Vector measurements require both magnitudeand phase data. Some typical examples are thecomplex reflection coefficient, the magnitudeand phase of the transfer function, and thegroup delay.

The seminar will cover four basic topics: (1) Areview of accuracy enhancement for one-portnetworks. (2) One-port calibration techniques.(3) Accuracy enhancement for two-portnetworks, and (4) two-port calibrationtechniques.

This section reviews the basic theory ofaccuracy enhancement for one-portnetworks.

The block diagram of the measurementsystem consists of a sweep oscillator, areflectometer consisting of two couplersconnected back-to-back, and the unknownone-port (ΓA), The direction of power flowthrough the system is indicated by the arrows.The measured reflection coefficient is definedas ΓM. The measurement system is not perfectand contributes errors to the measurement ofΓA. For example, a one-port network (anideal resistive termination which has a 40 dBreturn loss), has a well behaved frequencyresponse. But the measured ΓM Still typicallyhas 10 dB peak-to-peak variations withrespect to frequency. The variations are not inthe resistive device. An ideal resistor isincapable of causing this type of changingreflection coefficient. The errors in this caseare in the measurement system.

The following is a discussion of the variouserrors that are inherent in a reflectometer.

An imperfect reflectometer can be modeled bytaking all the linear errors of the system andcombining them into a fictitious two-port erroradapter between the reflectometer and theunknown one-port. This results in anabsolutely perfect reflectometer with no loss,no mismatch, and no frequency responseerrors. The fictitious two-port error adapterhas four error terms. When making ratiomeasurements, only three terms are needed.

The flow graph of the fictitious two-port erroradapter is connected to the unknown reflectioncoefficient (ΓA). A flow graph is a good way togive a practical feel for how signals are flowingin a microwave network. For example, suppose,ΓA is a perfect Z0 termination. When ΓM ismeasured, there are some residual signalscaused by the system directivity e00 . This signaldoes not flow to the device under test but flowsthrough the branch of the flow graph labelede00. The frequency response terms e10 and e01

can be best understood by measuring a short. Ashort has a reflection coefficient of -1 with nofrequency response variations. But, due to thecouplers, front end mixers, cables, and otherhardware there are definite frequency responsevariations modeled by the branches e10 and e01.Looking back into the test port of themeasurement system, it does not present aperfect Z0 source impedance. There is somereflection when a signal is incident on the testport. This is primarily due to the match of thecouplers and/or adapters. The equivalent portmatch is modeled by the branch e11.

ΓM is mathematically related to ΓA by three errorterms: directivity, frequency response, and portmatch. If these three terms are known from acalibration procedure, (discussed later), it ispossible to solve for the actual reflectioncoefficient ΓA.

This slide shows the return loss of a resistivetermination before and after error correction.The directivity of the measurement system isabout the same as the return loss of the resistivetermination. 40 dB is a respectable equivalentdirectivity for a measurement system. But, atsome frequencies, the directivity term adds inphase with the reflection coefficient of theactual device under test and yields a 6 dBhigher measurement. At other frequencies, thedirectivity and reflection coefficient terms areout of phase and cancel, causing deep dips inthe measurement data. Error correction does notremove all errors. Residual errors remaining inthe measurement system may be due toimperfect standards, repeatability, noise, etc.But the measurement accuracy is greatlyimproved. In fact, greater than 20 dBimprovement is typical.

Having shown what error correction canaccomplish, the problem of determiningthe error coefficients remains.

A process of calibration will now bediscussed. By a change of variables, theequation for the measured reflectioncoefficient can be converted to a bilinearequation with the error terms a, b and c asvariables. If there is a way to generate twomore of these equations, giving threeequations in the three unknowns, (a, b andc), then a system of equations results that canbe solved for the error terms. The approachused in error correction for one-portnetworks is to measure three different knownloads (ΓA). This generates three differentmeasured reflection coefficients (ΓM). Thisprocedure yields the necessary threeequations to solve for the error terms a, band c.

This is a graphic approach to the calibrationtechnique. From complex algebra, if three pointson the ΓA plane and three points on the ΓM planeare known, the terms a, b and c are uniquelydetermined for a bilinear transformation. Thisgraphic approach will be helpful to visualizedifferent calibration techniques.

These are four of the more popular calibrationtechniques. The first one basically goes back tothe mathematical treatment of three different,known offset shorts for ΓA. An offset short is ashort circuit of precisely known length down anair line. The frequency must be knownaccurately to determine the phase shift at theoffset short. The three offset shorts generatethree equations to solve for the three unknownerror terms. The phase of offset shorts isfrequency sensitive. Therefore, care must betaken to avoid lengths that are multiples of ahalf-wavelength, or the offset shorts will not beunique. This technique is useful when dealingwith microstrip and other transmission mediawhere it is very difficult to realize Z0 loads oropens.

The second technique has been used for manyyears by Hewlett Packard on the large,computer-based, automatic network analyzers.The technique consists of using a Z0 termination,a good short circuit, and a series of offset shortswhich are a quarter wavelength long in the centerof the band to approximate an open circuit. Itrequires a series of offset shorts over thefrequency range in order to approximate opencircuits. The offset shorts are not broadband, butprovide good accuracy.

The third technique is very similar to thesecond, but replaces the offset short with anopen circuit. The open circuit has a fringingcapacitance that needs to be modeled veryprecisely in order to achieve accuracy.

The last technique uses two different slidingterminations and a short circuit. Thisapproach is the fussiest and the mostmathematically involved, but has thepotential of achieving the best performance.

The last two methods will now be discussedin detail.

The technique has three steps. First, attacha Z0 termination (notice ΓA = 0) andmeasurer ΓM1. This yields the directivityterm e00 directly. Next, connect a shortand measurer ΓM2. The final step is toconnect a shielded open and measurerΓM3. Assume that the capacitance of theshielded open, which determines the phaseterm, is known. There are now twoequations with two unknowns (thefrequency response e10e01, and the portmatch e11) for which we can solve.

This is a graphic presentation of thepreceding technique.

The short circuit is at -1 on the ΓA plane, and thesliding load generates a circle, centered at theorigin. The center of the circle is determined bythe mechanical dimensions of the air line andyields an effective directivity of approximately 60dB. The radius of the circle is determined by theresidual reflection coefficient of the slidingelement. As the sliding element is movedphysically, it traces out the locus of a circle onthe ΓA plane. The open circuit at low frequenciesis at +1 on the ΓA plane, but as the frequencyincreases, the capacitive reactance starts to rotatethe position of the shielded open clockwise. Onthe ΓM plane the short's position is modified andthe sliding load is no longer centered at theorigin. The position of the shielded open is alsomodified.

The shielded open consists of an extension of theouter conductor of the coax beyond thetermination of the inner conductor. Thisextension is a circular waveguide operatingbelow its cut-off frequency. The impedance ofsuch a waveguide section is reactive (a shuntcapacitance) and cannot propagate power. Thereflection coefficient of a shunt capacitor is e-jβ.Notice that jβ is a function of frequency, so thefrequency has to be known accurately as well asthe value of the capacitance. The capacitance ofa waveguide increases the closer you get to thecut-off frequency. We can model thiscapacitance as a power series and determineempirically the power series coefficients. Noticethat the capacitance rises approximately 15% at18 GHz.

This slide shows the results of measuring thereflection coefficient of an APC-7 offset shortusing the open capacitance model. With thecapacitance equal to zero, there isapproximately an 8 dB peak-to-peak error,showing that a simple open circuit is not agood model for high frequency calibration. Ifthe dc capacitance term is modeled, the errorcan be reduced to approximately I dBpeak-to-peak error. By modeling thecapacitance as a function of frequency, theerror is reduced to less than .1 dBpeak-to-peak.

This second procedure shows how to use twodifferent sliding terminations and a short circuit forcalibration. Step 1 connects a short circuit on thetest port and measures ΓM1, Step 2 uses a sliding,low reflection termination to define a circle on theΓM plane with the center at Γ02 and the radius R2.Step 3 connects a high reflection sliding terminationto define a second circle with the center Γ03 andradius R3. Steps 1 through 3 yield three equationswhich can be solved for the three unknowns: a, b,and c. The center and radius of the two circles canbe determined by a circle fitting technique.

It should be noted that certain assumptions havebeen made about the sliding terminations: (1) Themagnitude of the reflection coefficient cannotchange as it rotates in phase. (2) Airline loss isnegligible.

This technique does not require an accurateknowledge of the frequency. The magnitude andphase of the sliding terminations are also notneeded. This technique can be very accurate, butit is more time consuming and not nearly as easyto apply as the previous approach.

This graphically shows how the circleson the ΓA plane are modified when theyare transformed to the ΓM plane.

In conclusion, this section demonstrates thatwith one-port error correction, the systemdirectivity is extended, response errors arereduced significantly, and port matchuncertainties are lowered. Calibration makesa marked improvement when measuring withadapters and cables since their effects areremoved. The use of cables requires carebecause their characteristics change as theyare flexed. If semi-rigid coax is used, iscarefully positioned, and remains stationary,then the errors due to these lines can begreatly reduced, offering alternative ways toconnect to the device under test.

Accuracy enhancement techniques for two-port networks will now be discussed. Thepresentation will follow the same develop-ment as the prior one-port case.

The two-port measurement system duplicatespart of the one-port measurement systemdiscussed earlier. A dual directional couplet isattached to the input of the unknown two-port,and another dual directional coupler is attachedto the output, A switch changes the direction ofthe incident power to the unknown two-port forforward and reverse measurements andterminates the unknown two-port in animpedance Z0. The problems are very similar tothose of the one-port case. There are directivityerrors, frequency response errors, and portmatch errors, for both ports. For example, thedotted line in the graph represents the actualfrequency response of a device. The solid linerepresents the measured result. Again, it is notthe device but the measurement system causingthe error.

Paralleling the previous one-port analysis, all thelinear errors of the imperfect reflectometer canbe combined into an error adapter. In this case,the fictitious error adapter must be a four-port.This error adapter has 16 error terms. Thenumber of error terms is the square of thenumber of ports.

There are additional problems because the erroradapter does not include switch repeatabilityerrors, or changing port match due to differentswitch positions. Errors caused by the switch canbe removed by a procedure that will bediscussed later. To resolve further difficulties,one of the following assumptions must be made.(1) There are four samplers or mixers which canbe connected at all times to the fourmeasurement ports of the couplers. Otherwise,additional switches would need to be included inthe model. (2) If additional switches areincluded, an assumption could be made that theisolation of the couplers is high enough toreduce the errors caused by this additionalswitching.

Here is another approach to the samemeasurement problem. A fictitious erroradapter has been created for the forwarddirection with the source and the loadconnected in the proper position. There areonly three measurement ports. There isanother separate error adapter for the reversedirection. By using two separate erroradapters, the switch problems and thenecessity of having four measurement portshas been removed. This is a typical modelused widely in error correction techniques.Each one of these error adapters has 12 errorterms. So, there is a total of 24. Two of thequestions that arise: Are all of the error termsmeaningful? And, what is the physicaldescription of each of the error terms? Thesewill be discussed next.

This is a pictorial flow graph indicating theerror terms for the forward configuration. Asmentioned earlier, there are 12 error terms.There are two port match terms (e11 and e22

and a directivity term (e00). For the frequencyresponse, there are: e10 and e01 for makingreflection measurements and e10 and e32 fortransmission measurements. The remainingsix terms are leakage terms between thevarious ports.

For a typical S-parameter test set using highdirectivity couplers with good isolation, theprimary terms are indicated by the solid linesand the less important terms by the dotted lines.The primary errors are: The two-port matcherrors (e11 and e22); The directivity error (e00);The frequency response errors (the reflectionfrequency response term e10e01 and thetransmission frequency response term e10e32);and the leakage term (e30). These are the sixcomplex error terms for the forward direction.The procedure can be repeated for the reverseconfiguration to obtain six additional errorterms. This leads to the "twelve term errormodel".

This is a flow graph for the forwardconfiguration with the assumptions thatwere made on the previous slide. Note thatS11M and S21M are functions of all fourS-parameters of the device under test [SA].

This is a flow graph of the reverseconfiguration. Note again, that themeasured S-parameters are functionsof all the S-parameters of the deviceunder test.

Four measurements have been made: TheS-parameters S11M, S21M, S12M and S22M.Each of these measured S-parameters is afunction of the four S-parameters of thedevice under test plus 12 error terms. Now,if these error terms can be determined by acalibration procedure, (described later),four equations in the four unknowns S11A,S21A, S22A and S12A result. This system ofequations can he solved for the actualS-parameters. Notice that the S21A

transmission coefficient is a function notonly of the measured S21M but of the otherthree measured S-parameters as well. Ittakes four measurements to calculate oneS-parameter.

Here are the solutions for the remainingactual S-parameters of the device undertest.

This slide shows the magnitude S21A for a"mismatched pad." The dotted line is thecorrected data with only the frequencyresponse errors removed. The solid lineshows the corrected data when using the12 term error model. By including theerror terms e11 and e22, the .5dB peakto-peak error is modified significantly.

The theory of error correction for two-port networks has been developed. It nowremains to develop the solutions for thevarious error terms.

Two calibration techniques will bediscussed in detail. One is a standardcalibration technique that has been usedfor many years at Hewlett Packard. Theother is a self-calibration technique whereprecise knowledge of the standards is notrequired.

Step (1) of the standard calibrationprocedure is to calibrate port 1 using theone-port procedure that was developedearlier. Place a short, an open, and a loadon port 1 and solve for the three errorterms: the equivalent directivity (e00), theport match (e11), and the reflectionfrequency response (el0e01). When one-portdevices are used on the two-portmeasurement system, the equation for thereflection coefficient of the two-port casereduces to that of the one-port case.

Step (2) is to determine the leakage.This can be accomplished by placingZ0 terminations on port 1 and port 2.In this case, the equation for S21M isequal to the leakage term e30.

Step (3) is to connect port 1 and port 2 ofthe measurement system together. Thisreduces the equation for S11M to anequation for a one-port measurementsystem. Since the three error terms for port1 were previously determined, the one-portmeasurement system can now be used tomeasure the port match of port 2 (e22).

S21M reduces to a simple expression forthe thru connection. The leakage term(e30) has been determined and the two portmatches (e11 and e22) are known, so theremaining transmission frequencyresponse term (e10e32) can be easilysolved.

This procedure can be repeated for thereverse direction to determine the remainingsix error terms. It should be noted that withthis calibration procedure, it is necessary toknow the Z0 termination accurately. It is alsonecessary to have a good short and to knowthe open circuit capacitance effectsaccurately. However, it is a very straightforward procedure when the characteristicsof the standards are known.

The self-calibration technique requires verylittle knowledge about the standards.Assume the leakage error terms can bemeasured by Step (2) of the standardcalibration procedure described earlier. Theerror adapter can then be separated into anX-error adapter and a Y-error adapter, andthere will be no leakage between adapters.The switching errors can be removed by atechnique to be described later. Then themeasurement system can be modified so thatit has an error adapter (X) on the port 1 sideand an error adapter (Y) on the port 2 sideof the device under test. This measurementapproach uses eight error terms.

The self-calibration technique is a three stepprocedure. Step (1), connect the two portstogether (thru connection) and measure thefour S-parameters; Step (2), connect anairline delay of unknown length and unknownpropagation constant (but known Z0) andmake four more S-parameter measurements;Step (3), connect an unknown high reflectiontermination, an open or short for example, toport 1 and port 2, and make the two reflectionmeasurements. A total of ten measurementshave now been made. Only eight unknownerror terms need to be determined. Theredundant information can be used tocalculate the propagation constant of theairline and the reflection coefficient of thehigh reflection termination that was placed onthe system in Step (3).

A flow graph can be drawn for themeasurement system with the erroradapter X, the unknown two-port, andthe error adapter Y. S-parameters canbe used to solve this system. However,it is mathematically more convenientto use cascade parameters orT-parameters because the overallmeasured T-parameter matrix is simplya multiplication of the individualT-parameter matrices.

Step (1) is the thru connection. Noticethat the T-parameters of the thruconnection [TAT] is the identity matrix.Step (2) is the delay connection. Forthe delay connection, the T-parameters[TAD] also form a diagonal matrixsince the airline is assumednon-reflective. The length and loss ofthe airline are defined by γl. The fourS-parameters are measured for the thruand delay connection, thentransformed to T-parameters. The tworesultant matrix equations [TMT] and[TMD] can then be solved for thedirectivity terms (e00 and e33), thepropagation constant (γl) of the airline,and two terms that are the ratio of thefrequency responses (e10e01) and(e23e32) to port match e11 and e22. Theport match terms e11 and e22 will bedetermined next.

Step (3) is to solve for the two port matchterms (e11 and e22) by using an unknowntermination (ΓA). First, place thetermination on port 1 and measure thereflection coefficient (ΓMX); then place ΓA

on port 2 and measure ΓMY. During thethru connection, use the port 1reflectometer to measure the port matche22 yielding ΓM1. These three equationscan be solved for the two port matches(e11 and e22) and the reflection coefficientof the unknown termination (ΓA). Thedirectivity, the frequency response, andthe port match have now been determinedfor both reflectometers. The propagationconstant (γl) of the airline and theunknown termination (ΓA) have also beendetermined. It is still necessary todetermine the frequency response terms(e10e32 and e23e01) in transmission. Thiscan be done using Step (3) of the standardcalibration procedure described earlier.

An airline does not have shunt losses.Therefore, the Z0 is defined as γl/jωCl. Thepropagation constant (γl) was calculated aspart of the calibration procedure. So, theonly unknown remaining is the totalcapacitance of the airline (Cl). Thiscapacitance is determined by actual physicaldimensions of the airline. These physicaldimensions become the standard.

It has been mentioned many times that it isnecessary to remove the switching errors.Typically, when S-parameters are measured,the ports are terminated with a Z0

termination, and the S-parameters aremeasured as a simple ratio of the reflectedto incident signal, etc. However, it is just asvalid to measure the S-parameter when anetwork is not terminated in its Z0

impedance. It simply takes moremeasurements. A set of S-parameterequations can be written for the forwarddirection and a set of S-parameter equationsfor the reverse direction. This proceduregenerates four equations that can be solvedfor the measured S-parameters. This yieldsthe general equations for the S-parameters ifthe system has not been terminated with Z0

impedances. The above technique eliminatesthe effect of the switch. It does this in apractical sense by "ratioing out" the switcherrors. For example, the source errors are"ratioed out" with the two reflectometers onthe source side, and the load errors are"ratioed out" with the two reflectometers onthe load side. Notice that the equationssimplify back to the standard form if thesystem is terminated by Z0 (a3=a’0=0).

There are many advantages to the selfcalibration technique. It is not necessary toknow all the characteristics of thestandards. The Z0 impedance of the airlineis the only critical parameter. Thefrequency does not need to be extremelyaccurate, only repeatable. This technique isalso applicable to unusual transmissionmedia. For example, this would be anexcellent calibration technique formicrostrip or stripline where the section ofline of an unknown length becomes thestandard. Good open and short circuits arealso hard to achieve in microstrip, but theself calibration technique does not requireprecise terminations. Switch repeatabilityerrors can also be removed, yieldingextremely accurate and repeatablemeasurements.

This seminar has not discussed all of theavailable techniques. It has discussed onlythose that currently have the most impactand are most widely used.