2. Modified "Stevenson" Theory

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1 RADIO SCIENCE Vo. (New Series), No., January 966 Sots in Dieectricay Loaded Waveguide! R. W. Larson and V. M. Powers Radiation Laboratory, University of Michigan, Ann Arbor, Mich. (Received March, 965; revised Juy 4, 965) Simpe modifications of formuas presented by A. A. Oiner are found to aow the determination of the impedance and resonant ength of a sot in rectanguar waveguide when the guide is fied with dieectric materia. The modifications are justified by reference to earier work by Oiner and others, and are verified by experimenta resuts. (It has been recenty earned that B. R. Cheo of New York University and H. Jones have coaborated on a very simiar study. Their work is in agreement and wi be reported in the near future.). Introduction This note describes a theoretica technique for the design of sots in rectanguar waveguide oaded with dieectric materia. It is found that simpe modifications of previous design formuas suffice to describe the sot characteristics. Experimenta verification is incuded. It is we known in the fied of array design that sot spacings greater than one free-space waveength between adjacent sots ead to grating obes [Eaton, Eyges, and Macfarane, 949, pp. 38 ff.]. It is common practice in array design to avoid grating obe probems by using haf-guide waveength spacing. Equa phase of the eements is then obtained by means of a 8 phase reversa between adjacent eements, introduced by aternating thir asymmetry. The. aternative approach, reducing the guide waveength beow a free-space waveength, has severa drawbacks [Eaton, Eyges, and Macfarane, 949, p. 39]. However, in the course of a design study for a dua-poarized sot array, where it was found impossibe to empoy phase reversa, dieectric oading seemed the best aternative. Since sot design formuas are universay presented for air-fied waveguides ony, it is beieved that presentation of formuas modified for dieectric-fied guides may prove usefu to others for the above appication, as we as to those interested in: () prevention of breakdown (especiay mutipactor) () reduction of waveguide size (3) increased fexibiity in sot ocation (4) reduction of sot ength (perhaps to aow simpe edge shunt sots to remain entirey in the narrow wa) (5) modification of the resonant resistance. The research reported in this paper was sponsored by the Aeronautica Systems Divi sion, AFSC, Wright-Patterson AFB, Ohio, under Contract AF33(65)-!45.. Modified "Stevenson" Theory Two approaches have been deveoped for the study of sotted waveguides. The first is that of Stevenson [948] which is presented in MKS units by Eaton, Eyges, and Macfarane [949]. The second, discussed in the next section, is that of Oiner [95n The Stevenson approach gives information on the magnitude of the resonant radiation resistance or conductance ony. The basic ideas of the Stevenson approach are to equate the radiated, refected, and transmitte<! power to the incident power and to obtain the radiated power from a knowedge of the radiation characteristics of an equivaent dipoe through appication of Babinet's principe [Booker 946]. It is apparent from a consideration of the fieds of the sot and the compementary dipoe that one must study a dipoe entirey in air (not at an air-dieectric interface) in order to anayze the sot radiating into an air hafspace. The magnetic fied of the dipoe can be the compement of the sot's radiated eectric fied ony if the dieectric is ignored. We must, therefore, evauate the dipoe resistance, RdiP for a ength a' not equa to A. /; RdiP can be obtained by the Poynting vector method [Booker, 946]. The far fied of a dipoe can be obtained by assuming a sinusoida current. The radiated power is obtained from a spatia integration of the Poynting vector and can be expressed in terms of sine and cosine integras, by using partia fraction expansions and integration by parts. When this power is set equa to I, the radiation resistance can be put in terms of powers of a'/a. The vaue obtained in this way is so cose to a simiar vaue given by Oiner that the coefficients of his expansion wi be incorporated into this resut [Oiner, 957, eq (4)]: (a') [ ( a') (a')4] Rd;p= A.o +.3 Ao. ()

2 3 For a haf-waveength dipoe, this reduces to the famiiar 73. n. The Stevenson approach assumes that the sot is "resonant," aowing the simpifications that foow from having a rea refection coefficient. The reader is reerred to. Eton, Eyges, and Macfarane [949] for tiat discussion so that the subject here may be confined to the modifications that are required when using an interior dieectric. Then using Babinet's principe (Eaton, Eyges, apd Macfarane, 949, p. 95; or Booker, 946, eq ()], _ f.lo Rsot_R- 4 -, dip Eo (note use of Eo here), and otherwise foowing the present!tion of Eaton, Eyges, and Macfarane, we eventuay obtain the modifications given in tabe, where - o )I/ A.o Yo-- - o A.f %An error that shoud be noted occurs in eq (5bJ on page 93 which shoud read (re pacing K by k): So="oJ(fJ oi)' k () (3) >._{}f.= Ao arson and Powers V Er-(A.o/a) (4) It shoud be noted that the poes of h and ft are canceed by corresponding zeros in the cosine terms. These resuts have been presented in a somewhat redundant manner to faciitate comparison with the Stevenson forms [Eaton, Eyges, and Macfarane, 949; Ehrich, 96]. They reduce to the air-fied guide resuts where Er=. 3. Modified "Oiiner'' Theory It is ony from the more compete anaysis presented by Oiner (tabe ) for broadwa sots that the resonant ength and reactance can be found. The Oiner resuts reduce to the Stevenson resuts for the specia case when the sot is a haf waveength ong (a'= A.o/) and the guide is air-fied. The resonant ength is defined in this study as the ength for which the reactance is zero. Because a of the Oiner cases depend on the resuts for the admittance of a centered broadwa series sot, it is enough to consider in detai ony the TABLE. Resonant resistance or conductance Sot type Equation Definitions BroadWa shunt g=f,.9-;:: b cos: T.; sin -; {[ >., (a) ("''')] (-rrz)} f< ao( -o.374(f)' +.3 (f)'] f,=f Broadwa incined series r=/:{.3 ()[/(8) sin 8+}(8) cos 8]'} cos () cos (";) /(8)= e -' cos () cos (9) }(8)=----- J-e -' Edge shunt BrnadwaJ dispaced series } _ { ( [in 8 cos (sins) g-f,.3 >)a b ( ", )' - -sin 8 >-,, = Za' cos 8 -sin 8 A,_ a <' Ao =-cos 8+-sin 8 f,=j..\, u,,-fo --- r _,- (i;;i] -(J Nute typographica error m Ehrich f96j. eq (9-7).

3 Sots in Oieectricoy oaded Waveguide 33 TABLE. Modifications of Oiner formuas G, I (Y')' G - -! _!j (I + tan:t K' t} n Yo Yo Equations Definitions = k :?' [ (f)'- +.3 (f.)'] I ab [r -(a'<' ]' -;;;= a'b' 4 cos (Ta'/a} I Y' {8,; Y; [ (Y')' (8';)' (G';)' ]. }! -.-(-tan K;t)+ :t tank fk_!!.j. n::t Yo Yo Yo Yo Yo Yo _ ) tan K't+!J--tan K' y.,-y.,+ (C ' [Y' 8 ]' Yo o Y,t Yo u = { ()' [c+-n YIK;Ib']+ sin k,a' +[I+()'] <S-- ()'} Yo A! ko koa' n' 3n' a' B; I 8, b [ r b' 3 ( b)' ] -=--+- n Y, Y, k,, 6 b k, 8, 4b [ rb' ( b )' ("b') ] -=- ncsc-+- - cos_. - Ykg bkg b - ()' os :,". I()' j'[ + (."b' )' ] In ( ) kg, '"" ra' ("')' '-', try b -' cos- -9- a a when K is imaginary: cot (K;t = i coth (!K:,It)) ko= r(ko K = tr(jo., = J.,A- ()' (I - y.,-"' C Cikott'+tr+Cijk,a'-,.j S. = Sikott' +tr-si(aoa' -,.} tr.r sin t J,. cos t Siu= -dt.ci.\= -dt o t,. I In y=.577, y=.78 ".=!! modifications of those formuas for which experimenta resuts are aso given beow. The formuas corresponding to Oiner's other cases foow directy. The basic Oiner approach is to obtain an equivaent circuit for the sot, as shown in figure. The radiating centered series sot is characterized as an -pane T junction (series susceptance Bi and transformer ratio ni: ), a short section of sma waveguide (characteristic admittance Y, and a radiating junction (admittance Gri+ jbrj). It is obvious in this circuit that the parameters associated with the radiating junction (fig. a) are unaffected by the materia inside the waveguide. This corresponds to the use, in the preceding section, of Babinet's principe with a dipoe radiating in free space rather than one radiating at an air-dieectric interface. Thus, the vaues of Bri and Gri are unchanged by the presence of the dieectric. Simiary, the section of transmission ine characterized in figure c by Y is unchanged. However, the parameters of the -pane T junction shown in figure b must be carefuy re-evauated to determine how they are atered. This method gives the resuts in the preceding section on conductance at resonance as a byproduct, but our main interest is the effect on the susceptance. Oiner [957] does not derive formuas for ni and Bj, but notes the resuts of an earier study [95]; scribe how the resuts are obtained from a of these sources and how they are modified by the presence of a dieectric. STRUCTURE this report in turn draws upon sti earier materia by Marcuvitz [949, 95]. We sha ony briefy de- -b'f- T.Lb. m T Haf-space b c. Radiating Junction E-pane T Junction The Thick. Centered Series Sot EQUIVALENT CIRCUIT Bri L FIGURE. Sot structures and their equivaent circuits. B. ) r

4 34 Severa assumptions can be made that greaty simpify the work. First is that of ignoring the shunt eements (not shown in fig. I) in the more genera pi-equivaent circuit. This was done by Oiner because the experimenta resuts showed it to be a vaid approximation. Aso (again foowing Oiner) we sha drop Xc and Xd of the T representation given by Marcuvitz [I95I, p. 336]. For further justification we can refer to the theoretica-experimenta agreement discussed in the sixth section. The derivation of ni as given by Oiner foows that of Marcuvitz [95I, pp. VI-4 ff.]; it is easiy shown that the vaue of tj obtained from a variationa formuation is independent of the guide dieectric. In fact, with their normaization, Er does not appear in the derivation of ni Oiner obtained an expression for Bj by using the assumptions given above and by further assuming that the rate of change of energy storage in the main guide is unaffected by the stub-guide dimensions (athough it is a strong function of the sot dimensions). It aso foows that the main guide energy storage can be assumed independent of the stub-guide dieectric (in this case, air). The susceptance vaue for the ongitudina broadwa sot is thus obtained from the vaue for an -pane junction of identica guides, which is in turn obtained from the theory of a transverse sot in a uniform guide, which in turn is obtained from the theory for a capacitive iris. The ongitudina sot differs from the transverse sot ony by incusion of image terms. This difference is obtained by comparison of ongitudina and transverse vaues given by Marcuvitz [I95, p. 9, p. 35]. Since these resuts are given in terms of Ag, it is ony necessary to use It is thus found that the expressions given by Oiner can be used directy in the anaysis of dieectricay oaded guides as ong as (denoting by- the repacement of a symbo by the one to be used when Er I inside the guide) (5) Larson and Powers cay fied waveguide than that of obtaining the sot ength a' which gives a tota susceptance Be= for each E, For RG-5/U waveguide (a=.9 in., b =.4 in.) and a square-ended broad wa series sot that is resonant near /=.7 Gc (a'=.5 in.) when Er=.O, a curve is given in figure showing the variation of the normaized resonant ength, a' /A.o, with e, This variation was potted from cacuations at a number of frequencies with Er =.O,.6,.5, and 3., using the Oiner formuas as modified by (6). An empirica fit to this curve is given by a'/a.o =.945- O.I6(Er -) +.3(Er -) (7) 5. Experimenta Verification Two types of verification were attempted. Using e,.=.o and dieectric sugs with nomina vaues of e,.=.6 and.55, the impedance characteristics were measured near resonance at severa frequencies. The measured resonant vaues of both ength and resistance are seen to compare favoraby with the theoretica vaues in figure. In accordance with Oiner's suggestion for comparing round- and squareended sots. these measurements used a round-ended sot that ws percent onger (a'=.53 in.) than the vaue (.5I in.), used in the cacuations. The second type of verification consisted of comparing the theoretica and experimenta variation with frequency for the case, Er =.6. This is shown in figure 3; the agreement is reasonaby good and seems to exhibit the same variations observed by Oirier for an air-fied guide. The measurement technique consisted of measuring the shift in nu positions in an air-fied, sotted section when the sot was covered and when it was radiating. A probe under the sot was monitored to aow positioning of the short-circuit guide termination to give a current maximum at the sot. Knowing the ength of dieectric between the generator and the sot (typicay about.5 in.) aowed a cacuation of the sot characteristics taking into account the impedance transformation at the dieectric interface. Nomina rather than experimenta vaues of Er were used in these cacuations. (6) K stays the same (not so if the dieectric extends up into the sot). 4. Resonant Length There does not seem to be a more direct method of obtaining the resonant ength of a sot in a dieectri- 6. Concusions It has been shown that unexpectedy simpe modifications of existing sot formuas suffice to predict sot characteristics in dieectricay oaded waveguides. Athough the sot ength resuts presented are ony for the broadwa series sot, their use is undoubtedy justified for other cases. The use of the modified Stevenson formuas (given in sec. ) in conjunction with (7) for the proper sot ength shoud aow the soution of most design probems for sots in rectanguar guide.

5 Sots in Oieectricay oaded Waveguide 35 t.! r --Theory Experiment t I :.-;:-e-. E --- _, <.9 3. N a: u N c ;. c a:.j "'.,., E.8. z..; <: c a: E z :::: ----""':--"""' f\b. <r o-- \\ \: ' r---.",.7l ='-:: ;'t.o t. 3. Reative Dieectric Constant, r FIGURE. Normaized resonant sot ength and resistance versus dieectric constant. FIGURE 3. Impedance and admittance parameters versus normaized sot ength. The authors thank R. E. Hiatt for suggesting the probem and for severa hepfu discussions, H. Hunter and M. Larudee for assistance in computation, and ]. Ferris and K. Young for assistance in performing the experimenta measurements. The authors aso acknowedge the reviewers' suggestion to present the resuts in tabuar form. 7. References Booker, H. G. (946), Sot aerias and their reation to compemen tary wire aerias (Babinet's principe), Proc. Inst. Eec. Engrs. (London) 93, Pt. 3A, 6Q-66. Eaton, J. E., L. J. Eyges, and G. G. Macfarane (949), Linear array antennas and feeds, Microwave Antenna Theory and Design, MIT Radiation Lab. Series,, ed. S. Siver, ch. 9 (McGraw Hi Book Co., Inc., New York). Ehrich, M. J. (96), Sot-antenna arrays, Antenna Engineering Handbook, ed. H. Jasik, ch. 9 (McGraw-Hi Book Co., Inc., New York). Marcuvitz, N. (949), The representation, measurement, and cacuation of equivaent circuitry for waveguide discontinuities w"ith appication to rectanguar sots, Report No. R-93-49, PIB-37, Microwave Res. Inst., Brookyn, N.Y. Marcuvitz, N. (95), Waveguide Handbook, MIT Radiation Lab. Series, (McGraw-Hi Book Co., Inc., New York). Oiner, A. A. (Aug. 95), Equivaent circuits for sots in rectanguar waveguide, Report No. 4.-6,, Microwave Res. Inst., Brookyn, N.Y. Oiner, A. A. (957), The impedance properties of narrow radiating sots in the broad face of rectanguar waveguide, I, Theory; II, Comparison with measurement, IRE Trans. Ant. Prop. AP-5, No., 4-. Stevenson, A. F. (Jan. 948), Theory of sots in rectanguar waveguides, J. App. Phys. 9, No., (Paper --6)