Scattering parameters or S-parameters are the elements of a scattering matrix or S-matrix which describe the steady state response of linear electrical networks to sinusoidal signals.
The parameters are useful for several branches of electrical engineering, including electronics, communication systems design, and especially for microwave engineering.
The S-parameters are members of a family of similar parameters, other examples being: Y-parameters and Z-parameters,Pozar, David M. (2012); pp. 175–178. H-parameters, T-parameters and ABCD-parameters.Pozar, David M. (2012); pp. 188–194.Morton, A. H. (1985); pp. 33–72. 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 terminations 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 phasors using essentially the same circuit as that used for the demodulation of digitally modulated wireless signals.
Many electrical properties of networks of components (inductors, capacitors, resistors) may be expressed using S-parameters, 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 S-parameters, 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, S-parameters are mostly used for networks operating at radio frequency (RF) and microwave frequencies. S-parameters in common use – the conventional S-parameters – are linear quantities (not power quantities, as in the below mentioned 'power waves' approach by ()). S-parameters change with the measurement frequency, so frequency must be specified for any S-parameter measurements stated, in addition to the characteristic impedance or system impedance.
Background The first published description of S-parameters was in the thesis of Vitold Belevitch in 1945.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. 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.Valkenburg, Mac Elwyn Van Circuit Theory: Foundations and Classical Contributions, p.334, Stroudsburg, Pennsylvania: Dowden, Hutchinson & Ross, 1974 . 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.Penfield, Jr., Paul "Noise in negative-resistance amplifiers", IRE Transactions on Circuit Theory, vol.7, iss.2, pp. 166–170, June 1960.Youla, D. C. "On scattering matrices normalized to complex port numbers", Proceedings of the IRE, vol.49, iss.7, p. 1221, July 1962. 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.Amakawa, Shuhei "Scattered reflections on scattering parameters", IEICE Transactions on Electronics, vol.E99-C, iss.10, pp. 1100–1112, October 2016. In his seminal paper,Kurokawa, Kaneyuki "Power waves and the scattering matrix", IEEE Transactions on Microwave Theory and Techniques, vol.13, iss.2, pp. 194–202, March 1965. 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.Marks, R. B. and Williams, D. F. "A general waveguide circuit theory", Journal of Research of National Institute of Standard and Technology, vol.97, iss.5, pp. 533–562, 1992.
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.Pozar, David M. (2012); p. 174.Morton, A. H. (1985); p. 33. 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 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 ), it may be displayed on an impedance or admittance Smith Chart normalised to the system impedance. The Smith Chart allows simple conversion between the 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', and respectively.
KurokawaKurokawa, K., "Power Waves and the Scattering Matrix", IEEE Trans. Micr. Theory & Tech., Mar. 1965, pp. 194–202 defines the incident power wave for each port as
and the reflected wave for each port is defined as
where is the impedance for port i, is the complex conjugate of , and are respectively the complex amplitudes of the voltage and current at port i, and
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
and
Note that as was pointed out by Kurokawa himself, the above definitions of and are not unique.
The relation between the vectors a and b, whose i-th components are the power waves and respectively, can be expressed using the S-parameter matrix S:
Or using explicit components:
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 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.Pozar, David M. (2005) (op. cit.); p. 173.
Lossless networks A lossless network is one which does not dissipate any power, or: . 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 , where is the conjugate transpose of and 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: . Thus , and is positive definite.S-Parameter Design; Application Note AN 154; Agilent Technologies; p 7
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 outgoing ('reflected'), incident waves and the S-parameter matrix is given by: .
Expanding the matrices into equations gives:
and .
Each equation gives the relationship between the outgoing (e.g. reflected) and incident waves at each of the network ports, 1 and 2, in terms of the network's individual S-parameters, , , and . If one considers an incident wave at port 1 () there may result from it waves exiting from either port 1 itself () or port 2 (). However, if, according to the definition of S-parameters, port 2 is terminated in a load identical to the system impedance () then, by the maximum power transfer theorem, will be totally absorbed making equal to zero. Therefore, defining the incident voltage waves as and with the outgoing/reflected waves being and , and .
Similarly, if port 1 is terminated in the system impedance then becomes zero, giving and
The 2-port S-parameters have the following generic descriptions: is the input port voltage reflection coefficient is the reverse voltage gain is the forward voltage gain 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 and reverse waves then and . 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 .
Using these definitions, the above matrix may be expanded to give the reflected waves and in terms of the incident waves and as
S-parameter properties of 2-port networks