Proposed new waveguide standardof reflection coefficient

A. L. Cullen and S.K. Judah

Indexing terms: Measurement standards, Microwave measurement, Permittivity measurement, Waveguides

Abstract: A new standard of reflection coefficient for accurate waveguide measurements is proposed. Itconsists of a short length of dielectric-filled waveguide. It is shown that such a device, backed by a matchedload, can have a reflection coefficient whose magnitude is almost constant over a 40% bandwidth. Permitti-vity values for dielectric materials may now be measured with an accuracy adequate for most applicationsof the new component. However, an alternative calibration technique is proposed in which a priori know-ledge of the permittivity of the dielectric medium is not required.

1 Introduction

A waveguide standard of reflection coefficient is a valuablecomponent in any microwave-measurement laboratory. Ofthe many devices which have been proposed, two will bementioned here. The first1 is the half-round inductive post(and the corresponding double half-round post). An exactanalysis of the half-round post has been given by Kerns,2

so that the theory of the device is completely understood,and the reflection coefficient can be calculated in terms ofits physical dimensions and the wavelength. From a prac-tical viewpoint, however, it has certain disadvantages.

The most fundamental is its relatively narrow bandwidth.For example, with a post radius of one-fifth of the broaddimension of the. guide the v.s.w.r. varies from 2-30 to1-15 as / / / c varies from 1-3 to 1-8. Although not essential,it is very desirable to have the reflection coefficient magni-tude more nearly constant over the normal waveguidefrequency band. A second disadvantage is that the reflec-tion coefficient is rather critically dependent on the radiusof the half-round post and its precise location; it is by nomeans a simple task to measure the physical dimensionsof a complete half-round post and waveguide assemblywith the required accuracy. The second device to be notedis the reduced-height waveguide proposed by Beatty.3

This consists of a short length of guide (about Xg/4 atmidband frequency) in which the narrow dimension hasbeen reduced to lower the voltage/current characteristicimpedance. Simple quarter-wave transformer theory gives alst-order description of the characteristics of the device,and, from this viewpoint, the relatively broadband per-formance is easy to understand. To use the device as astandard, however, a more complicated theory is needed,since the discontinuity susceptance at the two junctionsof the reduced-height guide with normal height must beallowed for. This effect is relatively small, however, and thetwo susceptances tend to cancel when the reduction inheight is not too great.

The device proposed in the present paper is rathersimilar in performance to the reduced-height guide. It hasalso very good broadband performance, and has the advan-tage over both of the two preceding devices that verysimple waveguide theory is adequate since only the funda-mental TEOi mode is required in the exact analysis of thedevice. At first sight, the use of a dielectric material may

seem to invalidate any claim that the new device is anabsolute standard. However, we shall show that, when morethan one of these devices is available, determination of thefrequency at which a stock of three or four identicaldevices becomes reflectionless, taken together with thephysical dimensions, will provide all the informationneeded to calculate the reflection coefficient of a singledevice at any frequency in the band. There is in fact noneed for accurate a priori knowledge of the permittivityof the dielectric material used.

o

11 illiIII f 1 i

o

k- «

o

III

O

' -H

i

->]

Fig. 1

2 Theory

Fig. 1 shows a length of rectangular waveguide, supportingthe TE10 mode, and filled with a low-loss dielectric ofrelative permittivity er. Let the phase constants of thedielectric-filled and empty sections of the guide be ]3 and0o, respectively. Then, if the guide to the right of theterminal plane T2 is perfectly matched, the magnitude|p| and the phase 0 of the reflection coefficient at Tican be formed from the following equations:

(1)

L2

(2)

Also we have

Paper T54M, received 27th January 1977Prof. Cullen and Dr. Judah are with the Department of Electronic& Electrical Engineering, University College, London, TorringtonPlace, London WC1E 7JE, England

= (erk*-k2cr

2(3)

so that |3 and j30 can be calculated when the permittivityof the dielectric, the cut-off frequency of the empty

120 MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3

waveguide, and the operating frequency are known. Thus,with the additional information of the length /, Ipl and0 can be calculated with the help of eqns. 1 and 2.

One might, at first, suppose that, as in the case of thereduced-height standard, the optimum choice for / wouldbe that which makes / = X /4 at the midband frequency.This is not the case, however, and a more nearly constantvalue of Ipl over the frequency band is obtained when theelectrical length of the dielectric-filled waveguide is sub-stantially less than this. It is remarkable, in fact, that forseveral values of permittivity of currently available low-lossmaterials, excellent broadband performance is obtained fora wide range of lengths below this value. This feature isillustrated in Figs. 2—4 where the reflection coefficientmagnitude is plotted as a function of frequency for anX-band version of the device. It will be observed that theprincipal effect of varying the permittivity or the lengthis to alter the reflection coefficient at midband withoutgreatly affecting its variation over the band. The physicalreason for the relatively constant magnitude of the reflec-tion coefficient for short specimens is as follows. There is,of course, a tendency for the reflection to increase withincreasing frequency as the electrical length 0/ increasesfrom a small initial value. On the other hand, there is atendency for the reflection to decrease with increasingfrequency as the characteristic impedance of the dielectric-filled and empty waveguides become more nearly equal.Most fortunately, the two effects balance very closely.When the electrical length of the dielectric is sufficientlyshort to justify the approximation, we can write the fre-quency-dependent part of eqn. 1 as

0o 00(4)

Differentiating with respect to frequency we find that y(and therefore Ipl) is independent of frequency when

0o 1 fr\

7 = 2 (5)

Note that this condition(i) is independent of er

(ii) is independent of / (provided that / is not too large)(iii) coincides with the normal criterion for midband

working in the empty guide.These three desirable features are clearly confirmed by Figs.2—4, as also is the loss of broadband performances when /approaches too closely to a quarter wavelength.

0-9

0-8

0-7

0-6

0-5

0-4

03

0 2 -

0-1 •

0Q60 70 80 90 100

frequency, GHz—•110 120

Fig. 2

The phase of the reflection coefficient has no particu-larly interesting properties. As 0/ •+ 0, 0 -• — 7r/2, the devicethen corresponding essentially to a shunt capacitance ona transmission line.

In using the device as a standard, numerical values ofIpl and 0, of course, are obtained with the help of a com-puter program. It is useful, however, to have simple form-ulas available for the errors in Ipl and 0 arising from errorsin er, / and kc. (We can assume that the frequency is knownwith negligible uncertainty). Error formulas are presentedin Section 3.

0-90

0-80

0-70

060

_ 0-50

— 0 40

030

0-20

010

0 060 70 80 90 100

frequency, GHz -*•110 120

Fig. 3

090

0-80

0 70

0-60

0 50

040

030

020

0 10

0 070 80 90 100

frequency, GHz-110 120

Fig. 4

3 Sources of error

First, we consider the effect of an error in er. The relevantformulas are

20der 2-.3/2

(6)

der -%v>«« (7)

A specific numerical example may be useful: let er = 2-54,/ = 0-1415cm and / = 9-276GHz. Then, taking the guideto be standard X-band waveguide, we find Ipl = 0-291.

MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3 121

Now suppose that there is an uncertainty of ±0-01 iner. The corresponding uncertainties in Ipl and 0 are

5lpl = ±0-0015

50 = 0-10°

Next, we consider the effect of an error in the length /.The relevant formulas for this case are

dlpl

dl

0 /0 0o— —2 \0o 0

cos

2T 3/2 (8)

0o0 / 0 0o\ . .— —) cos0/2 \0O 0 /

dl (9)

Using the same basic design data as before, we now find,for 5/ = ±0-0025cm, the results 5 Ipl = ±0-0043 and50 = 0-45°.

A comparison with the reduced-height standard will nowbe made. Taking lpl= 0-291 again, the height reductionratio needed is 0-741. Consider first the effect of an errorof ±0.0025 cm in the reduced height. We find that 5 Ipl = ±0-003 and 50 = 0-029°. Next, consider the effect of anerror of ±0-0025 cm in the length. We now find that5 lpl= ±0-00005, and 50 = 0-177°. In the reduced-heightcase, we have assumed that the length is precisely \g0/4,and that the discontinuity capacitance associated with thechange of height may be neglected.

Although the reduced-height waveguide has the ad-vantage over the dielectric-loaded guide that it is lesssensitive to dimensional imperfections, it is not too diffi-cult to meet the tolerances required by the dielectric-filled guide for standards use.

In Section 4, we shall see that the dielectric-filled guidedoes permit a calibration procedure not available to thereduced-height guide. This, coupled with the very simpletheory of the device may be regarded as an advantagesufficiently substantial to outweigh the somewhat in-creased sensitivity to dimensional accuracy.

In use, an additional source of error arises from resi-dual reflection from the matched-load termination. Thisapplies also to the half-round post and to the reduced-height waveguide. In all three cases, the worst-case error inthe desired reflection coefficient is of the same order ofmagnitude as the reflection coefficient corresponding tothe residual reflection.

4 Calibration technique

One obvious advantage of the dielectric-filled guide overthe reduced-height guide is the complete absence of dis-continuity capacitances. The corresponding elimination ofthe need to calculate these capacitances is, in itself, rela-tively minor.

A less obvious advantage arises from the fact that theimpedance ratio of the two guides is equal to the inverseof the phase-coefficient ratio. We assume that the wave-guide dimensions and the length / are accurately known.

122

At a given frequency j30 can then be calculated, and itremains only to find j3 to obtain the impedance ratio.

The following technique can be used: It is assumedthat several identical devices available. Suppose n are stackedtogether and inserted in a perfectly matched waveguide.The frequency is now adjusted until the system is oncemore perfectly matched. We then know that

= m-n (10)

at this particular frequency.

The integer n is, of course, known (3,4, 5 would be suit-able values). The permittivity will, in practice, be sufficientlywell known that inherent ambiguity in m can be removed(m = 1 is a suitable value). We therefore know j3 in terms of/ from eqn. 10, and hence |p| and 0 can be calculated fromeqns. 1 and 2 without the need to know er explicitly. Ofcourse, this technique is only directly applicable at certainspot frequencies at which eqn. 10 can be satisfied, and, inpractice, only two or three such frequencies can con-veniently be determined.

However, from eqns. 3 it follows that

2 _02 = ( e r -

••-»(?) (11)

Thus, if 0 and /30 are known at any one frequency / , er

can be deduced from eqn. 11. Then eqn. 11 (or eqn. 3)can be used to enable |3, and hence p, to be found at anydesired frequency.

Experimentally, all that is needed is a microwave bridgecapable of detecting very small reflections, and an accuratefrequency meter; e.g. a frequency counter.

When the procedure outlined here is adopted, oneadditional advantage is found. If there is a small airgapbetween the dielectric and the metal, the value of er de-duced from eqn. 11 is the effective permittivity. It can beshown that when a very small airgap exists, the effectivecharacteristic impedance of the partially filled guide willstill be inversely proportional to |3, so that this perturbedvalue of j3 is, in fact, the correct value to use in eqns. 1and 2. Although the effective er deduced from eqn. 11 willdepend on frequency when an airgap exists, this dependencewill normally be extremely small over the useful frequencyband. There will also be a discontinuity capacitance whenan airgap exists, but this will be entirely negligible for asmall airgap. In practice, it is not difficult, with the veryshort sections needed, to ensure an extremely good fit,and the value of er deduced from measurement on the newdevice differs from the value determined by the openresonator measurements by a very small amount, wellwithin the limits of experimental error, as will be shown inSection 5.

It should be mentioned that a trapped-mode TE2o-moderesonance is possible at 11-689 GHz for the / = (0-1210 in)device used in our experiments. This is unlikely to beexcited in a well-made device, but was, in fact, outside ourrange of measurements.

5 Experimental procedure and results

Several methods for measuring the reflection coefficient inphase and magnitude of a 2-port device were considered.The nodal shift method5 for loss-free devices offers satis-factory accuracy. However, it is laborious, for over a dozen

MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3

accurate measurements of position have to be made ateach frequency.

The method eventually employed uses a standing-waveindicator (s.wi.) and a short-circuit plunger. The schematicis shown in Fig. 5. A short circuit is first placed at referenceplane T and a convenient node T' found using the s.w.i.;this node serves as a reference plane. The dielectric slabis then placed in position, with the short circuit at planeT" and the shift in the node position dx in the s.w.i.measured.

Then

r

-S]2 = 0

T"

v v

! I! I

U-

I I

I I

where Sn and Sl2 are the scattering coefficients for thedielectric slab.

The short circuit is now moved back a quarter wave-length and the nodal position d2 in the s.w.i. measuredagain.

Then

Sn) + Sn(l-Sn) + S212 = 0

Eliminating Sn from the above two equations one has

11

From which-d2)

3 + 2(cos 2pd! - cos 2j3tf2) ~ cos 2/3 (dx -d2

and

= cos"1 -\Sn\+d2)

cos(l(dl —d2)

Essentially, the method is a nodal shift technique, althoughrequiring only two nodal shift measurements at each fre-quency. It should be pointed out that the test device neednot be lossfree and, furthermore, no further measurementsare necessary to determine Sn as well.

The accuracy of the method is essentially dependent onthe accuracy with which (dx — d2) is measured. The errorin Sn due to this error is then

and 50

tan

where D = di —d2.The permittivity was found at a spot frequency using

a magic-r junction as a bridge. The method is outlinedin Section 4. A stack of four identical dielectric slabs wereused in the measurement. A null was found at 8-739 GHz.Using the formula

Fig. 5

with / = 04806 in ± 0 0001 in we find

er = 2-537 ±0-008

This is consistent with the values ranging from 2-536 to2-538 given by Cullen et al* for the same dielectric material(polystyrene) using an open-resonator technique.

Table 1: Reflection coefficient of dielectric slab

Frequency I Pith \P\exp

GHz800058-25008-50069-00039-5000

10000010-5000011 0005

0-535 ± 00010-522 ± 00010-513 ±0-0010-501 ± 00010-494 ± 00010-491 ± 00010-489 i 00010-489 ± 0001

0-532 ±00100-514 ±00100-509 ± 00110-495 ± 00140-488 ±00170-483 ± 00210-490 i 00240-463 i 0030

—139-9.10-2-140-7 10-2-141-610-2-143-8 10-2-146-2±0-2-148-810-2-151-4 10-2-154-010-2

-1400 i 0-3-140-0 i 0-3-141-0 10-3-1430 i 0-4-145010-6-1490 i 0-8-1520 1 10-1530 1 1-3

er = 2-537 t 0008, ld = 0-1208 in ± 0001 in

MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3 123

Table I shows a comparison between theory and experi-ment for one of the dielectric slabs used in the stack offour. The theoretical values for the reflection coefficientwere found using the permittivity determined by thebridge technique.

At 8-50 GHz the discrepancy between calculated andmeasured phase angle </> is marginally greater than theestimated combined error limit. In general, however, thecalculated and measured reflection coefficients are in verysatisfactory agreement in both magnitude and phase.

6 Conclusions

We have shown that a short length of dielectric-filledwaveguide can be used as a standard of reflection coeffic-ient magnitude.

It has been shown that when several identical devicesare available, a calibration technique can be employedwhich removes the need for accurate knowledge of thepermittivity of the dielectric, and provides inherent com-pensation for the effect of any airgaps that may exist.This calibration technique requires only the accuratemeasurement of the frequency at which a reflection coeffic-ient is defined by the dimensions of the guide and thelength of the dielectric-filled section.

Our experiments support the theory, and confirm ourbelief that the device may prove to be a useful standardof reflection coefficient.

7 Acknowledgments

The authors wish to acknowledge their indebtedness toA.E. Bailey, of the National Physical Laboratory, for thebenefit of a most valuable discussion. Prof. Cullen is alsovery grateful to M. Jean Le Mezec, of Centre Nationald'Etudies des Telecommunications (Lannion), Brittany,France, for arranging for his period of sabbatical leave tobe spent in the stimulating CNET laboratories, duringwhich time part of the work described here was com-pleted.

8 References

1 BEATTY, R.W., and KERNS, D.M.: 'Recently developed micro-wave impedance standards and methods of measurement', IRETrans. 1958,1-7, p. 319

2 KERNS, D.M.: 'Half-round inductive obstacles in rectangularwaveguide',/. Res. Nat. Bur. Stds., 64B, pp. 113-139

3 BEATTY, R.W.: '2-port Xg/4 waveguide standard of voltagestanding-wave ratio', Electron. Lett. 1973, 9, (2), pp. 24-26

4 CULLEN, A.L., NAGENTHIRAM, P., and WILLIAMS, A.D.: 'Im-provement in open-resonator permittivity measurement', ibid.,1972, 8, pp. 577-578

5 FEENBERG, E.: 'Relation between nodal positions and standingwave ratio in a complex transmission system', /. Appl. Phys.,1946,17, p. 530

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124 MICROWAVES, OPTICS AND ACOUSTICS, APRIL 1977, Vol. 1, No. 3