The first published description of S-parameters was in the thesis of
Vitold Belevitch 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 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 (), 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' 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 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
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 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 into one terminal is equal to the current leaving the other. S-parameters are used at frequencies where the ports are often
coaxial or
waveguide connections. The S-parameter
matrix 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 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 S-parameter magnitude may be expressed in linear form or
logarithmic form. When expressed in logarithmic form, magnitude has the "
dimensionless unit" of
decibels. The S-parameter angle is most frequently expressed in
degrees but occasionally in
radians. Any S-parameter 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 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.
Kurokawa power-wave S-parameters 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{1}{2}\, k_i (V_i + Z_{i} I_i) and the reflected wave for each port is defined as : b_i = \frac{1}{2}\, k_i (V_i - Z_{i}^{*} 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{\left|\real\{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 : a_i = \frac{1}{2}\, \frac{(V_i + Z_{0} I_i)}{\sqrt{\left|\real\{Z_{0}\}\right|}} and : b_i = \frac{1}{2}\, \frac{(V_i - Z_{0}^{*} I_i)}{\sqrt{\left|\real\{Z_{0}\}\right|}} 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{b} = \mathbf{S} \mathbf{a} Or using explicit components: : \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 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.
Lossless networks A lossless network is one which does not dissipate any power, or: \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 (S)^\mathrm{H} (S) = (I), where (S)^\mathrm{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 outgoing (e.g. 'reflected') powers at all ports. It therefore dissipates power: \Sigma\left|a_n\right|^2 \ne \Sigma\left|b_n\right|^2. Thus \Sigma\left|a_n\right|^2 > \Sigma\left|b_n\right|^2, and (I) - (S)^\mathrm{H} (S) is
positive definite. == Two-port S-parameters ==