IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985
Precision 4-Port Junction for the Connection of CoaxialImmittance Standards in Parallel
DONALD WOODS
Abstract-Previously published work on the application of the 3-portjunction to immittance standardization is extended to the 4-port junc-tion which enables either three or two immittance standards to be con-nected in parallel to provide calculable 1- or 2-port S-parameters, respec-tively, having moduli > 0 and < 1. Calibration equations are given forprecise junction characterization.
I. INTRODUCTION
PRECISION reflectometers and 2-port network analyzersemploying either complex voltage-ratio detection or the
6-port technique are normally calibrated with modulus-onestandards, i.e., short- and open-circuits associated with preci-sion airlines. Many calibration procedures have been publishedand are too numerous to quote. Having calibrated an instru-ment in this manner it is necessary to evaluate the accuracy ofmeasurement at other points within the unit circle, both forSI, and S21 parameters. This of course, cannot be realized bythe series-connection of existing standards, and an accuratemethod of parallel-connection becomes necessary. For ex-ample, a calibrated 3-port junction in association with a matchedload and two airlines can be used to generate a reflection co-efficient anywhere within the unit circle [1]. Similarly, a 3-port junction terminated with a sliding short-circuit at port 3,for example, can be used to generate variable SI, and S21parameters over a limited range using the principle of reflectiveattenuation [2]. With a 4-port junction terminated in a resis-tive load and a variable short-circuit at ports 3 and 4, calculableSI, and S21 parameters can be generated employing both dis-sipative and reflective attenuation.An early theoretical analysis of the 3-port junction is given
in [3] and two of the methods proposed in that paper forjunc-tion characterization have recently been employed to charac-terize a precision symmetrical 3-port fitted with GR900 con-nectors [4] . These techniques have now been extended to the4-port junction and the relevant analysis and calibration equa-tions are given in the following sections.
II. THEORY: INPUT REFLECTION COEFFICIENTThe input reflection coefficient Plu of a mismatched sym-
metrical 4-port, terminated in reflection coefficients F2, r3and F4 at ports 2, 3, and 4 is readily obtained by matrix re-normalization [5].
Manuscript received August 20, 1984.The author is with the Department of Chemistry, University of Surrey,
P = (a + jo)l.I is the length of each port and reflection coefficients are nor-malized to the complex characteristic impedance ZO of theports. Complex normalization is readily taken into accountby a propagation correction term ¢ [6], [7].
1 1a b
8rrV/(uf) In b/awhere a and b are the conductor radii in centimeters, a theconductivity in siemens per centimeter, and fin gigahertz. Wethen have [7]Zo Roo(l + j¢) (4)P={3O +jio(1 + O)} (5)
where R. and B. are the characteristic resistance and phaseconstant of an ideal line of infinite conductivity. R., = 138.02loglob/a and B. = c/vol where v.0 = 2.9970 X 1010 cm/s, thevelocity of propagation in air [7].
It will be noticed from (5) that ¢ gives not only the attenua-tion in terms of the known parameter O.w, but also the phasecharacteristic 3 which is greater than joo because the velocityof propagation is less, being equal to v.4/(I + t).The parameters needed to characterize the junction are O.,
I and r to evaluate P in (5). In calibration we determine (axl)and (i1) by finding the frequency at which the junction gives3600 phase transformation for example, between ports 2, 3,and 4 short-circuited and port 1, in which case, from(l),(OI) =ir/2, i.e., the ports are X/4 long. Port attenuation (al) is ob-tained from the ratio M of two moduli, namely P and themodulus of a reference short-circuit. There are a number ofvariations of this technique and the equation for (al) in termsofM depends on the method adopted [4].From (2) and (5) we can now express l and ¢ in terms of (al),
(,BI), and f3. noting that in the imaginary part of (5) j34 +
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985
short-circuit. The two moduli are measured and their ratiodenoted as M. This procedure eliminates minor errors in themeasurement system which should have good discriminationand repeatability in the order of one or two parts in 1 04. Italso eliminates the finite impedance of the short-circuit and itsconnector contact impedance [7].For the above conditions Pu is given by putting r2 = 173 =
I74 = -1, e126 = ei6O = -1 and e180 = +1 in (1) so thattoo 0 S.. I-360 80S O.C. Ito 240
340
32o 16o
300 280
Fig. 1. Plot of Pl = (-ei20 + 3e166 - 2e180)/(2 - 3ej20 + eI6O)for values of 20 between 00 and 3600.
e-2 Cil -3e-6 al 2e-8allPit = 2+3e-2a1I- e-6al
Adopting the following approximateponentials
e-2 iI= I - 2o1 + 2(ol)2e-6cel= I - 6al + 18(al)'e-8a 1= - 8al + 32(a1)2
we obtain
-1 +8a1- 29(cil)2Pi 1-3(-1)2
(10)
expansions of the ex-
(1 Ia)
(llb)
(lIc)
(12)
(7) In practice axl is sufficiently small to ignore all terms of higherorder than (<xl)2 so that (12) becomes
Phase transformation through the 4-port is nonlinear as shownin Fig. 1 which is a plot of (1) with cal= 0 and P2 = r3 = 74 =
- 1, namely,
- ei2O + 3 e'60 - 2e80(8Pil = 3e520 eJ66 (8)
in which 0 = ,I.It will be seen that 360° phase transformation occurs when
when 26 = 1200, i.e., i31= rrf3. The second revolution of thevector Pi, is also shown in Fig. I by the degree scale outsidethe circle.
If we put 1= 0 then P = 0 and (1) becomes
-1 + r2r3 + I2r4 + 3r44+ 2r2r3rI4Pit =
2+1'2 +r3 +74
21'3 49which is the reflection coefficient at a very low frequencywhere the ports do not have significant attenuation or phasecharacteristics such as in low-frequency immittance bridge ap-plications. If either r2, 13, or I`4 is put equal to -1 in (9)then PII = -1. However, if we put two or three of the rs equalto -1 then. the equation is singular, i.e., both numerator anddenominator are zero. The reason for this is that two or more
short circuits in parallel cannot be distinguished from a singleshort-circuit. In other words a single short-circuit at the junc-tion isolates the other two ports. In practice, at higher fre-quencies, complete isolation does not occur owing to thefmite attenuation of the ports.
III. CALI BRATION
A. Method 1
With ports 2, 3, and 4 short-circuited, the frequency is foundwhich makes the phase angle of Pu equal to that of a reference
Pul = 1 - 8c1+ 32(al)2. (13)
Solving for cal and substituting M as the ratio of | to thatof a reference short-circuit we then have
(cl)= { -} Np. (14)
I and r are then given by (6) and (7) in which (B1) = ir/2 andI0. is known from the frequency of measurement so that P can
be evaluated from (5).The calibration applies to other frequencies because 1 is the
physical length and is independent of frequency. The effectiveelectrical length is taken into account by the propagation cor-
rection term ¢ which varies asfl/2 For ports 6.7-cm long thefrequency for this method will be about 1.1 GHz [4] .
B. Method 2
At audiofrequencies the port length 1 can be obtained bymeasuring the total capacitance of the junction as described in[3] using shielded open-circuits of known discontinuity capac-
itance [8], [9] in which allowance is made for protrusion ofthe inner conductor contact [4]. The relevant equation for a
4-port is
I= {R00VO(Cm - 2Cd)} (15)
where Cm is the measured change in capacitance between an
open-circuit and port 1 of the junction terminated in open-circuits at ports 2, 3, and 4. Cd is the discontinuity capacitance.Results of calibrations of a 3-port junction using Methods 1
and 2 are reported in [4] and show agreement to within 0.1mm in determining 1.
C. Method 3With ports 2, 3, and 4 short-circuited the frequency is found
which makes the phase angle of Pit = 00 so that, from Fig. 1,
-e-2&l/20 + 3e 6 1LZ60 - 2e&8ac1/80P =-- 2- 3e 2l/-20 +e6'I/-60-l (16)
Substituting for /-20 = /-80 = (-0.5 - jO.866) and /-Q =1 + jO and solving for pu we obtain
lpl I 1- 4al+ 8(cd1)2.
-,
(17)
Solving for (al) and substituting M for the ratio of IPI I tothat of a reference open-circuit we have
(ol)={1- F2M-1}Np. (18)
This formula is different form that given in the CPEM 84Digest and is slightly more accurate. For ports 6.7-cm long thefrequency for this method will be about 740 MHz. It is not soaccurate as Method 1 because the attenuation is less and thereference open-circuit will have a phase angle of a few degrees.Also, it does not eliminate any small error that may be causedby the finite impedance of the short-circuit and its associatedcontact impedance.
IV. THEORY: S'l AND S21 PARAMETERSThe scattering matrix of a symmetrical 4-port junct
given [3] by
Sll S12 S13 S14 2 2 2
S21 S22 S23 S24 2 2~~ =&e2P 22S31 S32 S33 S34 2 2
S41 S42S43 S44 2 2
in which P is defined by (2).To obtain the computed values of S', and S'1 from ;
reflection coefficient standards connected to ports 3 an(apply matrix renormalization as described in [10] , name
S[5 S21
S21 S22 (0034)
in which the subscript imphes 1, = F2 = 0 and we renor:at ports 3 and 4. Substituting from the appropriate tranmin [10] we obtain
From (19) we see that all main diagonal terms Sii(i=the same and similarly for all transmission terms Si(thus
1 e-2PSij(i )= e2.
From these equations we can evaluate the minor cofAi>, 6ii and the major cofactors C'1 as defined in [10]then find that
(a)
-I
(b)Fig. 2. (a) Loci of Si1 at three frequencies. (b) Loci of S', at the same
three frequencies.
'Aij(i= j) = 5ij(i= j) = O
,Ai>(i0 ) = 5 ii(i j) = -ePCii(i = j) = - e-6P
(23a)(23b)
(23c)
Cjj (i =A j) = - - e-6p. (23d)2~~~~~~~~~2dSubstituting these values into (21a)-(21c) we obtain the re-
quired solution of (20) when
sforms S = 2P e 4F3P4- I
l2 +e-2P(r3 +1r4)1()(21a)
S' = e-2P I + e-P(r3 + r4) + e-4]P3r4(21b)
22 + e"2p(r3 + r4) (z
(24a)
(24b)
(21c) It will be noticed that if we put '2 = 0 in (1) we obtain
,) are (24a).As an example of the range of variable S', and S', param-
Z 1 Z eters that can be generated by a 4-port junction, suppose port3 is terminated in a matched load and a X/2 variable short-cir-
(22a) cuit is connected to port 4. If we define the electrical length(22b) of the offset short-circuit as 4 degrees and assume that L is
zero, then r3 = 0, r4 = -e}2', and (24a), (24b) becomeactors / _-j2(
We = ~ j0j~)~)(25a)
21 1
IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-34, NO. 2, JUNE 1985
(25b)(I - ei2Oei2°e221 ~ 2 -e-i2Oe-i2 (R.)'The loci circles of these two parameters, whose origins are,
of course, at the center of the unit circle, are plotted in Fig. 2(a),(b) for three frequencies at which the junction port lengthsare X/8, X/4, and 3X/8. The values corresponding to a directshort-circuit, i.e., k = 0, are indicated by a dot and the direc-tion of rotation, as q increases, by an arrow. The rate of changearound the loci circles is very nonlinear for equal incrementsof p. Maximum rate of change occurs when the combinedlengths of port 4 and the variable short-circuit are X/2, i.e.,when a short-circuit occurs at the junction of the four ports.The radius of the S', and S'1 loci circles is 3 and the center ofthe former is displaced 32 from the center of the unit circle.
V. CONCLUSIONS
In view of the encouraging results obtained so far in the cali-bration of a precision 3-port junction [4] it is considered that3- and 4-port junctions are likely to play an important part inthe future in checking the accuracy of reflectometers andnetwork analyzers that are calibrated in terms of modulus-onestandards. In precision reflectometry it is necessary to definereflection coefficients and S-parameters normalized to a com-plex ZO [6]. A simple technique has already been described[7] that utilizes a single propagation correction term ¢ whichgives the attenuation, the phase characteristic, and the com-plex characteristic impedance in terms of the parameters of anideal line of infinite conductivity.
REFERENCES
[1 ] D. Woods, "Generation of reflection coefficient standards of anyvalue by means of a 3-port coaxial junction at microwave fre-quencies," Electron. Lett., vol. 10, no. 18, pp. 379-380, 1974.
[2] -, "Coaxial junction with 3-ports as a variable 2-port S-param-eter standard at microwave frequencies," Electron. Lett., vol. 11,no. 1,pp. 6-8, 1975.
[3] -, "Application of 3-port coaxial junction to r.f. immittancestandardization and measurements," Proc. Inst. Elect. Eng., vol.119, no. 2. pp. 261-268, 1972, and Errata, Proc. Inst. Elect. Eng.,vol. 119, no. 5, p. 636, 1972.
[4] -, "Calibration of a precision 3-port coaxial junction: a newcomponent for immittance standardization," Proc. Inst. Elect.Eng., pt. A, vol. 131, no. 5, pp. 302-306, 1984.
[5] , "Application of the 4-port junction to r.f. immittance stan-dardization and measurements," Proc. Inst. Elect. Eng., vol. 125,no. 8, pp. 715-716, 1978.
[6] , "Relevance of complex normalization in precision reflectom-etry," Electron. Lett., vol. 19, no. 15, pp. 596-598, 1983, andErratum,Electron. Lett., vol. 19, no. 17, p. 706, 1983.
[7] -, "Immittance transformation using precision air-dielectriccoaxial lines and connectors," Proc. Inst. Elect. Eng., vol. 118,no. 11, pp. 1667-1674, 1971, and Errata, Proc. Inst. Elect. Eng.,vol. 119, no. 6, p. 772, 1972.
[8] -, "Shielded-open-circuit discontinuity capacitance of a coaxialline," Proc. Inst. Elect. Eng., vol. 119, no. 12, pp. 1691-1692,1972.
[9] B. Bianco, A. Corana, L. Gogioso, and S. Ridella, "Open-circuitedcoaxial lines as standards for microwave measurements,"Electron.Lett.,vol. 16, no. 10, pp. 373-374, 1980.
[10] D. Woods, "Multiport network analysis by matrix renormaliza-tion: Extension to 4-ports," Proc. Inst. Elect. Eng., vol. 124, no.9, pp. 749-753, 1977, and Errata, Proc. Inst. Elect. Eng., vol.125, no.5, p. 376, 1978.
Redundant Circuits as Substitutes For Check StandardsTOM GULDBRANDSEN
Abstract-Some simple switched two-port networks are proposedwhose scattering elements for the different states contain a consid-erable amount of redundancy. The networks which can be made broad-banded and remotely controlled are very suitable for checking theaccuracy of vector network analyzers.
I. INTRODUCTIONM/tODERN automatic network analyzers (ANA's) are very
accurate instruments of high complexity. The accuracyis achieved by setting up an accurate model whose parametersare determined by means of a calibration procedure. This pro-
Manuscript received August 20, 1984.The author is with the Physics Laboratory III, Technical University
of Denmark, DK-2800 Lyngby, Denmark.
cedure consists in making measurements either on a numberof standards, i.e., one- or two-ports, whose scattering matricesare accurately known, or on a number of unknown compo-nents. In the latter case consistency requirements are appliedfor the calibration. When the calibration procedure is finished,the parameters are inserted in the model and the ANA is readyfor real measurements on a device under test.Commercial ANA's often produce only the results, i.e., the
scattering matrix of the device under test as a function of fre-quency but there is no indication of the magnitudes of sys-tematic and random errors. If the measurements give rise tosuspicion the normal way to check the ANA is to measure anumber of standards and compare the measured values withthe values that are already known from measurements made
