Bibliography

*Guillermo Gonzalez, "Microwave Transistor Amplifiers, Analysis and Design, 2nd. Ed.", Prentice Hall, New Jersey; *David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc.; *William Eisenstadt, Bob Stengel, and Bruce Thompson, "Microwave Differential Circuit Design using Mixed-Mode S-Parameters", Artech House; ;
"S-Parameter Design", Application Note AN 154, Keysight Technologies
or ttp://literature.cdn.keysight.com/litweb/pdf/5952-1130.pdf scan of Richard W. Anderson's original article*A. J. Baden Fuller, "An Introduction to Microwave Theory and Techniques, Second Edition, Pergammon International Library; *Ramo, Whinnery and Van Duzer, "Fields and Waves in Communications Electronics", John Wiley & Sons; *C. W. Davidson, "Transmission Lines for Communications with CAD Programs", Second Edition, Macmillan Education Ltd.; {{DEFAULTSORT:Scattering Parameters Electrical parameters Two-port networks Transfer functions