Application Note 1364-1

AgilentDe-embedding and EmbeddingS-Parameter Networks Using aVector Network Analyzer

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IntroductionTraditionally RF and microwave com-ponents have been designed in pack-ages with coaxial interfaces. Complexsystems can be easily manufacturedby connecting a series of these sepa-rate coaxial devices. Measuring theperformance of these componentsand systems is easily performed withstandard test equipment that usessimilar coaxial interfaces.

However, modern systems demand ahigher level of component integra-tion, lower power consumption, andreduced manufacturing cost. RF com-ponents are rapidly shifting awayfrom designs that use expensivecoaxial interfaces, and are movingtoward designs that use printed cir-cuit board and surface mount tech-nologies (SMT). The traditionalcoaxial interface may even be elimi-nated from the final product. Thisleaves the designer with the problemof measuring the performance ofthese RF and microwave componentswith test equipment that requirescoaxial interfaces. The solution is touse a test fixture that interfaces thecoaxial and non-coaxial transmissionlines.

The large variety of printed circuittransmission lines makes it difficultto create test equipment that can eas-ily interface to all the different typesand dimensions of microstrip andcoplanar transmission lines1 (Figure1). The test equipment requires aninterface to the selected transmissionmedia through a test fixture.

Accurate characterization of the sur-face mount device under test (DUT)requires the test fixture characteris-tics to be removed from the mea-sured results. The test equipmenttypically used for characterizing theRF and microwave component is thevector network analyzer (VNA) whichuses standard 50 or 75 ohm coaxialinterfaces at the test ports. The testequipment is calibrated at the coaxialinterface defined as the “measure-ment plane,” and the required mea-surements are at the point where thesurface-mount device attaches to theprinted circuit board, or the “deviceplane” (Figure 2). When the VNA iscalibrated at the coaxial interfaceusing any standard calibration kit, theDUT measurements include the testfixture effects.

Figure 1. Types of printed circuit transmissionlines

Figure 2. Test fixture configuration showingthe measurement and device planes

Over the years, many differentapproaches have been developed forremoving the effects of the test fix-ture from the measurement, whichfall into two fundamental categories:direct measurement and de-embed-ding. Direct measurement requiresspecialized calibration standards thatare inserted into the test fixture andmeasured. The accuracy of the devicemeasurement relies on the quality ofthese physical standards.2 De-embed-ding uses a model of the test fixtureand mathematically removes the fix-ture characteristics from the overallmeasurement. This fixture “de-embedding” procedure can producevery accurate results for the non-coaxial DUT, without complex non-coaxial calibration standards.

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The process of de-embedding a testfixture from the DUT measurementcan be performed using scatteringtransfer parameters (T-parameter)matrices.3 For this case, the de-embedded measurements can bepost-processed from the measure-ments made on the test fixture andDUT together. Also modern CAEtools such as Agilent EEsof AdvancedDesign System (ADS) have the abilityto directly de-embed the test fixturefrom the VNA measurements using anegation component model in thesimulation.3 Unfortunately theseapproaches do not allow for real-timefeedback to the operator because themeasured data needs to be capturedand post-processed in order toremove the effects of the test fixture.If real-time de-embedded measure-ments are required, an alternate technique must be used.

It is possible to perform the de-embedding calculation directly on the VNA using a different calibrationmodel. If we include the test fixtureeffects as part of the VNA calibrationerror coefficients, real time de-embedded measurements can be displayed directly on the VNA . Thisallows for real-time tuning of compo-nents without including the fixture aspart of the measurement.

The following sections of this paperwill review S-parameter matrices, signal flow graphs, and the error cor-rection process used in standard oneand two-port calibrations on allAgilent vector network analyzers suchas the E8358A PNA Series NetworkAnalyzer. The de-embedding processwill then be detailed for removing theeffects of a test fixture placed betweenthe measurement and device planes.Also included will be a description onhow the same process can be used toembed a hypothetical or “virtual” net-work into the measurement of theDUT.

S-parameters and signalflow graphs

RF and microwave networks areoften characterized using scatteringor S-parameters.4 The S-parametersof a network provide a clear physicalinterpretation of the transmission andreflection performance of the device.The S-parameters for a two-port net-work are defined using the reflectedor emanating waves, b1 and b2, as thedependent variables, and the incidentwaves, a1 and a2, as the independentvariables (Figure 3). The generalequations for these waves as a func-tion of the S-parameters is shownbelow:

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

Using these equations, the individualS-parameters can be determined bytaking the ratio of the reflected ortransmitted wave to the incidentwave with a perfect terminationplaced at the output. For example, todetermine the reflection parameterfrom Port 1, defined as S11, we takethe ratio of the reflected wave, b1 tothe incident wave, a1, using a perfecttermination on Port 2. The perfecttermination guarantees that a2 = 0since there is no reflection from anideal load. The remaining S-parame-ters, S21, S22 and S12, are defined in asimilar manner.5 These four S-para-meters completely define the two-portnetwork characteristics. All modernvector network analyzers, such as theAgilent E8358A, can easily the mea-sure the S-parameters of a two-portdevice.

Figure 3. Definition of a two-port S-Parameternetwork

Another way to represent the S-para-meters of any network is with a signal flow graph (Figure 4). A flowgraph is used to represent and analyzethe transmitted and reflected signalsfrom a network. Directed lines in theflow graph represent the signal flowthrough the two-port device. Forexample, the signal flowing from nodea1 to b1 is defined as the reflectionfrom Port 1 or S11. When two-portnetworks are cascaded, it can beshown that connecting the flow graphsof adjacent networks can be donebecause the outgoing waves from onenetwork are the same as the incomingwaves of the next.6 Analysis of thecomplete cascaded network can beaccomplished using Mason’s Rule6. It is the application of signal flowgraphs that will be used to develop the mathematics behind network de-embedding and modifying the errorcoefficients in the VNA.

Figure 4. Signal flow graph representation of atwo-port S-parameter network

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Defining the test fixtureand DUT

Before the mathematical process ofde-embedding is developed, the testfixture and the DUT must be repre-sented in a convenient form. Usingsignal flow graphs, the fixture anddevice can be represented as threeseparate two-port networks (Figure 5). In this way, the test fixtureis divided in half to represent thecoaxial to non-coaxial interfaces oneach side of the DUT. The two fixturehalves will be designated as Fixture Aand Fixture B for the left-hand andright-hand sides of the fixture respec-tively. The S-parameters FAxx (xx =11, 21, 12, 22) will be used to repre-sent the S-parameters for the left halfof the test fixture and FBxx will beused to represent the right half.

Figure 5. Signal flow graph representing thetest fixture halves and the device under test(DUT)

If we wish to directly multiply thematrices of the three networks, wefind it mathematically more conve-nient to convert the S-parametermatrices to scattering transfer matri-ces or T-parameters. The mathemati-cal relationship between S-parameterand T-parameter matrices is given inAppendix A. The two-port T-parame-ter matrix can be represented as [T],where [T] is defined as having thefour parameters of the network.Because we defined the test fixture

and DUT as three cascaded networks,we can easily multiply their respec-tive T-parameter networks, TA, TDUTand TB. It is only through the use ofT-parameters that this simple matrixequation be written in this form.

This matrix operation will representthe T-parameters of the test fixtureand DUT when measured by the VNAat the measurement plane.

General matrix theory states that if amatrix determinate is not equal tozero, then the matrix has an inverse,and any matrix multiplied by itsinverse will result in the identitymatrix. For example, if we multiplythe following T-parameter matrix byits inverse matrix, we obtain the iden-tity matrix.

It is our goal to de-embed the twosides of the fixture, TA and TB, andgather the information from the DUTor TDUT. Extending this matrix inversion to the case of the cascadedfixture and DUT matrices, we canmultiply each side of the measuredresult by the inverse T-parametermatrix of the fixture and yield the T-parameter for the DUT only. The T-parameter matrix can then be converted back to the desired S-para-meter matrix using the equations inAppendix A.

Using the S or T-parameter model ofthe test fixture and VNA measure-ments of the total combination of thefixture and DUT, we can apply theabove matrix equation to de-embedthe fixture from the measurement.The above process is typically imple-mented after the measurements arecaptured from the VNA. It is oftendesirable that the de-embedded mea-surements be displayed real-time onthe VNA. This can be accomplishedusing techniques that provide somelevel of modification to the errorcoefficients used in the VNA calibration process.

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Test Fixture Models

Before we can mathematically de-embed the test fixture from thedevice measurements, the S or T-parameter network for each fixturehalf needs to be modeled. Because of the variety of printed circuit typesand test fixture designs, there are no simple textbook formulations forcreating an exact model. Looking atthe whole process of de-embedding,the most difficult part is creating anaccurate model of the test fixture.There are many techniques that canbe used to aid in the creation of fixturemodels, including simulation toolssuch as Agilent Advanced DesignSystem (ADS) and Agilent HighFrequency Structure Simulator(HFSS). Often observation of thephysical structure of the test fixtureis required for the initial fixturemodel. Measurements made on thefixture can be used to optimize thefixture model in an iterative manner.Time domain techniques, available on most network analyzers, can alsobe very useful when optimizing thefixture model.2

Let’s examine several fixture modelsthat can be used in the de-embeddingprocess. We will later show that someof the simpler models are used in thefirmware of many vector networkanalyzers to directly perform theappropriate de-embedding withoutrequiring the T-parameter matrixmathematics.

The simplest model assumes that thefixture halves consist of perfect trans-mission lines of known electricallength. For this case, we simply shiftthe measurement plane to the DUTplane by rotating the phase angle of the measured S-parameters (Figure 6). If we assume the phaseangles, θA and θB, represent thephase of the right and left test fixturehalves respectively, then the S-para-meter model of the fixture can berepresented by the following equa-tions.

The phase angle is a function of thelength of the fixture multiplied by thephase constant of the transmissionline. The phase constant, β, is definedas the phase velocity divided by thefrequency in radians.This simplemodel assumes that the fixture is alossless transmission line that ismatched to the characteristic imped-ance of the system. An easy way tocalculate the S-parameter values forthis ideal transmission line is to use asoftware simulator such as AgilentADS. Here, each side of the test fix-ture can be modeled as a 50-ohmtransmission line using the appropri-ate phase angle and reference fre-quency (Figure 7). Once the simulatorcalculates all the S-parameters for thecircuit, the information can be savedto data file for use in the de-embed-ding process.

Figure 6. Modeling the fixture using an idealtransmission line

This model only accounts for thephase length between the measure-ment and device planes. In some cases,when the fixture is manufactured withlow-loss dielectric materials and useswell-matched transitions from thecoaxial to non-coaxial media, thismodel may provide acceptable mea-surement accuracy when performingde-embedding.

Figure 7. Agilent ADS model for the test fixtureusing an ideal two-port transmission line

An improved fixture model modifiesthe above case to include the inser-tion loss of the fixture. It can alsoinclude an arbitrary characteristicimpedance, ZA, or ZB, of the non-coaxial transmission line (Figure 8).The insertion loss is a function of thetransmission line characteristics andcan include dielectric and conductorlosses. This loss can be representedusing the attenuation factor, α, or theloss tangent, tanδ.

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Figure 8. Modeling the fixture using a lossytransmission line

To improve the fixture model, it maybe possible to determine the actualcharacteristic impedance of the testfixture’s transmission lines, ZA andZB, by measuring the physical charac-teristics of the fixture and calculatingthe impedance using the knowndielectric constant for the material.If the dielectric constant is specifiedby the manufacturer with a nominalvalue and a large tolerance, then theactual line impedance may vary overa wide range. For this case, you caneither make a best guess to the actualdielectric constant or use a measure-ment technique for determining thecharacteristic impedance of the line.One technique uses the time domainoption on the vector network analyzer.By measuring the frequency responseof the fixture using a straight sectionof transmission line, the analyzer willconvert this measurement into a TimeDomain Reflectometer (TDR) responsethat can be used to determine theimpedance of the transmission line.Refer to the analyzer’s User’s Guidefor more information.

Once again, a software simulator canbe used to calculate the required S-parameters for this model. Figure 9shows the model for the test fixturehalf using a lossy transmission linewith the attenuation specified usingthe loss tangent. For this model, theline impedance was modified to avalue of 48-ohms based on physicalmeasurements of the transmissionline width and dielectric thicknessand using a nominal value for thedielectric constant.

Figure 9. Agilent ADS model for the test fixture using a lossy two-port transmissionline

We will later find that many vectornetwork analyzers, such as theAgilent E8358A, can easily implementthis model by allowing the user toenter the loss, electrical delay andcharacteristic impedance directly intothe analyzers “calibration thru” definition.

The last model we will discussincludes the complex effects of thecoax-to-non-coaxial transitions aswell as the fixture losses and imped-ance differences we previously dis-cussed. While this model can be themost accurate, it is the hardest one tocreate because we need to include allof the non-linear effects such as dis-persion, radiation and coupling thatcan occur in the fixture. One way todetermine the model is by using acombination of measurements ofknown devices placed in the fixture(which can be as simple as a straightpiece of transmission line) and acomputer model whose values areoptimized to the measurements. Amore rigorous approach uses an elec-tromagnetic (EM) simulator, such asAgilent HFSS, to calculate the S-para-meters of the test fixture. The EMapproach can be very accurate aslong as the physical test fixture char-acteristics are modeled correctly inthe simulator.

As an example, we will show a modelcreated by optimizing a computersimulation based on a series of mea-surements made using the actual test fixture. We begin by modeling acoax-to-microstrip transition as alumped series inductance and shuntcapacitance (Figure 10). The valuesfor the inductance and capacitancewill be optimized using the measuredresults from the straight 50-ohmmicrostrip line placed in the test fix-ture. An ADS model is then createdfor the test fixture and microstrip lineusing this lumped element model.

Figure 10. Simplified model of a coax tomicrostrip transition

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The Agilent ADS model, shown inFigure 11, use the same lumped ele-ment components placed on eachside to model the two test fixturetransitions. A small length of coax isused to represent the coaxial sectionfor each coax-to-microstrip connec-tor. A microstrip thru line is placed inthe center whose physical and electri-cal parameters match the line mea-sured in the actual test fixture. Thismicrostrip model requires an accu-rate value for dielectric constant andloss tangent for the substrate materialused. Uncertainty in these values willdirectly affect the accuracy of themodel.

Figure 11. Agilent ADS model of test fixtureand microstrip line

Figure 12. Comparison of S11 for the measured and modeled microstrip thru line

S-parameters measurements are thenmade on the test fixture and themicrostrip thru line using a vectornetwork analyzer such as the Agilent E8358A. The four S-parame-ters can be directly imported into theADS software over the GPIB. Themodel values for inductance andcapacitance are optimized using ADSuntil a good fit is obtained betweenthe measurements and the simulatedresults. As an example, Figure 12shows the measured and optimizedresults for the magnitude of S11 usingthe test fixture with a microstrip thruline. All four S-parameters should be

optimized and compared to the measured S-parameters to verify theaccuracy of the model values. Becauseof non-linear effects in the transition,this simplified lumped element modelfor the transition may only be validonly over a small frequency range. Ifbroadband operation is required, animproved model must be implementedto incorporate the non-linear behaviorof the measured S-parameters as afunction of frequency.

Once the lumped element parametersare optimized, the S-parameters foreach half of the test fixture can besimulated and saved for use by the de-embedding algorithm. Keep inmind that it is necessary to includethe actual length of microstrip linebetween the transition and devicewhen calculating the S-parameters forthe test fixture halves.

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The de-embeddingprocess

Whether a simplified model, such as alength of ideal transmission line, or acomplex model, created using an EMsimulator, is used for the test fixture,it is now necessary to perform the de-embedding process using this S-parameter model. There are twomain ways the de-embedding processcan be implemented. The first tech-nique uses measured data from a net-work analyzer and processes the datausing the T-parameter matrix calcula-tions discussed in the previous section.The second technique uses the net-work analyzer to directly perform thede-embedding calculations, allowingthe user to examine the de-embeddingresponse in real-time. This techniqueis accomplished by modifying the cal-ibration error terms in the analyzer’smemory.

The Static Approach

This approach uses measured datafrom the test fixture and DUT gath-ered at the measurement plane. Thedata can be exported from the net-work analyzer or directly importedinto a simulation tool, such as ADS,over the GPIB. Using the fixturemodel, the de-embedding process isperformed using T-parameter matrixcalculations or the negation model inADS.3 Once the measurements arede-embedded, the data is displayedstatically on a computer screen orcan be downloaded into the analyzer’smemory for display.

There are five steps for the process of de-embedding the test fixture usingT-parameters:

Step 1: Create a mathematical modelof the test fixture using S or T-para-meters to represent each half of thefixture.

Step 2: Using a vector network ana-lyzer, calibrate the analyzer using astandard coaxial calibration kit andmeasure the S-parameters of thedevice and fixture together. The S-parameters are represented as com-plex numbers.

Step 3: Convert the measured S-para-meters to T-parameters.

Step 4: Using the T-parameter modelof the test fixture, apply the de-embedding equation to the measuredT-parameters.

Step 5: Convert the final T-parame-ters back to S-parameters and displaythe results. This matrix representsthe S-parameters of the device only.The test fixture effects have beenremoved.

The Real-Time Approach

This real-time approach will bedetailed in the following sections of this application note. For this technique, we wish to incorporate the test fixture S-parameter modelinto the calibration error terms in the vector network analyzer. In this way,the analyzer is performing all the de-embedding calculations, whichallows the users to view real-timemeasurements of the DUT withoutthe effects of the test fixture.

Most vector network analyzers arecapable of performing some modifica-tion to the error terms directly fromthe front panel. These include portextension and modifying the calibra-tion “thru” definition. Each of thesetechniques will now be discussed,including a technique to modify thetraditional twelve-term error model toinclude the complete S-parametermodel for each side of the test fixture.

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Simple corrections for fixture effects

Port extensions

The simplest form of de-embedding isport extensions, which mathematical-ly extends the measurement planetowards the DUT. This feature isincluded in the firmware of mostmodern network analyzers such asthe Agilent E8358A. Port extensionsassume that the test fixture looks likea perfect transmission line of someknown phase length. It assumes thefixture has no loss, a linear phaseresponse, and constant impedance.Port extensions are usually applied tothe measurements after a two-portcalibration has been performed at theend of the test cables. If the fixtureperformance is considerably betterthan the specifications of the DUT,this technique may be sufficient.

Port extension only adds or subtractsphase length from the measured S-parameter. It does not compensatefor fixture losses or impedance dis-continuities. In most cases, there willbe a certain amount of mismatchinteraction between the coax-to-fix-ture transition and the DUT that willcreate uncertainty in the measured S-parameter. This uncertainty typical-ly results in an observed ripple in theS-parameter when measured over awide frequency range. As an example,consider the measurements shown inFigure 13 of a short placed at the endof two different constant impedancetransmission lines: a high-qualitycoaxial airline (upper curve), and amicrostrip transmission line (lowercurve). Port extensions were used to move the measurement plane up to the short. However, as seen in thefigure, port extension does not com-pensate for the losses in the transmis-sion line. Also note that the airlinemeasurement exhibits lower ripple inthe measured S11 trace while thecoax-to-microstrip test fixture shows a

much larger ripple. Generally, the rip-ple is caused by interaction betweenthe discontinuities at the measure-ment and device planes. The largerripple in the lower trace results fromthe poor return loss of the microstriptransition (20 dB versus >45 dB forthe airline). This ripple can be reducedif improvements are made in thereturn loss of the transition section.

Figure 13. Port extension applied to a measurement of a short at the end of an airline (upper trace) and at the end of amicrostrip transmission line (lower trace)

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Modifying thru standard

A second method to modify the errorterms involves modification of theelectrical length, loss and impedanceof the definition for the “thru” calibra-tion standard. By including a modeledresponse for the fixture into the defi-nition of the coaxial standards, themodeled response will be effectivelyremoved through the built-in errorcorrection routine7. Three fixturecharacteristics can be entered intothe “thru” calibration standard oradapter standard. They are offsetdelay, offset loss and characteristicimpedance.

Offset delay The test fixture will have electricaldelay between the measurementplane and DUT due to the signaltransmission time through the fix-ture. For coaxial transmission lines,the delay can be obtained from thephysical length, propagation veloci-ty of light in free space and the per-mittivity constant.

Note the electrical delay for transmis-sion lines other than coax, such asmicrostrip, will require a modificationto the above equation due to thechange in the effective permittivity of the transmission media. Most RFsoftware simulators, such as the ADSLineCalc tool, will calculate the effective phase length and effectivepermittivity of a transmission linebased on the physical parameters ofthe circuit. The effective phase canbe converted to electrical delay usingthe following equation.

Offset lossThe network analyzer uses the offsetloss to model the magnitude loss dueto skin effect of a coaxial type stan-dard. Because the fixture is non-coax-ial, the loss as a function of frequencymay not follow the loss of a coaxialtransmission line, so the valueentered may only approximate thetrue loss of the fixture. The value ofloss is entered into the standard defi-nition table as gigohms/second orohms/nanosecond at 1 GHz.

The offset loss in gigohms/second canbe calculated from the measured lossat 1 GHz and the physical length ofthe particular standard by the follow-ing equation:

where:dBloss @ 1GHz =measured insertion

loss at 1 GHzZ0 = offset Z0

= physical length of the offset

Figure 14. Measurement of a microstrip “thru”line (lower trace) and the test fixture “thru”after the VNA was calibrated with a modifiedadapter loss (upper trace)

Figure 14 shows the true insertionloss of a microstrip thru line (lowertrace). This figure also shows a S21measurement of the fixture “thru”after modifying the adapter definitionto include the effects of the fixtureloss (top curve). The offset loss wasset to 10 Gohm/sec for the FR-4 mate-rial used. For this case, a 3-inchlength of microstrip 50-ohm transmis-sion line was used with an approxi-mate dielectric constant of 4.3. Thevalue of 10 Gohm/sec for the offsetloss is a good compromise across the300 kHz to 9 GHz frequency range.This value can easily be modified tooptimize the offset loss over the frequency range of interest.

Offset Loss = GΩs 10 log10(e) 1GHz

dBloss 1GHz c εr Z0

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Offset Z0The offset Z0 is the characteristicimpedance within the offset length.Modification of this term can be usedto enter the characteristic impedanceof the fixture.

Modification of the thru adapter iseasily performed on the E8358A usingthe Custom Adapter Length pull-downmenu. Figure 15 shows the adaptervalues entered for a test fixture usinga 50-ohm microstrip line. The calibra-tion is performed using a standardcoaxial full two-port calibration witha physical zero length coaxial thru.The test fixture is not used until afterthe calibration is complete but its offset electrical characteristics areentered in the pull-down menu priorto calibration. For this case, the thrudefinition has been modified to havean Offset Length of –650 psec thateffectively adds the test fixture elec-trical length to the calibration thru.The Offset Loss is modified to a nominal 10 Gohm/sec and theCharacteristic Impedance is set to 50 ohms.

Figure 15. Agilent E8358A PNA menu showinga modified adapter definition

Changing the thru definition will compensate for the linear phase shift,constant impedance, and somewhatapproximate the loss of the test fix-ture, but it assumes that these fixturecharacteristics are ideal and don’tchange with frequency. It also assumesthat any mismatches between thetransitions are solely created from animpedance discontinuity. Generally,the coaxial to non-coaxial transitioncannot be modeled in such a simplemanner and more elaborate modelsneed to be implemented for the testfixture. Introduc-tion of complexmodels for the fixture will requiremodifying the complete twelve-termerror model used by the VNA duringthe error correction process.

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Modifying the twelve-term error model

In this section, we review the twelve-term error model used in theVNA to remove the systematic errorof the test equipment. We then devel-op the de-embedding mathematics inorder to combine the test fixturemodel into the VNA’s error model. Inthis way, the VNA can directly displaymeasurement data without includingthe effects of the fixture.

VNA calibration and error models

Measurement errors exist in any network measurement. When using aVNA for the measurement, we canreduce the measurement uncertaintyby measuring or calculating the causes of uncertainty. During theVNA calibration, the system measuresthe magnitude and phase responsesof known devices, and compares themeasurement with actual device characteristics. It uses the results tocharacterize the instrument and effec-tively remove the systematic errorsfrom the measured data of a DUT.8

Systematic errors are the repeatableerrors that result from the non-idealmeasurement system. For example,measurement errors can result fromdirectivity effects in the couplers,cable losses and mismatches betweenthe test system and the DUT. A typicaltwo-port coaxial test system can bemodeled as having twelve errors thatcan be corrected. These errors arecharacterized during system calibra-tion and mathematically removedfrom the DUT measurements. As longas the measurement system is stableover time and temperature, theseerrors are repeatable and the samecalibration can be used for all subsequent measurements.

In a way, the VNA calibration processis de-embedding the system errorsfrom the measurement. Figure 16shows the three system errorsinvolved when measuring a one-portdevice. These errors separate theDUT measurement from an idealmeasurement system. Edf is the for-ward directivity error term resultingfrom signal leakage through the direc-tional coupler on Port 1. Erf is theforward reflection tracking termresulting from the path differencesbetween the test and reference paths.Esf is the forward source match termresulting from the VNA’s test portimpedance not being perfectlymatched to the source impedance.We can refer to this set of terms asthe Error Adapter coefficients of theone-port measurement system. Theseforward error terms are defined asthose associated with Port 1 of theVNA. There are another three termsfor the reverse direction associatedwith reflection measurements fromPort 2.

Figure 16. Signal flow diagram of a one-porterror adapter model

The VNA calibration procedurerequires a sufficient number ofknown devices or calibration stan-dards. In this case three standardsare used for a one-port calibration.The VNA calculates the terms in theError Adapter based on direct or rawmeasurements of these standards. Atypical coaxial calibration kit con-tains a short, open and load standardfor use in this step of the process.Once the Error Adapter is character-ized, corrected measurements of aDUT can be display directly on theVNA display. The equation for calcu-lating the actual S11A of the DUT isshown below. This equation uses thethree reflection error terms and themeasured S11M of the DUT.

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Expanding the above model of theError Adapter for two-port measure-ments, we find that there exists anadditional three error terms for measurements in the forward direction.Once again, we define forward mea-surements as those associated withthe stimulus signal leaving Port 1 ofthe VNA. The additional error termsare Etf, Elf, and Exf for forward trans-mission, forward load match and forward crosstalk respectively. Thetotal number of error terms is twelve,six in the forward direction and six inthe reverse. The flow diagram for theforward error terms in the two-porterror model is shown in Figure 17.This figure also shows the S-parametersof the DUT. This forward error modelis used to calculate the actual S11 andS21 of the DUT. Calculations of theactual S12 and S22 are performedusing the reverse error terms. Theerror model for the reverse directionis the same as in Figure 17, with all ofthe forward terms replaced by reverseterms.8

Figure 17. Signal flow diagram of the forwardtwo-port error terms

Modifying the error terms to include the fixtureerror

If we examine the forward errormodel and insert a test fixture beforeand after the DUT, we can create anew flow graph model that includesthe fixture terms. Figure 18 showsthe original forward calibration termsbeing cascaded with the fixture errorterms. This figure also shows how thetwo sets of error terms can be com-bined into a new set of forward errorterms denoted with a primed variable.The same holds true for the reverseerror model. It should be noted thatfor the VNA to correct for both thesystematic errors in the test equipmentand errors in the test fixture, thecombined error model must still fitinto a traditional twelve-term errormodel. This includes using a unityvalue for the forward transmissionterm on the left side of the flow diagram.9

Figure 18. Signal flow diagram of the combined test fixture S-parameters with theforward two-port error terms

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Renormalizing the modified errorterms, we find the set of equationsthat are used to represent the com-bined errors of the instrument andthe fixture. Once these terms areentered into the analyzer’s error cor-rection algorithm, the displayed mea-surements will be those of the DUTonly at the device plane. The test fix-ture effects will be de-embeddedfrom the measurement.

There are two error terms that arenot modified in the equations. Theyare the forward and reverse crosstalkerror terms, Exf and Exr. Here weassume that the leakage signal acrossthe test fixture is lower than the isolation error term inside the instru-ment. In some cases, this may not bea valid assumption, especially whenusing test fixtures with microstriptransmission lines. Often radiationwithin the fixture will decrease theisolation between the two test portsof the instrument. Before trying tomodel these isolation terms, we mustdecide if the terms will introduce sig-nificant error into the measurementof the DUT. If we are trying to mea-sure a DUT that has 10 dB insertionloss (S21) and the isolation of the fix-ture is 60 dB, the difference betweenthese two signals is 50 dB and theerror introduced into the S21 mea-surement is less than 0.03 dB and 0.2degrees. Unless the DUT measure-ment has a very large insertion loss,on the order of the test fixture orinstrument isolation, then the crosstalkerror terms can be used without mod-ification and often omitted altogetherfrom the VNA calibration.

Should the DUT have a very largeinsertion loss, it may be necessary to include the isolation term of thefixture in the de-embedding model.This term could be measured by plac-ing two terminations inside the testfixture and measuring the leakage orisolation of the fixture.

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Steps to perform de-embedding on the VNA

The de-embedding process beginswith creating a model of the test fix-ture placed on either side of the DUT.The accuracy of the model directlyaffects the accuracy of the de-embed-ded measurements on the DUT.

The next step is to perform a standardcoaxial two-port calibration on theVNA using any calibration type suchas SOLT (Short, Open, Load, Thru) orTRL (Thru, Reflect, Line). Becausethis technique uses the traditionaltwelve-term error model, networkanalyzers that use either three or fourreceivers can be used. This calibrationis then saved to the instrument memory. This same de-embeddingtechnique can be applied to one-portdevices by modifying only the firstthree error terms, namely Edf, Esf, Erffor Port 1 measurements and Edr, Esr,Err for Port 2.

Using a full two-port calibration, thetwelve error terms are then down-loaded into a computer program andmodified using the model(s) for eachside of the test fixture. The twelvemodified error terms are then placedback into the analyzer’s calibrationmemory. At this point, the VNA nowdisplays the de-embedded responseof the DUT. All four S-parameters canbe displayed without the effects ofthe test fixture in real-time.

Using the internal automation capability on the E8358A PNA SeriesNetwork Analyzer, this technique canbe easily implemented without requir-ing an external computer. This familyof network analyzers is PC-based,running the Windows® 2000 operatingsystem. The fixture model can bedownloaded into the instrument usingthe internal floppy drive, built-in LANinterface or any USB compatibledevice.

To show the effects of fixture de-embedding on the calibration errorcoefficients, let’s examine the direc-tivity error term. Figure 19 shows theforward directivity error term for theE8358A PNA series network analyzer.The lower trace is the directivityerror using a standard 3.5 mm coaxialcalibration kit. The upper trace is thesame term modified to de-embed theeffects of a coax-to-microstrip testfixture. Because the directivity is thesum of all the leakage signals appear-ing at the receiver input with a goodtermination placed at the test port,uncertainty in this error term willgenerally affect the reflection mea-surements of well-matched DUTs. Forthis example, unless de-embedding isperformed, the test fixture willdegrade the capability for measuringreturn loss down to the raw perfor-mance of the fixture, which is about15 dB in this case.

Figure 19. Forward directivity error term (1)using standard coaxial calibration (lowertrace) and (2) modified to include the effectsof a coax-to-microstrip test fixture (uppertrace)

As a measurement example, let’s measure the return loss and gain of asurface-mount amplifier placed in amicrostrip test fixture. We can comparethe measured results when calibratingthe analyzer using a standard two-portcoaxial calibration versus de-embed-ding the test fixture using a model ofthe coax-to-microstrip fixture. Figure 20shows the measured S11 of the ampli-fier over a 2 GHz bandwidth. For thecase when using a standard coaxialcalibration, the measured S11 showsexcessive ripple in the response due tomismatch interaction between the testfixture and the surface-mount amplifier.When the test fixture is de-embeddedfrom the measurement, the actual performance of the amplifier is shownwith a linear behavior as a function offrequency.

16

We can also examine the measuredS21 with and without the effects of de-embedding the test fixture. Figure 21shows the measured gain responseover the 2-4 GHz band. Much like theS11 response, the measurement usinga standard coaxial calibration showsadditional ripple in the S21 response.The overall gain is also reduced byabout 0.5 dB due to the additional fixture insertion loss included in themeasurement. Once the fixture is de-embedded, the measured amplifiergain displays more gain and lower ripple across the frequency band.

Figure 20. Measured S11 of a surface mountamplifier. The trace with the large ripple isthe response using a standard coaxial cali-bration. The linear trace shows the deembedded response

Figure 21. Measured S21 of a surface mountamplifier. The lower trace shoes the responseusing a standard coaxial calibration and the-upper trace shows the de-embedded response

Embedding a virtual network

The de-embedding process is used to remove the effects of a physical network placed between the VNA calibration or measurement planeand the DUT plane. Alternately, thissame technique can be used to inserta hypothetical or “virtual” networkbetween the same two planes. Thiswould allow the operator to measurethe DUT as if it were placed into alarger system that does not exist. Oneexample where this technique can bevery useful is during the tuningprocess of a DUT. Often the device isfirst pre-tuned at a lower circuit leveland later placed into a larger networkassembly. Due to interaction betweenthe DUT and the network, a seconditeration of device tuning is typicallyrequired. It is now possible to embedthe larger network assembly into the VNA measurements using the de-embedding process described inthis paper. This allows real-time mea-surements of the DUT including theeffects of the virtual network. Theonly additional step required is to create the “anti-network” model ofthe virtual network to be embedded.

The anti-network is defined as thetwo-port network that, when cascad-ed with another two-port network,results in the identity network. Figure 22 shows the cascaded net-works [S] and [SA], representing theoriginal network and its anti-network.These two networks, when cascadedtogether, create an ideal network thatis both reflectionless and lossless. Ifwe insert these two networks, [S] and[SA], between the measurement planeand DUT plane, we would find no dif-ference in any measured S-parameterof the DUT. The process of de-embed-ding moves the measurement planetoward the DUT plane. Alternatively,the process of embedding a networkwould move the measurement planeaway from the device plane (Figure 23). Therefore, movement ofthe measurement plane toward theDUT (de-embedding) through theanti- network, [SA], is the same asmovement of the measurement planeaway from the DUT (embedding)through the network, [S].

17

Figure 22. Definition of the identity matrixusing a network and the anti-network representations

Figure 23. Diagram showing the movement inthe measurement plane during de-embeddingand embedding of a two-port network

In order to embed a network, it isonly necessary to first calculate theanti-network and apply the same de-embedding algorithm that was previously developed. The equationsfor calculating the anti-network areshown below.

18

As an example of embedding a virtualnetwork into a measured response,let’s measure a surface-mount amplifi-er with a virtual bandpass filter net-work place after the amplifier. Figure 24 shows the measuredresponse before and after embeddingthe filter network. The upper traceshows the measured gain of theamplifier only. Here the amplifier gainis relatively flat over the 2 GHz band-width. The lower trace shows theresponse of the amplifier and embed-ded filter network. In this case, themeasured gain is modified to includethe effects of the bandpass filter net-work. By embedding a virtual filterinto the VNA’s error terms, we canoptimize the amplifier’s gain using amodel of the filter that is not physi-cally part of the actual measurement.

Figure 24. Measured S21 showing the effectsof embedding a virtual bandpass filter network into the measurement of a surfacemount amplifier. Upper trace is the measuredresponse of the amplifier only. Lower traceshows the response including an embeddedfilter network

SummaryThis application note described thetechniques of de-embedding andembedding S-parameter networkswith a device under test. Using theerror-correcting algorithms of thevector network analyzer, the errorcoefficients can be modified so thatthe process of de-embedding orembedding two port networks can beperformed directly on the analyzer inreal-time.

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Appendix A :Relationship of S and T parametermatricesTo determine the scattering transferparameters or T-parameters of thetwo-port network, the incident andreflected waves must be arranged sothe dependent waves are related toPort 1 of the network and the inde-pendent waves are a function of Port2 (see Figure A1). This definition isuseful when cascading a series oftwo-port networks in which the outputwaves of one network are identicalto the input waves of the next. Thisallows simple matrix multiplication tobe used with the characteristic blocksof two-port networks.

The mathematical relationship betweenthe T-parameters and S-parameters isshown below in Figure A2.

Figure A1. Definition of the two-port T-para-meter network

Figure A2. Relationship between the T-para-meters and S-parameters

References1) Edwards, T.C., Foundations of

Microstrip Circuit Design, John Wiley & Sons, 1981

2) In-Fixture Measurements Using

Vector Network Analyzers, Agilent Application Note 1287-9, May 1999

3) Use Agilent EEsof and a Vector

Network Analyzer to Simply

Fixture De-Embedding Agilent White Paper, part number 5968-7845E, October 1999.

4) S Parameter Design, Hewlett-Packard Application Note 154, April 1972 (CHECK THIS NOTE FOR THE LATEST REPRINT)

5) Understanding the Fundamental

Principles of Vector Network

Analyzers, Agilent Application Note 1287-1, May 1997

6) Hunton, J.K. Analysis of

Microwave Measurement

Techniques by Means of Signal

Flow Graphs, IRE Transactions on Microwave Theory and Techniques, March 1960. Availableon the IEEE MTT Microwave Digital Archive, IEEE Product Number JP-17-0-0-C-0

7) Specifying Calibration

Standards For The Agilent 8510

Network Analyzer Product Note 8510-5A, July 2000

8) Applying Error Correction to

Network Analyzer

Measurements, Agilent Application Note 1287-3, April 1999

9) Elmore, G. De-embedded

Measurements Using the 8510

Microwave Network Analyzer, Hewlett-Packard RF & MicrowaveSymposium, 1985.

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