Application Note 1364-1

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

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Introduction

Traditionally 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. RFcomponents are rapidly shiftingaway from designs that use expen-sive coaxial interfaces, and are mov-ing toward designs that use printedcircuit board and surface mounttechnologies (SMT). The traditionalcoaxial interface may even be elimi-nated from the final product. Thisleaves the designer with the prob-lem of measuring the performanceof these RF and microwave compo-nents with test equipment thatrequires coaxial interfaces. Thesolution is to use a test fixture thatinterfaces the coaxial and non-coax-ial transmission lines.

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 thesurface mount device under test(DUT) requires the test fixture char-acteristics to be removed from themeasured results. The test equip-ment typically used for characteriz-ing the RF and microwavecomponent is the vector networkanalyzer (VNA) which uses standard50 or 75 ohm coaxial interfaces atthe test ports. The test equipment iscalibrated at the coaxial interfacedefined as the “measurement plane,”and the required measurements areat the point where the surface-mount device attaches to the print-ed circuit board, or the “deviceplane” (Figure 2). When the VNA iscalibrated at the coaxial interfaceusing any standard calibration kit,the DUT measurements include thetest fixture effects.

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 standardsthat are inserted into the test fix-ture and measured. The accuracy ofthe device measurement relies onthe quality of these physical stan-dards.2 De-embedding uses a modelof the test fixture and mathematical-ly removes the fixture characteris-tics from the overall measurement.This fixture “de-embedding” proce-dure can produce very accurateresults for the non-coaxial DUT,without complex non-coaxial cali-bration standards.

Figure 1. Types of printed circuit transmission lines

Figure 2. Test fixture configuration showing the measurement and device planes

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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 EEsofAdvanced Design System (ADS)have the ability to directly de-embedthe test fixture from the VNA mea-surements using a negation compo-nent model in the simulation.3

Unfortunately these approaches donot allow for real-time feedback tothe operator because the measureddata needs to be captured and post-processed in order to remove theeffects of the test fixture. If real-time de-embedded measurementsare required, an alternate technique must be used.

It is possible to perform the de-embedding calculation directly on the VNA using a different calibra-tion model. If we include the testfixture effects as part of the VNAcalibration error coefficients, realtime de-embedded measurementscan be displayed directly on the VNA.This allows for real-time tuning ofcomponents without including thefixture as part of the measurement.

The following sections of this paperwill review S-parameter matrices, signal flow graphs, and the errorcorrection process used in standardone and two-port calibrations on allAgilent vector network analyzerssuch as the E8358A PNA SeriesNetwork Analyzer. The de-embed-ding process will then be detailedfor removing the effects of a test fix-ture placed between the measure-ment and device planes. Alsoincluded will be a description onhow the same process can be usedto embed a hypothetical or “virtual”network into the measurement ofthe DUT.

S-parameters and signalflow graphs

RF and microwave networks areoften characterized using scatteringor S-parameters.4 The S-parametersof a network provide a clear physi-cal interpretation of the transmis-sion and reflection performance ofthe device. The S-parameters for atwo-port network are defined usingthe reflected or emanating waves, b1and b2, as the dependent variables,and the incident waves, a1 and a2,as the independent variables (Figure 3). The general equations for these waves as a function of theS-parameters is shown below:

b1 = S11a1 + S12a2b2 = S21a1 + S22a2

Using these equations, the individ-ual S-parameters can be determinedby taking the ratio of the reflectedor transmitted wave to the incidentwave with a perfect terminationplaced at the output. For example,to determine the reflection parame-ter from Port 1, defined as S11, wetake the ratio of the reflected wave,b1 to the incident wave, a1, using aperfect termination on Port 2. Theperfect termination guarantees thata2 = 0 since there is no reflectionfrom an ideal load. The remaining S-parameters, S21, S22 and S12, aredefined in a similar manner.5 Thesefour S-parameters completely definethe two-port network characteris-tics. All modern vector network analyzers, such as the Agilent E8358A,can easily the measure the S-para-meters of a two-port device.

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 analyze the transmitted and reflect-ed signals from a network. Directedlines in the flow graph represent thesignal flow through the two-portdevice. For example, the signal flow-ing from node a1 to b1 is defined asthe reflection from Port 1 or S11.When two-port networks are cascad-ed, it can be shown that connectingthe flow graphs of adjacent networkscan be done because the outgoingwaves from one network are thesame as the incoming waves of thenext.6 Analysis of the complete cas-caded network can be accomplishedusing Mason’s Rule.6 It is the appli-cation of signal flow graphs that willbe used to develop the mathematicsbehind network de-embedding andmodifying the error coefficients inthe 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 fix-ture is divided in half to representthe coaxial to non-coaxial interfaceson each side of the DUT. The twofixture halves will be designated asFixture A and Fixture B for the left-hand and right-hand sides of the fix-ture respectively. The S-parametersFAxx (xx = 11, 21, 12, 22) will beused to represent the S-parametersfor the left half of the test fixtureand FBxx will be used to representthe 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 transfermatrices or T-parameters. The math-ematical relationship between S-parameter and T-parametermatrices is given in Appendix A.The two-port T-parameter matrixcan be represented as [T], where [T]is defined as having the four para-meters of the network.

Because we defined the test fixtureand DUT as three cascaded net-works, we can easily multiply theirrespective T-parameter networks,TA, TDUT and TB. It is only throughthe use of T-parameters that thissimple matrix equation be written inthis form.

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

General matrix theory states that ifa matrix 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 theidentity matrix.

It is our goal to de-embed the twosides of the fixture, TA and TB, andgather the information from theDUT or TDUT. Extending thismatrix inversion to the case of thecascaded fixture and DUT matrices,we can multiply each side of themeasured result by the inverse T-parameter matrix of the fixture andyield the T-parameter for the DUTonly. The T-parameter matrix canthen be converted back to thedesired S-parameter matrix usingthe equations in Appendix A.

Using the S or T-parameter model ofthe test fixture and VNA measure-ments of the total combination ofthe fixture and DUT, we can applythe above matrix equation to de-embed the fixture from the measure-ment. The above process is typicallyimplemented after the measure-ments are captured from the VNA. Itis often desirable that the de-embed-ded measurements be displayedreal-time on the VNA. This can beaccomplished using techniques thatprovide some level of modificationto the error coefficients used in theVNA 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 circuittypes and test fixture designs, thereare no simple textbook formulationsfor creating an exact model. Lookingat the whole process of de-embed-ding, the most difficult part is creat-ing an accurate model of the testfixture. There are many techniquesthat can be used to aid in the cre-ation of fixture models, includingsimulation tools such as AgilentAdvanced Design System (ADS) andAgilent High Frequency StructureSimulator (HFSS). Often observationof the physical structure of the testfixture is required for the initial fix-ture model. Measurements made onthe fixture can be used to optimizethe fixture model in an iterativemanner. Time domain techniques,available on most network analyz-ers, can also be very useful whenoptimizing the fixture model.2

Let’s examine several fixture modelsthat can be used in the de-embed-ding process. We will later show thatsome of the simpler models are usedin the firmware of many vector net-work analyzers to directly performthe appropriate de-embedding with-out requiring the T-parametermatrix mathematics.

The simplest model assumes thatthe fixture halves consist of perfecttransmission lines of known electri-cal length. For this case, we simplyshift the measurement plane to theDUT plane by rotating the phaseangle of the measured S-parameters (Figure 6). If we assume the phaseangles, θA and θB, represent thephase of the right and left test fix-ture halves respectively, then the S-parameter model of the fixture canbe represented by the followingequations.

The phase angle is a function of thelength of the fixture multiplied bythe phase constant of the transmis-sion line. The phase constant, β, isdefined as the phase velocity divid-ed by the frequency in radians.Thissimple model assumes that the fix-ture is a lossless transmission linethat is matched to the characteristicimpedance of the system. An easyway to calculate the S-parametervalues for this ideal transmissionline is to use a software simulatorsuch as Agilent ADS. Here, eachside of the test fixture can be mod-eled as a 50-ohm transmission lineusing the appropriate phase angleand reference frequency (Figure 7).Once the simulator calculates all theS-parameters for the circuit, theinformation can be saved to data filefor use in the de-embedding 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 somecases, when the fixture is manufac-tured with low-loss dielectric materi-als and uses well-matchedtransitions from the coaxial to non-coaxial media, this model may pro-vide acceptable measurementaccuracy when performing de-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 ofthe transmission line characteristicsand can include dielectric and con-ductor losses. This loss can be repre-sented using the attenuation factor,α, or the loss 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 char-acteristics of the fixture and calcu-lating the impedance using theknown dielectric constant for thematerial. If the dielectric constantis specified by the manufacturerwith a nominal value and a large tol-erance, then the actual line imped-ance may vary over a wide range.For this case, you can either make abest guess to the actual dielectricconstant or use a measurementtechnique for determining the char-acteristic impedance of the line.One technique uses the time domainoption on the vector network analyz-er. By measuring the frequencyresponse of the fixture using astraight section of transmission line,the analyzer will convert this mea-surement into a Time DomainReflectometer (TDR) response thatcan be used to determine theimpedance of the transmission line.Refer to the analyzer’s User’s Guidefor more information.

Once again, a software simulatorcan be used to calculate therequired S-parameters for thismodel. Figure 9 shows the model forthe test fixture half using a lossytransmission line with the attenua-tion specified using the loss tangent.For this model, the line impedancewas modified to a value of 48-ohmsbased on physical measurements ofthe transmission line width anddielectric thickness and using anominal value for the dielectric con-stant.

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 imple-ment this model by allowing theuser to enter the loss, electricaldelay and characteristic impedancedirectly into the analyzers “calibra-tion 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 oneto create because we need toinclude all of the non-linear effectssuch as dispersion, radiation andcoupling that can occur in the fix-ture. One way to determine themodel is by using a combination ofmeasurements of known devicesplaced in the fixture (which can beas simple as a straight piece oftransmission line) and a computermodel whose values are optimizedto the measurements. A more rigor-ous approach uses an electromag-netic (EM) simulator, such asAgilent HFSS, to calculate the S-parameters of the test fixture. TheEM approach can be very accurateas long as the physical test fixturecharacteristics are modeled correctlyin the simulator.

As an example, we will show amodel created by optimizing a com-puter simulation based on a seriesof measurements made using theactual test fixture. We begin by mod-eling a coax-to-microstrip transitionas a lumped series inductance andshunt capacitance (Figure 10). Thevalues for the inductance and capac-itance will be optimized using themeasured results from the straight50-ohm microstrip line placed in thetest fixture. An ADS model is thencreated for the test fixture andmicrostrip line using this lumpedelement 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 sec-tion for each coax-to-microstrip con-nector. A microstrip thru line isplaced in the center whose physicaland electrical parameters match theline measured in the actual test fix-ture. This microstrip model requiresan accurate value for dielectric con-stant and loss tangent for the sub-strate material used. Uncertainty inthese values will directly affect theaccuracy of the model.

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 arethen made on the test fixture andthe microstrip thru line using a vec-tor network analyzer such as the Agilent E8358A. The four S-parame-ters can be directly imported intothe ADS software over the GPIB.The model values for inductanceand capacitance are optimized usingADS until a good fit is obtainedbetween the measurements and thesimulated results. As an example,Figure 12 shows the measured andoptimized results for the magnitudeof S11 using the test fixture with a microstrip thru line. All four S-parameters should be optimized

and compared to the measured S-parameters to verify the accuracyof the model values. Because of non-linear effects in the transition, thissimplified lumped element model forthe transition may only be valid onlyover a small frequency range. Ifbroadband operation is required, animproved model must be implement-ed to incorporate the non-linearbehavior of the measured S-parame-ters as a function of frequency.

Once the lumped element parame-ters are optimized, the S-parametersfor each half of the test fixture canbe simulated and saved for use bythe de-embedding algorithm. Keep inmind that it is necessary to includethe actual length of microstrip linebetween the transition and devicewhen calculating the S-parametersfor the test fixture halves.

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The de-embeddingprocess

Whether a simplified model, such asa length of ideal transmission line,or a complex model, created usingan EM simulator, is used for the testfixture, it is now necessary to perform the de-embedding processusing this S-parameter model.There are two main ways the de-embedding process can be imple-mented. The first technique usesmeasured data from a network ana-lyzer and processes the data usingthe T-parameter matrix calculationsdiscussed in the previous section.The second technique uses the net-work analyzer to directly performthe de-embedding calculations,allowing the user to examine the de-embedding response in real-time.This technique is accomplished bymodifying the calibration errorterms in the analyzer’s memory.

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 modelin ADS.3 Once the measurementsare de-embedded, the data is dis-played statically on a computerscreen or can be downloaded intothe analyzer’s memory for display.

There are five steps for the process of de-embedding the test fixtureusing T-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-parameters to T-parameters.

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

Step 5: Convert the final T-parame-ters back to S-parameters and dis-play the results. This matrixrepresents the S-parameters of thedevice only. The test fixture effectshave been removed.

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 inthe vector network analyzer. In thisway, the analyzer is performing allthe de-embedding calculations,which allows the users to view real-time measurements of the DUTwithout the effects of the test fixture.

Most vector network analyzers arecapable of performing some modifi-cation to the error terms directlyfrom the front panel. These includeport extension and modifying thecalibration “thru” definition. Eachof these techniques will now be dis-cussed, including a technique tomodify the traditional twelve-termerror model to include the completeS-parameter model for each side ofthe test fixture.

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Simple corrections for fixture effects

Port extensions

The simplest form of de-embeddingis port extensions, which mathemat-ically extends the measurementplane towards the DUT. This featureis included in the firmware of mostmodern network analyzers such asthe Agilent E8358A. Port extensionsassume that the test fixture lookslike a perfect transmission line ofsome known phase length. Itassumes the fixture has no loss, alinear phase response, and constantimpedance. Port extensions are usu-ally applied to the measurementsafter a two-port calibration has beenperformed at the end of the testcables. If the fixture performance isconsiderably better than the specifi-cations of the DUT, this techniquemay be sufficient.

Port extension only adds or sub-tracts phase length from the mea-sured S-parameter. It does notcompensate for fixture losses orimpedance discontinuities. In mostcases, there will be a certainamount of mismatch interactionbetween the coax-to-fixture transi-tion and the DUT that will createuncertainty in the measured S-parameter. This uncertainty typi-cally results in an observed ripple inthe S-parameter when measuredover a wide frequency range. As anexample, consider the measure-ments shown in Figure 13 of a shortplaced at the end of two differentconstant impedance transmissionlines: a high-quality coaxial airline(upper curve), and a microstriptransmission line (lower curve). Portextensions were used to move themeasurement plane up to the short.However, as seen in the figure, portextension does not compensate forthe losses in the transmission line.

Also note that the airline measure-ment exhibits lower ripple in themeasured S11 trace while the coax-to-microstrip test fixture shows amuch larger ripple. Generally, theripple is caused by interactionbetween the discontinuities at themeasurement and device planes. The larger ripple in the lower traceresults from the poor return loss ofthe microstrip transition (20 dB versus >45 dB for the airline). Thisripple can be reduced if improve-ments are made in the return 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 calibration standards

During calibration of the vector network analyzer, the instrumentmeasures actual, well-defined standards such as the open, short,load and thru, and compares themeasurements to ideal models foreach standard. Any differencesbetween the measurements and themodels are used to compute theerror terms contained within themeasurement setup. These errorterms are then used to mathemati-cally correct the actual measurementsof the device under test. This calibration process creates a reference or calibration plane at thepoint where the standards are connected. As long as a precisemodel is known for each calibrationstandard, an accurate referenceplane can be established. For example, in some Agilent coaxialcalibration kits, the short standardis not a true short at the referenceplane, it is actually an “offset” short.The offset short consists of a smallpiece of coaxial transmission lineplaced between the connector andthe true short. When selecting a calibration kit (definition), you areinstructing the instrument to usethe correct model for the offsetshort. For example, when using theAgilent 85033C 3.5mm coaxial calibration kit, the short standarduses an offset length of 50 ohmtransmission line equal to a delay of16.695 psec. When the analyzer usesthe proper model for the offsetshort, then the reference will be correctly calculated.

Another way to implement the reference plane or port extensions,discussed in the previous section,would be to redefine the cal kit definitions for each of the calibrationstandards. For example, if we wantedto extend each reference plane avalue of 100 psec past the point ofcalibration, we can modify eachstandard definition to include this100 psec offset. This value would besubtracted from the original offsetdelay of the short, open and loadstandards. The “thru” definitionwould include the total delay of theextensions from each port. As anexample, when using the Agilent85033E cal kit, we would modify theshort definition to have an offsetdelay of –68.202 psec (calculatedfrom 31.798–100 psec, with the original 3.5mm short delay of 31.798 psec). The same approach is applied to the open and load standards. The “thru” definitionwould have an offset delay of –200 psec (0–100–100 psec, usingthe original thru delay of 0 psec).The thru definition requires thedelay contributions from the port 1and port 2 port extensions.

The calibration kit definition actually includes three offset characteristics for each standard.7

They are Offset Delay, Offset Lossand Offset Impedance (Z0). Thesethree characteristics are used toaccurately model each standard sothe analyzer can establish a referenceplane for each of the test ports.

Fixture de-embedding can be accom-plished by adjusting the calibrationkit definition table to include theeffects of the test fixture. In thisway, some of the fixture characteristics can be included inthe error terms determined duringthe coaxial calibration process.Once the calibration is complete, theanalyzer will mathematically removethe delay, loss and impedance of thefixture. It should be noted thatsome accuracy improvements wouldbe seen over the previously discussed port extension technique,but some assumptions made aboutthe fixture model will limit the overall measurement accuracy of thesystem. We will now discuss theimplementation and limitations ofmodifying the cal kit definition toinclude the characteristics of thetest fixture.

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Offset delayThe 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 velocityof light in free space and the permit-tivity constant.

Here we assumed that the relativepermeability, µr, equals one. Notethe electrical delay for transmissionlines other than coax, such asmicrostrip, will have a require amodification to the above equationdue the change in the effective per-mittivity of the transmission media.Most RF software simulators, suchas the Agilent ADS LineCalc tool,will calculate the effective phaselength and effective permittivity of atransmission line based on the phys-ical parameters of the circuit. Theeffective phase can be converted toelectrical delay using the followingequation.

The “thru” standard would be modi-fied to include the delay from thetotal fixture length. The short, openand load standards would be modi-fied to one half this delay, since weare extending the reference planeshalf way on each side. Here weassume that the device under test isplaced directly in the middle of thefixture. Note that adjustments canbe made to the cal kit definitiontable should the fixture be asymmet-rical. In this case two sets of shorts,opens and loads would be defined, aseparate set for each test port.

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Offset lossThe network analyzer uses the offsetloss to model the magnitude lossdue to skin effect of a coaxial typestandard. Because the fixture isnon-coaxial, the loss as a function offrequency may not follow the loss ofa coaxial transmission line so thevalue entered may only approximatethe true loss of the fixture. Thevalue of loss is entered into thestandard definition table asgigohms/second or ohms/nanosecond at 1 GHz.

The offset loss in gigohms/secondcan be calculated from the measuredloss at 1 GHz and the physicallength of the particular standard bythe following equation:

where:dBloss @ 1GHz = measured insertion

loss at 1 GHzZ0 = offset Z0

= physical length of the offset

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 cal kit definitionto include the effects of the fixtureloss (top curve). For this case, wewould expect the measured loss ofthe fixture “thru” (after calibration)to be a flat line with 0 dB insertionloss. The actual measurement showsa trade-off between the high and lowfrequencies by adjusting the offsetloss to be optimized in the middle ofthe band. The offset loss was set to10 Gohm/sec for the FR-4 materialused. For this case, a 3-inch lengthof microstrip 50-ohm transmissionline was used with an approximatedielectric constant of 4.3 and losstangent value of 0.012. The value of10 Gohm/sec for the offset loss is agood compromise across the 300 kHzto 9 GHz frequency range. Thisvalue can easily be modified to optimize the offset loss over the frequency range of interest.

Offset Loss = GΩs 10 log

10(e) 1GHz

dBloss 1GHz c εr Z0

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

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Offset Z0The offset Z0 is the characteristicimpedance within the offset length.Modification of this term can beused to enter the characteristicimpedance of the fixture.

Modifying the standards definitionModification of the cal kit standardsdefinition is easily performed on theE8358A using the Advanced ModifyCal Kit dialog. Figure 15 shows thedefinition table which is used tochange the offset delay, offset lossand offset impedance for the short,open, load and thru model defini-tions.

We begin the process by selecting thecoaxial cal kit that will be used tocalibrate the vector network analyz-er over the frequency range of inter-est. We also require the values foroffset delay, loss and impedance ofthe test fixture. As an example, wewill assume that the total “thru”delay of the fixture is 650 psec, theoffset loss is calculated as 10 Gohm/sand the offset impedance is 50 ohms.We will also assume that the fixtureis symmetrical and each half of thefixture introduces 325 psec of delay(this value will be used to modifythe short, open and load defini-tions). The selected coaxial calibration kit definition will now bemodified to include the characteris-tics of the test fixture.

Adjust the short delay to a value calculated by subtracting the delayintroduced by the fixture half, fromthe original definition. For example,when using the 85033E 3.5mm cali-bration kit, the original offset delayis defined as 31.798 psec. Changethis value to –293.202 psec (calculated from 31.798-325 psec, inour example). Modify the loss to 10 Gohm/s. The offset impedancecan remain at 50 ohms for thisexample. Perform the same adjustments for the open and loaddefinitions.

The thru delay is modified to avalue equal to original delay minusthe total delay thru the test fixture.For this example, the modified thrudelay would be set to –650 psec,since the original delay is 0 psec.The thru loss is also set to 10 Gohms/sec and Z0 is 50 ohms.

Once the standards definitions aremodified to include the test fixturecharacteristics, the updated cal kitcan be saved as a user kit on thenetwork analyzer. Give the new calkit a specific name to distinguish itfrom the other kits stored in theanalyzer memory. The E8358A PNAalso assigns Kit ID numbers to eachcal kit definition table.

The analyzer calibration can now beperformed using a standard coaxial,full two-port calibration with thenew cal definition selected for thecal kit type. The test fixture is notconnected until after the coaxial calibration is complete. The errorcorrection mathematics in the analyzer will include the effects ofthe test fixture loss, delay andimpedance in the calibration.

Changing the offset definitions willcompensate for the linear phaseshift, constant impedance and,somewhat approximate the loss ofthe test fixture. Here we are assum-ing that the loss of the fixture follows the skin effect loss of a coaxial transmission, which in mostcases is not exactly valid. It alsoassumes that any mismatchesbetween the transitions are solelycreated from an impedance disconti-nuity. Generally, the coaxial to non-coaxial transition cannot bemodeled in such a simple mannerand more elaborate models need tobe implemented for the test fixture.Introduction of complex models forthe fixture will require modifyingthe twelve-term error model used bythe VNA during the error correctionprocess. The next section describesthe error model used by the vectornetwork analyzer and the processthat can be used to modify the error terms stored in the analyzer’smemory.

Figure 15 E8358A PNA dialog showing the modified cal kit definition

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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 thendevelop the de-embedding mathe-matics in order to combine the testfixture model into the VNA’s errormodel. In this way, the VNA candirectly display measurement datawithout including the effects of thefixture.

VNA calibration and error models

Measurement errors exist in any network measurement. When usinga VNA for the measurement, we canreduce the measurement uncertain-ty by measuring or calculating the causes of uncertainty. During theVNA calibration, the system mea-sures the magnitude and phaseresponses of known devices, andcompares the measurement withactual device characteristics. It usesthe results to characterize theinstrument and effectively removethe systematic errors from 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 mismatchesbetween the test system and theDUT. A typical two-port coaxial testsystem can be modeled as havingtwelve errors that can be corrected.These errors are characterized dur-ing system calibration and mathe-matically removed from the DUTmeasurements. As long as the mea-surement system is stable over timeand temperature, these errors arerepeatable and the same calibrationcan be used for all subsequent measurements.

In a way, the VNA calibrationprocess is de-embedding the systemerrors from the measurement.Figure 16 shows the three systemerrors involved when measuring aone-port device. These errors sepa-rate the DUT measurement from anideal measurement system. Edf isthe forward directivity error termresulting from signal leakagethrough the directional coupler onPort 1. Erf is the forward reflectiontracking term resulting from thepath differences between the testand reference paths. Esf is the for-ward source match term resultingfrom the VNA’s test port impedancenot being perfectly matched to thesource impedance. We can refer tothis set of terms as the ErrorAdapter coefficients of the one-portmeasurement system. These forwarderror terms are defined as thoseassociated with Port 1 of the VNA.There are another three terms forthe 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 orraw measurements of these stan-dards. A typical coaxial calibrationkit contains a short, open and loadstandard for use in this step of theprocess. Once the Error Adapter ischaracterized, corrected measure-ments of a DUT can be displaydirectly on the VNA display. Theequation for calculating the actualS11A of the DUT is shown below.This equation uses the three reflec-tion error terms and the measuredS11M 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 direc-tion. Once again, we define forwardmeasurements as those associatedwith the stimulus signal leaving Port1 of the VNA. The additional errorterms are Etf, Elf, and Exf for for-ward transmission, forward loadmatch and forward crosstalk respec-tively. The total number of errorterms is twelve, six in the forwarddirection and six in the reverse. Theflow diagram for the forward errorterms in the two-port error model isshown in Figure 17. This figure alsoshows the S-parameters of the DUT.This forward error model is used tocalculate the actual S11 and S21 ofthe DUT. Calculations of the actualS12 and S22 are performed usingthe reverse error terms. The errormodel for the reverse direction isthe same as in Figure 17, with all ofthe forward terms replaced byreverse terms.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 fixturebefore and after the DUT, we cancreate a new flow graph model thatincludes the fixture terms. Figure 18shows the original forward calibra-tion terms being cascaded with thefixture error terms. This figure alsoshows how the two sets of errorterms can be combined into a newset of forward error terms denotedwith a primed variable. The sameholds true for the reverse errormodel. It should be noted that forthe VNA to correct for both the sys-tematic 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 errorcorrection algorithm, the displayedmeasurements will be those of theDUT only at the device plane. Thetest fixture effects will be de-embed-ded from the measurement.

There are two error terms that arenot modified in the equations. Theyare the forward and reversecrosstalk error terms, Exf and Exr.Here we assume that the leakage sig-nal across the test fixture is lowerthan the isolation error term insidethe instrument. In some cases, thismay not be a valid assumption,especially when using test fixtureswith microstrip transmission lines.Often radiation within the fixturewill decrease the isolation betweenthe two test ports of the instrument.Before trying to model these isola-tion terms, we must decide if theterms will introduce significanterror into the measurement of theDUT. If we are trying to measure aDUT that has 10 dB insertion loss(S21) and the isolation of the fixtureis 60 dB, the difference betweenthese two signals is 50 dB and theerror introduced into the S21 mea-surement is less than 0.03 dB and0.2 degrees. Unless the DUT mea-surement has a very large insertionloss, on the order of the test fixtureor instrument isolation, then thecrosstalk error terms can be usedwithout modification and oftenomitted altogether from the VNAcalibration.

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 byplacing two terminations inside thetest fixture and measuring the leak-age or isolation of the fixture.

17

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 theDUT. The accuracy of the modeldirectly affects the accuracy of thede-embedded measurements on theDUT.

The next step is to perform a stan-dard coaxial two-port calibration onthe VNA using any calibration typesuch as SOLT (Short, Open, Load,Thru) or TRL (Thru, Reflect, Line).Because this technique uses the tra-ditional twelve-term error model,network analyzers that use eitherthree or four receivers can be used.This calibration is then saved to theinstrument memory. This same de-embedding technique can be appliedto one-port devices by modifyingonly the first three error terms,namely Edf, Esf, Erf for Port 1 measurements and Edr, Esr, Err forPort 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 thenplaced back into the analyzer’s cali-bration memory. At this point, theVNA now displays the de-embeddedresponse of the DUT. All four S-parameters can be displayed with-out the effects of the test fixture inreal-time.

Using the internal automation capability on the E8358A PNA SeriesNetwork Analyzer, this technique canbe easily implemented withoutrequiring an external computer. Thisfamily of network analyzers is PC-based, running the Windows® 2000operating system. The fixture modelcan be downloaded into the instru-ment using the internal floppy drive,built-in LAN interface or any USBcompatible device.

To show the effects of fixture de-embedding on the calibrationerror coefficients, let’s examine thedirectivity error term. Figure 19shows the forward directivity errorterm for the E8358A PNA series net-work analyzer. The lower trace is thedirectivity error using a standard 3.5 mm coaxial calibration kit. Theupper trace is the same term modi-fied to de-embed the effects of a coax-to-microstrip test fixture. Because thedirectivity is the sum of all the leak-age signals appearing at the receiverinput with a good termination placedat the test port, uncertainty in thiserror term will generally affect thereflection measurements of well-matched DUTs. For this example,unless de-embedding is performed,the test fixture will degrade the capa-bility for measuring return loss downto the raw performance of the fixture,which is about 15 dB in this case.

As a measurement example, let’s measure the return loss and gain ofa surface-mount amplifier placed ina microstrip test fixture. We can com-pare the measured results when cal-ibrating the analyzer using astandard two-port coaxial calibra-tion versus de-embedding the testfixture using a model of the coax-to-microstrip fixture. Figure 20 showsthe measured S11 of the amplifierover a 2 GHz bandwidth. For thecase when using a standard coaxialcalibration, the measured S11 showsexcessive ripple in the response dueto mismatch interaction between thetest fixture and the surface-mountamplifier. When the test fixture isde-embedded from the measure-ment, the actual performance of theamplifier is shown with a linearbehavior as a function of frequency.

Figure 19. Forward directivity error term (1) using standard coaxial calibration (lower trace) and (2) modified to include the effects of acoax-to-microstrip test fixture (upper trace)

18

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 measurementusing a standard coaxial calibrationshows additional ripple in the S21response. The overall gain is alsoreduced by about 0.5 dB due to theadditional fixture insertion lossincluded in the measurement. Oncethe fixture is de-embedded, the measured amplifier gain displaysmore gain and lower ripple acrossthe frequency band.

Figure 20. Measured S11 of a surface mount amplifier. The trace with thelarge ripple is the response using a standard coaxial calibration. The linear trace shows the de-embedded response

Figure 21. Measured S21 of a surface mount amplifier. The lower traceshoes the response using a standard coaxial calibration and the uppertrace shows the de-embedded response

19

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 toinsert a hypothetical or “virtual”network between the same twoplanes. This would allow the opera-tor to measure the DUT as if it wereplaced into a larger system thatdoes not exist. One example wherethis technique can be very useful isduring the tuning process of a DUT.Often the device is first pre-tuned ata lower circuit level and later placedinto a larger network assembly. Dueto interaction between the DUT andthe network, a second iteration ofdevice tuning is typically required.It is now possible to embed the larg-er network assembly into the VNAmeasurements using the de-embed-ding process described in this paper.This allows real-time measurementsof the DUT including the effects ofthe virtual network. The only addi-tional step required is to create the“anti-network” model of the virtualnetwork to be embedded.

The anti-network is defined as the two-port network that, whencascaded with another two-portnetwork, results in the identity net-work. Figure 22 shows the cascadednetworks [S] and [SA], representingthe original network and its anti-network. These two networks, whencascaded together, create an idealnetwork that is both reflectionlessand lossless. If we insert these twonetworks, [S] and [SA], between themeasurement plane and DUT plane,we would find no difference in anymeasured S-parameter of the DUT.The process of de-embedding movesthe measurement plane toward theDUT plane. Alternatively, the processof embedding a network would movethe measurement plane away fromthe device plane (Figure 23).Therefore, movement of the measure-ment plane toward the DUT (de-embedding) through the anti- network, [SA], is the same as move-ment of the measurement plane awayfrom the DUT (embedding) throughthe network, [S].

20

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.

21

As an example of embedding a virtual network into a measuredresponse, let’s measure a surface-mount amplifier with a virtual band-pass filter network placed after theamplifier. Figure 24 shows the mea-sured response before and afterembedding the filter network. Theupper trace shows the measuredgain of the amplifier only. Here theamplifier gain is relatively flat overthe 2 GHz bandwidth. The lowertrace shows the response of theamplifier and embedded filter net-work. In this case, the measuredgain is modified to include theeffects 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 canbe performed directly on the analyz-er in real-time.

22

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 indepen-dent waves are a function of Port 2(see Figure A1). This definition is use-ful when cascading a series of two-port networks in which the outputwaves of one network are identical tothe 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-parameter network

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

23

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, Agilent 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. Available on 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 & Microwave Symposium, 1985.

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