# Scattering Parameters

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Scattering parameters - Wikipedia, the free encyclopedia

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Scattering parametersFrom Wikipedia, the free encyclopedia

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 electrical engineering, electronics engineering, and communication systems design, and especially for microwave engineering. The S-parameters are members of a family of similar parameters, other examples being: Yparameters,[1] Z-parameters,[2] H-parameters, T-parameters or ABCD-parameters.[3][4]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. Moreover, the quantities are measured in terms of power. 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 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, 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.

Contents1 Background 2 The general S-parameter matrix 2.1 Definition 2.2 Reciprocity 2.3 Lossless networks 2.4 Lossy networks 3 Two-Port S-Parameters 4 S-Parameter properties of 2-port networks 4.1 Complex linear gain 4.2 Scalar linear gain 4.3 Scalar logarithmic gain 4.4 Insertion loss 4.5 Input return loss 4.6 Output return loss 4.7 Reverse gain and reverse isolation 4.8 Voltage reflection coefficient

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2011/10/31

Scattering parameters - Wikipedia, the free encyclopedia

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4.9 Voltage standing wave ratio 5 4-Port S-Parameters 5.1 4-Port Mixed-Mode S-Parameters 6 S-parameters in amplifier design 6.1 Port loading conditions for an amplifier to be unconditionally stable 6.1.1 values for (output stability circle) 6.1.2 values for (input stability circle) 6.2 Scattering transfer parameters 7 1-port S-parameters 8 Higher-order S-parameter matrices 9 Measurement of S-parameters 9.1 Vector network analyzer 9.2 Calibration 9.3 Minimization of systematic errors 9.4 Output format of measured and corrected S-parameter data 9.5 List format 9.6 Graphical (Smith chart) 9.7 Graphical (polar diagram) 9.8 Measuring S-parameters of a one-port network 9.9 Measuring S-parameters of networks with more than 2 ports 10 References 11 Bibliography 12 See also

BackgroundHistorically, an electrical network would have comprised a 'black box' containing various interconnected basic electrical circuit components or lumped elements such as resistors, capacitors, inductors and transistors. 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 currents either enter or exit the network. Sometimes these are referred to as pairs of terminals[5][6] so for example a 2-port network is equivalent to a 4-terminal network, though this terminology is unusual with S-parameters, because most S-parameter measurements are made at frequencies where coaxial or waveguide connectors are more appropriate. 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

http://en.wikipedia.org/wiki/Scattering_parameters

2011/10/31

Scattering parameters - Wikipedia, the free encyclopedia

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(normalised) impedance (or admittance) 'seen' at that port. The following information must be defined when specifying any S-parameter: (1) The characteristic impedance (often 50 ). (2) The allocation of port numbers. (3) Conditions which may affect the network, such as frequency, temperature, control voltage, and bias current, where applicable.

The general S-parameter matrixDefinitionFor a generic multi-port network, each of the ports is allocated an integer 'n' ranging from 1 to N, where N is the total number of ports. For port n, the associated S-parameter definition is in terms of incident and reflected 'power waves', and respectively. Kurokawa[7] defines the incident power wave for each port as

and the reflected wave for each port is defined as

where is the diagonal matrix of the complex reference impedance for each port, is the elementwise complex conjugate of , and are respectively the column vectors of the voltages and currents at each port 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

For all ports the reflected power waves may be defined in terms of the S-parameter matrix and the incident power waves by the following matrix equation:

http://en.wikipedia.org/wiki/Scattering_parameters

2011/10/31

Scattering parameters - Wikipedia, the free encyclopedia

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where S is an N x N matrix the elements of which can be indexed using conventional matrix (mathematics) notation.

ReciprocityA 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. An interesting property of 3-port networks, however, is that they cannot be simultaneously reciprocal, loss-free, and perfectly matched.[8]

Lossless networksA 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 and is the identity matrix.

Lossy networksA 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 : . In this case , and is positive definite.

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.[9] In this case the relationship between the reflected, incident power waves and the S-parameter matrix is given by:

Expanding the matrices into equations gives:

http://en.wikipedia.org/wiki/Scattering_parameters

2011/10/31

Scattering parameters - Wikipedia, the free encyclopedia

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and

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 S-parameters, , , and . If one considers an incident power 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 to

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