,,

AN ANALYSISOF

VECTOR MEASUREMENTACCURACY ENHANCEMENT

TECHNIQUES

DOUG RYTTINGNETWORK MEASUREMENTS DIVISION

1400 FOUNTAIN GROVE PARKWAYSANTA ROSA, CALIFORNIA 95401

RF~ MicrowaveMeasurementSymposiumandExhibition

Flin- HEWLETTa:~ PACKARD

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AN ANALYSIS OFVECTOR MEASUREMENT

ACCURACY ENHANCEMENT TECHNIQUES

2

AGENDA• ACCURACY ENl-lANCEMENT f:OR ONE -PORT NETWORKS

• ONE-PORT CALIBRATION TECHNIQUES

• ACCURACY ENHANCEMENT FOR TWO- PORi NETWORKS

• TWO- PORT CALIBRATION TECHNIQUES

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Vector ~asurements require botn magnitudeand phase data. Some typical examplesare the complex reflection coefficient,the magnitude and phase of the transferfuqction, and the group delay.

The seminar will cover four basic topics:(1) A review of accuracy enhancement forone-port networks. (2) One-port calibrationtechniques. (3) Accuracy enhancement fortwo-port networks, and (4) two-portcalibration techniques.

This section reviews the basic theoryof accuracy enhancement for one-port networks.

ACCURACY ENHANCEMENTFOR

ONE-PORT NETWORKS

3

The block diagram of the measurement sys­tem consists of a sweep os~illator. a re­flectometer consisting of two cou~lers

conne~ted ba~k-to-back, and the unknownone-port (fA)' The direction of power flowthrough the system i~ indicated by the ar­row~. The measured refle~tion coefficientis defined as f H. The measurement systemis not perfe~t and contributes errors tothe measurement of ~A' For example. aone-port network (an ideal resistive ter­mination which has a 40 dB return loss).has a well behaved frequency response.But the measured f H still typically has10 dB peak-to-peak variations with re­spect to frequency. The variations 3re notin the resistive device. An ideal resi­stor 1s incapable of causing this type ofchanging reflection coefficient. The er­rors in this case are in the measurementsystem.

4

THE PROBLEMONE-PORTME~SORED DATA (rM)

INCIDENT REFLECTED

~ ~ b l

E9 X X - IrA- ~I, II I

REFLfCTOMETER -SWEEP al UNKNOWNOSGILLATOR REFLECTa> bo ONE-PORT

r... = NCIDENT ;!O

IrMId8

0 ------------40 -vv-vv

~

2

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5

ONE-PORT ERROR MODELING

The fcllowin~ is a dis,u~$i~n 0f th~ variouserrors that ar~ inherent in a refl ...-tometer.

An imp~rf ..ct reflenOI:lt'ler can b,' lIloJdeled·by·taking all th~ linear err0rs of the svstemand combining thee int0 a fictitious ~wo­port error adapter betw~en the reflectometerand the unknown one-port. This results inan absolutely perfect reflectometer with noloss, no mismatch. and no frequency responseerrors. The fictitious two-port error adap­ter has four error t~rms. When making ratiomeasurements. only three terms are needed.

ONE-PORT FLOW GRAPHAND 50LUTION

(

PERFECTSWEEP REFLECTONETER.

OSCillATOR

ERROR Al».PTERr------------.,I ao b l I

I elO :I II eoo ell II I

: eO! I

bO al IL .J

MEASURED

!!Qli: see~DI~ I FOR D£TAILS

UNKNOWNONE-PORT

eoo. OlRECTNIT'l'

~IOeOI).~ RfSf'ONSC

ell • PORT MAT~

ACTUAL

3

6

The flow graph of the fictitious two-porterror adapter is connected to the unknownreflection coefficient (fA)' A flow graphis a good way to give a practical feel forhow signals are flowing in a eicrowave net­work. For example, suppose, fA is a perfectZ termination. When f M is measured, therea9~ some residual signals caused by the sys­tem directivity eOO ' This signal does notflow to the device under test but flowsthrough the branch of the flow graph label­led e

OO' The frequency response terms elO

and eOI can be best understood by measuringa short. A short has a reflection coeffi­cient of -I with no frequency response va­riations. But, due to the couplers. front­end mixers. cables. and other hardware thereare definite frequency response variationsaodeled by the branches el0 and eOl- Look­ing back into the test port of the ceasure­ment system. it does not present a perfectZo source impedance. There is some reflec­t10n when a signal is incident on the testport. This is primarily due to the match ofthe couplers and/or adapters. The equivalentport match is 80deled by the branch ell'

fM is mathematically related to fA by threeerror terms: directivity. frequency responae,and port match. If these three terms areknown from a calibration procedure, (discussedlater), it is possible to solve for the ac­tual reflection coefficient fA'

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This slide shows the return loss of aresistive termination before and aftererror correction. The directivity ofthe measurement system is about the sameas the return loss of the resistive ter­mination. 40 dB is a respectable equi­valent directivity for a measurement system.But, at aome frequencies, the directivityterm adds in phase with the reflection co­efficient of the actual device under testand yields a 6 dB higher measurement. Atother frequencies, the directivity and re­flection coefficient terms are out of phaseand cancel. causing deep dips in the measure­ment data. Error correction does not re­move all errors. Residual errors remainingin the measurement system may be due toimperfect standards. repeatability, noise.etc.. But the measurement accuracy is great­ly improved. In fact, greater than 20 dBimprovement is typical.

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

ONE-PORTBEfORE' ~D ~ ~ON

0,......--------------,~l)

140'"""'=--------------::/1lXEl~--1-

ONE-PORTCALIBRATION TECHNlqUE5

4

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8

..

.' .9

ONE-PORT ERROR MODELCALlBR~TION

n(_e-,I:.O_e.:::OI:,-)_f'A<:.'"". e -+ -00 I-e\lr... -

a.. e1oeo l - eoo ell

b- eooc.. -ell

• WITH 3 DIFFERENT KNOWN l"A I MEASURE THE RESULTANT' l"M

• THIS 'fIELDS 3 E-QUATIONS TO SOLlIE FOR a,b ( c

l"AI a + b - rAI l"'MI C" l"'MI

l"A2 a + b - l"'A2 l"M2 C" l"M2

l"A3 a + b - l"A3 l"M3 C" l"M3

10

BILINEAR TRANSFORMATIONGENERAL CASE

A process of calibration will now be dis­cussed. By a change of variables, theequation for the measured reflection co­efficient can be converted to a bilinearequation with the error terms a. band cas variables. If there is a way to gene­rate two more of these equations, givingthree equations in the three unknowps,(a, b .and c), then a system of equationsresults that can be solved for the errorterms. The approach used in error correc­tion for one-port networks is to aeasurethree different kn~~ l~ads crA)' Thisgenerates three different .easured reflec­tion coefficients (rM). This procedureyields the necessary three equations tosolve for the error terms a, b .and c.

This is a graphic approach to the calibrationtechnique. From complex algebra, if threepoints on the rA plane and three points on therM plane are known, the terms a, band careuniquely determined for a bilinear trans­formation. This graphic approach will behelpful to visualize different calibrationtechniques.

5

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These are four of the more popular calibrationtechniques. The first one basically goes back~o the ..thematical treatment of three diffe­rent, known offset shorts for fA' An offsetshort is a short circuit of precisely knownlength down an air line, The frequency mustbe known accurately to determine the phaseshift at the offset short. The three offsetshorts generate three equations to solve forthe three unknown error terms. The phase ofoffset shorts is frequency senSitive. There­fore, care must be taken to avoid lengthsthat are multiples of a half-wavelength, orthe offset shorts will not be unique. Thistechnique is useful when dealing with micro­strip and other transmission media where itis very difficult to realize Zo loads oropens.

The second technique has been used for manyyears by Hewlett Packard on the large, com­puter-based, automatic network analyzers. Thetechnique consists of using a Zo termination,a good short circuit, and a series of offsetshorts which are a quarter wavelength long inthe center of the band to approximate an opencircuit. It requires a series of offset shortsover the frequency range in order to approxi­mate open circuits. The offset shorts are notbroadband, but provide good accuracy.

The third technique is very similar to thesecond, but re~laces the offset short withan open circuit. The open circuit has afringing capacitance that needs to be modeledvery precisely in order to achiev~ accuracy.

The last technique uses two differentsliding ter~inations and a short circuit.This approach is the fussiest and the mostmathematically involved, but has the po­tential of achieving the best performance.

The last two methods will now be discussedin detail.

11

ONE-PORT CALIBRATION METHODS

• 3 OR MORE DIFFERENT KNOWN OFFSET SHORTS

• lo TERMINATION (FIX.ED OR &L1DING-). SHORT. (OFFSET SHORT

• 10 TERMINATION (FIXED OR SLIDING-). SHORT, (OPEN (CAPACITANCE MODELED)

• 2 DIFFERENT SLIDING- TERMINATIONS ~ A SHORT

6

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The short circuit is at -Ion the fA plane,and the sliding load generates a circle, cen­tered at the origin. The center of the circleis determined by the mechanical dimensions ofthe air line and yields an effective direc­tivity of approximately 60 dB. The radius ofthe circle is determined by the residual re­flection coefficient of the aliding eleoent.As the sliding element is ftOved physically,it traces out the locus of a circle on thefA plane. The open circuit at low frequenciesis at +1 on the fA plane, but as the fre­quency increases, the capacitive reactancestarts to rotate the position of the shieldedopen clockwise. On the f H plane the short'sposition is modified and the sliding load isno longer centered at the origin. The positionof the shielded open is also modified.

The technique has three steps. First,attach a Zo termination (notice, r -D)and measur~ r HI' This yields the Ai­rectivity term eOO directly. Next,connect a short and measure r . Thefinal step is to connect a sh~~lded openand measure r H3' AssUllle that the cap­itance of the Ihielded open, whichdetermines the phase term, is known.There are now two equations with twounknowns (the frequency responseelO eOl ' and the port aatch e ) forwfiich we can solve. 11

This is a graphic presentation of the pre­ceding technique.

xSHIELDED

OPEN

SLIDING LOAD/(CENTER:eoo)SHORT

)(

SOl..VE 2 EQUATIONS IN 2. UNKNOWNS

FOR ell < (etO eOI)

13

STEP 2SHORT

rMtr~r trAs-1I IL J

.. (tIC eOI)(-1)'·MZ·eOO+

I-ell (-I)

STEP 3Sl4I!UlEO 0f'eNr---,

illJISl IrM! I I :::=I I '1'1.:L J le-j~

I' (e!Oeol)(le-j~)M!aeOO'" I-ell(le-jl!l)

\..-------.,------_....)

,,, SHIELDED

OPEN

/~IDtNG/ LOAD

c:)

r'A PLANE

,.., _ r"a+b'M-

rac+1!::!Q!§.: SEE APPENDIX l[ FOR CIRCLE ~ITTIN& PROCEDURE

12

7

SHORT

BILINEAR TRANSFORMATION~o TERMINATION (SLIDING), SHORT<OPEN (CAPACITANCE MODELED)

ONE-PORT CAUSRATION USIN~ lTERMINATION (FIXED OR SLIDIN'l;.)~

SHORT ~ OPEN (CAPACITANCE MODELED)

r (elOeOI) (0)MI·eOO+.:....:::<--:..::...:...­

l-ell(O)

rMI·eOO

OPEN CAPACITANCE MODEL

14

()G-----+:r .~

r..- le-JAPC·'

CAPACITANCE OF 6HIELOEO OPEN

SHIELDED OPEN

FOR APC-':

Co =.O'9pf

Ct w Opf/Hz

Cp <4.0Xll;U pf/Hz2

-If3~ 2TAN 21TfClo

The shielded open consists of an extensionof the outer conductor of the coax beyondthe termination of the inner conductor.This extension is a circular waveguideoperating below its cut-off frequency. Theimpedance of such a waveguide section is re­active (a shunt capacitance) and cannot pro­pagate power. The reflection coefficient ofa shunt capacitor is e-jB. Notice that j3is a function of frequency, so the frequencyhas to be known accurately as well as thevalue of the capacitance. The capacitanceof a waveguide increases the closer you getto the cut-off frequency. We can model thiscapacitance as a power series and deteroineempirically the power series co~fficients.

Notice that the capacitance rises approximate­ly 15X at 18 GHz.

NOTE: SEe APPENDIX m FOR DETAILS

This slide shows the results of measuringthe reflection coefficient of an APC-7 off­set short using the open capacitance model.With the capacitance 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.If the dc capacitance term is modeled, theerror can be reduced to approximately ) dBpeak-to-peak error. By modeling the capa­citance as a functi~n of frequency, theerror is reduced to less than .1 dB peak-to­peak.

RESULTS USING'OPEN CAPACITANCE MODEL

APC-7 OFFSET SHORT

15

f

432I

IrAI dB 0 ~--.:IN.a"""'~-I-2-3-4l..----------'2 GHz

c-O

C-.079pf

8

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16

ONE-PORT CALIBRATION USING' Z DIFFERENTSLIDING TERMINATIONS ~ A SHORT

~LVE 3 EQUATIONS IN 3 UNKNOWNS

~: seE APPENDI)( :ISr FOR DETAILS

17

BILINEAR TRAN5FORMATIONZDIFFERENT SLIDING' TERMINATIONS

~ A SHORT

This second procedure shows how to use twodifferent sliding terminations and a shortcircuit for calibration. Step I connectsa short circuit on the test port and .easuresf MI • Step 2 uses a sliding, low reflectiontermination to define a circle on the f

Mplane

with the center at f O and the radius R2

•Step 3 connects a higt reflection slidingtermination to define a second circle withthe center f 03 and radius R]. Steps 1 through3 yield three equations whi~h·can be solvedfor the three unknowns: a, b, and c. Thecenter and radius of the two circles can bedetermined by a circle fitting technique.

It should be noted that certain assumptionshave been made about the sliding terminations:(1) The magnitude of the reflection coeffi­cient cannot change as it rotates in phase.(2) Air line loss is negligeable.

This technique does not require an accurateknowledge of the frequency. The magnitudeand phase of the sliding terminations arealso not needed. This technique can be veryaccurate, but it is more time consuming andnot nearly as easy to apply as the previousapproach.

This graphically shoJS how the cir­cles on the rA plane are modified whenthey are transformed to the rM plane.

""SLIDING­LOWREFLECTION

SHORT

\

rA PLANE

SLIDING­HI6'HREFLECTION

q'" SLIDING­'LOW

REFLECTION

SHORT\x

SLIDING­HI6'HREFLEClIOtJ

r - rA a + bM - rA C + I

9

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In conclusion. this section demonstratesthat with one-port error correction, thesystem directivity is extended. responseerrors are reduced significantly. andport match uncertainties are lowered.Calibration makes a marked improvementwhen measuring with adapters and cablessince their effects are removed. The useof cables requires care because theircharacteristics change as they are flexed.If .emi-ri~id coax is used, is carefullypositioned. and remains stationary. thenthe errors due to these lines can begreatly reduced. offering alternativeways to connect to the device under test.

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

18

CONTRIBUTIONSOF ONE-PORT ERROR CORRECTION

• EXTENDS SYSTEM DIRECTIVITY

• REDUCES FREQUENCY RESPONSE ERRORS

• LOWERS PORT MATCH UNCERTAINTIES

• LESSENS EFFECT OF ADAPTERS ( CABLES

19

ACCURACY ENHANCEMENTFOR

TWO-PORT NETWORKS

10

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!

P3ralleling the previous on~-port analysis.all th~ lin~ar errors of th~ imperfect re­flector.l~ter can be cOr.lbined into an errorad3ptc-r. In this case, the fictitious erroradapter 1"'lU~t be II 1·~lI.t·-p\·rt. Tid:-; ,-rr,'r 11!.11'­ter h~s 16 error terms. The number of errorterms is the square of the number of port~.

The two-p0rt measurement sy~tem dupli-c3tes p3rt of the one-r~rt measure~nt systemdis,ussed earlier. A dU31 directional coup­ler is att~ched to the input of the un-kno~~ two-port, and 3nother dual directionalcoupler is ~ttached to the output. A switchchanges the direction of the incident powerto the unknown two-port for f0rward. and re­verse_measurements and terminat~s th~ unkno_~

n"'-port in an ir.lpcd3nce 20' The pr0bl emsare very simi1~r to those of the one-puTtcase. There are directivity errors, fre­quency response errors. and port 1'I.1t,·h errors.for b0th ports. For example, th~ dotted linein the graph represents the actual frequencyresponse of J device; The solid line repre­sents the measured result. Again, it is notthe device but the measurement system causingthe error.

Th~re are additional problems becau~e theerror adapter does not include switch repeat­abilitv errors, or chan~ing pnrt m3tch duetn different switch pnsitions. Errors causedby the switch can be remnved by a procedureth~t will be discussed later. To resolve fur­ther difficulties. one of the following as­~umpt ions must be mad". (l) There are foursampl~rs nr mixers which ,an be connectedat all times to the four meA~urement pnrtsnf the couplers. Otherwise. addition~l

switches would need tn be included in themodel. (2) II additional switches are in­cluded. an assur.lption could be made that theisn13tion of the couplers is high enough tDreduce the errDrs caused by this additionals~·itching.

21

UHKNOWNTWO-PORT

UNKNOWNlWO·PORT

UNKNOWNTWO· PORT

L....-"""T""~

PORT 2

DUALREFlECTOMETER

x X?~

16 ERROR TERMS

NOTE: SEE APPENDIX 'Y FOR DETAILS

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11

THE PROBLEMTWO-PORT~~x x PORT 1

20

TWO·PORT ERROR MODEL(WITH 4 MEASUREMENT PORTS)

, ao T Tbo ,r-v- ~ PORT1.!!.

boSliM = ao ,OSC SWITQoIED~

b3SliM: ao ,OSC SWITCHED FORWARD

b3SZ21o\ :"""7 , OSC 5w ITCHED REvERSe,

33b'

SIZM =~ , OSC SWITCHED REVERSE

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. Thereare only three measurement ports. Thereis another separate error adapter for thereverse direction. By using two separateerror adapters, 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 12error terms. So, there is a total of 24.Two of the questions that arise are: Areall of the error terms meaningful? And,what is the physical description of eachof the error terms' These will be discussed.next.

This is a pictorial flow graph indicatingthe error terms for the forward configura­tion. As mentioned earlier, there ar~ 12error terms. There are two port matchterms (ell and c22) and n directivity term(cOO), For the frequency response, thereare: elO and eOl for making reflectionmeasurements and elO and e32 for trans­mission measurements. The re~aining sixterms are leakage terms between the variousports.

For a typical S-parameter test set usinghigh directivity couplers with good iso­lation, the primary terms are indicated bythe solid lines and the less important termsby the dotted lines. The primary errors are:The two-port match errors (ell and e22); Thedirectivity error (eOO); The frequency re­sponse errors (the reflection frequency re­sponse term elO eOI and the transmission fre­quency response term elO e32); and the lea­kage term (e30)' These are the six complexerror terms for the forward direction. Theprocedure can be repeated for the reverseconfiguration to obtain six additional errorterms. This leads to the "twelve term errormodel".

22·

TWO-PORT ERROR MODEL(WITH 3 MEASUREMENT PORTS)

, 40 T Tbo. FORWARO~ ~~ ERROR ~

)+....L~_...L.~'"=~ ==-='--PERFECT b; APAPTER :-Ii

COUPLERS FOR ~- SIl~Stt -

~b! a~

! b3 '

~: SEe APPENDI" 1Zr FOR DETAIl.S

23

ALL ERROR TERMS PRIMARY EAAOR TERMS (SOLlO)PORT "'''Tal: 2 ell , eu 2 ell, ~uOlRECTNITV: I eoo I eooFREQ RESPONSE: 3 e'O,eOI,en 2 (elOeo.),~lOen)

LfAKAGr: __6_~ezo,eoz,e,..e2l,et2 I e~

ItTElUoIS imMsREPEAT FOR REVERSE CONFIGURATION

12

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. .

24

This is a flow graph for the forward con­figuration with the assumptions that weremade on the previous slide. Note thatSllM and S2lM are functions of all fourS-para~ters of the device under test[SA)'

TWO·PORT FORWARD FLOW GRAPH(WITH 3 MEASUREMENT PORTS)

a,bo

e30

1r------- -....·0 elO bl I 5ZIA ,a2 e~z

I I b3ISIIA [SA] SUA:too 11/ : e22

I

I S.ZA I b,L J

f'ORT 1 POItT 2ERROR TERMS

PORT 1 MATCH: III TRANS~ RESPONSE: (IIOI~Z)PORT 2 MATCH: 122 DIRECTIVITY: eooRER AAEO. RESPONSE: (tIOIol) LEAKA<7E : I~

• SlIM: bo • too+ (1IOeOI) 511A - 122 DET [SA)

80 1- III SIIA- 1'2 522A +e ll eu DET (5AJ

• S21M:~ • eJO+(elOe~2) S21Aao 1- ell 5 IIA - eZ2 522A+ell e 22 DeT [SA]

NOTE: SEE APPENDIX iIr FOR DETAILS DET[S.J: SIlA~22A-S2IA51V.

ltffLEtTED

25

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

b', I 'A I ~ II t

b~ It" i}SIIA [SA] S,2A I ti2 t;:

1to. " L__~___ Jl>, &h ,',I

f'ORT 1e~

PORT 2

TWO·PORT REVERSE FLOW GRAPH(WITH 3 MEASUREMENT PORTS)

r--s;----'" .'

13

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Four Deasurements have been made: The S­parameters SliM. S21M' S12:1 and S2:'ll' Eachof these M~asured S-parameters is a functionof the four S-parameters of the device undertest plus 12 error terms. Now. if these errorterms can be determined by a calibration pro­cedure. (described later). four equations inthe four unknowns SlIA' S21A' S22A and S12Aresult. This system of equations L?n hesolved for the actual S-parameters. Noticethat the S21A transmission coefficient is afunction not only of the measured S21M but ofthe other three S-parameters as well. Ittakes four measurements to calculate oneS-parameter.

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

SOLVING FOR [SA](WITH 3 MEASUREMENT PORTS)

~ARD St,... • FUNCTION OF [SUA, 5tJA,St2A,5uA,.II,.Z1,(elOeo.) ~ eoo]52\N& RJNCTION OF [SllA,StlA,5aA,Sz2A'.II,.Z2,(elOe~H ~J

REVERSE SztM& FUNCTION OF [$IIA,S2IA. 5aA,St2A,elh.it.~neiV~e,,]SttM· FUNCTION OF [SllA.StIA,S.U,5tu,e" .eU,Ceneo.H to\)

• IF we KNOW THE EAAOll TERMS BY CALIBRATION• THEN we MAVE .. EQUATIONS IN .. UNllHOWNS• SOLVE SIM~T~OUSLY TO YIELD SIlA,SZIA, SrzA <S22A

NOTE: SEE APPENDIX 1ZIr FOR DET.tJLS

SOLVING FOR [SAl CONT'D(WITH 3 MEASUREMENT PORTS)(SwIM-eoo\~+ ~.' :\-e (~\(SrzM-.Os)

.5111.= .lOeOI i\' ene1Z 'l2J Z1 !joen1 -Oep,

D

(S'il"-~O!)~+(SIl",-eoo\/c _ e ):1.ncO! ~ .lOeOI}1.: II II 'J

• SQ.= D

26

27

!!Q!£: SEE APPENDIX m FOR DETAILS

14

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28

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This slide shows the magnitude S21A fora "mismatched pad." The dot ted Ilne isthe corrected data with only the frequencyresponse errors removed. The solid lineshows the corrected data when using the12 term error model. By including theerror terms e

1and e

22, the .5dB peak­

to-peak error Is reduced significantly.

The theory of error correction for two­port networks has been developed. Itnow remains to develop the solutions forthe various error terms.

29

17

•••••••••••• FREQUENCVRESPONSE CORRECTION

TOTj.L.EAAOR COAAECTION

13 14 " 16

GHz!DIV

15

1&00 +--+--+--+--+--..,12

11.00

I~.OO

....00

II)II) 15.00o..J

CD"'t)

I 16.00

~

TWO-PORTBEFORE AND AFTER CORRECTION

FOR AMISMATCHED PAD

TWO-PORTCALIBRATION TECHNIQUES

J ,

(

Two calibration techniques will be dis­cussed in detail. One is a standardcalibration technique that has been usedfor many years at Hewlett Packard. Theother is a self-calibration techniquewhere precise knowledge of the standardsis not required.

Step (1) of the standard calibrationprocedure is to calibrate port 1 usingthe one-port procedure that was developedearlier. Place a short, an open, and aload on port 1 and solve for the threeerror terms: thE equivalent directivity(e ), the port match (ell)' and there~2ection frequency response (e

lOe

OI)'

When one-port devices are used on thetwo-port measure~ent system, the equationfor rhe reflection coefficient of thetwo-port case reduces to that of thE one­port case.

30

TWO-PORTCALIBRATION TECHNIQUES

1. STANDARD CALIBRATION PROCEDURE

1[, 5ELF- CALIBRATION PROCEDURE

31

STANDARD CALIBRATION PROCEOURE(FORWARO CONFI&URATION WITH 3 MEASUREMENT PORTS)STEP 1: CALIBRATE PORT 1 USING ONE-PORT PROCEDURE

e'ellSMORT, OPEN,L.Oo1U)

512"- 521,,-0

• USE /IN'( ~ THE FOUR APPROJ.CHES DESCRIBED IN THE ONE-PORT CALI­ElRJ.TION SECTION

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, ,32

5TANO~RD CALIBRATION PROCEDURE(FORWARD CONFIGURATION WITH 3 UEA5UREMENT PORTS)STEP z.: CONNECT Zo TERMINATIONS TO PORTS 1 ~ 2

• S & b5 s. -t(e e ) 521A •2"" a; ~ 1O!l2 1- ell SIIA- 822 SZ2A" ellen OEr [s~.] - 30

• I 8toS Szu.\ I

33

STANDARD CALIBRATION PROCEDURE(FORWARD CONFIWRATION WITH 3 MEASUREMENT PORTS)STEP 3: CONNECT PORTS 1 ~ 2 TOG€THER

c;~~~~ (.eZ2

bJ

8 S.IoII s :: - 800+ (elCeot) I- :~eZ2• WE IU\IOW eoo. e II ~ (elO eO!) :. CSOl.~v:=eC:FOR=~e=-Z-2""1

• S2lfo4sl:>3 - e!O+(elO e5Z) IClo I- ell e%2

• WE kNOW 830,e. ~ eZ2 :.1 SO\.VE FOR (.10 en) IftEPEAT FOR ReveRSE CONF-l6URATION

17

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Step (Z) is to determine the leakage.This can be accomplished by placing20 terminations on port I and port Z.In this case. the equation for SZIMis equal to the leakage term e

30.

Step (3) is to connect port 1 and port Zof the measurement system together. Thisr~duces the equation for SliM to an equa­tlon for a one-port measur!ment system.Since the three error terms for port Iwere previously determined. the one-portmeasurement system can now be used tomeasure the port match of port Z (e

2Z)'

S21M reduces to a simple expression fortfie thru connection. The leakage term(e30 ) has been determined and the twoport matches (ell and e2Z ) are known. sothe remaining transmissIon frequencyresponse term (e 10e3Z ) can be easilysolved.

This procedure can be repeated for thereverse direction to determine theremaining six error terms. It ahould benoted that with this calibration procedure.it is necessary to know the 20 terminationaccurately. It is also necesSary to havea good short and to know the open circuitcapacitance effects accurately. However,it is a very straight-forward procedurewhen characteristics of the atandardsare known.

The self-calibration t~cnniqu~ requiresvery little kno~ledge ab0ut the standards.Assume the leakage error tcrres can bemeasured by St~p (2) 0f th~ standardcalibration procedure ccscribed earlier.The error adapter can then b~ separatedinto an X-error adapter and a )-~rror

adapter, and there ~ill be nu leakagebet ...""n adapt"rs. The s~·itching errorscan be removed bv a techni~ue to bedescribed later. Til~n th~ rn~Jsurement

syster,; loan bt' modif icO so chat it hasan err"r adapt"r (X) c'n tll" 1'< rt 1 sideanc an error adap:er (Y) on th. ~ort 2siel ~lf till- d'='\'il"t' unG",-,r If-st. :hism{'asurt'm~nt appr"a,,:, us ... s \:'i~hl 'frort ..·rms.

34

SELF·CALlBRlTION MEASUREMENT 5Y5TEM'~ xAo

J t boX f ~...-----._X t-:_~-=-_

tf'""-~~ -" 0] U*NOWN!.. AlW'TER _bt SA TY«l·PORT

'---+-~""X""'a:-, -1-1"""b~-=-j--"';;b3:_Y---' azASSUME: LEAKA&E ERROR TeRMS CAN BE MEASUREO BY STEP 2 OF THESTJ.NOARO PROCEDURE, SWIT~ eRROftS CAN BE ·RATIOEO· our "., USlN&FOUR ME~eMENT PORTS.

b3 013

r-=~~ h;.?y......-..;._.-.i3

35

SELF-CALIBRATION PROCEOURE

The self-calibration technlque is athre" step procedure. Step (J), connectthe two ports to,:etller (tllru connection)anc reeasure th~ four S-param"ters; Stoep(2), connect an air lin~ d~lay 0f unkno~n

length and unknown propa~ati"n "onstant(but kno~'n Zo) and mai<e f"ur mor" S­parameter measureMents; Step (3),connt<ct an unknu....m jd~h rt.>flectiont~rnination, an op~n ~'r short for exampl~.

to port J and port 2, and mak" th,· t~','

reflection measurements. A total of tenrn~asurements has n"I\"," been macl,.. Onlyeight unkno~'l1 ~rror terms n"ed to b"d~l~r~in~~. Ttle redundant informationcan be used to calculate the propa~ati"n

constant of the air line and the reflectioncoefficient of the high reflectiontermination that ...as plac~d on the systemin Step ;3). STEP 2

$WITCH

POlU1 PORTZ

AIRLINE DELAYUNl(NOWN LENGTH,lJNlQlOWN U,-KNOWN Zo·

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STEP 3

SWIT~

PORT 1 PORT Z

36

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

Step (1) is the thru connection.Notice that the T-parameters of thethru connection [T T] is the identitymatrix. Step (2) ts the delay con­nection. For the delay connection,the T-parameters [TAD] also form adiagonal matrix since the air line isassumed non-reflective. The lengthand loss of the air line are definedby yt. The four S-parameters aremeasured for the thru and delayconnection, then transformed to T­parameters. The two resultant matrixequations [TMT.J and [TMP~ can thenbe solved for the direcllvity terms(enD and e ,3 ), the propagation constant(ytJ of thl air line, and two termsthat are the ratio of the frequenc)responses (e iO eOl ) and (e2l e ,2 ) toport match ell and e22 . Tfil pOrtmatch terms ell and e22 will bedetermined nex!.

10z .. IL !.JEtlROfl ADlt.P~R

Y

I IL j

UNKNOWNTWO-POF\.r

THE ABOVE TWO MATRIX EQUo\TIONSCAN BE SOl.veO~ :

~: SEe APPENDI)( 'mit !=OR oeTAILS

19

37

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CASCADE PARAMETERS[~]= [T~ [::J ' [=Ua rAJ [::J ~~} ~yJ [::]

I[TMl= [Tx] [rAJ [Ty] I

5TEP 1THRU CONNECTION

[TAT]- [~ ~I!MT] -I!X] [Ty]

~: SEE APPENOI){ 1lIIr ~R DETAILS

PORT 1 PORT 2r:------..., r------, r------..,1aO bl I I I I a2 b3 I

I too a IL .!..JERROfl ADAPTeR

X

THRU AND DElA'( CONNECTIONFOR SELF-CALIBRATION

1 elO I I5 21....

I I e32 II I I I I I1 eOO ell I I 511.... (S~Sm I 1 eu en I

II I

S12" I I I, eo, I I I I en I

ERROR MODEL FOR SELF-CALIBRATION

·.\

Step (3) is to solve for the two portmatch terms (e

1and e,,) by using an

unknown termination (f;r. First, placethe termination on port 1 dnd measurethe reflection coefficient (r~~); thenplace f on port 2 and measure r "During the thru connection, use rheport 1 reflectometer to measure the portmatch e

22yielding r~l' These three

equations can be solved for the twoport matches (e 1 and e

2) and the

reflection coefflcient ot the unknowntermination (fA)' The directivity,the frequency response, and the portmatch have now been determined forboth reflectometers. The propagationconstant (,i) of the air line and theunknown termination (~ ) have also beendetermined. It is stif) necessary todetermine the frequency response terms(e\O e 12 and e 21 en ) in transmission.This c~n be don! u~fng Step (3) of thestandard calibration pro,eduredescribed earlier.

An air line does not have shunt losses.Therefore, the 2

0is defined as ,1/jwC£.

The propagation constant (yi) was cal­culated as part of the calibrationprocedure. So, the only unknown remainingis the total capacitance of the air line(Cl). This capacitance is determined byactual physical dimensions of the airline. These physical dimensions becomethe standard.

38

TERMINATION CONNECTION FOR SELF-CAliBRATIONSTEP 3: SOLVE FOR ell AND eZ2 USING- TERMINATION

~KNOWN REFLECTION THRU LNKNOWN REFLECTION(rA) ON PORT 1 (rA) ON PORT Z

I' .. +(elOeOI)1'" n _ /~eOl)ep "MV.fJ3+{en~)I"'AIoIt too 1_ ell"" MI-eoo l-elieU l-eZZI'A

SO!.VE FOR "... SOLlIE FOR ell SOLVE FOR "...'" TERMS OF ell'" TERMS OF eZ2 IN TERMS OF eu

~E ABOIIE TMREE EQUATIONS SOl.IIEO FOR :

II'A,ell ,etz IUSE STANDARD CALIBRATION PROCEDURE STEP 3 TO SOLlIE:

I(eloe~) «eneol) I

~: SEE APPENDIX 1lm FOR DetAILS

39

=0 OF UNKNOWN AIR LINEFOR SELF-CAlIBR~TION

IF NO SHUNT LOSSES:

--IL~o - jwCi

WE CALCULATED 2S.R.

NEED ONLY KN()IN Ci!!

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• 4

40

FORWARD REVERSEtoo ~ SIIMaO+ SIZM1l3 bOo SliM 110 +SIZM a3b3 ~S2IMaO+ SZ?Ma, b30SZ,,,,ao + SZZM a'3

SOL.IIIN~ T~E ABOVE SET OF EQUATIONS YIEL.DS :

It has been mentioned many times thatit is necessary to remove the switchingerrors. Typically, when S-parametersare measured, the ports are terminatedwith a Z termination, 3nd the S-para­meters a~e measured as a simple ratioof the reflected to incident signal,etc ..However, it is just as valid to measurethe S-parameter when a network is notterminated in its Zo impedance. Itsimply takes more measurements. A setof S-parameter equations can be ·writtenfor the forward direction and a set ofS-paramater equations for the reversedirection. This procedure generatesfour equations that can be solved forthe measured S-parameters. This yieldsthe general equations for the S-parametersif the system has not been terminated~ith L

Oimpedances. The above technique

eliminates the effect of the switch. Itdoes this in a practical sense'by "ratioin&out" the switch errors. For example,the source errors are "ratioed out" withthe two reflectometers on the sourceside, and tne load errors are "ratioedout" with the two reflectometerson the load side, ~otice that theequations simplify back to the standardform if the system is terminated byLO (a 3-aO-O)'

~:SEE

APPENDI)( IXFOR DETAIL.S

boa', -boil, 5 bOao-boil"~= a a' Cl' 12M- ilo il3-a,Cl'oo ,-a~ 0

b,a3- t>3il, b3Clo-b,a~SZI~= , • SzZM= • ,

Cloil,-a,ilo aoil,-a,ilo

REMOVAL OF 5WITCHING ERRORS(WITH 4- MEASUREMENT PORTS)

~ 1~ f~r-----.!!.PER~CT bo ERROR r:-';;::aC:-,---==--

RERECTONETER~ ADAPTER ~

~ .x.s"b;'-----laza, 1 b3 '

IMPERFECTSWITCH IS

"RATlOEO" our

CONTRIBUTIONSOF TWO-PORT ERROR CORRECTION

USING" SELF-CALIBRATION TECHNIQUE

• REDUCES 'THE LINEAR SYSTEM ERRORS

• REMOVES SWITCH REPEATABILITY ERRORS

• CALIBRATION STANDARDS ARE NOT CRITICAL

41

There are many advantaRes to the self­calibration technique. It is not neces­sary to know all the characteristics ofthe standards. The Zo impedance of theair line is the only critical parameter.The frequency does not need to beextremely accurate, only repeatable.This technique is also applicable tounusual transmission media. For example,this would be an excellent calibrationtechnique for microstrip or striplinewhere the section of line of an unknownlength becomes the standard. Good openand short circuits are also hard toachieve in microstrip, but the self­calibration technique does not requireprecise terminations. Switch repeat­ability errors can also be removed,yielding extremely accurate and repeat­able measurements .

This seminar has not discussed all ofthe available techniques. It hasdiscussed only those that currently havethe most impact and are most Widelyused.

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MARCH '982

F'i;' HEWLETT~~ PACKARD

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