Simple, efficient, and accurate analysis of the flanged parallel-plate waveguide

The flanged parallel-plate waveguide is analysed based on the method of Kobayashi potential (KP) using Fourier function space. The presentation of the method is free from intricate mathematics. Standard integral identities are used for problem formulation, without direct use of Weber-Schafheitlin (WS) integrals. The Fourier function space is exploited for the construction of the governing linear equations instead of Jacobi polynomials. A simple strategy is suggested for the evaluation of the required improper integrals. Near-field results are validated through convergence analysis. Far-field patterns are compared with predictions of the surface equivalence theorem (SET).


Introduction
Radiation from a flanged waveguide is an old problem that is both of interest from electromagnetic (EM) and antenna theory standpoints.This problem can be handled with diverse methods including analytical, semi-analytical, and numerical (full-wave) [1][2][3][4][5].Analytical solutions, thus far reported, are all approximate and full-wave solutions suffer from computational cost.A mid-solution is the method of Kobayashi potential (KP); an interesting semi-analytical method that can solve a particular class of mixed boundary value problems (BVPs) in electromagnetics (EM) [6][7][8].This method exploits discontinues Weber-Schafheitlin (WS) integrals to satisfy the edge and discontinuous Dirichlet boundary conditions (BCs).If applicable, the KP method is more efficient compared to the method of moments (MoM) and the finite element method (FEM) when conducting edges and corners are present.As time is the main concern regarding numerical methods, mathematical labor is expected in analytical and semi-analytical methods.This is especially the case in almost all papers regarding the KP method [5,[9][10][11][12][13].
In this work, an analysis of the flanged parallel-plate waveguide is carried out using the KP method with mathematics at the undergraduate level.There is no need to be familiar with either hypergeometric functions or Jacobi polynomials.Instead of WS integrals, and instead of Jacobi function space, Fourier basis functions are used for the construction of governing algebraic equation which leads to improved convergence and less CPU time [14].A simple strategy is suggested for the evaluation of the improper integrals.The solution is validated based on convergence analysis [15].The obtained far-field patterns are compared to predictions of the surface equivalence theorem (SET) as an approximate analytical solution.

Statement of the problem
The cross-section of the flanged parallel-plate waveguide to be analyzed is depicted in Figure 1.All the boundaries of the structure are assumed to be a perfect electric conductor (PEC).It is assumed that a known traveling EM wave in +y direction, emanating from y = −∞ excites the structure.The polarization of the incident wave is TE or TM to x.The incidence and reflected waves can be expanded over the parallel-plate waveguide modes; i.e.
for TE and TM cases, respectively, wherein multiplicative constants are known.In addition, with k 2 1 = ω 2 μ 1 ε 1 .The restriction on h n ensures satisfying the Sommerfeld radiation condition at y = + ∞.Similar expressions can be developed for the reflected waves; i.e.
Following [5], the representation of diffracted fields makes use of a special class of WS integrals to automatically satisfy the required auxiliary conditions (Appendix A): wherein and Similar to the reflected fields, k 2 2 = ω 2 μ 2 ε 2 .The unknown constants can be determined by imposing the continuity of tangential fields at the junction.The formulation can be generalized to consider conducting losses by taking the advantage of impedance boundary conditions [13].

Deriving the governing equations
Here, the resulting equations from the imposition of continuity conditions are expanded over the Fourier function space.This approach is simple and efficient, and does not require being involved with Jacobi's polynomials and hypergeometric functions.Moreover, it reduces the number of Bessel function evaluations and diagonalizes some of the resulting matrices.Finally, assuming ξ as the integration variable, all integrands decays with ξ 5/2 that is ξ 1/2 order faster compared to those obtained using the Bessel function space [5].Throughout the paper, u = a −1 x, ξ = k x a, κ 2 = k 2 a, and κ y = k y a.

TE case
For TE-mode incidence, using imposing the continuity condition for the EM fields leads to: where Equation ( 9) is a set of four N × N systems of equations to be solved for the unknown coefficients.Note that corresponding matrices to A TE m and B TE m are diagonal.

TM case
For TM-mode incidence, using imposing the continuity condition for the EM fields leads to: where

Evaluation of integrals
For each of the TE and TM wave incidences, there are two sets of integrals with similar mathematical structures.For brevity, only one set for each case is studied here.

TE case
Consider the semi-infinite integral of the form: wherein Note that f TE J and f TE c are undefined at ξ = 0, (2n + 1)π/2.Since J ν (x) is zero at the origin, the singularity of the former is removable.This is the case for singularities of f TE c which can be verified by applying L'Hopital's rule.Thus, the said singularities are not troublesome.Also, f TE κ is bounded over the real axis, and thus the p.v. symbol can be omitted.To evaluate (14), let I TE = I TE 1 + I TE 2 with The value of κ TE is not determined, yet.Evaluation of I TE 1 can be carried out by standard quadrature rules.To determine a proper value for κ TE and to develop a method to properly approximate I TE 2 , consider (16b) and assume κ TE max[|κ 2 |, (2n + 1)π/2].Thus, the Bessel function can be replaced by its large-argument approximation [16]; i.e.

TM case
Consider the semi-infinite integral of the form: There is no concern with singularities of the integrand and the p.v. symbol can again be omitted.Although f TE κ is unbounded at ξ = κ 2 , it has a well-defined antiderivative there, since ξκ −1 y = −d(κ 2 2 − ξ 2 ) 1/2 /dξ .Thus, the integrand is weak-singular.To evaluate (22), let I TM = I TM 1 + I TM 2 with: wherein which can be evaluated analytically.

Numerical results
In this section, the claims of the paper are verified.It is assumed that the waveguide is airfilled and is excited by the first possible mode for each polarization.Validation is carried out based on the convergence analysis.The second norm of the electric field distribution over the junction is used as the convergence measure.All field distributions are normalized.For the second region, both lossless and lossy media are considered.For the former, and for the latter, The first set of results, reported in Figure 2, prove the superiority of the Fourier function space over the Bessel function space in the sense of convergence and computational cost, wherein 'Conv.' and 'Prop.' stand to, respectively, 'conventional' and 'proposed', and k 2 a = 5.In Figure 3, convergence curves corresponding to waveguides with k 2 a = 1, 10, radiating into lossless and lossy media, are reported.To show that the BCs at the waveguide opening is satisfied, the ratio of the EM field at the vicinity of the junction is reported in Figures 4 and 5 for TE and TM excitations, respectively.For further validation, far-field patterns obtained from the KP method are compared to what obtains from the application of the SET [3] (Figures 6 and 7).To demonstrate the capability of the method in handling high-order modes, the singular component    of the EM fields over the junction for the first three odd(even) modes corresponding to TE(TM) polarization is reported in Figure 8. Finally, the magnitude of the reflection coefficient vs. the aperture size is depicted in Figure 9, assuming lossless media.Simulations are   performed using a personal computer with an Intel ® Core TM i7-7500U processor.Curves including the CPU time are not reported for brevity.To provide an estimate, analysis with 6 modes lasts about 0.3 s.

Conclusion
Analysis of the flanged parallel-plate waveguide can be solved by the KP method without intricate mathematics.There is no need to be involved with the WS integrals and, thus, hypergeometric functions.Construction of the governing linear system of equations is possible using the Fourier function space.Evaluation of required improper integrals does not need complicated mathematics and can be carried out using standard quadrature rules and available mathematical packages.

Figure 1 .
Figure 1.Cross-section of the flanged parallel-plate waveguide.

Figure 4 .
Figure 4.The ratio of the boundary fields for TE excitation: (a) lossless, (b) lossy media.

Figure 5 .
Figure 5.The ratio of the boundary fields for TM excitation: (a) lossless, (b) lossy media.