rectangular-to-circular waveguide transitions …rectangular waveguide bends between two of the...

JHU/APL

TG 1375 AD-A213 925SEPTEMBER 1989

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RECTANGULAR-TO-CIRCULAR WAVEGUIDETRANSITIONS FOR HIGH-POWERCIRCULAR OVERMODED WAVEGUIDESWILLIAM A. HUTING

DTIC1 ELECTE

OCT33 11989

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I SEPTEMBER 1989

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II RECTANGULAR-TO-CIRCULAR WAVEGUIDEII TRANSITIONS FOR HIGH-POWER

CIRCULAR OVERMODED WAVEGUIDESIWILLIAM A. HUTING

IIIII

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1 TITLE (Include Security Classification)

Rectangular-to-Circular Waveguide Transitions for High-Power Overmoded Waveguides

12 PERSONAL AUTHOP(S)

Huting, William A.13a, TYPE OF REPORT 13b TIME COVERED 14. DATE OF REPORT IYear, Month, Day) 15. PAGE COUNT

Technical Memorandum FROM 0,00 TO o.o0 September 1989 3016 SUPPLEMENTARY NOTATION

17 COSATI CODES 18 SUBJECT TERMS

FIELD I GROJP SUB-GROUP Communication Pow~er WaveguidesMicrowaves RadarModes TransitionsOvermoded Transmission

19 ABSTRACT (Continue on reverse if necessary and identify by block number)

Circular overmoded waveguide is being investigated at The Johns Hopkins University Applied Physics Laboratory (JHU/APL) for low loss, high powermicrowave transmission near 3.0 GHz. Unfortunately, most microwave power sources and receivers are compatible with single-mode rectangular waveguiderather than overmoded circular waveguide. This problem is usually solved by using metallic waveguide transitions. Design objectives for such transitionsinclude high power-carrying capability, low mode conversion, low reflectivity, and low transmission loss. An overview of the theory of waveguide transi-tion operation is presented. It is concluded that a prerequisite for a comprehensive waveguide transition analysis is to solve the uniform waveguide problemfor the various cross sections of the transition. A review is included that describes some of the difficulties encountered in developing A computer programpackage that analyzes uniform waveguides with arbitrary cross sections. An overview is included of important waveguide design issues, several proposeddesigns, and several different tasks and problem areas one might encounter in a long-term study.

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THE JOHNS HOPKINS UNIVERSITY

APPLIED PHYSICS LABORATORYLAUREL, MARYLAND

5 ABSTRACT

Circular overmoded waveguide is being investigated at The Johns Hopkins Uni-versity Applied Physics Laboratory (JHU/APL) for low loss, high power micro-wave transmission near 3.0 GHz. Unfortunately, most microwave power sourcesand receivers are compatible with single-mode rectangular waveguide rather thanovermoded circular waveguide. This problem is usually solved by using metallicwaveguide transitions. Design objectives for such transitions include high power-carrying capability, low mode conversion, low reflectivity, and iov.' transmissionloss. An overview of the theory of waveguide transition operation is presented.It is concluded that a prerequisite for a comprehensive waveguide transition analy-sis is to solve the uniform waveguid, p,'o, for the various cross scctions of the

transition. A review is included that describes some of the difficulties encounteredin developing a computer program package that analyzes uniform waveguides witharbitrary cross sections. An overview is included of important waveguide designissues, several proposed designs, and several different tasks and problem areas one5 might encounter in a long-term study.

I

I

I Aaoesston For

M IS C__RA&t1DTIC TARiUlnn w ced 0JustJ titation.

I By_Dintrb ut Io /5 Availability COqOS

Avail a nd/orDist s peclel

I THE JOHNS HOPKINS UNIVERSITY


L ist of Illustrations ...................................................................... v

1. IN T RO D U C T IO N ................................................................... 1

2. WAVEGUIDE TRANSITION THEORY ..................................... 7

n 3. COMPUTER SOFTWARE DEVELOPMENT ............................ 17

4. FUTURE WORK ................................................................. 213 R eferences .................................................................................... 23

III ILLUSTRATIONS

I Sheathed-helix waveguide ..................................2 Multiport rectangular TE 0 to circular TE0 , mode transducer .............. 23 Desired electric field lines for the multiport rectangular TE,0

to circular TEO, m ode transducer .................................................. 34 M ode-transducing elbow .............................................................. 45 The M arie transducer ............................................................... 56 Desired electric fields for the Marie transducer .............................. 57 Generalized waveguide transition geometry ................................... 78 Waveguide transition geometry for electromagnetic field

boundary conditions .............................................................. 139 Waveguide cross section approximated as a union of triangles ............ 18

10 Nonuniform waveguide with transverse planes .............................. 1911 Computer program package for scattering matrix determination .......... 2012 Waveguide run with regions capable of supporting a spurious mode .... 2213 Waveguide run in which some of the spurious mode resonances

have been elim inated .............................................................. 22

I

THE JOHNS H("KINS UNIVERSITY


5 1. INTRODUCTION

The sheathed-helix waveguide (Fig. 1) that is being investigated at APL for lowloss, high power microwave transmission I consists of a closely wound insulated wiresurrounded by a two-layer jacket. The inner layer of the jacket is a lossy dielectricwhile the outer layer is a good conductor. Several short sections of this waveguidehave been built at APL for use at S band (Table 1). Some of these sections are curvedfor use in 90' bcds, while others are straight for use in straight waveguide runs.The purpose of the helix and the dielectric is to suppress all of the modes abovecutoff except for the TEO, mode.

Microwave transmission using the TEO, mode of a circular waveguide is charac-terized by an attenuation that decreases monotonically with frequency. In order toachieve low attenuation and to minimize phase distortion, it is necessary to operatewell above the TEO, cutoff frequency.) This implies t':at there are seveai modesabove cutoff in addition to the four modes with cutoff frequencies less than or equalto that of the TE0 mode. One major design goal for the sheathed-helix waveguide

Conducting~pipe h

Dielectricsheath

E=E0 Er

Helically -woundinsulated wire

Figure 1 Sheathed-helix waveguide.

Table 1u Characteristics of overmoded waveguides built at APL.

Parameter(see Fig. I) Value

I a 8.0 cmt 0.8 cm

Cr 5.2 - j0.0523.0-3.6 GHz

Wire gauge No. 14

I'W. A. Huting, J. A. Krill, and K. .1. Webb, "New Developnents in Circular OvermodedWaveguide," in 1988 URSI Radio Science Meeting Programs and Abstracts, International Unionof Radio Science. Syracuse, N.Y., p. 202 (1988)."H. E. Rowe and W. 1). Wartcrs, "Transmission in Multimodc Wa\ cguide with Random Imper-

fections," Bell .vst. Tech. J. 41, 1031-1170 (1962).U



is to avoid the transfer of energN from the TE 1 mode to the other modes. Theseother modes are usually called "unwanted moder" or "spurious modes." The energytransferred to the spurious modes cannot be recovered because of dispersion andmatching problems at the receiver. Therefore, significant spurious mode excitationcould greatly increase transmission loss.

There are at least three major sources of coupling to spurious modes that must

be considered in designing a run of overmoded waveguide. One of these sourcesis the presence of geometrical imperfections in the waveguide; an effort is now un-der way to investigate the effects of fabrication tolerances. L' The second majorsource of spurious mode excitation ("mode conversion") is the use of sharp bendsin the waveguide run. 3 Reference 3 gives a procedure for designing such bends forlow mode conversion. The third source of mode conversion occurs (1) within theinterface between the microwave power source and the waveguide run, and (2) withinthe interface between the waveguide run and the receiver. This additional mode con-version occurs because most microwave power sources and receivers are not com-patible with circular overmoded waveguide. Rather, they are compatible with thedominant TE10 mode in standard-size rectangular waveguide. 4 In order to matchcircular overmoded waveguide with such devices, it is necessary to use rectangular-to-circular waveguide mode transducers. Usually, such a mode transducer consistsof a nonuniform waveguide whose cross section changes gradually along the lon-gitudinal, or axial, dimension of the waveguide. In all nonuniform waveguides, somemode conversion is inevitable.-' 6 Quantification of this mode conversion is an es-sential step in the evaluation of waveguide system performance. Furthermore, modifi-cation of existing waveguide transition designs may be necessary to achieve acceptableloss and power levels.

One waveguide transition that may be suitable for high power use has been pro-posed by Zinger and Krill.' That transition (Fig. 2) is to be made out of copperor some other good conductor. The reason for the presence of four rectangularwaveguide ports instead of only one (as in most previous designs) is that the power-carrying capability of circular waveguide may be more fully utilized for the waveguidesizes under consideration. Figure 3 illustrates the operation of this waveguide tran-sition, showing the successive cross sections of the device, together with the electricfield lines of the desired waveguide mode.

Figure 2 Multiport rectangular TE10 to circular TE 01 mode transducer.

1H.-G. Unger, "Normal Mode Bends for Circular Electric Waves," BellSys'. Tech. J. 36, 1292-1307(1957).

'S. S. Saad, "Computer Analysis and Design of Gradually Tapered Wavcguide with ArbitraryCross Sections," Ph.D. dissertation, University College, London (Oct 1973).

'G. Reiter, "Generalized Telegraphist's Equation for Waveguides of Varying Cross Sections,"Proc. IEE 106B, Suppl. 13, 54-57 (1959).

'L. Solymar, "Spurious Mode Generation in Nonuniform Waveguide," IRE Trans. Microwave.Theory and Techniques MTT-7, 379-383 (1959).W. H. Zinger and J. A. Krll, "Multiport Rectangular TE1,, to Circular TEO, Mode TransducerHaving Pyramidal Shaped Transducing Means," U.S. Patent No. 4,628,287 (9 Dec 1986).2|



I ___ FATA 44, A

.

4*-

Figure 3 Desired electric field lines for the multiport rectangular TE 1o tocirculai TE O, mode transducer.

As previously mentioed, another prominent source of mode conversion is the

presence of sharp intentional bends in the waveguide. Tests at S band indicate thatthe helical waveguide bends made at APL are characterized by acceptable insertionlosses and reflectivity levels,.l Unfortunately, in order to avoid unsatisfactory lev-els of mode conversion, helical waveguide bends must be sufficiently gentle; i.e.,they must have large radii of curvature. 3 If applications should require sharperbends, then the heli,.al waveguide configuration will not be suitable. One solution,proposed by Irzinski, Krill, and Zinger,' is to insert four commercially availablerectangular waveguide bends between two of the transitions described above (Fig.4). Currently, construction and testing of this waveguide "elbow" and the multiportrectangular-to-circular waveguide transitions are under way.

E. P1, Irzinski, .1. A. Krill, and W. 11. Zingcr, "Sharp Mlode- ranlsduccr flcnd for OvermodedWavguide," U.S. Patent No. 4,679,008 (4 Aug 1987).



MtMulpiport rectangular TEoto circular TET 1 mode

transducer

Commercially availablercar waveguidetan e ie

Multiport rectangular TE M0to circular TEde modets etransducer

Figure 4 Mode-transdlucing elbow.

One rectangular-to-c ir cular waveguide transition, the Marie transducer, has beenbuilt, analyzed, and tested. This device, which has been described as "the besttransition to TEop e" r0 is shown in Fig. 5, and the electric field lines of the desiredmode are shown in Fig. 6. Both the Marie transducer and the multiport transitionconsist, in part, of a transition between circular and cross-shaped waveguide. Be-cause of thi, the Mari ravgde transition opera to deeo tia teci-tions described above. One characteristic of the Marie transducer d at may makeit unsuitable for high-power use is that it has only one rectangular port.

A study of waveguide transition operation is under way. Design objectives fortransitions built at APL include low levels of mode conversion and high power-carrying capability. In order to achieve these goals, it is necessary to thoroughly

un.andersn the theory of waveguide transition operation, to develop analytical tech-

niques for the prediction of waveguide transition performance, and to validate suchanalytical techniques through extensive tests. Section 2 of this report discusses previoustheoretical work on waveguide transitions. Section 3 briefly describes work done

'S. S. Saad, .1. B. Davies, and 0. J. Davies, "Computer Analysis of Gradually Tapered Waveguidewith Arbitrary Cross Sections," IEEE Trans. Mficrowave Theory and Techniques MFU.26, 437-440(May 1977).

1"T. N. Anderson, "L.ow Loss transmission Using Overnioded Waveguide: A Practical 1981 Re-view of the State of the Art," WEEE AP/MTT-S Phfiladelphia Section Benjamin Franklin 1981

Symp. on Adv. in Antenna and Microwave rechnol., Philadelphia (16 May 1981).

4

THE JOHNS HOPKINS UNIVFRSITY

APPLIED PHYSICS LABORATORYLAUREL, MARYLANr)

Section 1 Section 2 Stction 3

Figure 5 The Marie transducer

I

+ A

I

IiI

I--

Figure 6 Desired electric fields lines for the Marie transducer.

at APL on computer programs based on this theoretical work. Future reports willdiscuss validation of these computer programs using scattering parameter tests andwill describe a computer-aided design package for waveguide transitions. Specifictransitions to be treatcd include the Marie transducer, the multiport rectangularTE,1 to circular TE(, mode transducer, and the mode-transducing elbow. Section4 of the report discus.es future wvork av.i idcntifies several anticipated problem areas.

I5


APPLIED PHYSICS LABORATORYLAUREL. MARYLAND

3 2. WAVEGUIDE TRANSITION THEORY

One of the earliest theoretical treatments of nonuniform waveguides was givenby Reiter, who used the well-known fact that uniform waveguide modal fieldsform a complete orthogonal basis for physically realizable electromagnetic fields ina wvaveguide. ' Extendin, ,his concept to waveguide transitions (Fig. 7), he assert-ed that the transverse electromagnetic field (i.e., the x and y components) could bewkritten as a Sum of the transverse fields of the uniform modes corresponding tothe local waeguide transition cross section,

I F, (xvy) = V,,,(z) e,,., xv) (I

3 and

l 'lx~y,.) = l,,.(Z)h,,, (x,y~z) ,,,,(2)

where Er is the transserse electric field,H, is the transverse magnetic field,Se,,. is the trans,,erse electric field of the inth uniform wvaveguidc

trode (suitably nortnalizcd),h,,, is the transverse magnetic field of the inth uniform w.aveguide

mode (suitably normalized).V,,, (z) is the so-called "equivalent voltage," andI,, (z) is the so-called "equivalent current."I

m z = L plane --

II

I

z = 0 planey

m Figure 7 Generalized waveguide transition geometry

IR. I. ( oll , In', I! lhcorv o/ (iuded c',s. \.c( iriA -Iill Itook ( oilipall}, Nc%. '()I I, (146f))

*7



The uniform waveguide modal transverse fields satisfy the orthonormalityrelations I

i \ e, (x,y,:).e (x,y, z) (Ix di, = . (3)sIand!

and h, (x,v,.z) .h (x,y,z) dx dy = 6, (4)

where the integration is over the local waveguide transition cross section and

6, = I i-=j, I

6,, = 0 otherwise.

The equivalent voltages and currents are determined by an infinite set of ordi-nary differential equations:

d = -j ,,,Z,,,I,,, T,, Vn (5)dzn=

and

dl,,, _ l,

=- yT,,, , , (6)dz Z11

respectively.Both TE and TM modes are included in Eq. 5 and 6. The variable 0,,, denotes

the wave number of the mth mode, the variable Z,, denotes the wave impedance

of the mth mode, and the "transfer coefficients" T,,,, (which describe coupling be-tween the two modes m and n) are given by5

Tr,, (Z) = e,, dx dy . (7)

S

The integration is over the local waveguide cross section. Equations 5 and 6 are knownas the Generalized Telegraphist's Equations. Most methods of solving these equa-tions start with a change of variables. Solymar proposed 6

= 'Z,,, (a,, + a,,7 ) (8)

and

I/- (a,,, - a,,, ) , (9)

8

I THE JOHNS HOPKINS UNIVFRSITY

APPLIED PHYSICS LABORATORYgLAUREL. MARYLAND

where a,,,' and ct,, denote, respectikely, the for%%ard-traveling and the backward-traveling parts of the rmth waveguide mode. Substitution of Eqs. 8 and 9 into Eqs.5 and 6 results in a new system of ordinary differential equations. Solylnar foundan approximate solution to tHese equations for the case where no mode goes throughcutoff inside the transition.' Subsequently, Saad, Davies, and Davies used Soly-mar's solution to analyze ,he Marie transducer.-" Although several modes in theMarie transducer do, in fact, go through cutoff, thereby violating one of the keyassumptions of Solymar's solution, rough agreement between theory and experi-ment was achieved.

The reason that some previous treatments avoid the case where a mode goesthrough cutoff inside the transition can be seen by examining Eqs. 8 an 1 9. Inthe plane where the mode goes through cutoff, the parameter \;Z,, in Eq. 8 willexperience a singularity if the mode is TE, and the parameter 1\Z,,, in Eq.9 will experience a singularity if" the mode is TM. In order to circumvent this, ourapproach will be to solve Eqs. 5 and 6 directly. By truncating the series in Eqs. 5and 6, the equations are cast into a form amenable to well-known numerical tech-niques (e.g., Ref. 12).

Two studies have, in fact, examined the solution of Eqs. 5 and 6 in some de-tail. "-'4 Two Marie transducers are available for testing. Once such tests have beenused to verify the numerical methcds, it should be possible to develop a computer-aided design package for waveguide transitions. A prerequisite for such a study isthe ability to specify the uniform vaveguide modes associated with the various tran-sition cross sections. Specifically, the modal electromagnetic fields are needed to com-pute the transfer coeficients of Eq. 7. A computer program that determines modal

eigenvalues and ele. romagnetic fields is described in Section 3 of this report.We conclude the present discussions by reviewing Reiter's derivation of Eqs. 5

and 6.' (We present this derivation because Ref. 5 is not very widely available inthe United States.)

The first step in deriving the Gencraiized Telegraphist's Equations (Eqs. 5 and6) is to resolve the electromagnetic fields and the del operator into longitudinal andtransverse components:

I

I E= F', + zE

H = H, + zH. (IO)

V V, + zazI

R. W. I I a m n g, .%iuneru cal u lerh odso Si 'un51s and Ftneers, N I c( i ra\ I I Il 1Boo k (o, n pa-ji r, Ne% Yorl, (1962).

If. l. lgel and E'. Kuhtn, "-( ompuler-Aidcd .. naly,,kand IDc,,ip of Circular WVaveguide lapei,,"IIEt: Trum. krowave' ftuon' and lechniqis M'I11-36, 332 336 (1988).

"W. A. liing and K. .1. Wchb, "Niumcrical 'solution of the ('on iilnlou W\avcguide tranmilionProhlem," I11:1: LraT ts. Micro wave 1hcorv and Tec'hniques (o he published in Noellher 19.

I9



The longitudinal components can be eliminated from Maxwell's Equations to ob-tain (e.g., Ref. 15, p. 186)

- - = j p(Hxz) - - V,[VI-(Hxz)] (11)

and

-H = jwe(zxE,) - ,.(z x E ) . (12)az jio . I(2

The basic procedure in this derivation is to insert the series expansions for E, andH, (Eqs. I and 2) into Eqs. 11 and 12 and to take advantage of the orthonormali-ty relations given in Eqs. 3 and 4. The last two relationships allow us to determinethe equivalent voltages and currents,

Vm(Z) = IfE,(xy,z)'e,,(xy,z) dxdv (13)S

and

I (H,= (x~y,z).h,,(x1y,z) dxdy, (14)respectively. I

As before, the integrations are performed over the local waveguide transition crosssection.

The basis functions e,, and hm have several helpful properties. First of all, thesevectors are related by a simple vector cross product,

h,, (xy,z) = zxe, (xy,z) . (15) 1Also, by using Eqs. 41, 43, 44, and 46 (which are presented later), one obtains thefollowing equations for TM and TE uniform waveguide modes,

V,[ V, 'e, Tr (x,Y,z)] + hn, TeM ef (x,Y, Z) = 0 (16)

and I

V ,[V h,,rTL (x,y,z)I + h,,TI.- h,,,1 rt (x,y,z) = 0 (17)

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves, Prentice-Hall, Inc., Engle- Iwood Cliffs, N.J. (1973).

10



We now derive Eq. 6 for the case where the subscript in denotes a TM uniformwaveguide mode. Taking the dot product of Eq. 12 with h,,.rl, and using Eq. 13,one finds that

- o 'z 'h,,:. , dx dV

S

I=- ] E ,,T~j JI - V,[V, (zxE, ) h,,,. T1 dx dy . (18)

It will now be shown that the integral on the right-hand side of Eq. 18 vanishes.To do this, one applies the two-dimensional form of Green's theorem (e.g., Ref.15, p. 188),

ds (A.vV B - B .vv,-A)

1dl [(A.n)(V,-B) - (B.n)(V,.-A)] , (19)

C

where n is the outward unit normal on the contour C. Substituting Eq. 19 into theintegral on the right-hand side of Eq. 18 results in

! ffVj[IT"(zxE,)]h'",T ' dxdy

I S= (zxEt)V,(V,h,,,T.,) dxdy

+ (h,,,r .fn)V, .(zxE,)diC

- [(zxE,) n] V ,t-h,.T, dl . (20)

C

From the Maxwell divergence equation for B, the first and third integrals vanish.The second integral also vanishes because h,,,Tr satisfies the boundary conditionfor a perfect electrical conductor on C.

Inserting Eq. 2 into the left-hand side of Eq. 18, and taking note of Eq. 14, results

in

d l'n'rv = -j V ,, , - I,, h,,,. dx dy (21)

dz. dz

Remembering Eqs. 7 and 15, one can show that Eqs. 6 and 21 are equivalent bysubstituting the following expressions:

f(2 2J3,, = \= m p( - h,,, (22)

II



and

Zn ,T - (23)

Equation 22 applies to both TE and TM modes. Therefore, the subscript TMhas been dropped. The eigenvalue h,, (not to be confused with the modal magnet-ic field h,,) is given in Eqs. 16 and 17 and in Eqs. 41 and 44 (which appear later).We now seek to derive Eq. 5 for the case where m denotes a TM mode. Proceedingas before, Eqs. 11 and 14 are combined to yield:

- E .e,,," dx dy

S

= jWiIl,,,r.. - j o VIV, . (H, Xz)]'e,,,. 7 , dx dy . (24)

Unfortunately, the integral on the right-hand side of Eq. 24 does not vanish. UsingGreen's theorem (Eq. 19),

SIeTM'*V,, (H, xz)] dxdyS

= I'(H, Xz)V,(V,,e,,.TM) dxdy

+ (em..Tfn)V,-(Hxz) dlc

- [(H, xz).-n], -emM dl . (25)c

Using the simple vector identity A • (B C) = B (C x A), and using Eqs. 14,15, and 16. the first term on the right-hand side of Eq. 25 is seen to be

IH,xz. V,(V,.em M) dxdy h' .Tm InTM (26)

Applying the Maxwell curl equation for H to the second term on the right-handside of Eq. 25, one obtains

(erM.n)V,.(H, xz) d, = (e,,.TM n) jWEE d. (27)

C C

From Fig. 8, one obtains the electric field boundary condition,

E, = -(tan 0) E,. n . (28)

12



IZIM* m

I,I

Figure 8 Waveguide transition geometry for electromagnetic field boundary conditions.

Equation 28 follows from the requirement that the electric field tangent to a perfectconductor must vanish. Inserting Eq. 28 into Eq. 27, and noting that emTM is par-allel to n, one finds that

(e,, TV'n) V,.(H, xz) dl = -jwf (tan k) E, em. TM dl. (29)

C C

Finally, the third term on the right-hand side of Eq. 25 is seen to vanish if one ap-plies the Maxwell divergence equation to e.,TM and uses the electromagnetic bound-ary conditions for a perfect electrical conductor. Combining Eqs. 24, 25, 26, and29 yields

ITE- e, mdx dy

(jW + hTV TV + (tan 0) E,.e.. TV dl. (30)

We treat the term on the left-hand side of Eq. 30 by invoking the differentiationtheorem of surface integrals with variable surface,

d x(x,y,z) dx dy = dx dy + X (tan 0) dl. (31)

I 13



Using this theorem, the left-hand side of Eq. 30 becomes

- -"e,,. 4 dx dy d dz

+ fEt.eTdx dy + (tan 0) E, e,,.M dl. (32)S z c

Combining Eqs. 13, 30, and 32 results in

d + = " - -- T V, e, - - dx dy. (33)dzjEn=I S aZ

Equations 7, 22, and 23 can be used to show that this result is equivalent to Eq. 5.Reiter's derivation of Eqs. 5 and 6 for the case where m denotes a TE mode is

quite similar to the foregoing, and will not be given here. The expression for a TEmode wave impedance is

Z, -E = -, . (34)0mr. TE

Before concluding this section, let us consider the convergence of the seriesrepresentations of the electric and magnetic fields and how such convergence is af-fected by the electromagnetic field boundary conditions. At the boundary of thetransition, the fields satisfy the conditions (Fig. 8)

E x m = 0 (35)

and

H-m = 0. (36)

Decomposing these requirements into longitudinal and transverse parts, we find that

n x E, = 0 , (37)

E, = (tan 0) E,.-n ,(27)

and

H, = (cot 4,) H,.n . (38)

14

THE JOHNS HOPKINS UNIVERSITYf


The boundary conditions satisfied by the vector functiois e,, and h,,, (the modalfields of a uniform waveguide) are

n x e,,, = 0 (39)

and

n, h,,, = 0. (40)

I Thus boundary condition Eq. 37 for the transverse electric field is satisfied by theindividual terms of the series in Eq. 1. Therefore, following the approach of Ref.16, we assume that the expansion is valid both in the interior of S and on C (theboundary of S). This is not the case for the magnetic field. The individual termsof this series satisfy Eq. 40 but the actual magnetic field must satisfy Eq. 38. Againfollowing the approach of Ref. 16, the series in Eq. 2 is assumed to be a valid repre-sentation for H, in the interior of S but not on the boundary of S. In the unionof S and its boundary, the series of Eq. 2 will represent a discontinuous functionand hence will not converge uniformly. Because of this, term-by-term differentia-tion with respect to x or y will certainly make the series diverge. However, suchdifferentiation has been avoided in the above derivation. Term-by-term integrationhas not been avoided, but the conditions for the validity of this procedure are not

stringent (e.g., Ref. 17, pp. 40-43 and 345-347). These general conclusions-namely,that one field seri, - is valid on the boundary but that the other field series is not-have been noted in other treatments of nonuniform waveguide theory. 16.18

I

I

'6 A. F. Stevenson, "General Theory of Electromangetic Horns," J. Appl. Phys. 22, 1447-1460(1951).

17 E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, London (1939).18H.-G. Unger, "Circular Waveguide Taper of Improved Design," Bell Syst. Tech. J. 37, 899-912

(1958).I 15

I THE JOHNS HOPKINS UNIVERSITY


3. COMPUTER SOFTWARE DEVELOPMENT

The ability to calculate the modal fields and wave numbers associated with uni-form waveguides is a prerequisite for solving the system of differential equationsdescribed in Section 2. This has to be done for a wide variety of waveguide crosssections. This section lists the equations that determine the modal fields, describesnumerical techniques for solving these equations, and describes work done at APLin this area. For a TM mode, the starting point in specification of the fields is tosolve the two-dimensional Helmholtz equation with Dirichlet boundary conditions,

II hV, , + h ,. ,,, = 0 (41)

* and

0,,,c = 0 on C . (42)

the transverse field functions and the wave number may be found from

U e,,,.r~T = V ,,,.rk T , (43)

h,,, = zxe ,,, (15)

and

0 = ,.AIE h - (22)

The equations for the modal magnetic field h,, and the wave number 0,, apply toboth TE and TM modes. Therefore, the subscript TM has been dropped. For TEfields, one starts with the Helmholtz equations with Neumann boundary conditions,

V" 2 ,,,.TE + h. E Vn.TE = 0 (44)

andI 8I¢,,,r

an 0 on C. (45)an

I The wave number may be found from Eq. 22 and the field functions may be ob-tained through

Ih, = V, ,,.Tn: (46)

* 17



and

e h,,, xz . (47)

One procedure that can be used to solve the Helmholtz equation numerically isthe finite element method (FEM). An FEM package has been ordered from McGillUniversity and installed at APL on an Amdahl 5890 computer system. Validationof this package has been accomplished by solving waveguide problems with knownclosed-form solutions. For descriptions of the FLvI method, see Refs. 19-22. In or-der to run the McGill University FEM package, it is necessary to approximate thewaveguide cross section as a union of triangles (Fig. 9). This package does not have

Figure 9 Waveguide cross section approximated as a union of triangles.

19A. Konrad and P. P. Silvester, "Scalar Finite-Element Package for Two-Dimensional Field Prob-lems," IEEE Trans. Microwave Theory and Techniques MTI-19, 952-954 (1971).

20P. P. Silvester, "A General High-Order Finite-Element Waveguide Analysis Program," IEEETrans. Microwave Theory and Techniques MT-17, 204-210 (1969).21p. P. Silvester, "High-Order Polynomial Triangular Finite Elements for Potential Problems,"International J. Eng. Sci. 7, 849-861 (1969).

22p. P. Silvester and R. L. Ferrari, FinitL ElnL'nts for Electrical Enginc:rs, Cambridge UniversityPress, New York (1983).

18



a routine for generating these triangles but instead depends on user-generated in-put. For the large number of waveguide cross sections to be treated in our work,the triangle-generating process is complicated and formidable. Software that gener-ates such triangles has been ordered from Los Alamos National Laboratory andinstalled on the Amdahl system. This triangle-generator (or "mesh generator") ispart of a finite-difference package called POISSON/SUPERFISH, which was de-veloped for the Department of Energy for the purpose of solving magnetostatic andelectrostatic problems and RF cavity problems.2 3'24 The two routines that are rele-vant to our work are AUTOMESH and LATTICE. AUTOMESH reads in bound-ary data and mesh size specifications and prepares a preliminary mesh. LATTICEuses the method of Ref. 25 to refine the mesh. AUTOMESH has been successfullytested by running example problems from Ref. 23. Testing of the routine LATTICEis under way.

Once work is complete on the mesh generator and the FEM package, we will beable to analyze waveguide transitions by performing a uniform waveguide analysisfor some large number Al of waveguide transition cross sections (Fig. 10). The vec-tor functions e,,, Cadi L, lound by taking the gradient of the solutions to the Helm-holtz equation. Since the FENI package approximates these solutions as polynomialsin x and y, analytical instead of numerical differentiation will be used. Numericaldifferentiation will have to be used in differentiating e, with respect to z (Eq. 7)Finally, the integration of Eq. 7 and the ultimate solution of the truncated systemof differential equations will also have to be accomplished numerically.

Figure 11 shows a flow chart for computing the transmission and reflection prop-erties of a waveguide transition. To summarize, the first step in analyzing a waveguidetransition is to solve the uniform waveguide problem for some large number M oftransition cross sections. The routines AUTOMESH and LATTICE approximateeach cross section as a union of triangles. The FEM package then solves for theuniform waveguide modes. Finally, the transfer coefficients (Eq. 7) are integratednumerically, and the system of differential equations is truncated and solved. Todate, the mesh generation package and the FEM package have been installed at APLand tested. Future work includes implementation of routines to set up anJ solve

the system of differential equations. I3.14IN=1 2 3 M

Figure 10 Nonuniform waveguide with transverse planes.

2'M. T Menzel and H. K. Stokes, "User's Guide for the POISSONiSUPERFISH Group of Codes,"

Los Alamos National Laboratory LA-UR-87-115 (Jan 1987).24j. L. Warren, G. Boicourt, M. Foss, B. Tice, M. T. Menzel, and F. K. Stokes, "Reference Manual

for the POISSON/SUPERFISH Group of Codes," los Alamos National Laboratory LA-UR-87-126 (Jan 1987).

2'A. M. Winslow, "Numerical Solution of the Quasilinear Poisson Fquation for a NonuniformMesh," J. Computational Phyvs. 1, 149-172 (1966).

19



IStart

Do N. 1, M

Automesh

Lattice

utine

JFEM routie

Computation oftransfer coefficients

Differential equationsolver

Figure 11 Computer program package for scattering matrix determination.

20

THE JOHNS HOPKINS UNIVERSITYAPPLIED PHYSICS LABORATORY3 LAUREL, MARYLAND

4. FUTUJRE WORK

A~s previously mentioned, future work inClUdes completion of the computer Soft-ware development, experimental validation of the com1puter programs by scatteringparameter measurements, arid development of a com1puter-aided design procedure.Specific trar sitions ,o be treated include the Marie transducer, thle muh~liport transi-tion, and the iiode-transducing elbow. To date, the tolloss ju tasks and problemI areas have been identified:1. Uniform Waveguide Mode Compu71.tation

Development of' thle routine [.ATTl( n iust he finishcd. Once this is done, theuniform waveguide modes for two \Iarie t ransducers %k ill ble compu11.ted, along ss iththle associated transfer coefficients. ta qain

2.Solution of the Systemn of' Ordinary lDiffcrenta quiosIn order to specify the transmission and refle~ction properties of at \\m~ cuide Iran-I srtion, Eqs. 5 arid 6 mu1Lst be solved. I1li methods descibed in Ref's. 13 and 14 \\ ill

be Studied for possible applicability to the complicated transitions described in See-tioni .I 3. Mode conversion at a Met allicA' iclical Wa% eLuidc Interface

The ultimrate goal for this project is to efficient lv couple enrice into a round mser-ruoded sheathed-helix wNaveiguide. TheC unction bet ecn at metallic svas eguide tran-sition and a helical waveguide presents, a Sharp discontinuity to the elect romlacucticI waves, causing mode converston atid reflections.,(4. Computation of the Transmission and Reflect ion Properties of a Waveguide Tran-sitio n

The scattering matrix of a metllic 1wavegtuide transition is Specified bv solvingEqs. 5 and 6. Two Marie transdurcers are available to uIS for testing usingl art [iP8409C automatic network analyzer. In all tests, the Marie transducerS wil be con-nected to each other either by a plain copper circular waseguide or by a sheathed-I helix wkaveguide. We hope to achieve better agreement between theory and experi-ment than was possible in tile past."'5. Multiport Transition Analys is and D~esign

Analyze the multiport rectangular 1E1, to circurlar I', mode transducer andthe mode transduIcing elbow. As before, the arnal.%., wil! be accomplished by solv-ing Eqs. 5 and 6. The four rectangular berrds in the elbow\ will be analyzed by solv-ing the differential equations given by Schelkunoff. 2I 6. Spurious Mode Resonances

In a multrmodc ssavcguide system, thle presenice of' a region that earl Support Oneor more spurious modes -an lead to high transmission loss at frequencies for whichthe regzion is resonant for one of these modes. Previous designs of' the Marie tranls-I ducer have sougiit ~o eliminate such resonances by eliminating sonie of the spurriousmode-su~pporting regions (F'igs. 12 arid 13).42'I"" One such region that cannot be3 eliminated, of course, is thle circtular overmoded waseguide itself. Reference 2N in-

*6J W. Ltcehleider, 'Miode (on er~or1 ar the II Onct ion of 1-eid, wav e ad operip,2"Bell 5SVst. Tech. J. 38, 1- t17-1 329 (1959). -ieai oprPp.S. A. Schel kurnoff, ''(oml srsion ol Nlais wets I -quti'ir Irs into ( errealiied 1 clegraph st's [oqua-lions,'' Bell .S vst. Tech..4, 995-1043 t 1955).

- A. P. King and L. A. Marcatiti, "'! rarisoission loss due ro Resonance ofvIoosci Coupled Modesrn a Multimiode System, Btc/I S.ns. Tech. .1. 35, 99)-(X)6 (1956).29Y . N1. Isayen ko, 'H -1,, If,,[ ( orn, i n niros frans olrnt'r .' I'rmc. Third C olloquiip? )? ol,r ro-

w0~aw Cot,,,ncafuon, 13dap-fs Ibl,--~~' pp. 467 -475 (Ap 1906).'S. S. Saad, .t. 13. Davies. and 0. .1. tDajs C.'Arvi anld I )si (If a ( irL'ntar t[:,,, \ModeTransducer,' ,' J~:.. Alicrowai1's, ()/)I. and :Ictnwq. 1, 58 62 (1t977).

3 21



dicates that mode-suppressing circular waveguide tends to lower the quality factor,Q, of these resonances, thereby flattening frequency responses. In order to com-pletely eliminate this type of loss, spurious mode resonances must be located suffi-ciently far outside the operating band so that this flattening effect does not degradeperformance. I7. High Power Considerations

High power transmission can lead to two undesirable effects in waveguide transi-tions: excessive heat losses inside the wall of the transition, and electrical break-down within the air inside the transition. Design rules for maximizing power-carryingcapability are sought as part or this study.8. Design procedure for Waveguide TransitionN

One objective of this study is to deelop a computer-aided design package forrectangular-to-circular waveguide transitions such as the Marie transducer and the Imultiport transition. This package should be capable of dealing with tradeoffs amongthe goals of maximum power-carrying capability, low reflectivity, low transmissionloss, low mode conversion, minimum transition physical length, and location of the Ispurious mode resonances outside the operating band.

I ot/ \'- R fc//

Waveguide Circular waveguide Wavegu;detranson tranistion

Figure 12 Waveguide run with regions capable of supporting a spurious mode The spurious modecutoff frequency is shown as a function of position

Waveguide Circular waveguide Waveguidetransition transion

Figure 13 Waveguide run in which some of the spurious mode resonances have been eliminatedNote that two of the possible resonant regions are no longer present.

22

I THE JOH4NS HOPKINS UNIVERSITYAPPLIED PHYSICS LABORATORY

LAUREL. MARYLAN,

UThe author is eratul11 to D~r. Ke% in J. \\*ebb aftilh: L ixersir t fMaryland taor manyhelpf'ul discussions aiid suggestions.

WA. A.. J uin,.. A. Krill, and K. J1. \ ch. -Nex'. Dcx cloprncnms in Circular Over-maded \ax cuidc,'' in 19S L CRSI Radio Science W~eetill, Prog~ramis andl Ab-,tracts, Inlt ernatianal U'nian of' Radio Science. Syracuse. N.Y.. p. 202 (1988).11, E. Roee and W. D. \%arters. -Transmission in Niultimode Wavcuide w\ithI Random I mpert'ect ions," Bell .Svs. Tech. .. 41, 1031-1170 (1962).H. -G Unver, "Normial Made Bend, taor (Cirular Electric Waves,"' Bell .v. Tech.J1. 36, 1292-1307 (1957)IS. S. Saad, -'ompu)Lter N.nalxsis and Design of' CradUallx Tapered WaVe2uidexx ith Arbit rarv C~ross Sections,' Ph.D, dissertation. Uix ersitv- Collcee London(Oct 1973).GC. Reiter, -Generalized Telegraphist's Equation for Waxeguides of Varying CrossSections," Proc. lEE 106B, Suppt. 13, 54-57 (1959).

"I.. Solyrnar, "Spurious Mode Generation in Nonuniform Waveguide,'' IRE' Tran.

M.icro wave Theor ' and Techniques MITT-7, 379-383 (1959).\\. H. Zinger and J. A. Krill, "Multiport Rectangular TE,1 to Circular FE,Mode Transducer Having Pyramidal Shaped Transducing Means," U.S. PatentNo. 4,628,287 (9 Dec 1986).

'L. P. lrtinski, J. A. Krill, and W. H-. Zinger, "Sharp Mvode-Transducer Bendtaor Overmioded Waveguide," U.S. Patent No. 4,679,008 1(, Aug 1987).S. ~S.Saad 1.B.Daviesand 0.J.Daies, "Compter Anlyi ofGrdal

orv and Techniques VITT-26, 437-440 (May 1977)."T.* N. Anderson, "Low I ass Transmission Using Overmoded Waxeguide: A Prac-tical 1981 Reviex'. of the State at' the Art," IEEE AP/MTT-S Philadelphia Sec-tion Benjamin Franklin 1981 Sytnp. on Adv. in Antenna and Microwave Technol..Philadelphia (16 May 1981).H R. E. Collin, Fieldl Theor' of Guided B'ave.5, Mc~jraw%-Hill B~ook Comnpany. New

RW.Ilamming.,ueia Aetd o Scientists antd Eng'ineers, NlcGraxv -HillI Book Company, New York (1962)."H. Hlogel atid F. Kfiln, ''Comnputer-Aided A.nalysis and Design of Circular

WVaveguide Tapers,'' llLLi Trani.s. M1icrowave Theory and Techniques, MITT-36,I 332-336 (1988)."W. A. Iluting and K. .1. W\ebb. "Numerical Solution of the Continuous. WaveguideTransition Problem.'' submitted to IEE-E Trans. M1iicrowave TheorY an(I Tech-niques (to he published in Nox ember 1989).

II . B. Felsen and N. Marenyit, Radiation and .Scutterin) of If aye'., Prentice-Hall,Inc., ELnglex'.ood Cliffs, N.J. (1973).

I 23



"'A. F. Stevenson, "Gencrai Theorv of idlectromagnetic Horns, ' .1. Appl. Plhnr..22, 1447-1460 (1951).

'-E. C. Titchmarsh, The Tl'oir ojFTunclions, 2nd ed., Oxford University Press,London (1939).

'sH.-G. Unger, "Circular Waveguide Taper of Improved Design," Bell Svst. Tech.J. 37, 899-912 (1958).

I 9A. Konrad and P. P. Silvester, "Scalar Finite-Element Package for Two-Dimensional Field Problems," IEEE Trans. Microwave Theory and TechniquesMTT-19, 952-954 (1971).1". P. Silvester, "A General High-Order Finite-Element Waveguide Analysis Pro-grarn," IEEE Trans. Microwave Theor:v and Techniques MTT-17, 204-210 (1969).P. P. Silvester, "High-Order Polynomial Triangular Finite Elements for Poten-tial Problems," International J. Eng. Sci. 7, 849-861 (1969).

,2 P. P. Silvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Cam-bridge Uiversity Press, New York (1983).

'M. T. Menzel and H. K. Stokes, "User's Guide for the POISSON/SUPERFISHGroup of Codes," Los Alamos National Laboratory LA-UR-8'"-115 (Jan 1987).

2"J. L. Warren, G. Boicourt, M. Foss, B. Tice, M. T. Menzel, and H. K. Stokes,"Reference Manual for the POISSON/SUPERFISH Group of Codes," Los Ala-mos National Laboratory LA-UR-87-126 (Jan 1987).

2 A. M. Winslow, "Numerical Solution of the Quasilinear Poisson Equation fora Nonuniform Mesh," J. Computational Phys. 1, 149-172 (1966).26 J. W. Lechleider, "Mode Conversion at the Junction of Helix Waveguide andCopper Pipe," Bell Syst. Tech. J. 38, 1317-1329 (1959).

- S. A. Schelkunoff, "Conversion of Maxwell's Equations into GeneralizedTelegraphist's Equations," Bell SYst. Tech. J. 34, 995-1043 (1955).

" A. P. King and E. A. Marcatili, "Transmission Loss due to Resonance of Loose-ly Coupled Modes in a Multimode System," Bell Syst. Tech. J. 35, 899-906 (1956).

-Y. M. Isayenko, "H,,-Ho2 -H,) Continuous Transformer," Proc. Third Collo-quium on Microwave Communication, Budapest, Hungary, pp. 467-475 (Apr1966).

'S. S. Saad, J. B. Davies, and 0. J. Davies, "Analysis and Design of a CircularTE,, Mode Transducer," fEE J. Microwaves, Opt. and Acoust. 1, 58-62 (1977).

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