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Absfrrrcr - The natural frequency calculation for cylindrical cavities with helices touching their end plates is carried nut. The explicit expressions for the case of long cavity with one helix Contacting with both end plates and fnr the case of cavity with one short helix contacting with one end plate are obtained. The numerical calculations are carried out for snme set of dimensions of the one-helix structures and for the cavity with the both helices having close natural frequencies. Helices contained in cavity are widely used in the slow-wave structures. Their contact with the cavity end plate influences on the structure electrodynamic properties. A subject of the paper is electrodynamic calculation of the axially symmetric structure, which consists of the cylindrical cavity and inner anisotropically conducting cylinders touching one or both of the cavity end plates. The calculations are based on the method used in [I] and stated in [2] in detail. Applying the presented results to real helix structures one would take into account the difference between the helix and the anisotropically conducting cylinder, especially, for the single-wound large pitch helix. Also, near the contact with the end plate the wire direction usually changes becoming orthogonal to the end plate. I. MODEL AND FQUATIONS Any field in the bounded cylinder with helices is identical to some field in the infinite cylindrical waveguide with periodically disposed helices, which may be obtained from the initial bounded structure by means of successive mirror reflections in its end plates. In Fig.l(a), one period examples of two such structures are shown. To obtain the natural frequencies of such structure (and the field distributions) one can write the expressions for the electric and magnetic field considering the helix currents as known and using the known solution of the circular cylindrical waveguide excitation problem and then impose the condition of absence of the electric field strength projections on the helix conductivity direction. The helix current is proportional to the corresponding magnetic field strength projection jump. Assuming axial symmetry and dependence on time I in the form exp(-k/) (where c is the speed of light and xis the vacuum wave number for the given oscillation frequency, which will be called "frequency" below) and using the cylindrical coordinates {p,q,z) let us consider the field, which depends on z as exp(iwr) with the given w, obeys the Maxwell equations for fir and for pe(r, I), and whose elecmc component is normal to the surface pl. Let's introduce the designations i?(wp,v) and li(w,p,w) for the projections of the electric and magnetic field strength on the direction @(+e, cos p+5z sin v. determined with the angle v(& and 8, are the unit vectors). If the considered field has an equal limit values at (er, pcr) and at {pr, pr) for both tangent to the surface pr electric field projections and for the magnetic field projection on the direction &(vu) determined with the fixed angle fi then i?(w,p,W)=(2iK)-'~~(w,r, VU -90")[?Bu(v,r,p) sin yo shy+ 2 Bl(v,r,p)cosfi cosru] for any Y and pe(0. I). Here v(.$--I*L)IR, d(w,p, y)=lim(w, al[f?(w,p6, y)-fi(w,p6,y)]. and the functions &(v,p& (s=O,l) are determined with the equality B,(v.pl.p2)=iJ,(vp.) {H!"(vp*)-J~(vp+)[H!"oiJ,(v)l), in which p-=minb,fi), p+=max@l,fi), and J and H"' denote the Bessel and Hankel functions. Any field in the considered periodical structure may be written as the Fourier expansion F(sp,v)=Zi |
