- 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: 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. 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 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.
- 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.
Background
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.
Working
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.
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 thecurrent 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-Square 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 S-parameter magnitude may be expressed in linear form or logarithmic form. When expressed in logarithmic form, magnitude has the "dimensionless unit" of decibels.
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 reflected, incident power waves and the S-parameter matrix is given by:
Expanding the matrices into equations gives:
b1= (S11)(a1) + (S12)(a2),
b2=(S21)(a1)+(S22)(a2),
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 S11,S12 ,S21 , andS22 . If one considers an incident power wave at port 1 (a1) there may result from it waves exiting from either port 1 itself (b1 ) or port 2 (b2).
However if, according to the definition of S-parameters, port 2 is terminated in a load identical to the system
impedance (Z0 ) then, by the maximum power transfer theorem,b2 will be totally absorbed making a2 equal to zero.
Scattering parameters 4
zero. Therefore
S11=b1\a1
and
S21=b2\a1.
Similarly, if port 1 is terminated in the system impedance then becomes zero, giving
S12=b1\a2
and
S22=b2\a2
Each 2-port S-parameter has the following generic descriptions:
S11 is the input port voltage reflection coefficient
S12 is the reverse voltage gain
S21 is the forward voltage gain
S22 is the output port voltage reflection coefficient
S-Parameter properties of 2-port networks
An amplifier operating under linear (small signal) conditions is a good example of a non-reciprocal network and a matched attenuator is an example of a reciprocal network. In the following cases we will assume that the input and output connections are to ports 1 and 2 respectively which is the most common convention. The nominal system impedance, frequency and any other factors which may influence the device, such as temperature, must also be
specified.
Complex linear gain:- The complex linear gain G is given by
.G=S21
That is simply the voltage gain as a linear ratio of the output voltage divided by the input voltage, all values
expressed as complex quantities.
Scalar linear gain:-The scalar linear gain (or linear gain magnitude) is given by
|G| =|S21|
That is simply the scalar voltage gain as a linear ratio of the output voltage and the input voltage. As this is a scalar quantity, the phase is not relevant in this case.
Scalar logarithmic gain:- The scalar logarithmic (decibel or dB) expression for gain (g)is
g=20log|S21|dB.
This is more commonly used than scalar linear gain and a positive quantity is normally understood as simply a'gain'... A negative quantity can be expressed as a 'negative gain' or more usually as a 'loss' equivalent to its magnitude in dB. For example, a 10 m length of cable may have a gain of - 1 dB at 100 MHz or a loss of 1 dB at 100 MHz.
Insertion loss:-In case the two measurement ports use the same reference impedance, the insertion loss ( ) is the dB expression of the transmission coefficient . It is thus given by:
IL=-20log|S21|dB.
It is the extra loss produced by the introduction of the DUT between the 2 reference planes of the measurement.
Notice that the extra loss can be introduced by intrinsic loss in the DUT and/or mismatch. In case of extra loss the insertion loss is defined to be positive.
Input return loss
Input return loss ( ) is a scalar measure of how close the actual input impedance of the network is to the nominal system impedance value and, expressed in logarithmic magnitude, is given by
RLin=|20log|S11|| dB.
By definition, return loss is a positive scalar quantity implying the 2 pairs of magnitude (|) symbols. The linear part,|S11| is equivalent to the reflected voltage magnitude divided by the incident voltage magnitude.
Output return loss
The output return loss (RLout) has a similar definition to the input return loss but applies to the output port (port2) instead of the input port. It is given by
RLout=|20log|S22||dB.
Reverse gain and reverse isolation
The scalar logarithmic (decibel or dB) expression for reverse gain is:
g rev=20log|S12|dB.
Often this will be expressed as reverse isolation in which case it becomes a positive quantity equal to the magnitude of and the expression becomes:
Irev=|g rev|=|20log|S12||dB.
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 for the input port or for the output port.
At the input port, the VSWR ( ) is given by
Sin=(1+|S11|)\(1-|S11|);
At the output port, the VSWR ( ) is given by
Sout=(1+|S22|)\(1-|S22|);
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