Scattering parameters or Sparameters (the elements of a scattering matrix or Smatrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals.
The parameters are useful for electrical engineering, electronics engineering, and communication systems design, and especially for microwave engineering.
The Sparameters are members of a family of similar parameters, other examples being: Yparameters,^{[1]} Zparameters,^{[2]} Hparameters, Tparameters or ABCDparameters.^{[3]}^{[4]} They differ from these, in the sense that Sparameters do not use open or short circuit conditions to characterize a linear electrical network; instead, matched loads are used. These terminations are much easier to use at high signal frequencies than opencircuit and shortcircuit terminations. Moreover, the quantities are measured in terms of power.
Many electrical properties of networks of components (inductors, capacitors, resistors) may be expressed using Sparameters, such as gain, return loss, voltage standing wave ratio (VSWR), reflection coefficient and amplifier stability. The term 'scattering' is more common to optical engineering than RF engineering, referring to the effect observed when a plane electromagnetic wave is incident on an obstruction or passes across dissimilar dielectric media. In the context of Sparameters, scattering refers to the way in which the traveling currents and voltages in a transmission line 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.
Although applicable at any frequency, Sparameters are mostly used for networks operating at radio frequency (RF) and microwave frequencies where signal power and energy considerations are more easily quantified than currents and voltages. Sparameters change with the measurement frequency, so frequency must be specified for any Sparameter measurements stated, in addition to the characteristic impedance or system impedance.
Sparameters are readily represented in matrix form and obey the rules of matrix algebra.
Contents

Background 1

The general Sparameter matrix 2

Definition 2.1

Reciprocity 2.2

Lossless networks 2.3

Lossy networks 2.4

TwoPort SParameters 3

SParameter properties of 2port networks 4

Complex linear gain 4.1

Scalar linear gain 4.2

Scalar logarithmic gain 4.3

Insertion loss 4.4

Input return loss 4.5

Output return loss 4.6

Reverse gain and reverse isolation 4.7

Voltage reflection coefficient 4.8

Voltage standing wave ratio 4.9

4Port SParameters 5

4Port MixedMode SParameters 5.1

Sparameters in amplifier design 6

Port loading conditions for an amplifier to be unconditionally stable 6.1

\rho_{L}\, values for \left\rho_{in}\right = 1\, (output stability circle) 6.1.1

\rho_{S}\, values for \left\rho_{out}\right = 1\, (input stability circle) 6.1.2

Scattering transfer parameters 6.2

1port Sparameters 7

Higherorder Sparameter matrices 8

Measurement of Sparameters 9

Vector network analyzer 9.1

Calibration 9.2

Minimization of systematic errors 9.3

Output format of measured and corrected Sparameter data 9.4

List format 9.5

Graphical (Smith chart) 9.6

Graphical (polar diagram) 9.7

Measuring Sparameters of a oneport network 9.8

Measuring Sparameters of networks with more than 2 ports 9.9

See also 10

References 11

Bibliography 12
Background
The first published description of Sparameters was in the thesis of Vitold Belevitch in 1945.^{[5]} The name used by Belevitch was repartition matrix and limited consideration to lumpedelement networks. The term scattering matrix was used by physicist and engineer Robert Henry Dicke in 1947 who independently developed the idea during wartime work on radar.^{[6]}^{[7]} The technique was popularized in the 1960s by Kaneyuki Kurokawa^{[8]}
In the Sparameter approach, an electrical network is regarded as a 'black box' containing various interconnected basic electrical circuit components or lumped elements such as resistors, capacitors, inductors and transistors, which interacts with other circuits through ports. The network is characterized by a square matrix of complex numbers called its Sparameter matrix, which can be used to calculate its response to signals applied to the ports.
For the Sparameter definition, it is understood that a network may contain any components provided that the entire network behaves linearly with incident small signals. It may also include many typical communication system components or 'blocks' such as amplifiers, attenuators, filters, couplers and equalizers provided they are also operating under linear and defined conditions.
An electrical network to be described by Sparameters 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 into one terminal is equal to the current leaving the other.^{[9]}^{[10]} Sparameters are used at frequencies where the ports are often coaxial or waveguide connections.
The Sparameter matrix describing an Nport network will be square of dimension N and will therefore contain N^2\, elements. At the test frequency each element or Sparameter is represented by a unitless complex number that represents magnitude and angle, i.e. amplitude and phase. The complex number may either be expressed in rectangular form or, more commonly, in polar form. The Sparameter magnitude may be expressed in linear form or logarithmic form. When expressed in logarithmic form, magnitude has the "dimensionless unit" of decibels. The Sparameter angle is most frequently expressed in degrees but occasionally in radians. Any Sparameter may be displayed graphically on a polar diagram by a dot for one frequency or a locus for a range of frequencies. If it applies to one port only (being of the form S_{nn}\,), it may be displayed on an impedance or admittance Smith Chart normalised to the system impedance. The Smith Chart allows simple conversion between the S_{nn}\, 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 Sparameters:

The frequency

The 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 general Sparameter matrix
Definition
For a generic multiport network, the ports are numbered from 1 to N, where N is the total number of ports. For port n, the associated Sparameter definition is in terms of incident and reflected 'power waves', a_n\, and b_n\, respectively.
Kurokawa^{[11]} defines the incident power wave for each port as

a = \frac{1}{2}\, k (V + Z_{p} I)\,
and the reflected wave for each port is defined as

b = \frac{1}{2}\, k (V  Z_{p}^{*} I)\,
where Z_p\, is the diagonal matrix of the complex reference impedance for each port, Z_p^{*}\, is the elementwise complex conjugate of Z_p\,, V\, and I\, are respectively the column vectors of the voltages and currents at each port and

k = \left(\sqrt{\left\real\{Z_{p}\}\right}\right)^{1}\,
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 = \frac{1}{2}\, \frac{(V + Z_{0} I)}{\sqrt{\left\real\{Z_{0}\}\right}}\,
and

b = \frac{1}{2}\, \frac{(V  Z_{0}^{*} I)}{\sqrt{\left\real\{Z_{0}\}\right}}\,
For all ports the reflected power waves may be defined in terms of the Sparameter matrix and the incident power waves by the following matrix equation:

b = S a\,
where S is an N x N matrix the elements of which can be indexed using conventional matrix (mathematics) notation.
Reciprocity
A network will be reciprocal if it is passive and it contains only reciprocal materials that influence the transmitted signal. For example, attenuators, cables, splitters and combiners are all reciprocal networks and S_{mn} = S_{nm}\, in each case, or the Sparameter matrix will be equal to its transpose. Networks which include nonreciprocal materials in the transmission medium such as those containing magnetically biased ferrite components will be nonreciprocal. An amplifier is another example of a nonreciprocal network.
An interesting property of 3port networks, however, is that they cannot be simultaneously reciprocal, lossfree, and perfectly matched.^{[12]}
Lossless networks
A lossless network is one which does not dissipate any power, or: \Sigma\lefta_n\right^2 = \Sigma\leftb_n\right^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 Sparameter matrix is unitary, that is (S)^H (S) = (I)\,, where (S)^H\, is the conjugate transpose of (S)\, and (I)\, is the identity matrix.
Lossy networks
A lossy 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, or: \Sigma\lefta_n\right^2 \ne \Sigma\leftb_n\right^2\,. In this case \Sigma\lefta_n\right^2 > \Sigma\leftb_n\right^2\,, and (I)  (S)^H (S)\, is positive definite.^{[13]}
TwoPort SParameters
The Sparameter matrix for the 2port network is probably the most commonly used and serves as the basic building block for generating the higher order matrices for larger networks.^{[14]} In this case the relationship between the reflected, incident power waves and the Sparameter matrix is given by:

\begin{pmatrix}b_1 \\ b_2 \end{pmatrix} = \begin{pmatrix} S_{11} & S_{12} \\ S_{21} & S_{22} \end{pmatrix}\begin{pmatrix} a_1 \\ a_2 \end{pmatrix}\,.
Expanding the matrices into equations gives:

b_1 = S_{11}a_1 +S_{12}a_2\,
and

b_2 = S_{21}a_1 +S_{22}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 Sparameters, S_{11}\,, S_{12}\,, S_{21}\, and S_{22}\,. 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 Sparameters, port 2 is terminated in a load identical to the system impedance (Z_0\,) then, by the maximum power transfer theorem, 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_{11} = \frac{b_1}{a_1} = \frac{V_1^}{V_1^+} and S_{21} = \frac{b_2}{a_1} = \frac{V_2^}{V_1^+}\,.
Similarly, if port 1 is terminated in the system impedance then a_1\, becomes zero, giving

S_{12} = \frac{b_1}{a_2} = \frac{V_1^}{V_2^+}\, and S_{22} = \frac{b_2}{a_2} = \frac{V_2^}{V_2^+}\,
Each 2port Sparameter has the following generic descriptions:

S_{11}\, is the input port voltage reflection coefficient

S_{12}\, is the reverse voltage gain

S_{21}\, is the forward voltage gain

S_{22}\, 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_2 = V_2^. The Sparameters then take on a more intuitive meaning such as the forward voltage gain being defined by the ratio of the forward voltages S_{21} = V_2^+/V_1^+.
SParameter properties of 2port networks
An amplifier operating under linear (small signal) conditions is a good example of a nonreciprocal 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_{21}\,.
That is simply the voltage gain as a linear ratio of the output voltage divided by the input voltage, all values expressed as complex quantities.
Scalar linear gain
The scalar linear gain (or linear gain magnitude) is given by

\leftG\right = \leftS_{21}\right\,.
That is simply the scalar voltage gain as a linear ratio of the output voltage and the input voltage. As this is a scalar quantity, the phase is not relevant in this case.
Scalar logarithmic gain
The scalar logarithmic (decibel or dB) expression for gain (g) is

g = 20\log_{10}\leftS_{21}\right\, dB.
This is more commonly used than scalar linear gain and a positive quantity is normally understood as simply a 'gain'... A negative quantity can be expressed as a 'negative gain' or more usually as a 'loss' equivalent to its magnitude in dB. For example, a 10 m length of cable may have a gain of  1 dB at 100 MHz or a loss of 1 dB at 100 MHz.
Insertion loss
In case the two measurement ports use the same reference impedance, the insertion loss (IL) is the magnitude of the transmission coefficient S_{21} expressed in decibels. It is thus given by:^{[15]}
IL = 20\log_{10}\leftS_{21}\right\, dB.
It is the extra loss produced by the introduction of the device under test (DUT) between the 2 reference planes of the measurement. Notice that the extra loss can be introduced by 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.
Input return loss
Input return loss (RL_{in}) 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 decibels is given by

RL_\mathrm{in} = 10\log_{10}\left \frac{1}{S_{11}^2} \right =  20\log_{10} \left S_{11}\right\, dB.
Note that for passive twoport networks in which S_{11} ≤ 1, it follows that return loss is a nonnegative quantity: RL_{in} ≥ 0. Also note that somewhat confusingly, return loss is sometimes used as the negative of the quantity defined above, but this usage is, strictly speaking, incorrect based on the definition of loss.^{[16]}
Output return loss
The output return loss (RL_{out}) 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{out} =  20\log_{10}\leftS_{22}\right\, dB.
Reverse gain and reverse isolation
The scalar logarithmic (decibel or dB) expression for reverse gain (g_\mathrm{rev}\,) is:

g_\mathrm{rev} = 20\log_{10}\leftS_{12}\right\, dB.
Often this will be expressed as reverse isolation (I_\mathrm{rev}\,) in which case it becomes a positive quantity equal to the magnitude of g_\mathrm{rev}\, and the expression becomes:

I_\mathrm{rev} = \leftg_\mathrm{rev}\right = \left20\log_{10}\leftS_{12}\right\right\, dB.
Voltage reflection coefficient
The voltage reflection coefficient at the input port (\rho_\mathrm{in}\,) or at the output port (\rho_\mathrm{out}\,) are equivalent to S_{11}\, and S_{22}\, respectively, so

\rho_\mathrm{in} = S_{11}\, and \rho_\mathrm{out} = S_{22}\,.
As S_{11}\, and S_{22}\, are complex quantities, so are \rho_\mathrm{in}\, and \rho_\mathrm{out}\,.
Voltage reflection coefficients are complex quantities and may be graphically represented on polar diagrams or Smith Charts
See also the Reflection Coefficient 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_{11}\, for the input port or S_{22}\, for the output port.
At the input port, the VSWR (s_\mathrm{in}\,) is given by

s_\mathrm{in} = \frac{1+\leftS_{11}\right}{1\leftS_{11}\right}\,
At the output port, the VSWR (s_\mathrm{out}\,) is given by

s_\mathrm{out} = \frac{1+\leftS_{22}\right}{1\leftS_{22}\right}\,
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{1+\leftS_{kk}\right}{1\leftS_{kk}\right}\,
4Port SParameters
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.

\begin{pmatrix}S_{11} & S_{12} & S_{13} & S_{14} \\ S_{21} & S_{22} & S_{23} & S_{24} \\ S_{31} & S_{32} & S_{33} & S_{34} \\ S_{41} & S_{42} & S_{43} & S_{44} \end{pmatrix}
They are commonly used to analyze a pair of coupled transmission lines to determine the amount of crosstalk 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 4Port SParameters, for example the 10Gigabit Attachment Unit Interface (XAUI), SATA, PCIX, and InfiniBand systems.
4Port MixedMode SParameters
4Port mixedMode SParameters 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 4Port MixedMode SParameters.
4Port MixedMode SParameters



Stimulus




Differential

Common Mode




Port 1

Port 2

Port 1

Port 2

Response

Differential

Port 1

SDD11

SDD12

SDC11

SDC12

Port 2

SDD21

SDD22

SDC21

SDC22

Common Mode

Port 1

SCD11

SCD12

SCC11

SCC12

Port 2

SCD21

SCD22

SCC21

SCC22

Note the format of the parameter notation SXYab, where “S” stands for scattering parameter or Sparameter, “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 highspeed 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 singleended

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 crossmode quadrants. This is because they fully characterize any mode conversion occurring in the device under test, whether it is commontodifferential SDCab conversion (EMI susceptibility for an intended differential signal SDD transmission application) or differentialtocommon 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 commonmode signal SCCab propagating through the device under test. For a properly designed SDDab differential device there should be minimal commonmode output SCCab. However, the fourth quadrant commonmode response data is a measure of commonmode transmission response and used in a ratio with the differential transmission response to determine the network commonmode rejection. This common mode rejection is an important benefit of differential signal processing and can be reduced to one in some differential circuit implementations.^{[17]}^{[18]}
Sparameters in amplifier design
The reverse isolation parameter S_{12}\, 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_{21}\,. 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_{12}\, 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 \leftS_{12}\right\, is often called a buffer amplifier.
Suppose the output port of a real (nonunilateral or bilateral) amplifier is connected to an arbitrary load with a reflection coefficient of \rho_{L}\,. The actual reflection coefficient 'seen' at the input port \rho_\mathrm{in}\, will be given by^{[19]}

\rho_\mathrm{in} = S_{11} + \frac{S_{12}S_{21}\rho_L}{1S_{22}\rho_L}\,.
If the amplifier is unilateral then S_{12} = 0\, and \rho_\mathrm{in} = S_{11}\, 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{out}\, is the reflection coefficient seen at the output port and \rho_{s}\, is the reflection coefficient of the source connected to the input port.

\rho_{out} = S_{22} + \frac{S_{12}S_{21}\rho_s}{1S_{11}\rho_s}\,
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.^{[20]} 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:^{[21]}

\left\rho_S\right < 1\,

\left\rho_L\right < 1\,

\left\rho_\mathrm{in}\right < 1\,

\left\rho_\mathrm{out}\right < 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_{L}\, values for \left\rho_{in}\right = 1\, (output stability circle)
Radius r_L = \left \frac{S_{12}S_{21}}{\leftS_{22}\right^2\left\Delta\right^2} \right\,
Center c_L = \frac{(S_{22}  \Delta S_{11}^*)^*}{\leftS_{22}\right^2\left\Delta\right^2}\,
\rho_{S}\, values for \left\rho_{out}\right = 1\, (input stability circle)
Radius r_s = \left \frac{S_{12}S_{21}}{\leftS_{11}\right^2\left\Delta\right^2} \right\,
Center c_s = \frac{(S_{11}  \Delta S_{22}^*)^*}{\leftS_{11}\right^2\left\Delta\right^2}\,
where, in both cases

\Delta = S_{11}S_{22}  S_{12}S_{21}\,
and the superscript star (*) indicates a complex conjugate.
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{1\leftS_{11}\right^2\leftS_{22}\right^2+\left\Delta\right^2}{2\leftS_{12}S_{21}\right}\,
The condition of unconditional stability is achieved when K>1\, and \left\Delta\right<1\,
Scattering transfer parameters
The Scattering transfer parameters or Tparameters of a 2port network are expressed by the Tparameter matrix and are closely related to the corresponding Sparameter matrix. The Tparameter matrix is related to the incident and reflected normalised waves at each of the ports as follows:

\begin{pmatrix}b_1 \\ a_1 \end{pmatrix} = \begin{pmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{pmatrix}\begin{pmatrix} a_2 \\ b_2 \end{pmatrix}\,
However, they could be defined differently, as follows :

\begin{pmatrix}a_1 \\ b_1 \end{pmatrix} = \begin{pmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{pmatrix}\begin{pmatrix} b_2 \\ a_2 \end{pmatrix}\,
The RF Toolbox addon to MATLAB^{[22]} and several books (for example "Network scattering parameters"^{[23]}) 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 Tparameters compared to Sparameters is that they may be used to readily determine the effect of cascading 2 or more 2port networks by simply multiplying the associated individual Tparameter matrices. If the Tparameters of say three different 2port networks 1, 2 and 3 are \begin{pmatrix}T_1\end{pmatrix}\,, \begin{pmatrix}T_2\end{pmatrix}\, and \begin{pmatrix}T_3\end{pmatrix}\, respectively then the Tparameter matrix for the cascade of all three networks (\begin{pmatrix}T_T\end{pmatrix}\,) in serial order is given by:

\begin{pmatrix}T_T\end{pmatrix} = \begin{pmatrix}T_1\end{pmatrix}\begin{pmatrix}T_2\end{pmatrix}\begin{pmatrix}T_3\end{pmatrix}\,
As with Sparameters, Tparameters are complex values and there is a direct conversion between the two types. Although the cascaded Tparameters is a simple matrix multiplication of the individual Tparameters, the conversion for each network's Sparameters to the corresponding Tparameters and the conversion of the cascaded Tparameters back to the equivalent cascaded Sparameters, 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 2port networks.^{[24]}
From S to T:

T_{11} = \frac{\det \begin{pmatrix}S\end{pmatrix}}{S_{21}}\,

T_{12} = \frac{S_{11}}{S_{21}}\,

T_{21} = \frac{S_{22}}{S_{21}}\,

T_{22} = \frac{1}{S_{21}}\,
From T to S

S_{11} = \frac{T_{12}}{T_{22}}\,

S_{12} = \frac{\det \begin{pmatrix}T\end{pmatrix}}{T_{22}}\,

S_{21} = \frac{1}{T_{22}}\,

S_{22} = \frac{T_{21}}{T_{22}}\,
Where \det \begin{pmatrix}S\end{pmatrix}\, indicates the determinant of the matrix \begin{pmatrix}S\end{pmatrix}\,.
1port Sparameters
The Sparameter for a 1port network is given by a simple 1 x 1 matrix of the form (s_{nn})\, where n is the allocated port number. To comply with the Sparameter definition of linearity, this would normally be a passive load of some type.
Higherorder Sparameter matrices
Higher order Sparameters for pairs of dissimilar ports (S_{mn}\,), where m \ne \; n\, may be deduced similarly to those for 2port 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 2port case. Sparameters relating to single ports only (S_{mm}\,) 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_{mn} = \frac{b_m}{a_n}\,
and

S_{mm} = \frac{b_m}{a_m}\,
For example, a 3port network such as a 2way splitter would have the following Sparameter definitions

S_{11} = \frac{b_1}{a_1} = \frac{V_1^}{V_1^+}\,

S_{33} = \frac{b_3}{a_3} = \frac{V_3^}{V_3^+}\,

S_{32} = \frac{b_3}{a_2} = \frac{V_3^}{V_2^+}\,

S_{23} = \frac{b_2}{a_3} = \frac{V_2^}{V_3^+}\,
Measurement of Sparameters
The basic parts of a vector network analyzer
Vector network analyzer
The diagram shows the essential parts of a typical 2port vector network analyzer (VNA). The two ports of the device under test (DUT) are denoted port 1 (P1) and port 2 (P2). The test port connectors provided on the VNA itself are precision types which will normally have to be extended and connected to P1 and P2 using precision cables 1 and 2, PC1 and PC2 respectively and suitable connector adaptors A1 and A2 respectively.
The test frequency is generated by a variable frequency CW source and its power level is set using a variable attenuator. The position of switch SW1 sets the direction that the test signal passes through the DUT. Initially consider that SW1 is at position 1 so that the test signal is incident on the DUT at P1 which is appropriate for measuring S_{11}\, and S_{21}\,. The test signal is fed by SW1 to the common port of splitter 1, one arm (the reference channel) feeding a reference receiver for P1 (RX REF1) and the other (the test channel) connecting to P1 via the directional coupler DC1, PC1 and A1. The third port of DC1 couples off the power reflected from P1 via A1 and PC1, then feeding it to test receiver 1 (RX TEST1). Similarly, signals leaving P2 pass via A2, PC2 and DC2 to RX TEST2. RX REF1, RX TEST1, RX REF2 and RXTEST2 are known as coherent receivers as they share the same reference oscillator, and they are capable of measuring the test signal's amplitude and phase at the test frequency. All of the complex receiver output signals are fed to a processor which does the mathematical processing and displays the chosen parameters and format on the phase and amplitude display. The instantaneous value of phase includes both the temporal and spatial parts, but the former is removed by virtue of using 2 test channels, one as a reference and the other for measurement. When SW1 is set to position 2, the test signals are applied to P2, the reference is measured by RX REF2, reflections from P2 are coupled off by DC2 and measured by RX TEST2 and signals leaving P1 are coupled off by DC1 and measured by RX TEST1. This position is appropriate for measuring S_{22}\, and S_{12}\,.
Calibration
Prior to a VNA Sparameter measurement, the first essential step is to perform an accurate calibration appropriate to the intended measurements. Several types of calibration are normally available on the VNA. It is only in the last few years that VNAs have had the sufficiently advanced processing capability, at realistic cost, required to accomplish the more advanced types of calibration, including corrections for systematic errors.^{[25]} The more basic types, often called 'response' calibrations, may be performed quickly but will only provide a result with moderate uncertainty. For improved uncertainty and dynamic range of the measurement a full 2 port calibration is required prior to DUT measurement. This will effectively eliminate all sources of systematic errors inherent in the VNA measurement system.
Minimization of systematic errors
Systematic errors are those which do not vary with time during a calibration. For a set of 2 port Sparameter measurements there are a total of 12 types of systematic errors which are measured and removed mathematically as part of the full 2 port calibration procedure. They are, for each port:
1. directivity and crosstalk
2. source and load mismatches
3. frequency response errors caused by reflection and transmission tracking within the test receivers
The calibration procedure requires initially setting up the VNA with all the cables, adaptors and connectors necessary to connect to the DUT but not at this stage connecting it. A calibration kit is used according to the connector types fitted to the DUT. This will normally include adaptors, nominal short circuits (SCs), open circuits (OCs) and load termination (TERM) standards of both connector sexes appropriate to the VNA and DUT connectors. Even with standards of high quality, when performing tests at the higher frequencies into the microwave range various stray capacitances and inductances will become apparent and cause uncertainty during the calibration. Data relating to the strays of the particular calibration kit used are measured at the factory traceable to national standards and the results are programmed into the VNA memory prior to performing the calibration.
The calibration procedure is normally software controlled, and instructs the operator to fit various calibration standards to the ends of the DUT connecting cables as well as making a through connection. At each step the VNA processor captures data across the test frequency range and stores it. At the end of the calibration procedure, the processor uses the stored data thus obtained to apply the systematic error corrections to all subsequent measurements made. All subsequent measurements are known as 'corrected measurements'. At this point the DUT is connected and a corrected measurement of its Sparameters made.
Output format of measured and corrected Sparameter data
The Sparameter test data may be provided in many alternative formats, for example: list, graphical (Smith chart or polar diagram).
List format
In list format the measured and corrected Sparameters 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 2port Sparameter data obtained for a device is shown below:
! Created Fri Jul 21 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), Sparameters 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_{11}\,, S_{21}\,, S_{12}\, and S_{22}\, respectively in dB. Columns 3, 5, 7 and 9 are the angles of S_{11}\,, S_{21}\,, S_{12}\, and S_{22}\, respectively in degrees.
Graphical (Smith chart)
Any 2port Sparameter may be displayed on a Smith chart using polar coordinates, but the most meaningful would be S_{11}\, and S_{22}\, 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 2port Sparameter may be displayed on a polar diagram using polar coordinates.
In either graphical format each Sparameter 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. Many VNAs connect successive dots with straight lines for easier visibility.
Measuring Sparameters of a oneport network
The Sparameter matrix for a network with just one port will have just one element represented in the form S_{nn}\,, where n is the number allocated to the port. Most VNAs provide a simple oneport calibration capability for one port measurement to save time if that is all that is required.
Measuring Sparameters of networks with more than 2 ports
VNAs designed for the simultaneous measurement of the Sparameters 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 2port calibrated VNA with extra measurements followed by the correct interpretation of the results obtained. The required Sparameter 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 Sparameter 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_{11}\,, S_{12}\,, S_{21}\, and S_{22}\,) would be equivalent to S_{33}\,, S_{35}\,, S_{53}\, and S_{55}\, respectively, assuming that DUT ports 1, 2 and 4 were terminated in adequate 50 Ohm loads . This would provide 4 of the necessary 25 Sparameters.
See also
References

^ Pozar, David M. (2005); Microwave Engineering, Third Edition (Intl. Ed.); John Wiley & Sons, Inc.; pp. 170174. ISBN 0471448788.

^ Pozar, David M. (2005) (op. cit.); pp. 170174.

^ Pozar, David M. (2005) (op. cit.); pp. 183186.

^ Morton, A. H. (1985); Advanced Electrical Engineering; Pitman Publishing Ltd.; pp. 3372. ISBN 0273401726.

^ Belevitch, Vitold "Summary of the history of circuit theory", Proceedings of the IRE, vol.50, iss.5, pp. 848–855, May 1962.
Vandewalle, Joos "In memoriam – Vitold Belevitch", International Journal of Circuit Theory and Applications, vol.28, iss.5, pp. 429–430, September/October 2000.

^ Valkenburg, Mac Elwyn Van Circuit Theory: Foundations and Classical Contributions, p.334, Stroudsburg, Pennsylvania: Dowden, Hutchinson & Ross, 1974 ISBN 0879330848.

^ J. Appl. Phys. 18, 873 (1947); doi: 10.1063/1.1697561 A Computational Method Applicable to Microwave Networks R. H. Dicke

^ "Microwave Hall of Fame Part III". Microwaves 101. PN Designs. 2012. Retrieved December 9, 2012.

^ Pozar, David M. (2005) (op. cit.); p. 170.

^ Morton, A. H. (1985) (op. cit.); p. 33.

^ Kurokawa, K., "Power Waves and the Scattering Matrix", IEEE Trans. Micr. Theory & Tech., Mar. 1965, pp. 194202

^ Pozar, David M. (2005) (op. cit.); p. 173.

^ ; Application Note AN 154; Agilent Technologies; p 7SParameter Design

^ Choma J. & Chen W.K. (2007). Feedback networks: theory and circuit applications. Singapore: World Scientific. Chapter 3, p. 225 ff.

^ Collin, Robert E.; Foundations For Microwave Engineering, Second Edition

^ Trevor S. Bird, "Definition and Misuse of Return Loss", IEEE Antennas & Propagation Magazine, vol.51, iss.2, pp.166167, April 2009.

^ Backplane Channels and Correlation Between Their Frequency and Time Domain Performance

^ Bockelman, DE and Eisenstadt, WR "Combined differential and commonmode scattering parameters: theory and simulation," MTT, IEEE transactions volume 43 issue 7 part 1–2 July 1995 pages 15301539

^ Gonzalez, Guillermo (1997); Microwave Transistor Amplifiers Analysis and Design, Second Edition; Prentice Hall NJ; pp 212216. ISBN 0132543354.

^ J.M. Rollett, "Stability and PowerGain Invariants of Linear Twoports", IRE Trans. on Circuit Theory vol. CT9, pp. 2932, March 1962

^ Gonzalez, Guillermo (op. cit.); pp 217222

^ "RF Toolbox documentation".

^ R. Mavaddat. (1996). Network scattering parameter. Singapore: World Scientific.

^ ; Application Note AN 154; Agilent Technologies; p 14SParameter Design

^ ; Agilent Application Note AN 12873, Agilent Technologies; p6Applying Error Correction to Network Analyzer Measurements
Bibliography

Guillermo Gonzalez, "Microwave Transistor Amplifiers, Analysis and Design, 2nd. Ed.", Prentice Hall, New Jersey; ISBN 0135816467

David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc.; ISBN 0471170968

William Eisenstadt, Bob Stengel, and Bruce Thompson, "Microwave Differential Circuit Design usign MixedMode SParameters", Artech House; ISBN 1580539335; ISBN 9781580539333

"SParameter Design", Application Note AN 154, Agilent Technologies

"SParameter Techniques for Faster, More Accurate Network Design", Application Note AN 951, Agilent Technologies, PDF slides plus QuickTime video or scan of Richard W. Anderson's original article

A. J. Baden Fuller, "An Introduction to Microwave Theory and Techniques, Second Edition, Pergammon International Library; ISBN 0080242278

Ramo, Whinnery and Van Duzer, "Fields and Waves in Communications Electronics", John Wiley & Sons; ISBN 047170721X

C. W. Davidson, "Transmission Lines for Communications with CAD Programs", Second Edition, Macmillan Education Ltd.; ISBN 0333473981
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