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Ali El-Hajj, Karim Y. Kabalan, and George Nehmetallah

MOMENT METHOD SOLUTION OF A PARTIALLY FILLED PARALLEL PLATE FED THROUGH INFINITE CONDUCTING PLANE (TE CASE) Ali El-Hajj, Karim Y. Kabalan*, and George Nehmetallah Electrical and Computer Engineering Department American University of Beirut Beirut, Lebanon

‫اﻟﺨﻼﺻــﺔ‬ ‫ ﺣﯿﺚ إن اﻟﻮﺳﻂ‬، ‫ﯾﻬﺪف ﻫﺬا اﻟﺒﺤﺚ إﻟﻰ دراﺳﺔ ﺧﺼﺎﺋﺺ اﻟﺪﻟﯿﻞ اﻟﻤﻮﺟﻲ ذي اﻷﺳﻄﺢ اﻟﻤﺴﺘﻮﯾﺔ وﺗﺤﻠﯿﻠﻬﺎ‬ ‫ وذﻟﻚ ﻓﻲ ﺣﺎﻟﺔ‬، ‫ﻏﯿﺮ ﻣﺘﺠﺎﻧﺲ واﻟﺘﻐﺬﯾﺔ ﻋﺒﺮ ﺳﻄﺢ ﻣﻮﺻﻞ ﻻ ﻧﻬﺎﺋﻲ ﻣﺜـﻘﻮب ﺑﺤﺴﺐ أﺑﻌﺎد اﻟﺪﻟﯿﻞ اﻟﻤﻮﺟﻲ‬ . ‫ وﺑﺎﺳﺘﻌﻤﺎل ﻃﺮﯾﻘﺔ اﻟﻌﺰوم اﻟﻌﺪدﯾﺔ‬، ‫اﺳﺘـﻘﺒﺎل اﻟﻤﻮﺟﺎت اﻟﻜﻬﺮوﻣﻐﻨﺎﻃﯿﺴﯿّﺔ‬ (TE Case) ً‫وﻗﺪ ﺗﻤﺖ اﻟﺪراﺳﺔ ﺑﺘﻐﺬﯾﺔ اﻟﺪﻟﯿﻞ اﻟﻤﻮﺟﯿّﺔ ﻋﺒﺮ ﻣﻮﺟﺔ ﺳﺎﻗﻄﺔ ﻣﺴﺘﻮﯾﺔ ﻣﺴﺘﻌﺮﺿﺔ ﻛﻬﺮﺑﺎﺋﯿﺎ‬

. ‫ﺑﺎﺳﺘﻌﻤﺎل ﻣﻌﺎدﻟﺔ رﯾﺎﺿﯿﺔ ﺑﺤﺴﺎب اﻟﻤﺮﻛﺒﺎت اﻟﻤﻤﺎﺳﺔ ﻟﻠﻤﺠﺎل اﻟﻤﻐﻨﺎﻃﯿﺴﻲ ﻋﻠﻰ ﺳﻄﺢ اﻟﻔﺘﺤﺔ اﻟﻨﺎﺗﺠﺔ‬ ‫وﺗﻢ ﻛﺬﻟﻚ اﺳﺘﻌﻤﺎل ﻫﺬه اﻟﻤﻌﺎدﻟﺔ ﻟﺤﺴﺎب ﺷﺪة اﻟﺘﯿّﺎر اﻟﻤﻐﻨﺎﻃﯿﺴﻲ ﻋﻠﻰ ﺳﻄﻮح اﻟﻔﺘﺤﺔ ﻟﻠﺤﺼﻮل ﻋﻠﻰ اﻹﺷﻌﺎع‬ . ‫ وﻓﻲ ﻋﺪة ﺣﺎﻻت‬، ‫اﻟﻨﺎﺗﺞ ﻓﻲ ﺟﻤﯿﻊ اﻻﺗﺠﺎﻫﺎت‬ ‫ وﺑﻤﻘﺎرﺑﺔ ذﻟﻚ ﻣﻊ اﻟﻨﺘﺎﺋﺞ اﻟﻤﺘﻮﻓﺮة ﻓﻲ‬، ‫أﺧﯿﺮاً ﺗﻢ اﻟﺤﺼﻮل ﻋﻠﻰ ﻧﺘﺎﺋﺞ ﻋﺪدﯾﺔ ﻓﻲ ﺣﺎﻟﺔ وﺳﻂ ﻣﺘﺠﺎﻧﺲ‬ . ‫دراﺳﺎت أﺧﺮى‬ ABSTRACT In this paper, the moment method is used for the solution of the receive case of the partially filled parallel plate fed from a perforated infinite conducting plane. The waveguide is excited by an incident transverse electric (TE) plane wave. An operator equation is first formed using the tangential components of the magnetic fields at the aperture. This equation is used for the computation of the magnetic current on the apertures and the radiation pattern for different cases. Numerical results for the homogeneous case is also given and compared with other available data. Key words: Electrical Engineering, EM Scattering and radiation, Numerical Analysis.

* To whom correspondence should be addressed. Electrical and Computer Engineering Department American University of Beirut P.O. Box: 11-0236, Riad El-Solh 1107 2020, Beirut, Lebanon e-mail: [email protected] October 2003

The Arabian Journal for Science and Engineering, Volume 28, Number 2B.

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MOMENT METHOD SOLUTION OF A PARTIALLY FILLED PARALLEL PLATE FED THROUGH INFINITE CONDUCTING PLANE (TE CASE)

1. INTRODUCTION Electromagnetic coupling through two or more regions has been the subject of many research projects through the years due to its importance and its wide area of applications, such as aperture in conducting screens, waveguide-fed apertures, cavity-fed aperture, cavity to cavity, and many others. Historically, these problems were treated by analytical methods, but due to advances in numerical techniques and to the revolution in computer speed and capacity, many numerical techniques are now used in solving these kinds of problems, such as finite difference, finite element, and moment method. This problem is often encountered in assessing the effectiveness of a conducting shield and scattering. It has many applications in radar and communications systems. The problem of a flanged parallel plate waveguide has been addressed by many authors [1–18]; however, most of the previous work has been for cases where the medium inside the plate is homogeneous. In this paper, the case of a parallel plate filled with two different media fed from a perforated infinite conducting plane is considered. In 1976, Harrington and Mautz [19] devised a general formulation for aperture problems using the method of moments. In this formulation, the aperture characteristics are expressed in terms of two independent aperture admittance matrices that are used for deriving the fields at the aperture and the radiated power. This work considers the problem of a partially filled parallel plate fed through an infinite conducting plane. In this case, the geometry is assumed to be excited by a transverse electric (TE) (to the y-axis) plane wave. The problem is treated using the moment method [20]; as a result, an operator equation representing the continuity of the tangential components of the magnetic field across the aperture is obtained in terms of the magnetic current M . This operator equation is transformed, with the help of moment method, to a matrix equation whose solution, using the Galerkin’s method, determines the magnetic currents of the aperture. Once the magnetic current is obtained, it will be used for determining the fields at the aperture, and the radiation pattern.

2. PROBLEM ANALYSIS The geometry of the partially filled open ended parallel plate is excited by a (TE) to the y-axis plane wave, as shown in Figure 1. This waveguide is filled with one dielectric between x = 0 and x = d, and another between x = d and x = a. The dielectric characteristics are (ε1 , µ1 ) and (ε2 , µ2 ), respectively. The outside region a is free space and a source (E i , H i ) is assumed in this region. The equivalence principle [21] is used to divide the original problem in Figure 1 into an equivalent model valid in region a and an equivalent model valid in region b as shown in Figure 2. In Figure 2, the aperture is closed with perfect electric conductor and provided with hypothesized attached magnetic current sheets such that the electric field originally present at the aperture remains unchanged. The first boundary condition enforces the continuity of the tangential electric fields across the aperture in the original problem. Thus the equivalent magnetic current sheets are (+M1 , +M2 ) in region a and (−M1 , −M2 ) in region b. The second boundary condition enforces the tangential components of the magnetic fields to be continuous over the aperture in the original problem. Since the fields are TE to y, we shall choose in region b the wave functions Ψ1 and Ψ2 to represent the tangential components of the electric vector potential F in the waveguide region 1 (0 < x < d) and waveguide region 2 (d < x < a), respectively.

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Ψ1 = C1 cos(κx1 x) exp(−jκz1 z) Ψ2 = C2 cos[κx2 (a − x)] exp(−jκz2 z)

(1)

where κz1 = κz2 = κz and κx1 = κx2 . Since we have two different media in the x direction, their corresponding propagation constants are different, contrary to the z direction where the waveguide’s characteristics are uniform along this direction. Moreover, the separation equations in the two waveguides regions are: (κx1 )2 + (κz )2 = (ω)2 ε1 µ1 = (κ1 )2 (κx2 )2 + (κz )2 = (ω)2 ε2 µ2 = (κ2 )2 .

(2)

Infinite plane x Receive case x

Ei

Direction of propagation

.q

Hi

Region a

e2 , m2 z

Region b

e1 , m1

d

ea , ma

a

0

f Figure 1. Geometry of the problem.

M2

M2

M2

M1

M1

M1

Equivalent for region a

Equivalent for region b

Figure 2. The equivalent model. October 2003


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(1) can be used for the computation of both the electric and magnetic fields in region b. The boundary conditions on the problem are that the tangential components of E vanish on the conducting walls and are continuous together with the tangential components of H at x = d. Thus choose κx1 = κix1 ; κx2 = κix2 so that the boundary conditions on Ez and Hy are satisfied. Each integer i specifies a possible field or mode corresponding to pairs of solutions to Equation (2) as indicated in [21]. i Ez1 =

κix1 κiz C1 sin(κix1 x) exp(−jκiz z) ωε1

i = −κix2 κiz C2 sin(κix2 (a − x)) exp(−jκiz z) Ez2 i = −jκiz C1 cos(κix1 x) exp(−jκiz z) Hy1 i = −jκiz C2 sin(κix2 (a − x)) exp(−jκiz z) . Hy2

(3a)

Applying the continuity of the electric field and the magnetic field at x = d and rearranging, we obtain 1 1 i κ tan(κix1 d) = − κix2 tan(κix2 (a − d)) . ε 1 x1 ε2

(3b)

Every pair of solutions of Equations (2) and (3) corresponds to a mode of propagation in the parallel plate. In addition, the electric fields in region b can be expressed in terms of a series of functions by means of the representation [1]: Eb1t (M) =

Ai1 Y1i exp(−γ1i z)uz × ei1 =

C1 (κ21 − κix1 ) cos(κix1 x) exp(−jκiz z)ux jωε 1 i

Ai2 Y2i exp(−γ2i z)uz × ei2 =

C2 (κ22 − κix2 ) cos(κix2 (a − x)) exp(−jκiz z)ux . (4) jωε 2 i

i

Eb2t (M) =

i

In (4), t refers to the tangential components of the electric fields in the xy plane, γji are the modal propagation constants, Yji are the characteristic admittances, and Aij are the modal amplitudes. These parameters are related as in [1] by: γ1i = γ2i = γ i = jκiz Ai = jC κiz = γ i C Yi =

γi 1 = i jωε Z

= 1, 2 = 1, 2

(5)

with

γi =

224

 i 2  κx    1 − jκ   κ   2    κ  i   κx 1 − κix

κ > κix .

(6)

κ < κix


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In addition, ei1 = cos(κix1 x).uy ei2 = cos κix2 (a − x).uy

(7)

are the vectors chosen to satisfy the boundary conditions in the waveguide regions. Hence, the equivalent magnetic current in each region in the aperture (z = 0) is given by: 











− Ai1 Y1i ei1  Eb (M)    M1  i =  = uz ×  1t        i i i b E2t (M) M2 − A2 Y2 e2

  . 

(8)

i

Also, the magnetic fields in this region can be written as:   i A1 exp(−γ i z).ei1  Hb1t (M)    i  =    b H2t (M) Ai2 exp(−γ i z).ei2 

      =  

i

i

i

 −jκz C1 cos[κix1 x] exp(−jκz z)uy

−jκz C2 cos[κix2 (a − x)] exp(−jκz z)uy

  . (9) 

Where: Ai1 = C1 γ i Ai2 = C2 γ i ei1 = cos(κix1 x).uy ei2 = cos[κix2 (a − x)].uy .

(10)

In region a, the magnetic field is the sum of that due to impressed sources plus that due to the equivalent magnetic current Mi . The magnetic field due to the equivalent magnetic current Mi is given by:

Hat (M)

Ka =− . 2ηa

d

M1 (x

(2) )H0 (κa

0

Ka |x − x |)dx uy − . 2ηa

a

(2)

M2 (x )H0 (κa |x − x |)dx uy . (11)

d

Equation (11) states that the total magnetic field in region a is that due to the magnetic current over region 1 in the aperture plus the induced magnetic field due to the magnetic current over region 2 in the aperture. Moreover, the magnetic field due to the impressed sources over the aperture is given by: Hsc = 2 exp(−jκa x sin θi )uy .

(12)

Where the superscript (sc) means short circuited aperture and θi refers to the incident angle of the excitation fields. For a solution of the problem, the continuity of the magnetic field at the aperture is considered. This continuity results in the following equation: Hat = Hbt = Hbt (−M) = Hat (M) + Hsc t . October 2003

(13) The Arabian Journal for Science and Engineering, Volume 28, Number 2B.

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(13) can be rewritten in the form of an operator equation b a Y(M) = Hsc t = −Ht (M) − Ht (M).

(14)

In (14), the linearity of the operator H in medium b is used and Y(M) is called the admittance operator. In matrix form and taking into consideration the two aperture regions, we have: 







 Y1 (M)   −Ha1t (M) − Hb1t (M)  = .      Y2 (M) −Ha2t (M) − Hb2t (M)

(15)

Assume small apertures where d min(λ0 , λ1 , λ2 ) and consider the case of one mode approximation; that is, the principal mode, and using Equations (7) and (8), the equivalent magnetic currents are then given by: M1 = −A11 .Y11 e11 = −A11 Y11 cos(κx1 x)uy M2 = −A12 .Y21 e12 = −A12 Y21 cos [κx2 (a − x)] uy .

(16)

In (16), the modal amplitudes are determined from the boundary condition at the aperture and obtained to be: d A11

=

0

d 0

a

M1 .Y11 .e11 dx , (Y11 e11 )2 dx

A12

d

M2 .Y21 · e12 dx

= a d

.

(17)

(Y21 e12 )2 dx

Using (9) at z = 0 for i = 1, 2 and (11), and for 0 < x < d, (14) becomes:  d  a 2ηa  (2) (2) M1 (x )H0 (κa |x − x |)dx uy + M2 (x )H0 (κa |x − x |)dx uy  κa 0

d

+ A11 e11 = 2 exp(−jκa x sin θi )uy .

(18)

For d < x < a, (14) becomes:  a  d 2ηa  (2) (2) M2 (x )H0 (κa |x − x |)dx uy + M1 (x )H0 (κa |x − x |)dx uy  + κa 0

d

A12 e12 = 2 exp(−jκa x sin θi )uy .

(19)

The modal solution for the current M on the aperture can be obtained by using the characteristic currents as both expansion and testing functions in the method of moments. Following this procedure, M is assumed to be a linear combination of the characteristic currents: M1 (x ) =

P

Vp fp (x )uy

p=1

M2 (x ) =

Q

Wq gq (x )uy .

(20)

q=1

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October 2003


In (20), Vp and Wq are complex-coefficients to be determined, and fp (x ) and gq (x ) are expansion functions defined over the pth segment of region 1 of the aperture and the q th segment of region 2 of the aperture, respectively. Moreover, P and Q determine the number of divisions taken on each aperture region. Substituting (20) into (18) and (19), we obtain  d  P a Q 2ηa  (2) (2) Vp fp (x )H0 (κa |x − x |)dx uy + Wq gq (x )H0 (κa |x − x |)dx uy  κa p=1 q=1 0

d

+ A11 e11 = 2 exp(−jκa x sin θi )uy

(21)

and  a  Q d P 2ηa  (2) (2) Wq gq (x )H0 (κa |x − x |)dx uy + Vp fp (x )H0 (κa |x − x |)dx uy  κa q=1 p=1 0

d

+ A12 e12 = 2 exp(−jκa x sin θi )uy .

(22)

Equations (21) and (22) are complex-valued equality equations as indicated earlier. Galerkin’s solution is used by taking the inner product of (21) and (22) with each fs (x) and gt (x), where s = 1, 2, . . . , P and t = 1, 2, . . . , Q. Each inner product, out of the four, results in a complex-valued equation whose unknowns are the real parts and the imaginary parts of the complex valued coefficients Vp and Wq . These equations can be rearranged and regrouped in a matrix form as follows: 

 [M1Rps ]    [M  1Ips ]    [M  2Rpt ]   [M2Iptt ]

 



[M1Rqs ] −[M1Iqs ]  VRp   < 2 cos(κa sin θi ) · fs (x) >, 0 < x < d, 0 < x < d              [M1Rps ] [M1Iqs ] [M1Rqs ]   VIp  < −2 sin(κa sin θi ) · fs (x) >, 0 < x < d, 0 < x < d   . (23) =         i −[M2Ipt ] [M1Rqt ] −[M1Iqt ]  WRq   < 2 cos(κa sin θ ) · gt (x) >, d < x < a, d < x < a           i [M2Rpt ] [M1Iqt ] [M1Rqt ] WIq < −2 sin(κa sin θ ) · gt (x) >, d < x < a, d < x < a −[MiIps ]

In (23), VRp , VIp , WRq , and WIq are respectively the real parts and the imaginary parts of Vp and Wq . Moreover, the elements of the sub-matrices are determined as per follows: The [M1Rps ] in a (P × Q) matrix representing b (fp (x )). Similarly, the [M2Iqt ] in a (P × Q) matrix the real part of the inner product of fs (x) with Real H1t b (gq (x )). representing the imaginary part of the inner product of gt (x) with Real H2t

3. NUMERICAL RESULTS A computer program is written in MATLAB for solution of the problem. The first step in the calculation is to determine the elements of each of the sub-matrices of (23). In this calculation, a pulse expansion and testing functions are considered on each aperture. The singularities in the integrals are treated analytically. Once the elements of the matrices are obtained, the real parts and the imaginary parts of the coefficients of the series expansion of the magnetic current in both regions are computed. The value of these coefficients are used to determine the equivalents magnetic currents on the apertures, and the radiation pattern of the waveguide for various values of (a, εr1 , εr2 , θi , d) which are the width of the waveguide, the relative permittivity of regions, angle of incidence of the wave, and the width of the dielectric in region 1. A combination of the above parameters will give us a different pattern than the case of the homogenous waveguide. Figures 3 and 4 show October 2003


227


the magnitude of the magnetic current and the normalized magnitude of radiation pattern for the case where εr1 = εr2 = ε0 , a = 0.4λ0 , d = 0.1a, and θi = π3 .

MAGNETIC CURRENT

2

1.5

1

Real 0.5

0

Imaginary Image

-0.5

x/l0 -1

0

0.066

0.133

0.2

0.266

0.33

0.4

Figure 3. The magnitude of the magnetic current for the case where εr1 = εr2 = ε0 , a = 0.4λ0 , d = 0.1a, and θi = π3 . 90

1.0 60

120

H

0.75

2

0.5

150

30

0.25

180

0

210

330

240

300 270

Figure 4. The radiation pattern for the case where εr1 = εr2 = ε0 , a = 0.4λ0 , d = 0.1a, and θi = π3 where |H|2 is normalized to its maximum value.

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October 2003


Figures 5 and 6 show the magnitude of the magnetic current and the normalized magnitude of radiation pattern for the case where εr2 = 2εr1 = 2ε0 , a = 0.4λ0 , d = 0.9a, and θi = π3 . It is noticed that the difference between Figure 4 and Figure 6 is negligible and this may be referred to the small thickness of region 1 compared to that of region 2.

1.5

MAGNETIC CURRENT

1

Real 0.5 0

Imaginary Image

-0.5 -1 -1.5 -2

x/l0 0

0.066

0.133

0.2

0.33

0.266

0.4

Figure 5. The magnitude of the magnetic current for the case where εr2 = 2εr1 = 2ε0 , a = 0.4λ0 , d = 0.9a, and θi = π3 .

90

1.0 60

120

H

0.75

2

0.5

150

30

0.25

180

0

210

330

240

300 270

Figure 6. The radiation pattern for the case where εr2 = 2εr1 = 2ε0 , a = 0.4λ0 , d = 0.9a, and θi = π3 where |H|2 is normalized to its maximum value. October 2003


229


The real and the imaginary parts of the equivalent magnetic current and the normalized magnitude of radiation pattern are shown in Figures 7 and 8 for the case when εr1 = εr2 = ε0 , a = 0.4λ0 , d = 0.1a, and θi = 0. It is to note that the magnetic of the magnetic current currents are shown for x/λ ∈ [0, a = 0.4] and the radiation pattern is zero outside the range φ = [0, 180◦ ].

1.5

MAGNETIC CURRENT

1

Real 0.5

0

Imaginary Image -0.5

-1

x/l0 -1.5

0

0.066

0.133

0.2

0.266

0.333

0.4

Figure 7. The real and the imaginary parts of the equivalent magnetic current for the case when εr1 = εr2 = ε0 , a = 0.4λi , and θi = 0. 90

H

1.0 60

120

2

0.75 0.5

150

30

0.25

180

0

210

330

240

300 270

Figure 8. The radiation pattern for the case where εr1 = εr2 = ε0 , a = 0.4λ0 , d = 0.1a, and θi = 0 where |H|2 is normalized to its maximum value.

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From Figures 3–8, the following conclusion and remarks can be made. (1) If a decreases, the radiation pattern will be smoother. (2) For the case of a non-homogenous waveguide, the characteristic currents have a ripple at the interface between the two mediums, but the radiation pattern preserves the overall shape. (3) If zero angle of incidence, then the radiation pattern will be more directive and both the characteristic current and the radiation pattern are symmetric when the dielectric media inside the parallel plate are the same. The maximum lies at an angle φ = π2 for the radiation pattern. (4) For normal incidence the radiation pattern is more directive than for oblique incidence. (5) For the homogenous cases, the obtained results are in agreement with those of previous results obtained in [2].

4. CONCLUSION This paper presents a moment method solution of the receive case of the partially filled parallel plate excited by an incident transverse electric plane wave (to the y-axis) (TE). The magnetic current on the aperture and the radiation pattern are shown for different cases to indicate the effect of the partially filled waveguide parameters on the directivity and shape of the pattern. The homogenous case is also treated. This method is applicable to narrow slots where other modes are assumed to propagate in the waveguide but it will not affect the solution of the problem. For the case of a wider waveguide, other modes assumed to propagate in the waveguide must be taken into consideration. REFERENCES [1] K.Y. Kabalan and A. El-Hajj, “A Characteristic Mode Solution of the Parallel Plate-Fed Slot Antenna”, Radio Science, 30(2) (1995), pp. 353–360. [2] K.Y. Kabalan and A. El-Hajj, “Characteristic Mode Formulation of the Aperture-Fed Waveguide Problem”, AEU Journal of Electronics and Communication, 48(2) (1994), pp. 130–134. [3] C.M. Butler and R.D. Neuels, “Coupling Through a Slot in a Parallel-Plate Waveguide Covered by a Dielectric Slab”, AEU Journal of Electronics and Communication, 88(1) (1988), pp. 46–53. [4] C.M. Butler, C.C. Courtney, P.D. Manniko, and J.W. Silvestro, “Flanged Parallel Plate Waveguide Coupled to a Conducting Cylinder”, IEE Proc. H, 138(6) (1991), pp. 549–559. [5] A.Q. Howard, “On the Mathematical Theory of Electromagnetic Radiation from Flanged Waveguides”, J. Math. Phys., 13 (1972), pp. 482–490. [6] T. Itoh and R. Mittra, “A New Method of Solution for Radiation from a Flanged Waveguide”, Proc. IEEE, 59(3) (1971), pp. 1131–1133. [7] T. Itoh and R. Mittra, “TEM Reflection from a Flanged and Dielectric Filled Parallel-Plate Waveguide”, Radio Science, 9(10) (1974), pp. 849–855. [8] M.S. Leong, P.S. Kooi, and Chandra, “Radiation from a Flanged Parallel Plate Waveguide: Solution by Moment Method with Inclusion of Edge Condition”,IEE Proc. H. Microwaves, Antenna and Propagat., 135(4) (1988), pp. 249–255. [9] K. Hongo, Y. Ogawa, T. Itoh, and K. Ogasu, “Field Distribution in a Flanged Parallel-Plate Waveguide”, IEEE Trans. on Antennas and Propagat., 2(7) (1975), pp. 558–560. [10] K. Hongo, “Radiation from a Flanged Parallel-Plate Waveguide”, Radio Science, 32(7) (1972), pp. 955–964. [11] S.N. Sinha, D.K. Mehra, and R.P. Agarwal, “Radiation from a Waveguide Backed Aperture in an Infinite Ground Plane in the Presence of a Thin Conducting Plate”, IEEE Trans., AP-34 (1986), pp. 539–545. [12] R.W. Scharstein, “Two Numerical Solutions for the Parallel Plate Fed Slot Antenna”, IEEE Trans., AP-37(11) (1989), pp. 1415–1426. October 2003


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[13] D. Kinowski and M. Guglichmi, “Multimode Network Representations for the Scattering by an Array of Thick Parallel Plates”, IEEE Trans. on Antennas and Propagat., 45(4) (1997), pp. 608–613. [14] J.W. Lee, H.J. Eom, and J.H. Lee, “TM Wave Radiation from Flanged Parallel Plate into Dielectric Slab”, IEE Proc. Antennas Propagat., 143(3) (1996), pp. 207–210. [15] T.J. Park and H.J. Eom, “Analytic Solution for TE Mode Radiation from a Flanged Parallel Plate Waveguide”, IEE Proc. H, 140(5) (1993), pp. 387–389. [16] H. Cary and S. Waxman, “Wave Propagation Along a Fully or Partially Loaded Parallel Plate Chiro Waveguide”, IEE Proc. Microwaves Antennas Propagat., 141(4) (1994), pp. 299–306. [17] M. Mongiordo and T. Rozzi, “Singular Integral Equation Analysis of Flanged-Mounted Rectangular Waveguide Radiators”, IEEE Trans. on Antennas and Propagat., 41(5) (1993), pp. 556–565. [18] J.H. Lee, H.J. Eom, and J.W. Lee, “Scattering and Radiation from Finite Thick Slits in Parallel-Plate Waveguide”, IEEE Trans. on Antennas and Propagat., 44(2) (1996), pp. 212–216. [19] R.F. Harrington and J.R. Mautz, “A Generalized Network Formulation for Aperture Problems”, IEEE Trans. on Antennas and Propagation, AP 24(11) (1976), pp. 870–873. [20] R.F. Harrington, Field Computation by Moment Methods. New York: Macmillan Company, 1968. [21] R.F. Harrington, Time Harmonic Electromagnetic Fields. New York: McGraw-Hill book Company, 1961. Paper Received 21 April 2001; Revised 6 April 2002; Accepted 27 May 2002.

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