Daikovsky, A G

Portugalov, Yu I

Ryabov, A D

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Particle Accelerators1982 Vol. 12 pp. 59-640031-2460/82/1202-0059$06.50/0© Gordon and Breach, Science Publishers, Inc.Printed in the United States of AmericaPRUD-CODE FOR CALCULATION OF THE NONSYMMETRICMODES IN AXIAL SYMMETRIC CAVITIESA. G. DAIKOVSKY, YU. I. PORTUGALOV, and A. D. RYABOVInstitute for High Energy Physics, Serpukhov , USSR(Received September 11,1981; in final form January 5, 1982)This paper offers the method and brief description of the program package for calculating ~igenfrequencies andelectromagnetic fields with azimuthal variations in axial-symmetric cavities of an arbitrary ~hpe. The method isbased on the representation of the equations of electrodynamics in variables phlf>' pelf>. Apart from frequencies andfields, the accumulated energy, distribution of losses in the metal, other characteristics important for applicationare also computed. The package offers wide possibilities for graphic representation of the field topology, facilitatingthe analysis, and optimization of complicated accelerating structures.1. BASIC EQUATIONS. THE CHOICE OFINDEPENDENT VARIABLES(2)(1)rot h = Ke div h = 0rot rot e = K2e,or in components using cylindrical coordinates(p, 'P, z),2 2 ahq; 1Ilh p + K hp = "2 - + "2 hpp a<p p2 _ 2 ahp 1Ilh'f) + K h'f) - -2- + 2 hq; (3)p a<p prote = Kh dive = O.The system of eqs. (1) can be written in the formof pair of second order equations for hand erot rot h = K2h1.1. We solve the problem of finding eigenfre-quencies and their corresponding electromag-netic fields in an axially symmetric cavity withsymmetry axis z and an arbitrary boundary p(z);only those modes that vary in the cyclic coor-dinate 'P are considered.For time-harmonic fields let us with no loss ofgenerality put H = h Cos wI and E = e Sin wI.With K = w/c the Maxwell equations in free spacebecomeAxially symmetric cavities are utilized in manyradio-frequency devices. In particular, they formthe basis of the accelerating structures in linearand cyclic particle accelerators.Accelerating structures commonly utilize sym-metric waves (having no variation in 'P) as "work-ing" ones. To compute such fields, the well-known programs SUPERFISH,I,2 LANS,3 andothers are used. .Waves having no axial symmetry are used lesscommonly and in many cases are treated as"stray" ones. Nevertheless, as far as rf separa-tors and H-wave structures are concerned cal-culations of such waves would be useful.In conventional structures, azimuthally vary-ing waves may be present for many reasons: tech-nology-affected violation of axial symmetry,presence of coupling or tuning elements, asym-metric beam excitation, etc. Therefore to com-plete the analysis of accelerating structures, onemust provide for the analysis of azimuthally vary-ing fields. The problems associated with suchmodes are related to the presence of all six com-ponents of the electromagnetic field.This paper presents a brief description of thePRUD program package designed to computeresonance frequencies and azimuthally varyingfields in an axially symmetric cavity of essentiallyarbitrary cross section. The description of themethod, package facilities, typical accuracy, ex-amples and applications are given.5960 A. G. DAIKOVSKY, YD. I. PORTUGALOV, AND A. D. RYABOVThe equations for (e p , eq>' ez) are similar. Hered is the Laplace operator in cylindrical coordi-nates.Applic~tion of the method of separation of var-iables to the system in Eq. (3) together with divh = 0 yieldshp = 71Cp (p, z) (nq»where(nq» = Sin nq> (nq» = Cos nq>or (5)<1>' (nq» = Cos nq> <1>' (nq» = - Sin nq>.For the electric field one obtainsep = ~p(p, z) <1>' (nq»eq> = ~q>(p, z) (nq» (6)ez = ~z(p, z) <1>' (nq».1.2 It is common knowledge4 that any free-spaceelectromagnetic field can be expressed as a sumof solutions derived from two scalar functions.If the system possesses an axis, then the fieldsderived from the two solutions are generally re-ferred to as the TE and TM modes with respectto that axis. In general each mode will have fivecomponents of E, H. If the boundaries possessrotational symmetry about the axis, the numberof field components is not reduced. But if thefield is rotationally symmetric, the number offield components is reduced to three for eachmode.We consider the case of a structure possessingrotational symmetry. Write out Eq. (1) in cylin-drical coordinates and focus attention on the twopairs of arrowed equationsHere we have taken into account the fact thatdouble differentiation of any component with re-spect to and combine it with the other mem-ber of the pair. This can be done in four ways toyield hz, hCfJ , ez and ep in terms of phCfJ and peCfJ(8)(9)1 ( dphCfJ )h.l. = - V.l.-- + KpV.l. x peCfJx d<p1 ( dpeCfJ )e.l. = - V.l. -- + KpV.l. x phCfJ ,x dCfJ(K 2 _ n:) hz= ~ a2 ph<p + ~ ape<pp p dCfJdz P dp(K 2 _ n2) h =.! a2 ph<p _ ~ ape<pp2 p p2 dCfJd p P dz(K 2 _ n:) ez= ~ a2 pe<p + ~ aph<pp p dCfJdz P dp(K 2 _ n:) ep= ~ a2 pe<p _ ~ aph<p.p p dcpdp P dzwhere x = K 2p2 - n2 and V == (~ ~)..l. dp'dZThus in this case the two scalar·functions arephCfJ and peCfJ' For n =1= 0, each function generatesfive field components of E, H. However, if n= 0, phCfJ by itself generates only (e p , hCfJ , ez) with(h p , eCfJ , hz) equal to zero and peCfJ by itself gen-erates only (h p , eCfJ , hz) with (e p , hCfJ , ez) equal tozero.From Eq. (9) it follows that the general prob-lem of eigenfrequencies and fields of an axiallysymmetric cavity can be reduced to a two-di-mensional problem, where the two functions are(4)hq> = 71Cq>(p, z) (nq»hz = 71Cz(p, z) (nq» ,We shall show below that th~ choice of thesevariables is the most natural also from the view-point of satisfying the boundary conditions.Let us write another system which followsfrom Eq. (3), (4), (9) and relates ~CfJ and ~CfJ; itwill be helpful in the following.(7)iJ (p, z) = P~cp(p, z)E (p, z) = P~CfJ(p, z).(10)CALCULATION OF NONSYMMETRIC MODES 61(15)(14)Putting (13) into Eq. (11), we obtain a recursionrelation from which follow series containingterms of the same parity. The series begins withm = n - 1. That is,Using Eq. (8) and (14), we obtain immediatelyc,c,FIGURE 1 Results in a cylindrical cavity. The surface Ctis a conducting boundary and C2 is a plane of symmetry. (a)Isolines of Bcf>. (b) Isolines of Ecf>. (c) Vector field h in the <p= 0 plane.(16)2. OBTAINING A SYSTEM OF DISCRETEEQUATIONS2.1. To obtain a system of discrete equationswith r~spect to the variables H(p, z) = P~<p(p,z) and E(p, z) = pcg<p(p, z), let us do the following.Let there be set a triangulation5 of the cavitycross section in the plane (z, pi and basis {4Ji(Z,p)}; here i is the node number of the triangulation.Multiply Eq. (1) by a basis function 4Ji and by a<p-oriented area element ds and integrate the re-sult over the cross-sectional area. Thenf ljJi rot e · ds = Kf ljJih · dsf ljJiroth· ds = Kf ljJieds.Applying Stokes' theorem, write Eq. (16) in theformf [V'ljJi x el'l'ds + Kf ljJih'l'ds = <l>ljJieTdCf [V'i\Ji x hl'l'ds + Kf ljJie'l'ds = <l>ljJihTdC.(17)with <X~-l = <Xh-l, f3~-1 = - f3h-l.Thus with n > 1 all the field components areequal to zero on the axis. When n = 1 with ac-count of (4), (5) ~~, ~P' cg<p, cgp yield in the (p,<p)-plane a pair of nonzero mutually perpendicularvectors hand e. For the variables H, E, we havezero boundary conditions on the axis for any n2:: 1.(13)(12)(11)e-r = OlC2 or h-r = O/Cz,the latter depending on the symmetry of themodes sought.To estimate the order of smallness of the fieldcomponents in the axis neighbourhood, let us put'Jf~ and cg~ in the form1.3 The fields should also satisfy some boundaryconditions besides Eq. (1). Figure 1 shows theboundaries C1 and Cz in the plane (z, p): C1 is theboundary of an ideally conductive metal, Cz isthe symmetry plane. For free oscillations to exist,the boundary values H-r or E-r should be equal tozero. That is,e-r = Olc]On performing simple transformations and using62 A. G. DAIKOVSKY, YU. I. PORTUGALOV, AND~A. D. RYABOVrelations (4)-(6), (9) and the expansion(18)NE= ~EilfJi'iwe obtain a system of 2N equationsand impose the related continuity conditions onthis line. In this case, both the form of Eq. (19)and uniformity in computing the coefficients aremaintained.Let us bring formulas for calculation of someintegrals over a triangle which are present in (20)Ji = r 2 Kp 2 2 ds h = r 2 n 2 2 ds.la n - K P la n - K PNote that the matrix Aij is symmetric, the matrixBij is antisymmetric, and the complete matrix(19) is also symmetric. The final system of equa-tions will be derived from Eq. (19) with theboundary conditions taken into account.1. For the interior nodes of a triangle mesh,the contour integrals are equal to zero because~i = Ole2. For the nodes on the Z axis, Eqs. (19) arereplaced by iIi = 0, Ei = O.3. For the nodes on the metal CI, the contourintegral of the first equation is equal to zero be-cause ~T = 0, and the second equation is replacedby Ei = O.N~ AijHj + BijE; = f IjIiET dCjN~ -BijHj +'AijEj = fljli';JCTdC,JwhereAij = J[K IjI~IjIj - ~ (Vljli' Vljlj) JdSBij = J~ [Vljli x Vljlj]<pds.(19)(20)IfG1 = J 1 ds G2 = J 1 dsn + Kp n - Kpare introduced, then J I = (G2 - G1)/2, J2 = (G2+ G1)/2. To compute the integrals G1 and G2 itis convenient to apply the Stokes formular~ p(z, p)ds = f p(z, p)dpla az cand reduce integration over the square to inte-gration over the contour. The contribution to theintegrals G1 and G2 from the side of the triangle(Zl, PI) - (Z2' P2) is1 K~ - naJG1 = - (Z2 - Zl) + 2K Kx In I n + KP2!n + KPIx In I n - KP21 'n - KPIwhere a and ~ are the coefficients of the equationfor the given sideWe treat the nodes on the C2 boundary in similarfashion: depending on the type of the symmetryof the modes sought, we put either Ei = 0, ~T:::;: 0 or iIi = 0, 'JeT = O. The derivation performedcorresponds completely to the standard proce-dure of the finite-element method and for the casen = 0 leads to the equations coinciding with theflux scheme used in SUPERFISH.2.2 Equations (19) have been obtained for-mally without consideration for X taking the valuezero on the P* = n/K line, dividing the problemregion into two parts. It would be natural to putdown the equations for each subregion separatelyZ = ap + ~.3. EVALUATION OF THEEIGENFREQUENCYThe system of finite equations (19) has singular-ities which do not make it possible to considerit as a standard eigenvalue problem. One of thesingularities lies in the nonlinear dependence ofthe coefficients (20) of the matrix on K. Anothersingularity is the implicit dependence of thewhole matrix on K and reflects the fact that theCALCULATION OF NONSYM~fETRICMODES 63cebd•z--c--_Z__ d __FIGURE 3 Re:sults in a side-coupled cavity cell. (a) Tri-angular mesh. (b) Isolines for Bet> for the E120 mode. (c) Is-olines of Eet> for the E120 mode. (d) Isolines of Bet> for the E130mode. (e) Electric-field component ez along z = 0 for the E 130mode.FIGURE 2 Results in a diaphragmed waveguide. (a) Tri-angular mesh. (c) Isolines of Eet>. (c) Isolines of Bet>.4. THE PACKAGE FACILITIESThe package facilities a uniform computation onany type of oscillations with azimuthal variationsin the specified frequency band in axially sym-metric cavities of arbitrary shape. Below we shallpresent package applications to cavities of var-ious shapes. One can compute resonance fre-quencies, all six components of the fields e, h,the derived quantities, Le., the cavity character-istics: the stored energy, distribution of losses inthe metal walls, etc. If required by the user, thelist of derived quantities can easily be madelonger. The package is furnished with extensivefacilities of graphic representation of the field to-pology, which are helpful in mode analysis andoptimization of accelerating structures. Datapreparation for the package programs takes up5-20 minutes. The solution to the problem on themesh containing about 1000 nodes takes up 3-5minutes of CPU time of a computer with thespeed of about 1 million operations per second.The general features of the computing proce-dure are as follows. A user specifies the cavitygeometry in the (z, p)-plane, the form of boundarytriangle mesh necessary to calculate the coeffi-cients should be consistent with K in terms ofSection 2.2. To determine the resonance values,the method of exciting current l should be gen-eralized. This method applied in such a fashionlooks as follows.For some value of K a triangle mesh consistentwith it is constructed. In one of its nodes, say i,the so-called driving point, we put iI; = 1 (or E;= 1). Then Eqs. (19) are constructed; in this caseiI; = 1 (E; = 1) are treated as a boundary con-dition. The system of the equations obtained issolved with respect to the remaining node un-known values. It is always solvable, includingthe resonance values of K. Then one can calculateiI;* (E;*) and evaluate the residual function I(K)= 1 - iI;* (I(K) = 1 - E;*) from the first (andcorrespondingly the second provided E; = 1)equation of (19) written in the i-th node. It iseasily seen that I(K) = 0 for resonance and onlyfor resonance values of K. Thus the problem isreduced to finding the zeroes of I(K). The residualI(K) being equal to zero is interpreted simply ina physics way: resonance excitation of the fieldby vanishingly small intensity current (electric ormagnetic current depending on the type of os-cillation) .64 A. G. DAIKOVSKY, YU. I. PORTUGALOV, AND A. D. RYABOVconditions on separate parts of the boundary: e'T= 0 on the surface of ideally conductive metal,e'T = 0 or h'T = 0 on the symmetry planes de-pending on the type of modes sought, the numberof azimuthal variations, the mode ,of operationwith the program.The program works in two modes, a mode ofestimating the frequency spectrum in the speci-fied interval and a mode of accurate computationof a specific frequency and related fields and de-rived quantities.In the first mode, the program computes theresidual function I(K) whose zeroes specify res-onance frequencies. The distribution of these zer-oes are initial informafion for the user and serveas the initial approximations in precise compu-tation in the second mode. The frequency bandand resonance-tracing step are set up by a user.In the second mode, the initial approximationof frequency is specified more precisely by it-eration and if required by the user, the derivedquantities are computed and, if necessary, theinformation is dumped for subsequent operation,for example, for graphic representation of thedata.All the processes such as construction of tri-angle meshes, optimal enumeration of the nodeswith view to minimize the matrix bandwidth, der-ivation and solution of the systems of equations,the choice of the optimal position of the drivingpoint, testing the mesh against the current valueof the wave number, data control are carried outautomatically. However, the operational effi-ciency of the package is dependent on user's ex-perience and knowledge.5. EXAMPLESThe examples presented below are aimed to showthe package facilities for typical solutions and theform of their representation. The computationalaccuracy was estimated by testing the numericalsoJution against the analytic one for a cylindricalcavity and against the experimental data for a.diaphragmed waveguide.Example 1. A Cylindrical Cavity.Each of Figs. la,b,c shows a quarter of thecavity dimension in the z-direction with the struc-ture of H 332-type oscillations. The frequency ob-tained from the analytic solution is I = 6188.2MHz, and the one computed with the help of thePRUD package is I = 6191.9 MHz. Thus therelative inaccuracy is ilIlf = 6 · 10- 4 • Figuresla,b show the isolines of de'P' ~'P' Fig. lc showsthe vector field h in the 'P = 0 plane; C1 is themetal surface, C2 is the h'T = 0 symmetry plane.Example 2. A Cell of a Diaphragmed Wave-guide.The cell parameters have been borrowed fromRef. 6 and are as follows: d = 2.5 cm, a = 2.759cm, alb = 0.5, t = 0.56 cm. The oscillation fre-quency EH11 of the 7r/2 type, according to thedata from Ref. 6, is I = 2997.93 MHz in this celland the one computed by the technique describedon a mesh containing about 1500 nodes is I =2999.1 MHz. Thus the discrepancy is ilIlf = 4· 10- 4 • Figure 2a shows the typical triangle meshof the cross section, Fig. 2b shows the isolinesof ~'P and Fig. 2c those of de'P.Example 3. An Accelerating Structure Cavitywith Side-Coupled Cells.The parameters are a = 10 cm, C = 3.261 cm,d = 2.761 cm, t = 1.7 cm. The boundary con-dition is C2 - e'T = O. Figure 3 shows two possiblemodes in this structure. The former is similar tothat of the £120 type with the frequency I = 3212MHz and the latter to that of the £130 type withthe frequency I = 4606.4 MHz. Figure 3a pre-sents the typical triangle mesh of the cavity re-gion, Figs. 3b,c show the isolines of de'P and ~'Pfor the £120 mode, Fig. 3d shows the isolines forde'P for the £130 mode, Fig. 3e shows the com-ponent ez along the z = 0 line for the £130 mode.In conclusion the authors would like to expresstheir deep gratitude to V.A. Teplyakov and B.K.Shembel for their constant interest in and supportof the present work.REFERENCES1. K. Halbach and R. Holsinger. Particle Accelerators. 7,213(1976).2. A. G. Daikovsky et.al., Preprint IHEP 80-19, Serpukhov,1980.3. M. M. Karliner et.al., Preprint INP SO AN 79-59, No-vosibirsk, 1979.4. A. N. Tikhonov, and A. A. Samarsky, Equations ofMath-ematical Physics ("Nauka", USSR 1966,).5. G. Strang and G. J. Fix, An Analysis ofthe Finite ElementMethod (Prentice-Hall, Inc., Englewood Cliffs, N.J.,1973).6. O. A. Valdner et.al., Diaphragmed Waveguide ReferenceManual ("Atomizdat", USSR 1977).

Prud Code For Calculation Of The Nonsymmetric Modes In Axial symmetric cavities

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