524 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-36, NO.2, JUNE 1987

On-Line Accuracy Assessment for the Dual Six-PortANA: Extension to Nonmating Connectors

CLETUS A. HOER, MEMBER, IEEE, AND GLENN F. ENGEN, FELLOW, IEEE

TRL Calibration, Dual 6-port or 4-port

Gives A,. B ,• C ,. A 2 • B 2 • C 2 • and "'11 of line.

(a)

~2 I IrJ~A2':2.C2

LRL Calibration

.£1

~ Line!

----1~ ~r- Reflect

.J..2-1 ~ Line 2

(b)

Fig. 1. (a) TRL technique for calibrating a dual six-port or four-port re­ftectometer. (b) LRL calibration technique where the thru is replacedwith a short length of line.

II. LRL EXAMPLES

A. Two Pairs of Lines

An example of the LRL calibration technique is shownin Fig. 2(a) where both test port connectors are identicalmale connectors. When the calibration using two lines andone or more terminations with female connectors is com­plete, the S-parameters of two-port devices that have fe­male connectors on both ends can be measured.

Fig. 2(b) shows an example where adapters have beenadded to obtain two female test port connectors. The LRLcalibration can be applied to these new test ports usinglines and terminations with male connectors. Then two­port devices that have male connectors on both ends canbe measured.

To measure two-port devices that have one male andone female connector, adapter C (or D) is removed toobtain the test port configuration shown in Fig. 2(c). Theparameters for reflectometer A are known from the cali­bration in Fig. 2(a), and the parameters for reflectometerconfiguration F are known from the calibration in Fig.2(b). The repeatability of the connection between A and

A,r , + B,w, =- - ­

C , r , + 1

----1~ ~r- REFLECT

-1~---4~ LINE

Manuscript received June 23, 1987.The authors are with the National Bureau of Standards, Boulder, CO

80303.IEEE Log Number 8613604.

I. INTRODUCTION

THE "thru-reflect-line" (TRL) technique [I] for cali­brating a dual six-port or four-port reflectometer has

been modified so that the scattering parameters of two­port devices having any combination of connectors can bemeasured.

The thru, reflect, and line measurements made in theTRL calibration are shown in Fig. I(a). The complexsidearm ratios WI and W2 are measured for the three mea­surement conditions shown. The test ports are connectedtogether, then one or more highly reflecting terminationsare connected to one test port and then to the other testport. Finally, a length of precision transmission line isconnected between the two test ports. For a dual six-port,WI and W2 are each obtained from four power measure­ments made on the four sidearms of each six-port [I]. Fora dual four-port, WI and W2 are each obtained from a de­tector which measures the complex sidearm ratio directly.

The TRL calibration yields the parameters A I, B], andCI of four-port A, and A2 , B2 , and C2 of four-port B whichrelate W] and W2 to I', and r 2, respectively, as shown inFig. l(a). Also obtained from the TRL solution are thereflection coefficients of all the terminations used in thereflect measurements, and 'Ylof the precision transmissionline used in the line measurement, where 'Y is the propa­gation constant of the line, and l is its physical length. Asshown in Fig. l(a), the TRL technique can be appliedonly to reflectometers having identical sexless connectorsat the test ports. If the requirement for making a thru con­nection is replaced by a measurement with a short lengthof line as shown in Fig. 1(b), then the calibration tech­nique can be applied to a pair of reflectometers havingidentical connectors of any type, not just sexless connec­tors. This line-reflect-line (LRL) calibration technique isthe subject of this paper.

Abstract-A technique is described for calibrating a dual six-port orfour-port automatic network analyzer (ANA) so that the scattering pa­rameters of two-port devices having any combination of connectors canbe measured. The technique is a generalization of the "thru-reflect­line" (TRL) calibration technique in which the thru is replaced with asecond length of precision transmission line. Expressions for errorsassociated with the second line are derived.

U.s. Government work not protected by U.S. copyright

HOER AND ENGEN: ACCURACY ASSESSMENT FOR DUAL SIX-PORT ANA: EXTENSION TO NONMATING CONNECTORS 525

LRL is valid for any connector.

M-M Reference Planes:

~ ~

~ Linel

~ ~Reflect

---+~ Line 2

(a)

F-F Reference Planes:Add F-F Adapters

~>--ID-< )-{[}--<f-{[}-

-ill-<~>--CO--<0 ~>-

-<~>-(b)

M-F Reference Planes:Remove AdapterC (or D).

~>--ID-< )-{[}--<f-{[}-

~ )-{D-

~(c)

Fig. 2. (a)-(c) Using a set of female standards and a set of male standardsto do a complete LRL calibration so that a two-port device with anycombination of connectors can be measured.

adapter C (or between Band D) is not important whenthis sequence of measurements is used.

The connector types used in Fig. 2(a) and (b) can becompletely different. For example, those in Fig. 2(a)could be type N, while the connectors in Fig. 2(b) couldbe SMA. Two-ports that have a female type N on one endand a male SMA on the other end can be measured as inFig. 2(c). The LRL calibration technique can also be ap­plied if one set of connectors is waveguide, and the otherset is coax.

B. One Pair of Lines

Only one pair of lines is needed to calibrate the tworeflectometers if the connectors on the two-port under testare of the same type. For example, to measure two-portdevices that have any combination of type-N connectors,the reflectometers are first calibrated with type-N maleconnectors as shown in Fig. 3(a). Then a female-femaletype-N adapter C is measured as shown in Fig. 3(b), andleft connected to one of the reflectometers, say to A. Two­port devices that have male-female type-N connectors cannow be measured between C and B. To measure two-portdevices with two male type-N connectors, one may re­move adapter C and measure the parameters of anothersimilar adapter D. As shown in Fig. 3(c), leaving adapterC on reflectometer A, and reconnecting adapter D to re­flectometer B provides a pair of female test ports betweenwhich two-port devices that have male connectors can bemeasured.

III. LRL CALIBRATION

The computations and software used in the TRL solu­tion can also be used in the LRL solution with only slight

LRL 'withonly 1 set of standards

M-M Reference Planes:

~ ...IINV--<f.- Reflect

~~f.-. Line2

(a)

M-F Reference Planes:Measure adapter C.

Use C & B as new reference planes.

F-F Reference Planes:Remove Adapter C, measure adapter D.

Reconnect adapter C.

~ >--{[}--<~

(c)

Fig. 3. (a)-(c) Using one set of standards to do a complete LRL calibrationwhen the connectors on the device under test are the same type (but anycombination of sexes).

modifications. If a line of length II is used in the TRLcalibration instead of a thru (II = 0) connection, the fol­lowing modifications to the four-port parameters are ob­tained (see Appendix B):

B I = BIt (1)

CI/AI = Clt/Alt (2)

B2 = B2t (3)

C2/A2 = C2t/A21 (4)

AI = Alle'Yllt (5)

A2 = A2t e'Y I lt (6)

"(2 /2 - 'YI/I = ("(1)( (7)

The terms on the left of each equation are the desiredparameters. The corresponding parameters on the right ofeach equation with a subscript t are those obtained fromthe TRL solution. The first four equations show that thefour-port parameters B I, C I / AI, B2 and C2/A2are exactlyequal to those obtained from the TRL solution. Equations(5) and (6) show that the four-port parameters AI and A2are scaled by e 'Ylll .

In the TRL calibration, the optimum electrical lengthof the line is 90° (or an odd multiple of 90°). Electricallengths near 0° or 180° (or even multiples of 90°) mustbe avoided or the solution becomes ill conditioned. Equa­tion (7) shows that when two lines are used, it is the dif-ference in the electrical lengths of the two lines that isoptimally 90° and that must not be near 0° or 180 0

•

IV. LRL FOR SEXLESS CONNECTORS

At higher frequencies the optimum length of line for theTRL calibration can become physically too short to be

526 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, YOLo IM-36, NO.2, JUNE 1987

practical. However, if two lines are used, II can be someconvenient length and 12 slightly longer so that the differ­ence in length is electrically 90° in the center of the fre­quency band. For this reason, one may want to use theLRL calibration even for sexless connectors at higher fre­quencies.

Substituting (14) in (7) and solving for I'lll gives

1'1/1 = K(I'I)( - Kd1'12 (15)

or

V. EQUIVALENT REFERENCE PLANE

Equations (1)-(6) lead to

r = e-'Y1lJr1 (I

where

K' = K(1 _ d1'12)( 1'1 ) ( .

(17)

(18)

(8)

which say that the reflection coefficient of each tennina­tion used in the reflect measurements, or any other I' mea­surement, will differ from r(, or r(2 by the factor e-'Y1lJ

where r(, and r(2 are the reflection coefficients obtainedfrom A It, BIt, and Cit at test port 1, or A2(, B2(, and e2( attest port 2.

Equation (8) shows that r ratios can be measured ateither test port without knowing 1'1/1. If (8) is comparedto the following equation for transforming r L through anonreflecting line of length I

r = e-2'Y1r L (9)

we see that the effective reference plane of I', is at 11/2

which is at the center of the line of length II. This locationagrees with our intuition as to where the reference planeshould be since TRL assumes 11 = o.

Comparing (15)-( 17) with (12)-( 13) shows that the lastterm in (15) or (17) represents the error in 1'1 11 when 1'2=1= 1'1.

B. From a Known I'

Another way of obtaining 1'1/1 is from (8). If r is theknown reflection coefficient of a highly reflecting tenni­nation connected to either test port 1 or 2, then (8) gives

r(1'1 /1 = loge r.

A matched termination cannot be used because if randI', are both zero, 1'1/1 cannot be determined.

C. From l/; ofr

If the highly reflecting termination is a short or an offsetshort with a precisely known phase angle l/;s' then I'lll canbe determined from l/;s alone. Ir, I need not be known. Toshow this, define the real and imaginary parts of (l'l ) ( as

VI. DETERMINING 1'1/1

To complete the LRL calibration, 1'1/1 must be deter­mined. Four methods for determining 1'1/1 are outlined be­low.

A. From II and 12

If the lines 12 and II are made from the same stock andtheir cross-sectional dimensions are sufficiently the same,we can assume that (21)

(19)

1 - 1'1/1'2

12 - 111'1/ 1'2.

dl' _ 1'2 - 1'1

(1'1)( - 1'2 /2 - 1'1 /1

Then substituting (16) and (19) in (8) gives

Ir, Iej~s = e-K'xe-jK'y \ r, \ej~(. (20)

From (17) we see that K' is real if dl' = 0 or if dl' / (1'1 ) (is real. From (14) and (7)

Although 1'1 and 1'2 are complex, the ratio 1'1 /1'2 will bereal (or very nearly so) if the two lines are made from thesame stock. Then K' is real and (20) expands into thefollowing two equations:( 11 )

(10)1'2 = 1'1·

Then (7) becomes

which gives

(12)

[r.] = e-K'xlr(\

l/;s = -K'y + l/;(.

(22)

(23)

(24)K'l/;( - l/;s

y

Then Ir, I can be calculated from (22). Thus only thephase angle l/;s of the short needs to be known. For a flatshort

Since (l'l)( is mostly imaginary, y is known much moreaccurately than x. For this reason we choose (23) to cal­culate K'(13)

where

IIK == l2 - l) .

The physical lengths II and l2 of the two lines can be mea­sured to obtain a value for K.

The uncertainty in I'll1due to the assumption that 1'2 =

1'1 can be determined as follows. Let

1'2 = 1'1 + dl'. (14) l/;s = 180 - 12fo deg (25)

HOER AND ENGEN: ACCURACY ASSESSMENT FOR DUAL SIX-PORT ANA: EXTENSION TO NONMATING CONNECTORS 527

(27)

New Error Box AC

(33)

(32)

(34)

(30)

15

h+iiand

1512 - II = --- cm

t, + i2where il and i2 are in gigahertz. The corresponding dif­ferences in phase shifts ~cPfl and ~cPf2 ei f, andf2 are givenby

180dePf' = 180 - dePh = 1 + lz/fJ deg. (31)

If the ratio i2 IiI is so large that ~cPft becomes too small,say less than 20 0

, then the range f) and i2 may be brokeninto two ranges I) to I. and /; to ii, where I. is an inter­mediate frequency chosen such that

Three lines are required to cover these two ranges. Thelengths I I, 12 , and 13 of the three lines must satisfy

tween measurement planes WI and r 2 as shown in Fig. 4.The parameters AI, BI, and C{ are readily obtained fromWI and r2. Since the uncertainties associated with WI andr2 have been determined in the TRL calibration, the un­certainties of the parameter A I, B I , and C{ are also readilydetermined in the same way as for a two-port inserted be­tween measurement planes 1 and 2.

Lengths II and 13 are used from I. to /;, and lengths II arid12are used from /; to 12. The corresponding differences inphase shift are obtained from (31) using the appropriatefrequencies.

Equations similar to (30) and (31) can be derived for a

VIII. OPTIMUM LINE LENGTHS

The first line of length II can be any convenient length;however, it should not be longer than necessary. As ex­plained in Section III, the length 12 of the second lineshould be such that the difference in phase shift ~cP

through the two lines is 90 0 at the center of the operatingfrequency range il to i2. This condition when applied tocoaxial lines leads to [7]

w1 ,r2 w2

~~~W, Lr---Ac-1 t;; h w2

~ ~Fig. 4. Considering adapter C and error box A as a new two-port error box

between measurement planes WI and r 2 to get the parameters A;, B;, andC{ of AC directly.

(26)

where i is the frequency in gigahertz and 0 is the skindepth in centimeters into the face of the short [2]. Equa­tions for the phase angle of an offset short [3] or an open­circuit termination [4] are given in the literature.

VII. UNCERTAINTY IN f

Elsewhere in this TRANSACTIONS [5], [6] techniques aredescribed for propagating random and systematic errorsthrough the six-port calibration and measurement processwhen the TRL calibration technique is used. When theLRL calibration technique is used, these errors apply di­rectly to I', but not to f. Differentiating (8) leads to thefollowing expression for the worst-case uncertainty df i

in I', at test port i,

dri = e-"(Iltdrti - fid('Yt/I), i = 1,2. (29)

where drti is the uncertainty In f ti obtained from propa­gating errors through the TRL calibration and the mea­surement process, and d ( 'Y )/1 ) is the uncertainty in 'Y t /) •

As r approaches zero, the uncertainty in r due to d ( 'Y III )also approaches zero. This means that the error in 'Yt/)causes negligible additional error in the measurement oflow reflecting terminations.

Equations for estimating d ( 'Y )1 1 ) are given in AppendixA. These equations lead to the conclusion that method din Section VI is probably the most accurate method fordetermining 'Y II) •

Once the errors associated with r I and r2 at both testports are determined, the errors in the S-parameters of anytwo-port measured between these test ports are readilycalculated [5].

A special case occurs when the two-port being mea­sured is an adapter such as the female-female adapter Cin Fig. 3(b). In this case we are not so much interested inthe parameters and errors associated with the adapter C,but in the parameters and errors associated with the com­bination of error box A and adapter C. One way to deter­mine directly the parameters Ai, Bi, and C{ of the com­bination A + C, and the errors associated with theseparameters is to consider A + C as a new two-port be-

If line I t is then inserted between the test port and I'L' (8)gives for this second measurement

D. From r Ratio

A fourth way of obtaining 'Y )/) is from the ratio of twor measurements. Measure a highly reflecting terminationwith unknown reflection coefficient r L on one of the testports of the automatic network analyzer (ANA). Thenfrom (8) the reflection coefficient at that test port can bewritten

f' = e-2~(I11 fL

= e-"(\ll f;.

Taking the ratio of (27) to (26) gives

r' = e- 2-Y liJ r; (28)f f t

Since I', 1ft is known, 'Yt/t can be calculated from thisequation.

528 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-36, NO.2, JUN°E 1987

TABLE IOPTIMUM LENGTHS FOR RECTANGUI.AR WAVEGUIDE

WAVEGUIDE CUTOFF £1 £2 12- 11 A~£1

SIZE, WR FREQ GHz GHz em DEG

284 2.078 2.600 3.950 3.048 57.1187 3.152 3.950 5.850 2.052 58.6137 4.301 5.850 8.200 1.370 65.2112 5.259 7.050 10.000 1.136 64.0

90 6.556 8.200 12.400 0.971 57.462 9.486 12.400 18.000 0.644 61.742 14.047 18.000 26.500 0.445 60.128 21.081 26.500 40.000 0.300 57.722 26.341 33.000 50.000 0.240 57.419 31.357 40.000 60.000 0.197 58.815 39.863 50.000 75.000 0.160 58.010 59.010 75.000 110.000 0.108 59.9

7 90.840 110.000 170.000 0.073 54.34 137.520 170.000 260.000 0.047 56.13 173.280 220.000 325.000 0.037 59.4

Optimum difference 12 - II in two lengths of waveguide are calculatedfrom (38) to have a phase difference A<I> (or 180 - A<I» equal to or greaterthan A<P./I over the complete frequency band frorn f to!2' The shorter lengthI, can be any convenient length including zero.

(A2)

(A3)

In this expression dr is the total uncertainty with whichr is known.

with a short length of line whose ')'1/, is determined byone of the four methods outlined in this paper, it is pos­sible to use all of the TRL software to calibrate a dual six­port or four-port with any type of connector. Thus onlyone set of software is needed for any type of connector.

The uncertainty d (')'1 ) t in (')'1) t is determined from theTRL solution [5], [6]. From (13)

dK = K 2 !1 (dl l _ d12 )I, I, 12

ApPENDIX A

The uncertainty d ( ')']/]) in 'Y III depends on how 'Y ]/1 isdetermined. The worst-case uncertainties associated witheach of the four methods considered in Section VI for de­termining 'Y,/, are derived in this appendix.

A. From II and 12

Differentiating (15) and keeping only first-order termsleads to the following expression for the uncertainty in'Y]/] when 'Y]/I is determined from the physical lengths I]

and 12•

where dl, and dl2 are the uncertainties in measuring I] and12 •

B. From a Known rWhen a known reflection coefficient is used to obtain

'Y]/], (18) gives

(39)180

1 +

----;::::========- deg.Ii - f~

f? - f~

Table I gives values of 12 - I] and ~(j>fl for different wave­guide sizes.

IX. CONCLUSION

Considerable effort has been expended by the NationalBureau of Standards (NBS) to develop the TRL calibra­tiorr techniques to provide on-line estimates of the randomand systematic errors associated with all of the six-portcalibration parameters and the measured values of r or S­parameters. Currently, this technique has applied only tosexless connectors. By replacing the TRL thru connection

rectangular waveguide as follows. The phase shift througha rectangular waveguide can be written [8]

et> = 12/~/2 - I~ deg (35)

where I is the frequency in gigahertz, I is the physicallength in centimeters, and Ie is the cutoff frequency ingigahertz. For two lines, the difference in phase is

~1> = 12(/2 - 1,)~/2 - I~ deg. (36)

Choose the lengths of line so that the differences in phaseshift ~(j>f' and ~(j>f2 at the band edges I] and 12 satisfy

~(j>fl = 180 - ~(j>f2· (37)

This condition keeps the phase differences equally as farfrom 0 0 at II as it is from 180 0 at/2 • Substituting (36) in(37) and solving for 12 - II gives

I - I - 15 (38)2 I - ~!? _!:. + ~!i _!; em.

The difference in phase shifts at II and 12 are obtained bysubstituting (38) in (36)

HOER AND ENGEN: ACCURACY ASSESSMENT FOR DUAL SIX-PORT ANA: EXTENSION TO NONMATING CONNECTORS 529

C. From l/; ofr

When "I,l] is determined from the known phase angleof a termination using (16) and (24), d( "I]ll) is given bydifferentiating (16) to get

and the cascading matrix for the single line is replaced.with

(B2)

(Bl )

ApPENDIX B

The derivation of (1)-(7) is the same as that given byEngen and Hoer [1] except that the identity matrix whichdescribes the cascading parameters of the thru connectionin (20) of that paper is replaced with

is more accurate than method B or C. Method A is prob­ably the least accurate, but the easiest to apply.

(AS)

(A4)

(A6)

Let

where from (24)

1dK' = - (dl/;I - dl/;s - K'dy).

Y

Combining (A4) and (AS)

I ( "Il )I'd( "I]l,) = - (dl/;I - dl/;s)

y

+ K' (dht) - ("It) dY) .

I I Y

(A7) in (21) and (25) of that paper. The rest of the derivationis the same and leads to (1)-(7) of this paper.

(A9)

(A10)

Then, since x « y, (A6) reduces approximately to

d( "I]l]) =: K'dx + j(dl/;I - dl/;s). (A8)

D. From r Ratio

If "I]l] is obtained by measuring a termination with andwithout the line l, inserted in front of it, the uncertaintyin "I ]1 1 obtained from (28) is

I (dfl df:)d('Yttd = 2 ~ - r;

di', dlf t / •

=:: ~ = 1fT + JdY;/.

E. Comparing Methods

Assuming that all errors are roughly the same magni­tude, (AI), (A3), (A8), and (A10) imply that method D

REFERENCES

[1] G. F. Engen and C. A. Hoer, '" "Thru-rcflect-line ': An improved tech­nique for calibrating the dual six-port ANA," IEEE Trans. MicrowaveTheory Tech., vol. MTT -27, pp. 987-993, Dec. 1979.

[2] P. I. Somlo, "Recession depth in metalic conductors at low frequen­cies," Electron. Lett., pp. 776-777, Dec. 1971.

[3] H. B. Sequeira and B. C. Yates, "Approach for evaluating effects ofwall losses on quater-wave short-circuit impedance standards," IEEETrans. Microwave Theory Tech., vol. MTT-33, pp. 1106-1109, Nov.1985.

[4] B. Bianco et al., "Open-circuited coaxial lines as standards for micro­wave measurements," Electron. Lett., pp. 373-374, May 1980.

[5] R. M. Judish, "On-line accuracy assessment for the dual six-port ANA:Statistical methods for random errors," pp. 507-513, this issue.

[6] C. A. Hoer, "On-Iine accuracy assessment for the dual six-port ANA:Treatment of systematic errors," pp. 514-519, this issue.

[7] C. A. Hoer, "Choosing line lengths for calibrating network analyz­ers," IEEE Trans. Microwave Theory Tech., vol. MTT-31, pp. 76­78, Jan. 1983.

[8] S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Com­munication Electronics. New York: Wiley, 1956, p. 402.