Multi-port Scattering parameters
Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical 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, Z- parameters, H-parameters, T-parameters or ABCD-parameters. 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 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 or system impedance.
S-parameters are readily represented in matrix form and obey the rules of matrix algebra.
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', ai and bi respectively.
Kurokawa defines the incident power wave for each port as
and the reflected wave for each port is defined as
Note that as was pointed out by Kurokawa himself, the above definitions of ai and bi are not unique. The relation between the vectors a and b, whose i-th components are the power waves ai and bi respectively, can be expressed using the S-parameter matrix S: b=Sa
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 Smn=Snm 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: 
. 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, that is 
, where
is the conjugate transpose of （S） and is the 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: 
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.
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.
From Wikipedia
