Scattering parameters

Scattering parameters or S-parameters (the elementsof a scattering matrix or S-matrix) describe the elec-trical behavior of linear electrical networks when under-going various steady state stimuli by electrical signals.The parameters are useful for electrical engineering,electronics engineering, and communication systems de-sign, and especially for microwave engineering.The S-parameters are members of a family of similarparameters, other examples being: Y-parameters,[1] Z-parameters,[2] H-parameters, T-parameters or ABCD-parameters.[3][4] They differ from these, in the sense thatS-parameters do not use open or short circuit conditions tocharacterize a linear electrical network; instead, matchedloads are used. These terminations are much easier touse at high signal frequencies than open-circuit and short-circuit terminations. Moreover, the quantities are mea-sured in terms of power.Many electrical properties of networks of components(inductors, capacitors, resistors) may be expressed usingS-parameters, such as gain, return loss, voltage standingwave ratio (VSWR), reflection coefficient and amplifierstability. The term 'scattering' is more common to opticalengineering than RF engineering, referring to the effectobserved when a plane electromagnetic wave is incidenton an obstruction or passes across dissimilar dielectricmedia. In the context of S-parameters, scattering refersto the way in which the traveling currents and voltagesin a transmission line are affected when they meet adiscontinuity caused by the insertion of a network intothe transmission line. This is equivalent to the wavemeet-ing an impedance differing from the line’s characteristicimpedance.Although applicable at any frequency, S-parameters aremostly used for networks operating at radio frequency(RF) and microwave frequencies where signal power andenergy considerations are more easily quantified than cur-rents and voltages. S-parameters change with the mea-surement frequency, so frequency must be specified forany S-parameter measurements stated, in addition to thecharacteristic impedance or system impedance.S-parameters are readily represented in matrix form andobey the rules of matrix algebra.

1 Background

The first published description of S-parameters was in thethesis of Vitold Belevitch in 1945.[5] The name used by

Belevitch was repartition matrix and limited considera-tion to lumped-element networks. The term scatteringmatrix was used by physicist and engineer Robert HenryDicke in 1947 who independently developed the idea dur-ing wartime work on radar.[6][7] The technique was pop-ularized in the 1960s by Kaneyuki Kurokawa[8]

In the S-parameter approach, an electrical network is re-garded as a 'black box' containing various interconnectedbasic electrical circuit components or lumped elementssuch as resistors, capacitors, inductors and transistors,which interacts with other circuits through ports. Thenetwork is characterized by a square matrix of complexnumbers called its S-parameter matrix, which can be usedto calculate its response to signals applied to the ports.For the S-parameter definition, it is understood that a net-work may contain any components provided that the en-tire network behaves linearly with incident small signals.It may also include many typical communication systemcomponents or 'blocks’ such as amplifiers, attenuators,filters, couplers and equalizers provided they are also op-erating under linear and defined conditions.An electrical network to be described by S-parametersmay have any number of ports. Ports are the points atwhich electrical signals either enter or exit the network.Ports are usually pairs of terminals with the requirementthat the current into one terminal is equal to the currentleaving the other.[9][10] S-parameters are used at frequen-cies where the ports are often coaxial or waveguide con-nections.The S-parameter matrix describing an N-port networkwill be square of dimension N and will therefore containN2 elements. At the test frequency each element or S-parameter is represented by a unitless complex numberthat represents magnitude and angle, i.e. amplitude andphase. The complex number may either be expressed inrectangular form or, more commonly, in polar form. TheS-parameter magnitude may be expressed in linear formor logarithmic form. When expressed in logarithmicform, magnitude has the "dimensionless unit" of decibels.The S-parameter angle is most frequently expressed indegrees but occasionally in radians. Any S-parametermay be displayed graphically on a polar diagram by a dotfor one frequency or a locus for a range of frequencies.If it applies to one port only (being of the form Snn ), itmay be displayed on an impedance or admittance SmithChart normalised to the system impedance. The SmithChart allows simple conversion between the Snn param-eter, equivalent to the voltage reflection coefficient and

1

2 3 TWO-PORT S-PARAMETERS

the associated (normalised) impedance (or admittance)'seen' at that port.The following information must be defined when speci-fying a set of S-parameters:

1. The frequency

2. The characteristic impedance (often 50 Ω)

3. The allocation of port numbers

4. Conditions which may affect the network, suchas temperature, control voltage, and bias current,where applicable.

2 The general S-parameter matrix

2.1 Definition

For a generic multi-port network, the ports are numberedfrom 1 to N, where N is the total number of ports. Forport n, the associated S-parameter definition is in termsof incident and reflected 'power waves’, an and bn re-spectively.Kurokawa[11] defines the incident power wave for eachport as

a =1

2k(V + ZpI)

and the reflected wave for each port is defined as

b =1

2k(V − Z∗

pI)

whereZp is the diagonal matrix of the complex referenceimpedance for each port, Z∗

p is the elementwise complexconjugate of Zp , V and I are respectively the columnvectors of the voltages and currents at each port and

k =

(√|ℜ{Zp}|

)−1

Sometimes it is useful to assume that the referenceimpedance is the same for all ports in which case the def-initions of the incident and reflected waves may be sim-plified to

a =1

2

(V + Z0I)√|ℜ{Z0}|

and

b =1

2

(V − Z∗0 I)√

|ℜ{Z0}|

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

b = Sa

where S is an N x N matrix the elements of which canbe indexed using conventional matrix (mathematics) no-tation.

2.2 Reciprocity

A network will be reciprocal if it is passive and it con-tains only reciprocal materials that influence the transmit-ted signal. For example, attenuators, cables, splitters andcombiners are all reciprocal networks and Smn = Snm

in each case, or the S-parameter matrix will be equal to itstranspose. Networks which include non-reciprocal mate-rials in the transmission medium such as those contain-ing 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, isthat they cannot be simultaneously reciprocal, loss-free,and perfectly matched.[12]

2.3 Lossless networks

A lossless network is one which does not dissipate anypower, or: Σ |an|2 = Σ |bn|2 . The sum of the incidentpowers at all ports is equal to the sum of the reflectedpowers at all ports. This implies that the S-parametermatrix is unitary, that is (S)H(S) = (I) , where (S)His the conjugate transpose of (S) and (I) is the identitymatrix.

2.4 Lossy networks

A lossy passive network is one in which the sum of theincident powers at all ports is greater than the sum of thereflected powers at all ports. It therefore dissipates power,or: Σ |an|2 ̸= Σ |bn|2 . In this case Σ |an|2 > Σ |bn|2 ,and (I)− (S)H(S) is positive definite.[13]

3 Two-Port S-Parameters

See also: Two-port networkThe S-parameter matrix for the 2-port network is proba-bly themost commonly used and serves as the basic build-ing block for generating the higher order matrices forlarger networks.[14] In this case the relationship betweenthe reflected, incident power waves and the S-parametermatrix is given by:

3

a 1

a 2 b 1

b 2

(b1b2

)=

(S11 S12

S21 S22

)(a1a2

)Expanding the matrices into equations gives:

b1 = S11a1 + S12a2

and

b2 = S21a1 + S22a2

Each equation gives the relationship between the reflectedand incident power waves at each of the network ports, 1and 2, in terms of the network’s individual S-parameters,S11 , S12 , S21 and S22 . If one considers an inci-dent power wave at port 1 ( a1 ) there may result fromit waves exiting from either port 1 itself ( b1 ) or port2 ( b2 ). However if, according to the definition of S-parameters, port 2 is terminated in a load identical to thesystem impedance ( Z0 ) then, by the maximum powertransfer theorem, b2 will be totally absorbed making a2equal to zero. Therefore, defining the incident voltagewaves as a1 = V +

1 and a2 = V +2 with the reflected waves

being b1 = V −1 and b2 = V −

2 ,

S11 = b1a1

=V −1

V +1

and S21 = b2a1

=V −2

V +1

.

Similarly, if port 1 is terminated in the system impedancethen a1 becomes zero, giving

S12 = b1a2

=V −1

V +2

and S22 = b2a2

=V −2

V +2

The 2-port S-parameters have the following generic de-scriptions:

S11

S12

S21

S22

If, instead of defining the voltage wave direction relativeto each port, they are defined by their absolute directionas forward V + and reverse V − waves then b2 = V +

2 anda2 = V −

2 . The S-parameters then take on a more intu-itive meaning such as the forward voltage gain being de-fined by the ratio of the forward voltages S21 = V +

2 /V +1

.

4 S-Parameter properties of 2-portnetworks

An amplifier operating under linear (small signal) condi-tions is a good example of a non-reciprocal network anda matched attenuator is an example of a reciprocal net-work. In the following cases we will assume that the inputand output connections are to ports 1 and 2 respectivelywhich is the most common convention. The nominal sys-tem impedance, frequency and any other factors whichmay influence the device, such as temperature, must alsobe specified.

4.1 Complex linear gain

The complex linear gain G is given by

G = S21

That is simply the voltage gain as a linear ratio of theoutput voltage divided by the input voltage, all values ex-pressed as complex quantities.

4.2 Scalar linear gain

The scalar linear gain (or linear gain magnitude) is givenby

|G| = |S21|

That is simply the scalar voltage gain as a linear ratio ofthe output voltage and the input voltage. As this is a scalarquantity, the phase is not relevant in this case.

4.3 Scalar logarithmic gain

The scalar logarithmic (decibel or dB) expression for gain(g) is

g = 20 log10 |S21|

This is more commonly used than scalar linear gain anda positive quantity is normally understood as simply a'gain'... A negative quantity can be expressed as a 'nega-tive gain' or more usually as a 'loss’ equivalent to its mag-nitude in dB. For example, a 10 m length of cable mayhave a gain of - 1 dB at 100 MHz or a loss of 1 dB at 100MHz.

4.4 Insertion loss

In case the twomeasurement ports use the same referenceimpedance, the insertion loss (IL) is the magnitude of the

4 5 4-PORT S-PARAMETERS

transmission coefficient |S21| expressed in decibels. It isthus given by:[15]

IL = −20 log10 |S21| dB.It is the extra loss produced by the introduction of thedevice under test (DUT) between the 2 reference planesof the measurement. Notice that the extra loss can be in-troduced by intrinsic loss in the DUT and/or mismatch.In case of extra loss the insertion loss is defined to be pos-itive. The negative of insertion loss expressed in decibelsis defined as insertion gain.

4.5 Input return loss

Input return loss (RLᵢ ) can be thought of as a measure ofhow close the actual input impedance of the network isto the nominal system impedance value. Input return lossexpressed in decibels is given by

RLin = 10 log10∣∣∣∣ 1

S211

∣∣∣∣ = −20 log10 |S11|

Note that for passive two-port networks in which |S11| ≤ 1,it follows that return loss is a non-negative quantity: RLin≥ 0. Also note that somewhat confusingly, return loss issometimes used as the negative of the quantity definedabove, but this usage is, strictly speaking, incorrect basedon the definition of loss.[16]

4.6 Output return loss

The output return loss (RLₒᵤ ) has a similar definition tothe input return loss but applies to the output port (port2) instead of the input port. It is given by

RLout = −20 log10 |S22|

4.7 Reverse gain and reverse isolation

The scalar logarithmic (decibel or dB) expression for re-verse gain ( grev ) is:

grev = 20 log10 |S12|

Often this will be expressed as reverse isolation ( Irev )in which case it becomes a positive quantity equal to themagnitude of grev and the expression becomes:

Irev = |grev| = |20 log10 |S12||

4.8 Voltage reflection coefficient

The voltage reflection coefficient at the input port ( ρin )or at the output port ( ρout ) are equivalent to S11 andS22 respectively, so

ρin = S11 and ρout = S22 .

As S11 and S22 are complex quantities, so are ρin andρout .Voltage reflection coefficients are complex quantities andmay be graphically represented on polar diagrams orSmith ChartsSee also the Reflection Coefficient article.

4.9 Voltage standing wave ratio

The voltage standing wave ratio (VSWR) at a port, repre-sented by the lower case 's’, is a similar measure of portmatch to return loss but is a scalar linear quantity, the ra-tio of the standing wave maximum voltage to the standingwave minimum voltage. It therefore relates to the magni-tude of the voltage reflection coefficient and hence to themagnitude of either S11 for the input port or S22 for theoutput port.At the input port, the VSWR ( sin ) is given by

sin =1 + |S11|1− |S11|

At the output port, the VSWR ( sout ) is given by

sout =1 + |S22|1− |S22|

This is correct for reflection coefficients with a magni-tude no greater than unity, which is usually the case. Areflection coefficient with a magnitude greater than unity,such as in a tunnel diode amplifier, will result in a nega-tive value for this expression. VSWR, however, from itsdefinition, is always positive. A more correct expressionfor port k of a multiport is;

sk =1 + |Skk||1− |Skk| |

5 4-Port S-Parameters

4 Port S Parameters are used to characterize 4 port net-works. They include information regarding the reflectedand incident power waves between the 4 ports of the net-work.

5

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

They are commonly used to analyze a pair of coupledtransmission lines to determine the amount of cross-talkbetween them, if they are driven by two separate singleended signals, or the reflected and incident power of a dif-ferential signal driven across them. Many specificationsof high speed differential signals define a communicationchannel in terms of the 4-Port S-Parameters, for exam-ple the 10-Gigabit Attachment Unit Interface (XAUI),SATA, PCI-X, and InfiniBand systems.

5.1 4-Port Mixed-Mode S-Parameters

4-Port mixed-Mode S-Parameters characterize a 4 portnetwork in terms of the response of the network to com-mon mode and differential stimulus signals. The follow-ing table displays the 4-Port Mixed-Mode S-Parameters.Note the format of the parameter notation SXYab, where“S” stands for scattering parameter or S-parameter, “X”is the response mode (differential or common), “Y” isthe stimulus mode (differential or common), “a” is theresponse (output) port and b is the stimulus (input) port.This is the typical nomenclature for scattering parame-ters.The first quadrant is defined as the upper left 4 param-eters describing the differential stimulus and differentialresponse characteristics of the device under test. This isthe actual mode of operation for most high-speed differ-ential interconnects and is the quadrant that receives themost attention. It includes input differential return loss(SDD11), input differential insertion loss (SDD21), out-put differential return loss (SDD22) and output differen-tial insertion loss (SDD12). Some benefits of differentialsignal processing are;

• reduced electromagnetic interference susceptibility

• reduction in electromagnetic radiation from bal-anced differential circuit

• even order differential distortion products trans-formed to common mode signals

• factor of two increase in voltage level relative tosingle-ended

• rejection to common mode supply and ground noiseencoding onto differential signal

The second and third quadrants are the upper right andlower left 4 parameters, respectively. These are also

referred to as the cross-mode quadrants. This is be-cause they fully characterize any mode conversion oc-curring in the device under test, whether it is common-to-differential SDCab conversion (EMI susceptibility foran intended differential signal SDD transmission applica-tion) or differential-to-common SCDab conversion (EMIradiation for a differential application). Understandingmode conversion is very helpful when trying to optimizethe design of interconnects for gigabit data throughput.The fourth quadrant is the lower right 4 parametersand describes the performance characteristics of thecommon-mode signal SCCab propagating through the de-vice under test. For a properly designed SDDab dif-ferential device there should be minimal common-modeoutput SCCab. However, the fourth quadrant common-mode response data is a measure of common-mode trans-mission response and used in a ratio with the differ-ential transmission response to determine the networkcommon-mode rejection. This common mode rejectionis an important benefit of differential signal processingand can be reduced to one in some differential circuitimplementations.[17][18]

6 S-parameters in amplifier design

The reverse isolation parameter S12 determines the levelof feedback from the output of an amplifier to the inputand therefore influences its stability (its tendency to re-frain from oscillation) together with the forward gain S21

. An amplifier with input and output ports perfectly iso-lated from each other would have infinite scalar log mag-nitude isolation or the linear magnitude of S12 would bezero. Such an amplifier is said to be unilateral. Mostpractical amplifiers though will have some finite isolationallowing the reflection coefficient 'seen' at the input to beinfluenced to some extent by the load connected on theoutput. An amplifier which is deliberately designed tohave the smallest possible value of |S12| is often called abuffer amplifier.Suppose the output port of a real (non-unilateral or bi-lateral) amplifier is connected to an arbitrary load with areflection coefficient of ρL . The actual reflection coef-ficient 'seen' at the input port ρin will be given by[19]

ρin = S11 +S12S21ρL1− S22ρL

If the amplifier is unilateral then S12 = 0 and ρin = S11

or, to put it another way, the output loading has no effecton the input.A similar property exists in the opposite direction, in thiscase if ρout is the reflection coefficient seen at the out-put port and ρs is the reflection coefficient of the sourceconnected to the input port.

6 6 S-PARAMETERS IN AMPLIFIER DESIGN

ρout = S22 +S12S21ρs1− S11ρs

6.1 Port loading conditions for an ampli-fier to be unconditionally stable

An amplifier is unconditionally stable if a load or sourceof any reflection coefficient can be connected withoutcausing instability. This condition occurs if the mag-nitudes of the reflection coefficients at the source, loadand the amplifier’s input and output ports are simultane-ously less than unity. An important requirement that isoften overlooked is that the amplifier be a linear networkwith no poles in the right half plane.[20] Instability cancause severe distortion of the amplifier’s gain frequencyresponse or, in the extreme, oscillation. To be uncondi-tionally stable at the frequency of interest, an amplifiermust satisfy the following 4 equations simultaneously:[21]

|ρS | < 1

|ρL| < 1

|ρin| < 1

|ρout| < 1

The boundary condition for when each of these values isequal to unity may be represented by a circle drawn onthe polar diagram representing the (complex) reflectioncoefficient, one for the input port and the other for theoutput port. Often these will be scaled as Smith Charts.In each case coordinates of the circle centre and the as-sociated radius are given by the following equations:

6.1.1 ρL values for |ρin| = 1 (output stability cir-cle)

Radius rL =∣∣∣ S12S21

|S22|2−|∆|2

∣∣∣Center cL =

(S22−∆S∗11)

∗

|S22|2−|∆|2

6.1.2 ρS values for |ρout| = 1 (input stability circle)

Radius rs =∣∣∣ S12S21

|S11|2−|∆|2

∣∣∣Center cs = (S11−∆S∗

22)∗

|S11|2−|∆|2

where, in both cases

∆ = S11S22 − S12S21

and the superscript star (*) indicates a complex conjugate.

The circles are in complex units of reflection coeffi-cient so may be drawn on impedance or admittancebased Smith Charts normalised to the system impedance.This serves to readily show the regions of normalisedimpedance (or admittance) for predicted unconditionalstability. Another way of demonstrating unconditionalstability is by means of the Rollett stability factor ( K), defined as

K =1− |S11|2 − |S22|2 + |∆|2

2 |S12S21|

The condition of unconditional stability is achieved whenK > 1 and |∆| < 1

6.2 Scattering transfer parameters

The Scattering transfer parameters or T-parameters of a2-port network are expressed by the T-parameter matrixand are closely related to the corresponding S-parametermatrix. The T-parameter matrix is related to the incidentand reflected normalised waves at each of the ports asfollows:

(b1a1

)=

(T11 T12

T21 T22

)(a2b2

)However, they could be defined differently, as follows :

(a1b1

)=

(T11 T12

T21 T22

)(b2a2

)The RF Toolbox add-on to MATLAB[22] and severalbooks (for example “Network scattering parameters”[23])use this last definition, so caution is necessary. The“From S to T” and “FromT to S” paragraphs in this articleare based on the first definition. Adaptation to the sec-ond definition is trivial (interchanging T11 for T22, andT12 for T21). The advantage of T-parameters comparedto S-parameters is that they may be used to readily deter-mine the effect of cascading 2 or more 2-port networks bysimply multiplying the associated individual T-parametermatrices. If the T-parameters of say three different 2-port networks 1, 2 and 3 are

(T1

),(T2

)and

(T3

)re-

spectively then the T-parameter matrix for the cascade ofall three networks (

(TT

)) in serial order is given by:

(TT

)=

(T1

)(T2

)(T3

)As with S-parameters, T-parameters are complex val-ues and there is a direct conversion between the twotypes. Although the cascaded T-parameters is a sim-ple matrix multiplication of the individual T-parameters,the conversion for each network’s S-parameters to the

7

corresponding T-parameters and the conversion of thecascaded T-parameters back to the equivalent cascadedS-parameters, which are usually required, is not trivial.However once the operation is completed, the complexfull wave interactions between all ports in both directionswill be taken into account. The following equations willprovide conversion between S and T parameters for 2-port networks.[24]

From S to T:

T11 =− det

(S)

S21

T12 =S11

S21

T21 =−S22

S21

T22 =1

S21

From T to S

S11 =T12

T22

S12 =det

(T)

T22

S21 =1

T22

S22 =−T21

T22

Where det(S)indicates the determinant of the matrix(

S).

7 1-port S-parameters

The S-parameter for a 1-port network is given by a simple1 x 1 matrix of the form (snn) where n is the allocatedport number. To comply with the S-parameter definitionof linearity, this would normally be a passive load of sometype.

8 Higher-order S-parameter ma-trices

Higher order S-parameters for pairs of dissimilar ports( Smn ), where m ̸= n may be deduced similarly tothose for 2-port networks by considering pairs of portsin turn, in each case ensuring that all of the remaining(unused) ports are loaded with an impedance identical tothe system impedance. In this way the incident power

wave for each of the unused ports becomes zero yieldingsimilar expressions to those obtained for the 2-port case.S-parameters relating to single ports only (Smm ) requireall of the remaining ports to be loaded with an impedanceidentical to the system impedance therefore making allof the incident power waves zero except that for the portunder consideration. In general therefore we have:

Smn =bman

and

Smm =bmam

For example, a 3-port network such as a 2-way splitterwould have the following S-parameter definitions

S11 =b1a1

=V −1

V +1

S33 =b3a3

=V −3

V +3

S32 =b3a2

=V −3

V +2

S23 =b2a3

=V −2

V +3

9 Measurement of S-parameters

The basic parts of a vector network analyzer

9.1 Vector network analyzer

The diagram shows the essential parts of a typical 2-portvector network analyzer (VNA). The two ports of thedevice under test (DUT) are denoted port 1 (P1) and port

8 9 MEASUREMENT OF S-PARAMETERS

2 (P2). The test port connectors provided on the VNAitself are precision types which will normally have to beextended and connected to P1 and P2 using precision ca-bles 1 and 2, PC1 and PC2 respectively and suitable con-nector adaptors A1 and A2 respectively.The test frequency is generated by a variable frequencyCW source and its power level is set using a variableattenuator. The position of switch SW1 sets the direc-tion that the test signal passes through the DUT. Initiallyconsider that SW1 is at position 1 so that the test signalis incident on the DUT at P1 which is appropriate formeasuring S11 and S21 . The test signal is fed by SW1to the common port of splitter 1, one arm (the referencechannel) feeding a reference receiver for P1 (RX REF1)and the other (the test channel) connecting to P1 via thedirectional coupler DC1, PC1 and A1. The third port ofDC1 couples off the power reflected from P1 via A1 andPC1, then feeding it to test receiver 1 (RX TEST1). Sim-ilarly, signals leaving P2 pass via A2, PC2 and DC2 toRX TEST2. RX REF1, RX TEST1, RX REF2 and RX-TEST2 are known as coherent receivers as they share thesame reference oscillator, and they are capable of mea-suring the test signal’s amplitude and phase at the test fre-quency. All of the complex receiver output signals arefed to a processor which does the mathematical process-ing and displays the chosen parameters and format on thephase and amplitude display. The instantaneous value ofphase includes both the temporal and spatial parts, butthe former is removed by virtue of using 2 test channels,one as a reference and the other for measurement. WhenSW1 is set to position 2, the test signals are applied toP2, the reference is measured by RX REF2, reflectionsfrom P2 are coupled off by DC2 and measured by RXTEST2 and signals leaving P1 are coupled off by DC1and measured by RX TEST1. This position is appropri-ate for measuring S22 and S12 .

9.2 Calibration

Prior to a VNA S-parameter measurement, the first es-sential step is to perform an accurate calibration ap-propriate to the intended measurements. Several typesof calibration are normally available on the VNA. It isonly in the last few years that VNAs have had the suffi-ciently advanced processing capability, at realistic cost,required to accomplish the more advanced types of cal-ibration, including corrections for systematic errors.[25]The more basic types, often called 'response' calibrations,may be performed quickly but will only provide a resultwithmoderate uncertainty. For improved uncertainty anddynamic range of the measurement a full 2 port calibra-tion is required prior to DUT measurement. This willeffectively eliminate all sources of systematic errors in-herent in the VNA measurement system.

9.3 Minimization of systematic errors

Systematic errors are those which do not vary with timeduring a calibration. For a set of 2 port S-parameter mea-surements there are a total of 12 types of systematic er-rors which are measured and removed mathematically aspart of the full 2 port calibration procedure. They are, foreach port:1. directivity and crosstalk2. source and load mismatches3. frequency response errors caused by reflection andtransmission tracking within the test receiversThe calibration procedure requires initially setting up theVNA with all the cables, adaptors and connectors nec-essary to connect to the DUT but not at this stage con-necting it. A calibration kit is used according to the con-nector types fitted to the DUT. This will normally in-clude adaptors, nominal short circuits (SCs), open cir-cuits (OCs) and load termination (TERM) standards ofboth connector sexes appropriate to the VNA and DUTconnectors. Even with standards of high quality, whenperforming tests at the higher frequencies into the mi-crowave range various stray capacitances and inductanceswill become apparent and cause uncertainty during thecalibration. Data relating to the strays of the particularcalibration kit used are measured at the factory traceableto national standards and the results are programmed intothe VNA memory prior to performing the calibration.The calibration procedure is normally software con-trolled, and instructs the operator to fit various calibra-tion standards to the ends of the DUT connecting cablesas well as making a through connection. At each step theVNA processor captures data across the test frequencyrange and stores it. At the end of the calibration proce-dure, the processor uses the stored data thus obtained toapply the systematic error corrections to all subsequentmeasurements made. All subsequent measurements areknown as 'corrected measurements’. At this point theDUT is connected and a corrected measurement of itsS-parameters made.

9.4 Output format of measured and cor-rected S-parameter data

The S-parameter test data may be provided in many alter-native formats, for example: list, graphical (Smith chartor polar diagram).

9.5 List format

In list format the measured and corrected S-parametersare tabulated against frequency. The most common listformat is known as Touchstone or SNP, where N is thenumber of ports. Commonly text files containing this in-formation would have the filename extension '.s2p'. An

9

example of a Touchstone file 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 −13.457.2 51 −15.8 103.2 10.7 177.4 −33.1 9.6 −12.4 63.452 −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 54−16.6109.3 10.6 187.8 −38.1 9.6 −15.3 71.4Rows beginning with an exclamation mark contains onlycomments. The row beginning with the hash symbolindicates that in this case frequencies are in megahertz(MHZ), S-parameters are listed (S), magnitudes are indB log magnitude (DB) and the system impedance is 50Ohm (R 50). There are 9 columns of data. Column 1 isthe test frequency in megahertz in this case. Columns 2,4, 6 and 8 are the magnitudes of S11 , S21 , S12 and S22

respectively in dB. Columns 3, 5, 7 and 9 are the anglesof S11 , S21 , S12 and S22 respectively in degrees.

9.6 Graphical (Smith chart)

Any 2-port S-parameter may be displayed on a Smithchart using polar co-ordinates, but the most meaningfulwould be S11 and S22 since either of these may be con-verted directly into an equivalent normalized impedance(or admittance) using the characteristic Smith Chartimpedance (or admittance) scaling appropriate to the sys-tem impedance.

9.7 Graphical (polar diagram)

Any 2-port S-parameter may be displayed on a polar di-agram using polar co-ordinates.In either graphical format each S-parameter at a particulartest frequency is displayed as a dot. If the measurementis a sweep across several frequencies a dot will appear foreach. Many VNAs connect successive dots with straightlines for easier visibility.

9.8 Measuring S-parameters of a one-portnetwork

The S-parameter matrix for a network with just one portwill have just one element represented in the form Snn ,where n is the number allocated to the port. Most VNAsprovide a simple one-port calibration capability for oneport measurement to save time if that is all that is re-quired.

9.9 Measuring S-parameters of networkswith more than 2 ports

VNAs designed for the simultaneous measurement of theS-parameters of networks with more than two ports are

feasible but quickly become prohibitively complex andexpensive. Usually their purchase is not justified since therequired measurements can be obtained using a standard2-port calibrated VNAwith extrameasurements followedby the correct interpretation of the results obtained. Therequired S-parameter matrix can be assembled from suc-cessive two port measurements in stages, two ports at atime, on each occasion with the unused ports being termi-nated in high quality loads equal to the system impedance.One risk of this approach is that the return loss or VSWRof the loads themselves must be suitably specified to beas close as possible to a perfect 50 Ohms, or whatever thenominal system impedance is. For a network with manyports there may be a temptation, on grounds of cost, toinadequately specify the VSWRs of the loads. Some anal-ysis will be necessary to determine what the worst accept-able VSWR of the loads will be.Assuming that the extra loads are specified adequately, ifnecessary, two or more of the S-parameter subscripts aremodified from those relating to the VNA (1 and 2 in thecase considered above) to those relating to the networkunder 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 VNAis connected with VNA port 1 to DUT port 3 and VNAport 2 to DUT port 5, the measured VNA results ( S11 ,S12 , S21 and S22 ) would be equivalent to S33 , S35 ,S53 and S55 respectively, assuming that DUT ports 1, 2and 4 were terminated in adequate 50 Ohm loads . Thiswould provide 4 of the necessary 25 S-parameters.

10 See also

• Admittance parameters

• Impedance parameters

• Two-port network

• X-parameters, a non-linear superset of S-parameters

• Belevitch’s theorem

11 References

[1] Pozar, David M. (2005); Microwave Engineering, ThirdEdition (Intl. Ed.); JohnWiley & Sons, Inc.; pp. 170-174.ISBN 0-471-44878-8.

[2] Pozar, David M. (2005) (op. cit.); pp. 170-174.

[3] Pozar, David M. (2005) (op. cit.); pp. 183-186.

[4] Morton, A. H. (1985); Advanced Electrical Engineering;Pitman Publishing Ltd.; pp. 33-72. ISBN 0-273-40172-6.

10 12 BIBLIOGRAPHY

[5] Belevitch, Vitold “Summary of the history of circuit the-ory”, Proceedings of the IRE, vol.50, iss.5, pp. 848–855,May 1962.Vandewalle, Joos “In memoriam – Vitold Belevitch”, In-ternational Journal of Circuit Theory and Applications,vol.28, iss.5, pp. 429–430, September/October 2000.

[6] Valkenburg, Mac Elwyn Van Circuit Theory: Foundationsand Classical Contributions, p.334, Stroudsburg, Penn-sylvania: Dowden, Hutchinson & Ross, 1974 ISBN 0-87933-084-8.

[7] J. Appl. Phys. 18, 873 (1947); doi: 10.1063/1.1697561A Computational Method Applicable to Microwave Net-works R. H. Dicke

[8] “Microwave Hall of Fame Part III”. Microwaves 101. PNDesigns. 2012. Retrieved December 9, 2012. Externallink in |publisher= (help)

[9] Pozar, David M. (2005) (op. cit.); p. 170.

[10] Morton, A. H. (1985) (op. cit.); p. 33.

[11] Kurokawa, K., “Power Waves and the Scattering Matrix”,IEEE Trans. Micr. Theory & Tech., Mar. 1965, pp. 194-202

[12] Pozar, David M. (2005) (op. cit.); p. 173.

[13] S-Parameter Design; Application Note AN 154; AgilentTechnologies; p 7

[14] Choma J. & Chen W.K. (2007). Feedback networks: the-ory and circuit applications. Singapore: World Scientific.Chapter 3, p. 225 ff. ISBN 981-02-2770-1.

[15] Collin, Robert E.; Foundations For Microwave Engineer-ing, Second Edition

[16] Trevor S. Bird, “Definition and Misuse of Return Loss”,IEEE Antennas & Propagation Magazine, vol.51, iss.2,pp.166-167, April 2009.

[17] Backplane Channels and Correlation Between Their Fre-quency and Time Domain Performance

[18] Bockelman, DE and Eisenstadt, WR “Combined differen-tial and common-mode scattering parameters: theory andsimulation,” MTT, IEEE transactions volume 43 issue 7part 1–2 July 1995 pages 1530-1539

[19] Gonzalez, Guillermo (1997); Microwave Transistor Am-plifiers Analysis and Design, Second Edition; Prentice HallNJ; pp 212-216. ISBN 0-13-254335-4.

[20] J.M. Rollett, “Stability and Power-Gain Invariants of Lin-ear Twoports”, IRE Trans. on Circuit Theory vol. CT-9,pp. 29-32, March 1962

[21] Gonzalez, Guillermo (op. cit.); pp 217-222

[22] “RF Toolbox documentation”.

[23] R. Mavaddat. (1996). Network scattering parameter. Sin-gapore: World Scientific. ISBN 978-981-02-2305-2.

[24] S-Parameter Design; Application Note AN 154; AgilentTechnologies; p 14

[25] Applying Error Correction to Network Analyzer Measure-ments; Agilent Application Note AN 1287-3, AgilentTechnologies; p6

12 Bibliography• Guillermo Gonzalez, “Microwave Transistor Am-plifiers, Analysis and Design, 2nd. Ed.”, PrenticeHall, New Jersey; ISBN 0-13-581646-7

• David M. Pozar, “Microwave Engineering”, ThirdEdition, John Wiley & Sons Inc.; ISBN 0-471-17096-8

• William Eisenstadt, Bob Stengel, and BruceThompson, “Microwave Differential Circuit Designusign Mixed-Mode S-Parameters”, Artech House;ISBN 1-58053-933-5; ISBN 978-1-58053-933-3

• “S-Parameter Design”, Application Note AN 154,Agilent Technologies

• “S-Parameter Techniques for Faster, More AccurateNetwork Design”, Application Note AN 95-1, Agi-lent Technologies, PDF slides plus QuickTime videoor scan of Richard W. Anderson’s original article

• A. J. Baden Fuller, “An Introduction to MicrowaveTheory and Techniques, Second Edition, Pergam-mon International Library; ISBN 0-08-024227-8

• Ramo,Whinnery andVanDuzer, “Fields andWavesin Communications Electronics”, John Wiley &Sons; ISBN 0-471-70721-X

• C. W. Davidson, “Transmission Lines for Commu-nications with CAD Programs”, Second Edition,Macmillan Education Ltd.; ISBN 0-333-47398-1

11

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