1% EEE TRANSACTIONS ON ELECTROMAGNETIC COMPATBILITY, VOL. 36, NO. 3, AUGUST 1994

Scattering from and Penetration into a Dielectric- Filled Conducting Cylinder with Multiple Apertures

Ali El-Hajj and Karim Y. Kabalan

Abstract-The theory of characteristic modes for aperture problems is used in this paper to solve the equivalent magnetic current and aperture fields due to a conducting cylinder with multiple slots. It is assumed that these slots are illuminated by either a transverse electric (TE) or a transverse magnetic (TM) plane wave and the media inside and outside the cylinder exhibit different electromagnetic properties. The formulation is given for the general case and numerical results for a limited number of slots are presented.

I. INTRODUCTION The problem of electromagnetic coupling between different

media through apertures has many applications and has been lately the center of several works. The literature on these problems is extensive and many of them are listed in [l]. This paper discusses the problem of scattering and radiation from a conducting cylinder with multiple slots where the one-slot problem was studied in [2]. This problem is tackled with the use of the characteristic mode theory for aperture problems [3]. This theory has been widely used to treat large numbers of electromagnetic coupling between different media through apertures [4], [51. At each of these apertures, a vector equation, whose unknown is the equivalent magnetic current, is obtained. These equations are then transformed, with the use of the moment method, into a matrix equation for a numerical solution of the problem. Specialization of the method to a conducting cylinder with a limited number of slots is discussed. Numerical results for this particular problem when the slots are illuminated by a transverse-electric (TE) and a transverse-magnetic (TM) plane waves are given. It is assumed that the media inside and outside the cylinder exhibit different electromagnetic properties.

A. Geometry and Formulation of the Problem An infinitely thin perfectly conducting cylinder, of radius

a, with multiple slots is considered. A cylindrical coordinate system ( p , 4, z ) is erected at the center of the cylinder whose axis is in the z-direction. The ith slot is denoted by Si and its angular width is denoted by q5i. The mediums outside and inside the cylinder are denoted by (1) and (2), respectively. The general formulation is based on dividing the problem into two decoupled parts using the equivalent theorem [6] and by

Manuscript received April 28, 1993; revised December 27, 1993. This work was supported by the Research Board of the American University of Beirut.

The authors are with the Electrical Engineering Department, Faculty of Engineering and Architecture, American University of Beirut, Beirut, Lebanon.

IEEE Log Number 92165127.

placing magnetic sheets of different signs Mk and -Mk on the apertures, for the outside and inside problems, respectively, to ensure the continuity of the electric field. In this case, the field inside the cylinder is the sum of the fields due to the magnetic sheets -M, whereas the field outside the cylinder is the sum of the total field with aperture short-circuited and the fields due the magnetic sheets Mk. The formula representing the continuity of the magnetic field across the apertures is used as an operator equation on the magnetic sheets i&. This formula is given by

B?Ip=a + E a (1) ( M k ) I p = a = (2) ( - & ) l p = a . (l) k k

In (l), denotes the total field with aperture short-circuited, the subscript t refers to the tangential component in the z- direction and the superscript (1) and (2) refer to media (1) and (2), respectively. This equation is valid for 41 5 4 5 42,

4 3 5 4 5 q54, etc. Equation (1) can be rearranged in the form of an operator equation on the magnetic currents Mk.

In the TE plane wave case, the z-component of the total field is given by

n=w

In (2) K1 denotes the wave number of medium (1). The tangential component of the magnetic field due to a slot Mk in medium (1) and the tangential component of the magnetic field due to a slot -Mk in medium (2) are given by

In (3), K2 is the wave number of medium (2), 711 is the intrinsic impedance of medium (l), and 712 is the intrinsic impedance of medium (2). K1, Kz. 771. and 772 are all real because media (1) and (2) are considered to be perfect dielectrics which is the

0018-9375/94$04.00 0 1994 JEEE

ELHAJJ AND KABALm SCATTE,RING FROM AND PENETRATION INTO A CONDUCTING CYLINDER

case of most physical problems such as material measurements and biomedical engineering. Using Wronskian equality

in (2), it becomes

Substituting (3) and ( 5 ) into (l), the following operator equa- tion is then obtained:

(6)

Similarly, the operator equation for the TM case is given by

(7)

The left-hand side of (6) and (7) can be decomposed into a real part G ( M ) and an imaginary part B ( M ) , and thus can be written in the form

G ( M ) + j B ( M ) = I . (8)

Equation (8) is used in the next section to solve the equivalent magnetic currents Mk.

11. SOLUTION OF THE OPERATOR EQUATION

The characteristic currents of this problem are defined as the solution of the eigenvalue equation

B ( M n ) = b,G(Mm) (9)

where

Mm = u Mm,k. (10) k

In (9) and (lo), b, is real, Mm,k is the mth characteristic current of the kth slot, and UI, denotes the union over all mth characteristic magnetic currents. G(Mm) and B(M,) are obtained from (6) and (7) after leaving out the term (2a) as follows:

A. TE Case

n=m

B ~ ~ ( M ~ ) = [A , '~D, - B , T ~ c ~ ]

197

(11)

B. TM

where

Case

n=cc

n=-m k m.=m

BTM(Mm) = -2- [ATM& - BnTMCn] (13) n=-w k

A:M, and BTM are given by:

HI. NUMERICAL SOLUTION OF THE PROBLEM

An analytical solution of the problem is very difficult. To obtain an approximate solution, each slot is partitioned uniformly and expansion functions are taken on each partition. Therefore, each characteristic current is expressed in the form

J .

k j=1

Jk represents the number of divisions of the kth slot, f k j is a real known function, and umkj are constants to be determined in the solution. Each expansion function is defined on a given partition and vanishes on the remainder of the slot. This is taken into consideration when (15) is substituted into (1 1) and

198 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 3, AUGUST 1994

w - n - 3 - t u - z - 9 n - w - 2- E - 0 -

-1 z -

Fig. 1. Geometry of the two-slot cylinder problem.

-

-

,c

Fig. 2. The fmt four characteristic currents for the two-slot problem for TE excitation.

TABLE I T€E CONVERGENCE OF THE CHARACTERISTIC VALES FOR TE hCJDENCE WAVE

N bi b2 b3 b4

6 3.069 4.099 33.338 234.905 8 3.017 4.056 3 1.705 219.941

10 2.986 4.03 1 30.946 213.008 12 2.962 4.01 1 30.434 208.388 14 2.939 3.993 30.045 204.892 16 2.918 3.916 29.716 201.990

(13). After this substitution, the result is substituted into (9), and the following G operator is obtained.

n=m

A n = {;it for for TE Th4 case }. (17)

()( 4 0 W-SOO -

. . . . . . . . . . . . . . . . .

Fig. 3. two-slot problem and for TE excitation.

The normalized magnitude of the equivalent magnetic current for the

Fig. 4. The first four characteristic currents for the two-slot problem for TM excitation.

The same is true for Bn. The operator B(Mm) is similarly obtained. In order to transform this eigenvalue equation into a matrix form, an inner product is defined where the testing functions are the same as the expansion functions. These functions are denoted by $pq where p denotes the slot number and q denotes the number of divisions on the qth slot. Taking this inner product, the elements of the matrix G become:

Pulse and triangular expansion functions are considered for

199 EL-HAJJ AND KABALAN: SCAlTERING FROM AND PENETRATION INTO A CONDUCTING CKINDER

....... n- 1 n- 2 -.-.-.- n- 3 - n- 4

- - - -

0 - 0"

Fig. 5. The first four characteristic magnetic fields inside the cylinder for "E excitation.

\ \ \ 1.0

> I

+ - oo

6 -18.650 -20.516 -2270.977 -11769.406 8 -17.377 -19.040 -1682.856 -8676.840

10 -16.783 -18.357 -1476.514 -7597.020 12 -16.380 -17.896 -1366.319 -7022.355 14 -16.077 -17.546 -1291.440 -6632.448 16 - 16.054 - 17.250 - 1233.212 -6327.350

I and

I ipq f p q cos (n4) & f k j sin (n+> & k3

+ ipq f p q sin (n4) d 4

\ / \ /

. - - - A

Fig. 6. Merent angles of incidences for TM excitation.

f k j cos (n4) d 4 = 0, for v n The scattered fields at a distance twice the radius of the cylinder for k3

(20)

(18), the elements of the matrix G, becomes

1 the TE and TM cases, respectively. This is done to satisfy

equal and greater than zero using the identities

k j TE n=m

n=l the boundary condition of the problem. Integrating (18) for n [Gpq] = 4 ;EZA;~COS - n4kjl

x [1 - cos (n64)] + (21)

Similarly, the elements of the matrix B are found to be

1 n

n=m ipq fpq cos (.d'> d4 f k j cos &

k 3

-k lp, f p q sin (d) d4 [ ~ 3 ~ ~ = -4 7 ~ , T E COS - n4kjl

f k j sin (n4) d 4 n=l

k 3 x [I - cos (TI&$)] - ~ T ~ ( 6 4 ) ~ . (22)

In (19), (21), and (22) 64 is the width of partition and is ( 2/n2) [ cos (ndPq - n4kj)l [I - cos ( n ~ 4 ) ] , for n # o

(19) assumed to be the same for all slots. For Th4 polarization, the

elements of the matrices G and B are found to be 1 n=m

[GZlTM = 8

[B;{lTM = -8

,.A:M cos b 4 P - q - n h l

,.8.TM cos b 4 P - q - n4kjl

n=l

x [l - cos (nS4)I2 + ArM(S4)4 1 n=m

n=l

x [l - cos ( T L S ~ ~ ) ] ~ - BFM(S~)4. (23) The above equation is used in order to construct the matrices G and B needed for the matrix eigenvalue equation. Once the eigenvectors umkj of the matrix equation are determined, and since fkj’s are known, one can construct the characteristic currents Mm using (15).

IV. NUMERICAL RESULTS

A general computer program has been written to treat large numbers of slots of different width, of different angular distances between them, and for different polarization. Conver- gence of the solution is tested by increasing the number of slots and divisions on each slot. Numerical results for the two-slot problem are shown for simplicity. Each slot is of 15’ angular width separated by a 30’ angle, and the radius of the cylinder is defined by Ka = 3.456 as shown in Fig. 1. The medium outside the cylinder is assumed to be free space and that of the inside is characterized by E, = 2. The convergence for this case is shown in Tables I and I1 for TE and Th4 excitation, respectively. As can be seen, the convergence is monotone, and the convergence of the lower order mode is faster than that of the higher order modes. The highest values of N in Tables I and I1 determine the number of divisions used in the numerical computation. Fig. 2 shows the normalized magnitude for the first four characteristic currents of the problem for TE excitation. First, the right behavior of these currents at the edges is clearly seen even though such a behavior was not forced or assumed in the solution as it was done for the Th4 case. Second, the symmetry of these currents is clearly noticed when the S-1 slot is compared to the S+, slot. The normalized magnitude for the equivalent magnetic current for this problem and for TE excitation is shown in Fig. 3 for different angles of incidence. Fig. 4 shows the normalized magnitude for the first four characteristic currents of the problem when the slots are illuminated by a transverse-magnetic plane wave. The symme- try of the problem is also shown with the assumed boundary condition. In order to minimize the number of figures, it was decided to show the first four characteristic magnetic fields at a radius equal to 0.3a, where a is the radius of the cylinder and for TE excitation. This result is shown in Fig. 5. It was also decided to show the scattered fields at a distance twice the radius of the cylinder for different angles of incidences and for TM excitation. This result is shown in Fig. 6.

V. CONCLUSION This paper presents a solution for the multiple-slot problem

in an infinite thin perfectly conducting cylinder. It has been assumed that the mediums inside and outside the cylinder exhibit different electromagnetic properties and that the slots are illuminated by either a transverse-electric or a transverse-

200 IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 36, NO. 3, AUGUST 1994

~

magnetic plane wave. The solutions for the charactesstic currents and fields are used to solve for other quantities as obtained in this paper. The procedure is simple, however, it requires advanced numerical techniques.

ACKNOWLEDGMENT The authors wish to thank Prof. R. F. Harrington for

providing the atmosphere conductive to research by inviting us to spend the Summer 1991 at Syracuse University where this research was initiated. We also wish to thank Prof. Harrington and all referees for their valuable suggestions which were incorporated into this paper.

REFERENCES

[ l ] C. M. Butler, Y. Rahmat-Sami, and R. Mittra, “Electromagnetic pene- tration through apertures in conducting surfaces,” IEEE Trans. Antennas Propagat., vol. AP-26, no. 1, pp. 82-93, Jan. 1978.

[2] A. El-Hajj, K. Y. Kabalan, and R. F. Harrington, “Characteristic modes of a slot in a conducting cylinder and their use for penetration and scattering, TE case,” IEEE trans. Antennas Propagat., vol. AP-40, no. 2, pp. 156-161, Feb. 1992.

[3] R. F. Harrington and J. R. Mautz, “Characteristic modes for aperture problems,” IEEE Trans. Microwave Theory Tech., vol. MlT-33, no. 6, pp. 500-505, June 1985.

[4] K. Y. Kabalan, A. El-Hajj, and R. F. Harrington, “Characteristic mode analysis of a slot in a conducting plane separating different media,” IEEE Trans. Antennas Propagat., vol. AP-38, no. 4, pp. 476481, Apr. 1990.

[5] T. Wang, J. R. Mautz, and R. F. Harrington, “Characteristic modes and diDole reuresentations of small amrtures,” Radio Sci. , vol. 22, no. 7, pi. 12861297, Dec. 1987.

[6] R. F. Harrington, Time-Harmonic Electromagnetic FieMs. New York

[7] K. Y. Kabalan, A. El-Hajj, and R. F. Harrington, “Scattering and penetration characteristic mode for a dielectric filled conducting cylinder with longitudinal slot,” AEU, vol. 47, no. 3, pp. 137-142, 1993.

McG~~w-H~II, 1961.

Ali El-Hajj was born in Aramta, Lebanon, in 1959. He received the B.S. degree in physics from the Lebanese University, Lebanon, in 1979, the “Inge- nieur” degree from L’Ecole Superieure d’Electricite, France, in 1981, and the “Doctor Ingenieur” de- gree from the University of Rennes I, France, in 1983.

From 1983 to 1987 he was an Assistant Professor of Electrical Engineering at the Lebanese Univer- sity. Currently, he is an Associate Professor of ElLtrical Engineering at the Electrical Engineering

Department, Faculty of Engineering and Architecture, American University of Beirut, Lebanon. His research interests are numerical solution of electro- magnetic field problems and software development.

Grim Y. Kabalan was born in Jbeil, Lebanon. He received the B.S. degree in physics from the Lebanese University, Labanon, in 1979, and the M.S. and Ph.D. degrees in electrical engineering from Syracuse University, Syracuse, NY, in 1983 and 1985, respectively.

During the 1986 Fall semester, he was a Visit- ing Assistant Professor of Electrical Engineering at Syracuse University. Currently, he is an Associate Professor of Electrical Engineering in the Electrical

I I

Engineering Department, Faculty of Engineering and Architecture, American University of Beirut, Lebanon. His research interests are numerical solution of electromagnetic field problems and software development.