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', ${\displaystyle a_{i}\,}$ and ${\displaystyle b_{i}\,}$ respectively.

Kurokawa^[15] defines the incident power wave for each port as

${\displaystyle a_{i}={\frac {1}{2}}\,k_{i}(V_{i}+Z_{i}I_{i})\,}$

and the reflected wave for each port is defined as

${\displaystyle b_{i}={\frac {1}{2}}\,k_{i}(V_{i}-Z_{i}^{*}I_{i})\,}$

where ${\displaystyle Z_{i}\,}$ is the impedance for port i, ${\displaystyle Z_{i}^{*}\,}$ is the complex conjugate of ${\displaystyle Z_{i}\,}$, ${\displaystyle V_{i}\,}$ and ${\displaystyle I_{i}\,}$ are respectively the complex amplitudes of the voltage and current at port i, and

${\displaystyle k_{i}=\left({\sqrt {\left|\Re \{Z_{i}\}\right|}}\right)^{-1}\,}$

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

${\displaystyle a_{i}={\frac {1}{2}}\,{\frac {(V_{i}+Z_{0}I_{i})}{\sqrt {\left|\Re \{Z_{0}\}\right|}}}\,}$

and

${\displaystyle b_{i}={\frac {1}{2}}\,{\frac {(V_{i}-Z_{0}^{*}I_{i})}{\sqrt {\left|\Re \{Z_{0}\}\right|}}}\,}$

Note that as was pointed out by Kurokawa himself, the above definitions of ${\displaystyle a_{i}}$ and ${\displaystyle b_{i}}$ are not unique.

The relation between the vectors a and b, whose i-th components are the power waves ${\displaystyle a_{i}}$ and ${\displaystyle b_{i}}$ respectively, can be expressed using the S-parameter matrix S:

${\displaystyle \mathbf {b} =\mathbf {S} \mathbf {a} \,}$

Or using explicit components:

${\displaystyle {\begin{pmatrix}b_{1}\\\vdots \\b_{n}\end{pmatrix}}={\begin{pmatrix}S_{11}&\dots &S_{1n}\\\vdots &\ddots &\vdots \\S_{n1}&\dots &S_{nn}\end{pmatrix}}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}}$

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 ${\displaystyle S_{mn}=S_{nm}\,}$ in each case, or the S-parameter matrix will be equal to its transpose. 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.^[16]

Lossless networks

A lossless network is one which does not dissipate any power, or: ${\displaystyle \Sigma \left|a_{n}\right|^{2}=\Sigma \left|b_{n}\right|^{2}\,}$. The sum of the incident powers at all ports is equal to the sum of the outgoing (e.g. 'reflected') powers at all ports. This implies that the S-parameter matrix is unitary, that is ${\displaystyle (S)^{H}(S)=(I)\,}$, where ${\displaystyle (S)^{H}\,}$ is the conjugate transpose of ${\displaystyle (S)\,}$ and ${\displaystyle (I)\,}$ 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 outgoing (e.g. 'reflected') powers at all ports. It therefore dissipates power: ${\displaystyle \Sigma \left|a_{n}\right|^{2}\neq \Sigma \left|b_{n}\right|^{2}\,}$. Thus ${\displaystyle \Sigma \left|a_{n}\right|^{2}>\Sigma \left|b_{n}\right|^{2}\,}$, and ${\displaystyle (I)-(S)^{H}(S)\,}$ is positive definite.^[17]

Complex linear gain

The complex linear gain G is given by

${\displaystyle G=S_{21}={\frac {b_{2}}{a_{1}}}\,}$.

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 ${\displaystyle |G|>1}$ . 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

${\displaystyle \left|G\right|=\left|S_{21}\right|\,}$.

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:

${\displaystyle g=20\log _{10}\left|S_{21}\right|\,}$ 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 (IL) is the reciprocal of the magnitude of the transmission coefficient |S₂₁| expressed in decibels. It is thus given by:^[19]

${\displaystyle IL=-20\log _{10}\left|S_{21}\right|\,}$ dB.

It is the extra loss produced by the introduction of the device under test (DUT) between the 2 reference planes of the measurement. 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 (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

${\displaystyle RL_{\mathrm {in} }=10\log _{10}\left|{\frac {1}{S_{11}^{2}}}\right|=-20\log _{10}\left|S_{11}\right|\,}$ dB.

Note that for passive two-port networks in which |S₁₁| ≤ 1, it follows that return loss is a non-negative 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.^[20]

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

${\displaystyle RL_{\mathrm {out} }=-20\log _{10}\left|S_{22}\right|\,}$ dB.

Reverse gain and reverse isolation

The scalar logarithmic (decibel or dB) expression for reverse gain (${\displaystyle g_{\mathrm {rev} }\,}$) is:

${\displaystyle g_{\mathrm {rev} }=20\log _{10}\left|S_{12}\right|\,}$ dB.

Often this will be expressed as reverse isolation (${\displaystyle I_{\mathrm {rev} }\,}$) in which case it becomes a positive quantity equal to the magnitude of ${\displaystyle g_{\mathrm {rev} }\,}$ and the expression becomes:

${\displaystyle I_{\mathrm {rev} }=\left|g_{\mathrm {rev} }\right|=\left|20\log _{10}\left|S_{12}\right|\right|\,}$ dB.

Reflection coefficient

The reflection coefficient at the input port (${\displaystyle \Gamma _{\mathrm {in} }\,}$) or at the output port (${\displaystyle \Gamma _{\mathrm {out} }\,}$) are equivalent to ${\displaystyle S_{11}\,}$ and ${\displaystyle S_{22}\,}$ respectively, so

${\displaystyle \Gamma _{\mathrm {in} }=S_{11}\,}$ and ${\displaystyle \Gamma _{\mathrm {out} }=S_{22}\,}$.

As ${\displaystyle S_{11}\,}$ and ${\displaystyle S_{22}\,}$ are complex quantities, so are ${\displaystyle \Gamma _{\mathrm {in} }\,}$ and ${\displaystyle \Gamma _{\mathrm {out} }\,}$.

The 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 ${\displaystyle S_{11}\,}$ for the input port or ${\displaystyle S_{22}\,}$ for the output port.

At the input port, the VSWR (${\displaystyle s_{\mathrm {in} }\,}$) is given by

${\displaystyle s_{\mathrm {in} }={\frac {1+\left|S_{11}\right|}{1-\left|S_{11}\right|}}\,}$

At the output port, the VSWR (${\displaystyle s_{\mathrm {out} }\,}$) is given by

${\displaystyle s_{\mathrm {out} }={\frac {1+\left|S_{22}\right|}{1-\left|S_{22}\right|}}\,}$

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;

${\displaystyle s_{k}={\frac {1+\left|S_{kk}\right|}{|1-\left|S_{kk}\right||}}\,}$

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.^[21][22]

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.^[24] 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:^[25]

${\displaystyle \left|\Gamma _{s}\right|<1\,}$

${\displaystyle \left|\Gamma _{L}\right|<1\,}$

${\displaystyle \left|\Gamma _{\mathrm {in} }\right|<1\,}$

${\displaystyle \left|\Gamma _{\mathrm {out} }\right|<1\,}$

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:

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:

${\displaystyle {\begin{pmatrix}b_{1}\\a_{1}\end{pmatrix}}={\begin{pmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{pmatrix}}{\begin{pmatrix}a_{2}\\b_{2}\end{pmatrix}}\,}$

However, they could be defined differently, as follows :

${\displaystyle {\begin{pmatrix}a_{1}\\b_{1}\end{pmatrix}}={\begin{pmatrix}T_{11}&T_{12}\\T_{21}&T_{22}\end{pmatrix}}{\begin{pmatrix}b_{2}\\a_{2}\end{pmatrix}}\,}$

The RF Toolbox add-on to MATLAB^[26] and several books (for example "Network scattering parameters"^[27]) 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 ${\displaystyle {\begin{pmatrix}T_{1}\end{pmatrix}}\,}$, ${\displaystyle {\begin{pmatrix}T_{2}\end{pmatrix}}\,}$ and ${\displaystyle {\begin{pmatrix}T_{3}\end{pmatrix}}\,}$ respectively then the T-parameter matrix for the cascade of all three networks (${\displaystyle {\begin{pmatrix}T_{T}\end{pmatrix}}\,}$) in serial order is given by:

${\displaystyle {\begin{pmatrix}T_{T}\end{pmatrix}}={\begin{pmatrix}T_{1}\end{pmatrix}}{\begin{pmatrix}T_{2}\end{pmatrix}}{\begin{pmatrix}T_{3}\end{pmatrix}}\,}$

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.^[28]

From S to T:

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

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

${\displaystyle T_{21}={\frac {-S_{22}}{S_{21}}}\,}$

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

Where ${\displaystyle \det {\begin{pmatrix}S\end{pmatrix}}\,}$ indicates the determinant of the matrix ${\displaystyle {\begin{pmatrix}S\end{pmatrix}}}$,

${\displaystyle \det {\begin{pmatrix}S\end{pmatrix}}\ =S_{11}\cdot S_{22}-S_{12}\cdot S_{21}}$.

From T to S

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

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

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

${\displaystyle S_{22}={\frac {-T_{21}}{T_{22}}}\,}$

Where ${\displaystyle \det {\begin{pmatrix}T\end{pmatrix}}\,}$ indicates the determinant of the matrix ${\displaystyle {\begin{pmatrix}T\end{pmatrix}}\,}$.

${\displaystyle \det {\begin{pmatrix}T\end{pmatrix}}\ =T_{11}.T_{22}-T_{12}.T_{21},}$

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 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 ${\displaystyle S_{11}\,}$, ${\displaystyle S_{21}\,}$, ${\displaystyle S_{12}\,}$ and ${\displaystyle S_{22}\,}$ respectively in dB. Columns 3, 5, 7 and 9 are the angles of ${\displaystyle S_{11}\,}$, ${\displaystyle S_{21}\,}$, ${\displaystyle S_{12}\,}$ and ${\displaystyle S_{22}\,}$ respectively in degrees.

Graphical (Smith chart)

Any 2-port S-parameter may be displayed on a Smith chart using polar co-ordinates, but the most meaningful would be ${\displaystyle S_{11}\,}$ and ${\displaystyle S_{22}\,}$ 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 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 ${\displaystyle S_{nn}\,}$, 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 (${\displaystyle S_{11}\,}$, ${\displaystyle S_{12}\,}$, ${\displaystyle S_{21}\,}$ and ${\displaystyle S_{22}\,}$) would be equivalent to ${\displaystyle S_{33}\,}$, ${\displaystyle S_{35}\,}$, ${\displaystyle S_{53}\,}$ and ${\displaystyle S_{55}\,}$ respectively, assuming that DUT ports 1, 2 and 4 were terminated in adequate 50 Ohm loads . This would provide 4 of the necessary 25 S-parameters.