Scattering parameters
Encyclopedia
Scattering parameters or S-parameters (the elements of a scattering matrix or S-matrix) describe the electrical behavior of linear
electrical network
s 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 load
s are used. These termination
s 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 volt
ages 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.
' 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 linear
ly with incident small signals. It may also include many typical communication system components or 'blocks' such as amplifier
s, 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 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 decibel
s. The S-parameter angle is most frequently expressed in degrees
but occasionally in radian
s. 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 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.
and 
respectively.
Kurokawa 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:
where S is an N x N matrix the elements of which can be indexed using conventional matrix (mathematics)
notation.
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.
. 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
.
. In this case 
, and 
is positive definite
.
Expanding the matrices into equations gives:
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 totally absorbed making 
equal to zero. Therefore
and 
Similarly, if port 1 is terminated in the system impedance then
becomes zero, giving
and 
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
.
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.
.
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.
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.
) is the dB expression of the transmission coefficient 
. It is thus given by:
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.
) 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
dB.
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.
) 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
dB.
) is:
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:
dB.
) 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
See also the Reflection Coefficient
article.
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
This is correct for reflection coefficients with a magnitude no greater than unity, which is usually the case. A reflection coefficient with a magnitude greater than unity, such as in a tunnel diode amplifier, will result in a negative value for this expression. VSWR, however, from its definition, is always positive. A more correct expression for port k of a multiport is;
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.
Note the format of the parameter notation SXYab, where “S” stands for scatttering 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;
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.
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 buffer amplifier
.
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
.
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.
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:
Radius 
Center
Radius 
Center
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 
However, they could be defined differently, as follows :
The RF Toolbox add-on to MATLAB and several books (for example "Network scattering parameters") use this last definition, so caution is necessary. The "From S to T" and "From T to S" paragraphs in this article are based on the first definition. Adaptation to the second definition is trivial (interchanging T11 for T22, and T12 for T21).
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.
From S to T:
From T to S
Where
indicates the determinant
of the matrix
.
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.
), 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:
and
For example, a 3-port network such as a 2-way splitter would have the following S-parameter definitions
(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 coupler DC1, 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 processor which does the mathematical processing and displays the chosen parameters and format on the phase and amplitude display. The instantaneous value of phase includes both the temporal
and spatial parts, 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 
.
appropriate 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. 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 range
of 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.
1. directivity
and crosstalk
2. source and load mismatches
3. frequency response
errors 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 standards and 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.
or pole figure
).
listing 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.
using 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.
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.
, 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.
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.
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
electrical network
Electrical network
An electrical network is an interconnection of electrical elements such as resistors, inductors, capacitors, transmission lines, voltage sources, current sources and switches. An electrical circuit is a special type of network, one that has a closed loop giving a return path for the current...
s when undergoing various steady state
Steady state
A system in a steady state has numerous properties that are unchanging in time. This implies that for any property p of the system, the partial derivative with respect to time is zero:...
stimuli by electrical signals.
The parameters are useful for electrical engineering
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...
, electronics engineering, and communication systems
Communications system
In telecommunication, a communications system is a collection of individual communications networks, transmission systems, relay stations, tributary stations, and data terminal equipment usually capable of interconnection and interoperation to form an integrated whole...
design, and especially for microwave engineering
Microwave Engineering
Microwave engineering pertains to the study and design of microwave circuits, components, and systems. Fundamental principles are applied to analysis, design and measurement techniques in this field. The short wavelengths involved distinguish this discipline from Electronic 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
Open-circuit voltage
Open-circuit voltage is the difference of electrical potential between two terminals of a device when there is no external load connected, i.e. the circuit is broken or open. Under these conditions there is no external electric current between the terminals, even though there may be current...
or short circuit
Short circuit
A short circuit in an electrical circuit that allows a current to travel along an unintended path, often where essentially no electrical impedance is encountered....
conditions to characterize a linear electrical network; instead, matched load
Impedance matching
In electronics, impedance matching is the practice of designing the input impedance of an electrical load to maximize the power transfer and/or minimize reflections from the load....
s are used. These termination
Electrical termination
Electrical termination of a signal involves providing a terminator at the end of a wire or cable to prevent an RF signal from being reflected back from the end, causing interference...
s 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
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...
, return loss
Return loss
In telecommunications, return loss or reflection loss is the loss of signal power resulting from the reflection caused at a discontinuity in a transmission line or optical fiber. This discontinuity can be a mismatch with the terminating load or with a device inserted in the line...
, voltage standing wave ratio (VSWR), reflection coefficient
Reflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...
and amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
stability. The term 'scattering' is more common to optical engineering
Optical engineering
Optical engineering is the field of study that focuses on applications of optics. Optical engineers design components of optical instruments such as lenses, microscopes, telescopes, and other equipment that utilizes the properties of light. Other devices include optical sensors and measurement...
than RF engineering, referring to the effect observed when a plane electromagnetic wave
Plane wave
In the physics of wave propagation, a plane wave is a constant-frequency wave whose wavefronts are infinite parallel planes of constant peak-to-peak amplitude normal to the phase velocity vector....
is incident on an obstruction or passes across dissimilar dielectric
Dielectric
A dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in an electric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their average equilibrium positions causing dielectric...
media. In the context of S-parameters, scattering refers to the way in which the traveling currents
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...
and volt
Volt
The volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
ages in a transmission line
Transmission line
In communications and electronic engineering, a transmission line is a specialized cable designed to carry alternating current of radio frequency, that is, currents with a frequency high enough that its wave nature must be taken into account...
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
Electrical impedance
Electrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
differing from the line's characteristic impedance
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
.
Although applicable at any frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
, S-parameters are mostly used for networks operating at radio frequency
Radio frequency
Radio frequency is a rate of oscillation in the range of about 3 kHz to 300 GHz, which corresponds to the frequency of radio waves, and the alternating currents which carry radio signals...
(RF) and microwave
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
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
Characteristic impedance
The characteristic impedance or surge impedance of a uniform transmission line, usually written Z_0, is the ratio of the amplitudes of a single pair of voltage and current waves propagating along the line in the absence of reflections. The SI unit of characteristic impedance is the ohm...
or system impedance
Nominal impedance
Nominal impedance in electrical engineering and audio engineering refers to the approximate designed impedance of an electrical circuit or device...
.
S-parameters are readily represented in matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
form and obey the rules of matrix algebra.
Background
Historically, an electrical network would have comprised a 'black boxBlack box
A black box is a device, object, or system whose inner workings are unknown; only the input, transfer, and output are known characteristics.The term black box can also refer to:-In science and technology:*Black box theory, a philosophical theory...
' 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 linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
ly with incident small signals. It may also include many typical communication system components or 'blocks' such as amplifier
Amplifier
Generally, an amplifier or simply amp, is a device for increasing the power of a signal.In popular use, the term usually describes an electronic amplifier, in which the input "signal" is usually a voltage or a current. In audio applications, amplifiers drive the loudspeakers used in PA systems to...
s, attenuators
Attenuator (electronics)
An attenuator is an electronic device that reduces the amplitude or power of a signal without appreciably distorting its waveform.An attenuator is effectively the opposite of an amplifier, though the two work by different methods...
, filters
Electronic filter
Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both...
, 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 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
Coaxial
In geometry, coaxial means that two or more forms share a common axis; it is the three-dimensional linear analogue of concentric.Coaxial cable, as a common example, has a wire conductor in the centre a circumferential outer conductor and an insulating medium called the dielectric separating...
or waveguide
Waveguide (electromagnetism)
In electromagnetics and communications engineering, the term waveguide may refer to any linear structure that conveys electromagnetic waves between its endpoints. However, the original and most common meaning is a hollow metal pipe used to carry radio waves...
connectors are more appropriate.
The S-parameter matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
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 numberComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
that represents magnitude
Magnitude (mathematics)
The magnitude of an object in mathematics is its size: a property by which it can be compared as larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
and angle
Angle
In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
, i.e. amplitude
Amplitude
Amplitude is the magnitude of change in the oscillating variable with each oscillation within an oscillating system. For example, sound waves in air are oscillations in atmospheric pressure and their amplitudes are proportional to the change in pressure during one oscillation...
and phase
Phase (waves)
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...
. 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
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...
. When expressed in logarithmic form, magnitude has the "dimensionless unit
Dimensionless quantity
In dimensional analysis, a dimensionless quantity or quantity of dimension one is a quantity without an associated physical dimension. It is thus a "pure" number, and as such always has a dimension of 1. Dimensionless quantities are widely used in mathematics, physics, engineering, economics, and...
" of decibel
Decibel
The decibel is a logarithmic unit that indicates the ratio of a physical quantity relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities...
s. The S-parameter angle is most frequently expressed in degrees
Degree (angle)
A degree , usually denoted by ° , is a measurement of plane angle, representing 1⁄360 of a full rotation; one degree is equivalent to π/180 radians...
but occasionally in radian
Radian
Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...
s. Any S-parameter may be displayed graphically on a polar diagram by a dot for one frequency or a locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
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 ChartSmith chart
The Smith chart, invented by Phillip H. Smith , is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency engineering to assist in solving problems with transmission lines and matching circuits...
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 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.
Definition
For 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 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 andSometimes 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:
where S is an N x N matrix the elements of which can be indexed using conventional matrix (mathematics)
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
notation.
Reciprocity
A network will be reciprocal if it is passive and it contains only reciprocalReciprocity (electromagnetism)
In classical electromagnetism, reciprocity refers to a variety of related theorems involving the interchange of time-harmonic electric current densities and the resulting electromagnetic fields in Maxwell's equations for time-invariant linear media under certain constraints...
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 transposeTranspose
In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
. Networks which include non-reciprocal materials in the transmission medium such as those containing magnetically biased
Biasing (electronics)
Biasing in electronics is the method of establishing predetermined voltages and/or currents at various points of an electronic circuit to set an appropriate operating point...
ferrite
Ferrite (magnet)
Ferrites are chemical compounds consisting of ceramic materials with iron oxide as their principal component. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.Many ferrites are spinels with the...
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.
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 transposeConjugate transpose
In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
of
and is the identity matrixIdentity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
.
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, or :. In this case , and is positive definitePositive-definite matrix
In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form ....
.
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:
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 theoremMaximum power theorem
In electrical engineering, the maximum power transfer theorem states that, to obtain maximum external power from a source with a finite internal resistance, the resistance of the load must be equal to the resistance of the source as viewed from the output terminals...
,
will be totally absorbed making equal to zero. Therefore and Similarly, if port 1 is terminated in the system impedance then
becomes zero, giving and 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 coefficientS-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.
Scalar linear gain
The scalar linear gain (or linear gain magnitude) is given by.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 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: 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 dB.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 dB.Reverse gain and reverse isolation
The scalar logarithmic (decibel or dB) expression for reverse gain () is: 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: dB.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
See also the Reflection Coefficient
Reflection coefficient
The reflection coefficient is used in physics and electrical engineering when wave propagation in a medium containing discontinuities is considered. A reflection coefficient describes either the amplitude or the intensity of a reflected wave relative to an incident wave...
article.
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 byAt the output port, the VSWR (
) is given byThis is correct for reflection coefficients with a magnitude no greater than unity, which is usually the case. A reflection coefficient with a magnitude greater than unity, such as in a tunnel diode amplifier, will result in a negative value for this expression. VSWR, however, from its definition, is always positive. A more correct expression for port k of a multiport is;
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.| Stimulus | ||||||
|---|---|---|---|---|---|---|
| Differential | Common Mode | |||||
| Port 1 | Port 2 | Port 1 | Port 2 | |||
| Response | Differential | Port 1 | SDD11 | SDD12 | SDC11 | SDC12 |
| Port 2 | SDD21 | SDD22 | SDC21 | SDC22 | ||
| Common Mode | Port 1 | SCD11 | SCD12 | SCC11 | SCC12 | |
| Port 2 | SCD21 | SCD22 | SCC21 | SCC22 | ||
Note the format of the parameter notation SXYab, where “S” stands for scatttering 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.
S-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 buffer amplifierBuffer amplifier
A buffer amplifier is one that provides electrical impedance transformation from one circuit to another...
.
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.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. An important requirement that is often overlooked is that the amplifier be a linear network with no poles in the right half plane. 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: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)
Radius Center
values for 
(input stability circle)
Radius Center
where, in both cases
and the superscript star (*) indicates a complex conjugate
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
.
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 asThe condition of unconditional stability is achieved when
and Scattering 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:However, they could be defined differently, as follows :
The RF Toolbox add-on to MATLAB and several books (for example "Network scattering parameters") use this last definition, so caution is necessary. The "From S to T" and "From T to S" paragraphs in this article are based on the first definition. Adaptation to the second definition is trivial (interchanging T11 for T22, and T12 for T21).
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.
From S to T:
From T to S
Where
indicates the determinantDeterminant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...
of the matrix
.1-port S-parameters
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:and
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 analyzerNetwork analyzer (electrical)
A network analyzer is an instrument that measures the network parameters of electrical networks. Today, network analyzers commonly measure s–parameters because reflection and transmission of electrical networks are easy to measure at high frequencies, but there are other network parameter...
(VNA). The two ports of the device under test
Device under test
Device under test , also known as unit under test , is a term commonly used to refer to a manufactured product undergoing testing.-In semiconductor testing:...
(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
Carrier wave
In telecommunications, a carrier wave or carrier is a waveform that is modulated with an input signal for the purpose of conveying information. This carrier wave is usually a much higher frequency than the input signal...
source and its power level is set using a variable attenuator
Attenuator (electronics)
An attenuator is an electronic device that reduces the amplitude or power of a signal without appreciably distorting its waveform.An attenuator is effectively the opposite of an amplifier, though the two work by different methods...
. 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 coupler DC1, 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 coherentCoherence (physics)
In physics, coherence is a property of waves that enables stationary interference. More generally, coherence describes all properties of the correlation between physical quantities of a wave....
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 processor which does the mathematical processing and displays the chosen parameters and format on the phase and amplitude display. The instantaneous value of phase includes both the temporal
Time
Time is a part of the measuring system used to sequence events, to compare the durations of events and the intervals between them, and to quantify rates of change such as the motions of objects....
and spatial parts, 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 .Calibration
Prior to making a VNA S-parameter measurement, the first essential step is to perform an accurate calibrationCalibration
Calibration is a comparison between measurements – one of known magnitude or correctness made or set with one device and another measurement made in as similar a way as possible with a second device....
appropriate 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. The more basic types, often called 'response' calibrations, may be performed quickly but will only provide a result with moderate uncertainty
Uncertainty
Uncertainty is a term used in subtly different ways in a number of fields, including physics, philosophy, statistics, economics, finance, insurance, psychology, sociology, engineering, and information science...
. For improved uncertainty and dynamic range
Dynamic range
Dynamic range, abbreviated DR or DNR, is the ratio between the largest and smallest possible values of a changeable quantity, such as in sound and light. It is measured as a ratio, or as a base-10 or base-2 logarithmic value.-Dynamic range and human perception:The human senses of sight and...
of 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:1. directivity
Directivity
In electromagnetics, directivity is a figure of merit for an antenna. It measures the power density the antenna radiates in the direction of its strongest emission, versus the power density radiated by an ideal isotropic radiator radiating the same total power.An antenna's directivity is a...
and crosstalk
2. source and load mismatches
3. frequency response
Frequency response
Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input...
errors 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 standards and 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 ChartSmith chart
The Smith chart, invented by Phillip H. Smith , is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency engineering to assist in solving problems with transmission lines and matching circuits...
or pole figure
Pole figure
A pole figure is a graphical representation of the orientation of objects in space. For example, pole figures in the form of stereographic projections are used to represent the orientation distribution of crystallographic lattice planes in crystallography and texture analysis in materials...
).
List format
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 fileTouchstone File
A Touchstone file was originally a proprietary file format for the eponymous frequency-domain linear circuit simulator from EEsof, launched in 1984 and now owned by Agilent Technologies....
listing 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 chartSmith chart
The Smith chart, invented by Phillip H. Smith , is a graphical aid or nomogram designed for electrical and electronics engineers specializing in radio frequency engineering to assist in solving problems with transmission lines and matching circuits...
using 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.See also
- Admittance parametersAdmittance parametersAdmittance parameters or Y-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal response of non-linear networks...
- Impedance parametersImpedance parametersImpedance parameters or Z-parameters are properties used in electrical engineering, electronic engineering, and communication systems engineering to describe the electrical behavior of linear electrical networks. They are also used to describe the small-signal response of non-linear networks...
- Two-port networkTwo-port networkA two-port network is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network...
- X-parametersX-parametersX-parameters are a mathematical superset of S-parameters and are used for characterizing the amplitudes and relative phase of harmonics generated by nonlinear components under large input power levels.-Description:...
, a non-linear superset of S-parameters - Belevitch's theoremBelevitch's theoremBelevitch's theorem is a theorem in electrical network analysis due to the Russo-Belgian mathematician Vitold Belevitch . The theorem provides a test for a given S-matrix to determine whether or not it can be constructed as a lossless rational two-port network.Lossless implies that the network...