- Scattering parameters
Scattering parameters or S-parameters are properties used in
electrical engineering, electronics engineering, and communication systems engineering describing the electrical behavior of linear electrical networks when undergoing various steady state stimuli by small signals.
They are members of a family of similar parameters used in electronics engineering, other examples being:
Y-parameters, [Pozar, David M. (2005); "Microwave Engineering, Third Edition" (Intl. Ed.); John Wiley & Sons, Inc.; pp 170-174. ISBN 0-471-44878-8.] Z-parameters, [Pozar, David M. (2005) (op. cit); pp 170-174.] H-parameters, T-parametersor ABCD-parameters. [Pozar, David M. (2005) (op. cit);pp 183-186.] [Morton, A. H. (1985); " Advanced Electrical Engineering";Pitman Publishing Ltd.; pp 33-72. ISBN 0-273-40172-6] .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 and unmatched 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 or components may be expressed using S-parameters, such as
gain, return loss, voltage standing wave ratio(VSWR), reflection coefficientand amplifierstability. The term 'scattering' is more common to optical engineeringthan RF engineering, referring to the effect observed when a plane electromagnetic waveis incident on an obstruction or passes across dissimilar dielectricmedia. In the context of S-parameters, scattering refers to the way in which the traveling currents and voltages in a transmission lineare affected when they meet a discontinuitycaused 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 microwavefrequencies where signal power and energy considerations are more easily quantified than currents and voltages. S-parameters change with the measurement frequency so this must be included for any S-parameter measurements stated, in addition to the characteristic impedanceor system impedance. S-parameters are readily represented in matrix form and obey the rules of matrix algebra.
Historically, 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' [Pozar, David M. (2005) (op. cit);p170.] [Morton, A. H. (1985) (op. cit.); p 33] 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
coaxialconnectors 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, thus representing magnitude and angle, or amplitudeand phase. The complex number may either be expressed in rectangularform 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 Chartnormalised 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 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 matrix
For a generic multi-port network definition, it is assumed that 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 [Kurokawa, K., "Power Waves and the Scattering Matrix", IEEE Trans. Micr. Theory & Tech., Mar. 1965, pp194-202] 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
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:
where S is an N x N matrix the elements of which can be indexed using conventional
The phase part of an S-parameter is the "spatial" phase at the test frequency, not the temporal (time-related) phase.
The S-parameter matrix for the 2-port network is probably the most common and it serves as the basic building block for generating the higher order matrices for larger networks.cite book
author=J Choma & WK Chen
title=Feedback networks: theory and circuit applications
page= Chapter 3, p. 225 ff
url=http://books.google.com/books?id=EWAIYfilhfwC&pg=PA225&dq=isbn:9810227701&source=gbs_toc_r&cad=0_0&sig=WooLpzenKx-2biNlOJLlxgRQzzQ] 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:
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 totally absorbed making equal to zero. Therefore
Similarly, if port 1 is terminated in the system impedance then becomes zero, giving
Each 2-port S-parameter has 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
A network will be reciprocal if it is passive and it contains only isotropic 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. All networks which include anisotropic materials in the transmission medium such as those containing ferrite components will be non-reciprocal. Although it doesn't necessarily contain ferrites, an amplifier is also an 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. [Pozar, David M. (2005) (op. cit); p 173.]
The loss-free network
A loss-free 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, or , where is the
conjugate transposeof and is the identity matrix.
lossypassive 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 .
-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
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.
calar linear gain
The scalar linear gain (or linear gain magnitude) |G| is given by
That is simply the scalar voltage gain as a linear ratio of the output voltage divided by the input voltage. As this is a scalar quantity, the phase is not relevant in this case.
calar logarithmic gain
The scalar logarithmic (decibel or dB) expression for gain (g) is
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 (), usually also represented in dB, is given by:
Since insertion loss is, by definition, a loss (or, negative gain), the leading negative sign is often neglected. Insertion loss is commonly mistaken for (above), and deserves special consideration here. The difference between them is that penalizes the device for mismatch at the input, while insertion loss is not a function of the input or source impedances. This can be further expressed as:
: where is the available power from the source
: where is the power that makes it into port 1 of the network
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
By definition, return loss is a positive scalar quantity implying the 2 pairs of magnitude (|) symbols. The linear part, is equivalent to the reflected voltage magnitude divided by the incident voltage magnitude.
Output return loss
The output return loss () has a similar definition to the input return loss but applies to the output port (port 2) instead of the input port. It is given by
Reverse gain and reverse isolation
The scalar logarithmic (decibel or dB) expression for reverse gain () is:
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:
Voltage reflection coefficient
The voltage reflection coefficient at the input port () or at the output port () are equivalent to and respectively, so
: and .
As and are complex quantities, so are and .
Voltage reflection coefficients are complex quantities and may be graphically represented on polar diagrams or Smith Charts
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
At the output port, the VSWR () is given by
-parameters in amplifier design
The reverse isolation parameter determines the level of feedback from the output of an amplifier to the input and therefore influences its stability (its tendency to refrain from oscillation) together with the forward gain . An amplifier with input and output ports perfectly isolated from each other would have infinite scalar log magnitude isolation or the linear magnitude of would be zero. Such an amplifier is said to be unilateral. Most practical amplifiers though will have some finite isolation allowing the reflection coefficient 'seen' at the input to be influenced to some extent by the load connected on the output. An amplifier which is deliberately designed to have the smallest possible value of is often called a
Suppose the output port of a real (non-unilateral or bilateral) amplifier is connected to an arbitrary load with a reflection coefficient of . The actual reflection coefficient 'seen' at the input port will be given by [Gonzalez, Guillermo (1997); "Microwave Transistor Amplifiers Analysis and Design, Second Edition"; Prentice Hall NJ; pp 212-216. ISBN 0-13-254335-4.]
If the amplifier is unilateral then and or, to put it another way, the output loading has no effect on the input.
A similar property exists in the opposite direction, in this case if is the reflection coefficient seen at the output port and is the reflection coefficient of the source connected to the input port.
Port loading conditions for an amplifier to be unconditionally stable
An amplifier is unconditionally stable if a load or source of "any" reflection coefficient can be connected without causing instability. This condition occurs if the magnitudes of the reflection coefficients at the source, load and the amplifier's input and output ports are simultaneously less than unity. Instability can cause severe distortion of the amplifier's gain frequency response or, in the extreme, oscillation. To be unconditionally stable at the frequency of interest, an amplifier must satisfy the following 4 equations simultaneously: [Gonzalez, Guillermo (op. cit.); pp 217-222]
The boundary condition for when each of these values is equal to unity may be represented by a circle drawn on the polar diagram representing the (complex) reflection coefficient, one for the input port and the other for the output port. Often these will be scaled as Smith Charts. In each case coordinates of the circle centre and the associated radius are given by the following equations:
= values for (output stability circle)=
= values for (input stability circle)=
where, in both cases
and the superscript star (*) indicates a complex conjugate.
The circles are in complex units of reflection coefficient so may be drawn on impedance or admittance based Smith Charts normalised to the system impedance. This serves to readily show the regions of normalised impedance (or admittance) for predicted unconditional stability. Another way of demonstrating unconditional stability is by means of the Rollet stability factor (), defined as
The condition of unconditional stability is achieved when and
cattering (transfer) parameters
The Scattering transfer parameters or T-parameters of a 2-port network are expressed by the T-parameter matrix and are closely related to the corresponding S-parameter matrix. The T-parameter matrix is related to the incident and reflected normalised waves at each of the ports as follows:
The advantage of T-parameters compared to S-parameters is that they may be used to readily determine the effect of cascading 2 or more 2-port networks by simply multiplying the associated individual T-parameter matrices. If the T-parameters of say three different 2-port networks 1, 2 and 3 are , and respectively then the T-parameter matrix for the cascade of all three networks () in serial order is given by:
As with S-parameters, T-parameters are complex values and there is a direct conversion between the two types. Although the cascaded T-parameters is a simple matrix multiplication of the individual T-parameters, the conversion for each network's S-parameters to the corresponding T-parameters and the conversion of the cascaded T-parameters back to the equivalent cascaded S-parameters, which are usually required, is not trivial. However once the operation is completed, the complex full wave interactions between all ports in both directions will be taken into account. The following equations will provide conversion between S and T parameters for 2-port networks. [ [http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf "S-Parameter Design"; Application Note AN 154; Agilent Technologies; p 14] ]
From S to T:
From T to S
Where indicates the
determinantof the matrix .
The S-parameter for a 1-port network is given by a simple 1 x 1 matrix of the form where n is the allocated port number. To comply with the S-parameter definition of linearity, this would normally be a passive load of some type.
Higher-order S-parameter matrices
Higher order S-parameters for pairs of dissimilar ports (), where may be deduced similarly to those for 2-port networks by considering pairs of ports in turn, in each case ensuring that all of the remaining (unused) ports are loaded with an impedance identical to the system impedance. In this way the incident power wave for each of the unused ports becomes zero yielding similar expressions to those obtained for the 2-port case. S-parameters relating to single ports only () require all of the remaining ports to be loaded with an impedance identical to the system impedance therefore making all of the incident power waves zero except that for the port under consideration. In general therefore we have:
For example, a 3-port network such as a 2-way splitter would have the following S-parameter definitions
Measurement of S-parameters
Vector network analyzer
The diagram shows the essential parts of a typical 2-port vector network analyzer (VNA). The two ports of the
device under test(DUT) are denoted port 1 (P1) and port 2 (P2). The test port connectors provided on the VNA itself are precision types which will normally have to be extended and connected to P1 and P2 using precision cables 1 and 2, PC1 and PC2 respectively and suitable connector adaptors A1 and A2 respectively.
The test frequency is generated by a variable frequency CW source and its power level is set using a variable
attenuator. The position of switch SW1 sets the direction that the test signal passes through the DUT. Initially consider that SW1 is at position 1 so that the test signal is incident on the DUT at P1 which is appropriate for measuring and . The test signal is fed by SW1 to the common port of splitter 1, one arm (the reference channel) feeding a reference receiver for P1 (RX REF1) and the other (the test channel) connecting to P1 via the directional couplerDC1, PC1 and A1. The third port of DC1 couples off the power reflected from P1 via A1 and PC1, then feeding it to test receiver 1 (RX TEST1). Similarly, signals leaving P2 pass via A2, PC2 and DC2 to RX TEST2. RX REF1, RX TEST1, RX REF2 and RXTEST2 are known as coherent receivers as they share the same reference oscillator, and they are capable of measuring the test signal's amplitude "and" phase at the test frequency. All of the complex receiver output signals are fed to a processorwhich does the mathematical processing and displays the chosen parameters and format on the phase and amplitude display. The instantaneousvalue of phase includes both the temporaland spatialparts, but the former is removed by virtue of using 2 test channels, one as a reference and the other for measurement. When SW1 is set to position 2, the test signals are applied to P2, the reference is measured by RX REF2, reflections from P2 are coupled off by DC2 and measured by RX TEST2 and signals leaving P1 are coupled off by DC1 and measured by RX TEST1. This position is appropriate for measuring and .
Prior to making a VNA S-parameter measurement, the first essential step is to perform an accurate
calibrationappropriate to the intended measurements. Several types of calibration are normally available on the VNA. It is only in the last few years that VNAs have had the sufficiently advanced processing capability, at realistic cost, required to accomplish the more advanced types of calibration, including corrections for systematic errors. [ [http://cp.literature.agilent.com/litweb/pdf/5965-7709E.pdf "Applying Error Correction to Network Analyzer Measurements"; Agilent Application Note AN 1287-3, Agilent Technologies; p6] ] The more basic types, often called 'response' calibrations, may be performed quickly but will only provide a result with moderate uncertainty. For improved uncertainty and dynamic rangeof the measurement a full 2 port calibration is required prior to DUT measurement. This will effectively eliminate all sources of systematic errors inherent in the VNA measurement system.
Minimization of systematic errors
Systematic errors are those which do not vary with time during a calibration. For a set of 2 port S-parameter measurements there are a total of 12 types of systematic errors which are measured and removed mathematically as part of the full 2 port calibration procedure. They are, for each port:
2. source and
frequency responseerrors caused by reflection and transmission tracking within the test receivers
The calibration procedure requires initially setting up the VNA with all the cables, adaptors and connectors necessary to connect to the DUT but not at this stage connecting it. A calibration kit is used according to the connector types fitted to the DUT. This will normally include adaptors, nominal short circuits (SCs), open circuits (OCs) and load termination (TERM) standards of both connector sexes appropriate to the VNA and DUT connectors. Even with standards of high quality, when performing tests at the higher frequencies into the microwave range various stray capacitances and inductances will become apparent and cause uncertainty during the calibration. Data relating to the strays of the particular calibration kit used are measured at the factory traceable to
national standardsand the results are programmed into the VNA memory prior to performing the calibration.
The calibration procedure is normally software controlled, and instructs the operator to fit various calibration standards to the ends of the DUT connecting cables as well as making a through connection. At each step the VNA processor captures data across the test frequency range and stores it. At the end of the calibration procedure, the processor uses the stored data thus obtained to apply the systematic error corrections to all subsequent measurements made. All subsequent measurements are known as 'corrected measurements. At this point the DUT is connected and a corrected measurement of its S-parameters made.
Output format of measured and corrected S-parameter data
The S-parameter test data may be provided in many alternative formats, for example: list, graphical (
Smith Chart) and graphical ( polar diagram).
In list format the measured and corrected S-parameters are tabulated against frequency. The most common list format is known as Touchstone or SNP, where N is the number of ports. Commonly text files containing this information would have the filename extension '.s2p'. An example of a
Touchstone Filelisting for the full 2-port S-parameter data obtained for a device is shown below:
! Created Fri Jul 21 14:28:50 2005 # MHZ S DB R 50 ! SP1.SP 50 -15.4 100.2 10.2 173.5 -30.1 9.6 -13.4 57.2 51 -15.8 103.2 10.7 177.4 -33.1 9.6 -12.4 63.4 52 -15.9 105.5 11.2 179.1 -35.7 9.6 -14.4 66.9 53 -16.4 107.0 10.5 183.1 -36.6 9.6 -14.7 70.3 54 -16.6 109.3 10.6 187.8 -38.1 9.6 -15.3 71.4
Rows beginning with an exclamation mark contains only comments. The row beginning with the hash symbol indicates that in this case frequencies are in megahertz (MHZ), S-parameters are listed (S), magnitudes are in dB log magnitude (DB) and the system impedance is 50 Ohm (R 50). There are 9 columns of data. Column 1 is the test frequency in megahertz in this case. Columns 2, 4, 6 and 8 are the magnitudes of , , and respectively in dB. Columns 3, 5, 7 and 9 are the angles of , , and respectively in degrees.
Graphical (Smith chart)
Any 2-port S-parameter may be displayed on a
Smith chartusing polar co-ordinates, but the most meaningful would be and since either of these may be converted directly into an equivalent normalized impedance (or admittance) using the characteristic Smith Chart impedance (or admittance) scaling appropriate to the system impedance.
Graphical (polar diagram)
Any 2-port S-parameter may be displayed on a polar diagram using polar co-ordinates.
In either graphical format each S-parameter at a particular test frequency is displayed as a dot. If the measurement is a sweep across several frequencies a dot will appear for each. Many VNAs connect successive dots with straight lines for easier visibility.
Measuring S-parameters of a one-port network
The S-parameter matrix for a network with just one port will have just one element represented in the form , where n is the number allocated to the port. Most VNAs provide a simple one-port calibration capability for one port measurement to save time if that is all that is required.
Measuring S-parameters of networks with more than 2 ports
VNAs designed for the simultaneous measurement of the S-parameters of networks with more than two ports are feasible but quickly become prohibitively complex and expensive. Usually their purchase is not justified since the required measurements can be obtained using a standard 2-port calibrated VNA with extra measurements followed by the correct interpretation of the results obtained. The required S-parameter matrix can be assembled from successive two port measurements in stages, two ports at a time, on each occasion with the unused ports being terminated in high quality loads equal to the system impedance. One risk of this approach is that the return loss or VSWR of the loads themselves must be suitably specified to be as close as possible to a perfect 50 Ohms, or whatever the nominal system impedance is. For a network with many ports there may be a temptation, on grounds of cost, to inadequately specify the VSWRs of the loads. Some analysis will be necessary to determine what the worst acceptable VSWR of the loads will be.
Assuming that the extra loads are specified adequately, if necessary, two or more of the S-parameter subscripts are modified from those relating to the VNA (1 and 2 in the case considered above) to those relating to the network under test (1 to N, if N is the total number of DUT ports). For example, if the DUT has 5 ports and a two port VNA is connected with VNA port 1 to DUT port 3 and VNA port 2 to DUT port 5, the measured VNA results (, , and ) would be equivalent to , , and respectively, assuming that DUT ports 1, 2 and 4 were terminated in adequate 50 Ohm loads . This would provide 4 of the necessary 25 S-parameters.
*Guillermo Gonzalez, "Microwave Transistor Amplifiers, Analysis and Design, 2nd. Ed.", Prentice Hall, New Jersey; ISBN 0135816467
*David M. Pozar, "Microwave Engineering", Third Edition, John Wiley & Sons Inc.; ISBN 0471170968
* [http://cp.literature.agilent.com/litweb/pdf/5952-1087.pdf "S-Parameter Design", Application Note AN 154, Agilent Technologies]
* [http://contact.tm.agilent.com/Agilent/tmo/an-95-1/index.html "S-Parameter Techniques for Faster, More Accurate Network Design", Application Note AN 95-1, Agilent Technologies, PDF slides plus QuickTime video] or [http://cp.literature.agilent.com/litweb/pdf/5952-1130.pdf scan of Richard W. Anderson's original article]
*A. J. Baden Fuller, "An Introduction to Microwave Theory and Techniques, Second Edition, Pergammon International Library; ISBN 0-08-024227-8
*Ramo, Whinnery and Van Duzer, "Fields and Waves in Communications Electronics", John Wiley & Sons; ISBN 0-471-70721-X
*C. W. Davidson, "Transmission Lines for Communications with CAD Programs", Second Edition, Macmillan Education Ltd.; ISBN 0-333-47398-1
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