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Scattering by inductive post in uniformly curvedrectangular waveguide
I.V. Petrusenko and V.P. Chumachenko
Abstract: A rigorous solution to the problem of wave scattering by a single inductive post in acontinuously curved rectangular waveguide is presented. The position of the post and its radius arearbitrary. The waveguide curvature can be varied over wide limits. An efficient mathematicalmodel of the unit is based on the domain-product technique. The theory developed is applied to thenormalised reactances of the discontinuity and the induced surface current. The results obtainedlend support to the validity of the known approximate solution for a thin post as far as thisapproximation goes. It is found that data for a centrally placed post deviate slightly from those fora straight guide in a broad interval of curvature variation. The off-centre shift affects thecharacteristics appreciably. Over full waveguide bandwidth, the paper provides the data that areessential in the design of devices containing uniform bends, such as bandpass filters or transmissionresonators.
1 Introduction
The analysis of wave scattering by inductive obstacles in awaveguide has traditionally attracted considerable attentionin microwave theory [1–12]. A cylindrical post is a subject ofconstant interest in the design of manifold filters [1, 2, 4–12]and many other microwave devices [1–3]. Therefore, thecorrect analysis of inductive posts in various waveguidingstructures is of great practical importance.
The circular post in a straight rectangular waveguide hasbeen studied in detail [4–11]. The configuration has beenanalysed by a variety of methods. These include finite- andboundary-element methods [4, 5], the technique of equiva-lent sources [6, 7], the variational approach [8], the Rayleighhypothesis [9], the singular integral equation method [10],and the domain-product technique [11].
Scattering properties of the post in a curved guide are lesswell-known, although the problem is important forapplications as well. For example, transmission resonatorsand bandpass filters containing sections with inductive postsin both straight rectangular waveguides and continuouslycurved ones have been proposed. Such filters offer certainadvantages in the waveband away from the centralfrequency [12].
The cylindrical post in the curved waveguide is difficult toanalyse using traditional techniques in view of thecomplicated geometry of the problem. Owing to thesimultaneous presence of convex and concave boundaries,the use of full numerical techniques is not very efficient withrespect to computational effort and accuracy.
This paper presents a novel rigorous solution to theproblem of a circular post in a uniformly curved rectangular
waveguide. The domain-product technique (DPT) [13]is applied to calculate the normalised reactances of thediscontinuity and the induced surface current for a widerange of curvature variations, radii and arbitrary locationof the post in the interior of the waveguide bend. Thisapproach is highly efficient compared to full numericalmethods because DPT reduces the problem to anequation of the Fredholm type with a nuclear matrixoperator. This ensures a correct and accurate mathema-tical model, the absence of spurious solutions, thevalidity of the truncation procedure, the fast convergenceof the computed solution to the exact one and robustcalculations.
Here we deal only with a single inductive perfectlyconducting (PEC) cylinder, but the approach can easily beextended to the penetrable post with typical losses, or postarray placed parallel to the narrow or broad wall of theguide.
2 Problem specification and mathematicalmodelling
The configuration of interest and the co-ordinate systemsused are shown in Fig. 1. We consider a circular post ofradius r, placed across the guide parallel to the narrow PEC
Fig. 1 Geometry of the problem and pertinent co-ordinate systems
I.V. Petrusenko is with the Electronics Engineering Department, Gebze Instituteof Technology, Gebze, Kocaeli 41400, Turkey
V.P. Chumachenko is with the Zaporizhzhia National Technical University, 64Zhukovsky Str., Zaporizhzhia 69063, Ukraine
r IEE, 2003
IEE Proceedings online no. 20030773
doi:10.1049/ip-map:20030773
Paper first received 12th October 2002 and in revised form 16th June 2003.Online publishing date: 17 October 2003
498 IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003
wall and centred at r¼Rp¼R1+d in the uniformly curvedwaveguide of the width 2a¼R2–R1. The geometry is a two-dimensional one and only the electric field has non-vanishing Ey-component. The guide is filled with ahomogeneous lossless medium and terminated in matchingloads. The wave incident upon the post is the dominantLM10 mode travelling in the positive y-direction. Theconvention of time dependence is exp(jot) and k¼ 2p/l isthe wavenumber.
Let us divide the interior of the bend into regularwaveguide regions I and III, and interaction region II(Fig. 1). In regions I and III, the field can be represented interms of LMm0–mode series expansions
Eð1Þy ¼ c1ðkrÞe�jn1ðyþaÞ þ
X1m¼1
CRmcmðkrÞejnmðyþaÞ ð1Þ
Eð3Þy ¼
X1m¼1
CTmcmðkrÞe�jnmðy�aÞ ð2Þ
with the reflection CmR and transmission Cm
T coefficients tobe found. The radial distributions of the modes are definedby the cross–eigenfunctions
cmðkrÞ ¼ Pnmðkr; kR2ÞjjPnm jj�1;m ¼ 1; 2; . . . ð3Þ
where
Pnðkr; kR2Þ ¼ JnðkrÞNnðkR2Þ � JnðkR2ÞNnðkrÞ ð4Þis the cross-product Bessel function with norm 77Pn77, Jnand Nn are respectively the Bessel and Neumann functions.The needed properties of cm(kr) are described in theAppendix, Section 6.
The symmetry of the geometry makes it possible tosubdivide the initial problem into two subproblemscorresponding to the symmetric and antisymmetric excita-tions, subsequently referred to as ‘s’ and ‘a’. Henceforth,both types of excitation are presented simultaneously.
According to DPT, let us imagine the interaction regionII as the common part of five domains, namely
� the semi-infinite sector {(r, y): r4R1,�aoyoa};
� the first guide {(r, y):R1oroR2, yoa};
� the sector of annulus {(r, y): *RoroR2, �aoyoa},
where the value of radius *RA(0, R1) is virtually of noimportance;
� the second guide {(r, y):R1oroR2, y4 �a};
� the exterior to the post {(r0, y0}: r04r, �poy0rp}.
The angle aop is assumed arbitrary, but greater than theviewing angle of the post.
Because of the linearity of the Helmholtz equation, itssolution Ey
(2) can be represented in the form
Eð2Þy ¼
X5
i¼1
ui ð5Þ
where all the functions satisfy the same equation. Separatingvariables, we obtain
u1 ¼X1n¼0
bð1Þn jnðyÞHHnðkrÞ ð6Þ
u3 ¼X1n¼0
bð3Þn jnðyÞWWnðkrÞ ð7Þ
jnðyÞ ¼ cos mnðyþ aÞ; HHn ¼H ð2Þ
mn ðkrÞH ð2Þ
mn ðkR1Þ; mn ¼
np2a
under qualification qu1/qy¼ qu3/qy¼ 0 at y¼7a and alsosbð1Þ2n�1 ¼ sbð3Þ2n�1 ¼ 0; abð1Þ2n ¼ abð3Þ2n ¼ 0. Here Hm
(2) symbo-
lises the Hankel function. In (7), #Wn may be any function,which is a regular solution of the Bessel equation in the
interval *RrrrR2. One appropriate choice of this functionis described in the Appendix, Section 6.
Assuming that u2 and u4 vanish at r¼R1, R2, we get
u2 ¼X1n¼1
bð2Þn cnðkrÞejnnðy�aÞ ð8Þ
u4 ¼X1n¼1
bð4Þn cnðkrÞe�jnnðyþaÞ ð9Þ
with sbð2Þn ¼ sbð4Þn and abð2Þn ¼ �abð4Þn . Lastly, u5 is taken inthe form
u5 ¼X1n¼0ð1Þ
xsn cos ny0
xan sin ny0
� �H ð2Þn ðkr0ÞH ð2Þn ðkrÞ
ð10Þ
where xns, n¼ 0, 1,y, and xn
a, n¼ 1, 2,y, are the expansioncoefficients sought.
On the PEC surfaces, the boundary conditions can bewritten as
u1 þ u3 þ u5 ¼ 0; r ¼ R1;R2 ð11Þ
X5
i¼1
ui ¼ 0; r0 ¼ r ð12Þ
In the interval R1oroR2 the continuity conditions for thetangential electric and magnetic fields are
X5
i¼1
ui ¼Eð1Þy ; y ¼ �a
Eð3Þy ; y ¼ þa
(ð13Þ
@
@yðu2 þ u4 þ u5Þ ¼
@Eð1Þy
@y; y ¼ �a
@Eð3Þy
@y; y ¼ þa
8>><>>: ð14Þ
On satisfying the conditions (11)–(14) and using theorthogonality of the functions jn, we find an infinite systemof linear algebraic equations (SLAE):
ðIþ AÞx ¼ t ð15Þ
with the nuclear matrix operator A (details of the procedureof such algebraisation are given in [11]). Here I is anidentity. The structure of A, the right-hand vector t, and theproof of correctness of the mathematical model obtainedare given in the Appendix, Section 6.
On truncating the systems (15) and subsequent inverting,
the xs, xa are found. Let AN ;M ¼deffaNmngMm;n¼1 be a truncation
of the nuclear operator (N is the number of modes retainedin the regions I, III) and xN,M be the solution of a‘truncated’ counterpart of SLAE (15). Then the relativeerror of approximation
dðN ;MÞ ¼ jjx� xN ;M jjjjxjj � jjðIþ AÞ�1jjjjA� AN ;M jj ð16Þ
tends to zero with M, N-N because the first factor in theright-hand part of inequality is a bounded constant, whilethe second one is decreasing [14]. The rate of decay of thefunction d(N, M) can be taken as the cost of the algorithm
IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003 499
[15]. As an alternative estimate of truncation errors, thefunction
DðKÞ ¼ jjxK � xKþ1jjjjxK jj ;K ¼ N ;M ð17Þ
can be used too [16]. The behaviours of the functionsd(N, M) and D(K) are illustrated in next section.
Scattering parameters, with z¼�0,+0 as terminalplanes, can be expressed through expansion coefficients as
S11 ¼S22 ¼ ðsCR1 þ aCR
1 Þej2n1a
S12 ¼S21 ¼ ðsCT1 þ aCT
1 Þej2n1að18Þ
In a single-mode band, the normalised parameters ofan equivalent T-circuit of the post junction can be found asin [6]
XL ¼�j2S21
½ð1 � S11Þ2 � S221�
;XC ¼ j1 þ S11 � S21
1 � S11 þ S21ð19Þ
3 Numerical results
The algorithm proposed is a very efficient one in thenumerical implementation. Table 1 illustrates convergenceof the algorithm subject to geometrical parameters of theunit. The rate of stabilisation is extremely fast with respectto the truncation number M. It is also seen that the unitaryproperty of the S-matrix holds with high precision.
The error functions (16) and (17) are shown in Fig. 2aand 2b. In computing d, the data corresponding to M¼ 20,N¼ 60 have been fixed as the reference values. One can seethat it is enough to take a few equations to achieve anaccuracy sufficient for engineering needs. As a result,N¼ 14 and the 4� 4 SLAE can be proposed for theevaluation of physical characteristics. For this size oftruncated matrix the accuracy is do5� 10�4. Figure 2balso shows typical values of the condition number, whichare fairly close to unity. In all the cases, we cite the greatervalues of errors and condition numbers found in thenumerical process for the two subproblems considered.
According to known approximate approaches, a strip isused to simulate the thin post in a straight guide [10]. The
Table 1: Scattering characteristics subject to the truncationnumbers M, N and the parameters d, r and R1/R2 for k/
2a¼ 1.4286 (PCL¼ 7S1172+7S217
2, ORT¼S11%S21+S12
%S22)
d/a¼0.2, r/a¼0.1, N¼40
R1/R2 M 7S117 arg S11 PCL ORT
1 0.088746 1.705305 1.000003 0.000004
2 0.088546 1.706329 1.000003 0.000004
0.1 4 0.088536 1.706341 1.000003 0.000004
8 0.088536 1.706341 1.000003 0.000004
12 0.088536 1.706341 1.000003 0.000004
1 0.139468 1.730135 0.999999 �0.000001
2 0.138933 1.730920 0.999999 �0.000001
0.4 4 0.138932 1.730921 0.999999 �0.000001
8 0.138932 1.730921 0.999999 �0.000001
12 0.138932 1.730921 0.999999 �0.000001
1 0.151754 1.733333 1.000002 0.000002
2 0.151394 1.733933 1.000001 0.000001
0.9 4 0.151406 1.733935 1.000001 0.000001
8 0.151406 1.733935 1.000001 0.000001
12 0.151406 1.733935 1.000001 0.000001
d/a¼0.2, r/a¼0.1, M¼10
R1/R2 N 7S117 arg S11 PCL ORT
2 0.096980 1.531685 1.026847 0.029461
5 0.087857 1.695090 1.001607 0.001853
0.1 10 0.088511 1.705070 1.000170 0.000223
20 0.088533 1.706186 1.000025 0.000031
40 0.088536 1.706341 1.000003 0.000004
2 0.135502 1.728173 1.001399 �0.000119
5 0.138662 1.733175 0.999324 �0.000705
0.4 10 0.138918 1.731010 0.999976 �0.000030
20 0.138930 1.730933 0.999996 �0.000005
40 0.138932 1.730921 1.000000 �0.000001
2 0.148575 1.715790 1.004367 0.004387
5 0.151204 1.734106 0.999877 �0.000121
0.9 10 0.151390 1.733814 1.000034 0.000034
20 0.151396 1.733921 1.000005 0.000005
40 0.151396 1.733935 1.000001 0.000001
d/a¼ 1, r/a¼0.5, N¼ 40
R1/R2 M 7S117 arg S11 PCL ORT
1 0.999999 �1.994400 0.999998 0.000001
2 0.999770 �1.918815 0.999999 0.000001
0.1 4 0.999889 �1.911811 0.999999 0.000001
8 0.999889 �1.911795 0.999999 0.000001
12 0.999889 �1.911795 0.999999 0.000001
1 0.997720 �1.924362 0.999999 0.000000
2 0.999985 �1.851008 0.999999 0.000000
0.9 4 0.999999 �1.844055 0.999999 0.000000
8 0.999999 �1.844053 0.999999 0.000000
12 0.999999 �1.844053 0.999999 0.000000
d/a¼ 1, r/a¼0.5, M¼10
R1/R2 N 7S117 arg S11 PCL ORT
2 0.986429 �1.912672 0.973209 0.024541
5 0.999654 �1.911519 0.999518 0.000377
0.1 10 0.999844 �1.911772 0.999907 0.000079
20 0.999884 �1.911792 0.999988 0.000010
40 0.999889 �1.911795 0.999999 0.000001
2 0.992587 �1.847440 0.985337 0.005329
5 0.999732 �1.843906 0.999464 �0.000156
0.9 10 0.999963 �1.844051 0.999927 0.000008
20 0.999995 �1.844052 0.999990 0.000000
40 0.999999 �1.844053 0.999999 0.000000
Table 1: (continued)
R1/R2 M 7S117 arg S11 PCL ORT
500 IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003
admissibility of this substitution for a curved guide wasindicated in [12, 17]. Using Lewin’s method for the first-order approximation [10], the relations
XC �0
XL �n1
c1ðRpÞc1ðRp þ rÞX1m¼2
1
2jnmcmðRpÞcmðRp þ rÞ
ð20Þhave been obtained [17]. Comparison with the latter as wellas with the limiting case of a straight guide (R1-N,a¼ const) is given in Fig. 3. Here, the normalised reactances(19) are exhibited as a function of r/a for the centred post(d¼ a), various curvatures and a fixed operating frequency.One can take the view that the thin-strip approximation (20)is more suitable for a large curvature than a small one. Onecan see that a slight deviation from the straight guide datatakes place even for a sharp bend with R1oa (R1/R2o1/3).
Figure 4 presents the normalised reactances as a functionof r/a for the off-centred post (d¼ 0.6a) and various ratiosR1/R2. In contrast to the previous case, the reactance XC
depends strongly on the curvature and the curves begin todisperse for thin posts (rE0.05a). The effect is correlatedwith a noticeable shift of Ey-distribution from the post, as isillustrated in Fig. 5. In Fig. 4, the XL curve behaviourindicates that the range within which the approximation(20) is valid, is sufficiently smaller.
Fig. 2 The error functions d and D against number of modes N inthe curved waveguide, matrix truncation number M and conditionnumber of the matrix approximation.r¼ 0.25a, d¼ a, l/2a¼ 1.4, R1/R2¼ 0.5a Error functions against Nb Error functions against M (left-hand logarithamic axis) and condi-tion number (right-hand linear axis)
Fig. 3 Circuit parameters against radius size for centred postl/2a¼ 1.4
Fig. 4 Circuit parameters against radius size for off-centred postd¼ 0.6a, l/2a¼ 1.4
Fig. 5 Transverse distribution of the Ey-component of the incidentwave and relative position of off-centred posts of various sizesl/2a¼ 1.4
Fig. 6 Circuit parameters against position of the postr¼ 0.3a, l/2a¼ 1.2
IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003 501
Figures 6 and 7 present the dependence of reactances onposition of the post centre, operating wavelength andcurvature. The asymmetry of the characteristics about themedial line of the waveguide is clearly visible. It is seen thatthere are positions of the post and frequency points whereXL or XC is practically independent of the curvature.
Figures 8 and 9 deal with the current induced on the postsurface. The current can be derived from the electric field asfollows:
J ¼� 1
jom0
½uur0 � ðr� EyÞp0¼r
¼ uuy
j120pk
X5
i¼1
@ui@r0
� �r0¼r
ð21Þ
where #ur0 and #uy denote the unit vectors in the r0 and ydirections, respectively. Figures 8 and 9 show the currentdistributions against azimuth y0 for the centred and off-centred posts respectively. Here, the intervals (�1801, 0) and(0, 1801) correspond to the ‘illuminated’ portion of the postsurface and the ‘shadow’ one. Because of non-zerocurvature, the current has asymmetric distribution in bothregions even for the centred-post structure. In the case of asufficiently smooth bend, the results are in good agreementwith data from [7, 11].
4 Conclusions
The problem of a circular inductive post in a uniformlycurved rectangular waveguide has been solved using theDPT. All cases of thin and large (0oroa) arbitrarily placed(R1oRpoR2) posts have been considered over a broadrange of curvature variation (0.1rR1/R2o1). Taking
Fig. 7 Circuit parameters against l/2a for off-centred large postd¼ 0.6a, r¼ 0.3a
Fig. 8 Real and imaginary part of the surface current on thecentred postl/2a¼ 1.2a Realb Imaginary
Fig. 9 Real and imaginary part of the surface current on the off-centred postd¼ 0.6a, l/2a¼ 1.2a Realb Imaginary
502 IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003
advantage of the physical symmetry plane, the problem hasbeen partitioned into two independent subproblems differ-ing in the mode of excitation to get maximum efficiencyfrom the computational procedure.
The initial boundary value problem has been reduced toan infinite SLAE. It has been proved analytically that thematrix equation is of the Fredholm type with a nuclearoperator. A rapidly convergent numerical algorithm hasbeen developed to obtain the scattering matrix, theparameters of the equivalent T-network and the currentinduced on the post surface. The new data have beendetermined at low computational cost, and at the same timethe accuracies achieved are excellent with respect to anyengineering needs.
The results computed show good correlation with theapproximate solution, which has been derived by the Lewinmethod. All data obtained for a smooth bend (R1/R2Z0.9)agree well with data for the limiting case of a straight guide.
Over the full waveguide bandwidth, the method providesdata that are essential in the design of devices containinguniform bends with posts, such as transmission resonatorsor bandpass filters.
5 References
1 Marcuvitz, N.: ‘Waveguide handbook’ IEE Electromagnetic WavesSeries 21 (Peter Peregrinus, England, 1986)
2 Esteban, H., Cogollos, S., Boria, V.E., Blas, A.S. and Ferrando, M.:‘A new hybrid mode-matching/numerical method for the analysis ofarbitrarily shaped inductive obstacles and discontinuities in rectan-gular waveguides’, IEEE Trans. Microw. Theory Tech., 2002, 50, (4),pp. 1219–1224
3 Reiter, J.M., and Arndt, F.: ‘Rigorous analysis of arbitrarily shapedH- and E-plane discontinuities in rectangular waveguides by a full-wave boundary contour mode-matching method’, IEEE Trans.Microw. Theory Tech., 1995, 43, (4), pp. 796–801
4 Abdulnour, J., and Marchildor, L.: ‘Boundary elements and analyticexpansions applied to H-plane waveguide junction’, IEEE Trans.Microw. Theory Tech., 1994, 42, (6), pp. 1038–1045
5 Ise, K., and Koshiba, M.: ‘Numerical analysis of H-plane waveguidejunctions by combination of finite and boundary elements’, IEEETrans. Microw. Theory Tech., 1988, 36, (9), pp. 1343–1351
6 Leviatan, Y., Li, P.G., Adams A. T., and Perini, J.: ‘Single-postinductive obstacle in rectangular waveguide’, IEEE Trans. Microw.Theory Tech., 1983, 31, (10), pp. 806–811
7 Leviatan, Y., Shau, D.H., and Adams A. T.: ‘Numerical study of thecurrent distribution on a post in a rectangular waveguide’, IEEETrans. Microw. Theory Tech., 1984, 32, (10), pp. 1411–1415
8 Schwinger, J., and Saxon, D.S.: ‘Discontinuities in waveguides’(Gordon and Breach, New York, 1968)
9 Nielsen, E.D.: ‘Scattering by a cylindrical post of complex permittivityin a waveguide’, IEEE Trans. Microw. Theory Tech., 1967, 17, (3),pp. 148–153
10 Lewin, L.: ‘Theory of waveguides’ (Newnes Butterworths, London,1975)
11 Chumachenko, V.P., and Petrusenko, I.V.: ‘Domain product solutionfor scattering by cylindrical obstacle in rectangular waveguide’,Electromagnetics, 2002, 22, (6), pp. 473–486
12 Vol’man, V.I., and Knizhnik, A.I.: ‘Analysis of inductive postin continuously curved waveguide’, Radiotehnika, 1980, 35, (8),pp. 72–75 (in Russian)
13 Chumachenko, V.P.: ‘The integral equation method for solving a classof electrodynamics problems’, Izv. Vyssh. Ushebn. Zaced. Radiofiz.,1978, 21, (7), pp. 1004–1010 (in Russian)
14 Reed, M., and Simon, B.: ‘Methods of modern mathematical physics,v.1: Functional analysis’ (Academic Press, New York, 1978)
15 Nosich, A.I.: ‘The method of analytical regularisation in wave-scattering and eigenvalue problems: foundations and review ofsolutions’, IEEE Antennas Propag. Mag., 1999, 41, (3), pp. 34–49
16 Nosich, A.I., Okuno, Y., and Shiraishi, T.: ‘Scattering and absorptionof E- and H-polarised plane waves by a circularly curved resistivestrip’, Radio Sci., 1996, 31, (6), pp. 1733–1742
17 Petrusenko, I.V., Kodak, S.N., and Essert, V.E.: ‘Single-post inductiveobstacle in continuously curved waveguide’. Bulletin ‘‘Calculation andmeasurement of parameters of microwave components’’, Dniprope-trovs’k State University, 1988, pp. 62–66 (in Russian)
18 Morse, P.M., and Feshbach, F.: ‘Methods of theoretical physics’(McGraw-Hill, 1953)
19 Watson, G.N.: ‘A treatise on Bessel functions’ (Cambridge UniversityPress, 1944, 2nd edn.)
20 Petrusenko, I.V.: ‘High-order approximation of wave propagationconstants in the uniformly curved waveguide’, IEE Proc. Microw.Antennas Propag., 2001, 148, (5), pp. 280–284
6 Appendix
In a cylindrical frame (Fig. 1), the solution of the Helmholtzequation can easily be found via the variable separationmethod. Considering LMm0, m¼ 1, 2,y, modes ascirculating waves, an appropriate form for theirEy-component is
Ey ¼ unðkrÞe�jny ð22ÞHere un(kr) is an eigenfunction of the Sturm–Liouvilleproblem for Bessel’s equation of order n. On the finiteinterval 0oR1rrrR2, the set of these eigenfunctionsforms a complete orthonormal set of square integrablefunctions [18].
On the two curved walls Ey must vanish, and this leads tothe cross-product Bessel function
unðkrÞ ¼ c½JnðkrÞNnðkR2Þ � JnðkR2ÞNnðkrÞ�� cPnðkr; kR2Þ ð23Þ
and to the modal equation
PnmðkR1; kR2Þ ¼ 0 ð24ÞIn (23), the norming quantity c is found according to therelationship between solutions of the Bessel equation [19]Z
unðkrÞumðkrÞdrr
¼ rn2 � m2
@un@r
um � un@um@r
�ð25Þ
from which followsZ R2
R1
Pnðkr; kR2ÞPmðkr; kR2Þdrr
¼ dnmr2n
@Pnðkr; kR2Þ@r
@Pnðkr; kR2Þ@n
�r¼R1
ð26Þ
Here dnm is the Kronecker delta.In this paper, we use the special norm
jjPnjj ¼rsn
@Pnðkr; kR2Þ@r
@Pnðkr; kR2Þ@n
�1=2
r¼R1
s ¼ lnR2
R1
� ð27Þ
according to the orthogonality condition for the cross-eigenfunctions (3)Z R2
R1
cmðkrÞcnðkrÞdrr
¼ s2dmn ð28Þ
This provides the required quasistatic approximation formc1
cmðkrÞ ’ sinmps
lnR2
r
� �ð29Þ
The high-order approximation to the angular propagationconstant nm, m¼ 1,2,y, and k�n diagram for the modalequation (24) are given in [20].
In order to avoid division by zero during computations,let us construct the solution of the Bessel equation
WWnðkrÞ ¼ AJmnðkrÞ þ BNmnðkrÞ; ~RR � r � R2 ð30Þwhich does not vanish for any real value of the
wavenumber. For this purpose #Wn must satisfy
� boundary condition of the impedance type
WWnðk~RRÞ þ j~RR@WWnðk~RRÞ
@~RR¼ 0
� normalisation condition #Wn (kR2)¼ 1;
IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003 503
� additional requirement limn!1
WWnðkrÞ ¼ 0; ~RRoroR2 for
robust calculations.
The solution sought takes the form
WWnðkrÞ ¼WmnðkrÞWmnðkR2Þ
ð31Þ
WmðkrÞ ¼ Pmðk~RR; krÞ þ j~RR@Pmðk~RR; krÞ
@~RRð32Þ
Below the designations
p ¼2m
2m� 1
� �; q ¼
2n
2n� 1
� �; csnðyÞ
¼cos ny
sin ny
� �; f enðyÞ ¼ 2e�inna
cos nny
i sin nny
� �ð33Þ
are used to make formulae less cumbersome. Hereinafter,upper and lower symbols are connected with the symmetricand antisymmetric case respectively.
It can be shown in analogy to the result of paper [11] thatin (15) the matrix operator has the structure
A ¼ T1 �1
2TT2F1
� D1 þ T3 �
1
2TT2F3
� D3
� 1
2TT2ðF2 �GÞ ð34Þ
Let us introduce an auxiliary vector
sðnÞðr; yÞ ¼ ðHHqðkrÞjqðyÞ;cnðkrÞfenðyÞ;WWqðkrÞjqðyÞÞ
ð35Þ
then the elements of Ti, i¼ 1,2,3, and #T2 can be expressed inthe form
tðiÞmn ¼2
emp
Z p
0
½tðnÞi ðr; yÞ�r0¼rcsmðy0Þdy0;
ttð2Þmn ¼n1=2tð2Þmn ; em ¼2;m ¼ 0
1;m � 1
�ð36Þ
The elements of Fi, i¼ 1,2,3, and G are described by theformulae
f ð1Þmn ¼ 2m�1=2
sðm2q � n2
mÞ
� 2
pHHqðkR2ÞjjPnm jj
�1 þ r@cmðkrÞ
@r
�r¼R1
( )ð37Þ
f ð2Þmn ¼ 2
sm1=2
Z R2
R1
csnðy0ÞH ð2Þn ðkr0ÞH ð2Þn ðkrÞ
" #y¼�a
� cmðkrÞdrr
ð38Þ
f ð3Þmn ¼ 2m�1=2
sðm2q � n2
mÞ
� 2
pjjPnm jj
�1 þ r@cmðkrÞ
@r
�r¼R1
WWqðkR1Þ( )
ð39Þ
gmn ¼2m�1=2
jsnm
Z R2
R1
@
@ycsnðy0Þ
H ð2Þn ðkr0ÞH ð2Þn ðkrÞ
" #( )y¼�a
� cmðkrÞdrr ð40Þ
Matrices Di, i¼ 1,3, are defined by the elements
dð1Þmn ¼L�1p ½d�mn � WWpðkR1Þdþmn�
dð3Þmn ¼L�1p ½dþmn � HHpðkR2Þd�mn� ð41Þ
where
Lp ¼ 1 � HHpðkR2ÞWWpðkR1Þ ð42Þ
dþ;�mn ¼ � 2
epa
Z a
0
csnðy0ÞH ð2Þn ðkr0ÞH ð2Þn ðkrÞ
" #r¼R2;R1
jpðyÞdy
ð43Þ
Finally, the right-hand part of (15) takes the form
t ¼ f�tð2Þm1=2g.The proof of the correctness of the proposed model is
based on the Fredholm property of (15). It is sufficient toshow that A is a compact matrix operator in the Hilbertspace
h1 ¼def xn :X1n¼0
ðnþ 1Þjxnj2o1( )
ð44Þ
and the right-hand part tAh1 [14].Compactness of the operator (34) follows from asymp-
totic estimates of the integrals (36), (38), (40) and (43),which are Fourier coefficients of functions differentiableinfinitely many times. Namely, the estimation formulae
jtð1;3Þmn j ¼O e�mql1;3
m!mq
rRp
� m �
jttð2Þmn j ¼Oe�npl2
m!nmþ1=2 pr
sRp
� m �;m; n � 1; r � Rp
1 ¼ lnRpR1
� ;as; ln
R2
Rp
� � ð45Þ
give us tAh1 and lead to the inequalities
X1m;n
mjtð1;3Þmn j2 ¼ O½ð1 � z1;3Þ�b�o1
X1m;n
mjttð2Þmn j2 ¼ O½ð1 � z2Þ�t�o1
z ¼ rd;r
aRp;
r2a� d
� ; b4t40 ð46Þ
under conditions zio1, i¼ 1,2,3. Hence, T1,3, #T2 are theHilbert–Schmidt (H–S) operators h1-h1 [14] provided thatthe post does not touch the boundaries of the interactionregion II. Under the same condition, we get
X1m;n
jf ð1;3Þmn j2o1;
X1m;n
1
njxðm;nÞi j2o1; i ¼ 1; 2; 3
nðm;nÞ ¼ ðf ð2Þmn ; gmn; d
ð1;3Þmn Þ ð47Þ
Therefore, D1, D3, G, Fi, i¼ 1,2,3, present the H–Soperators as well. Thus, A: h1-h1 is the nuclear operatoras a sum of products of the H–S operators [14].
504 IEE Proc.-Microw. Antennas Propag., Vol. 150, No. 6, December 2003