ELECTRONICS LETTERS 7th June 2001 Vol. 37 No. 12 Iterative FEM for characterising radiating slot in broad w all of rectangular w aveguide Jongkuk Park, Jungwon Lee, Heeduck Chae and Sangwook Nam It is shown that the iterative finite element method with the radiation- type boundary condition can give an efficient and accurate solution to the radiation problem. The proposed method is applied to the characterisation of a radiating slot on the broad wall of a rectangular waveguide. The result is compared with those of other conventional methods, and shows a good agreement. Introduction: Recently an efficient iterative finite element method (itera- tive FEM) has been proposed [1], applied to a two-dimensional scatter- ing problem [2], and extended to the characterisation of three- dimensional discontinuities in a rectangular waveguide [3]. In the above cases, when there is a small number of meshes around a scatterer, a finite element method has been shown to give good results. However, the Dirichlet condition used in the above method has been found to be unsuitable for characterising the scattering caused by resonant slots or large cavities due to the internal resonance problem. Thus, to eliminate this phenomenon, a radiation-type boundary condition has been pro- posed [4] and extended to a three-dimensional scattering problem [5]. In this Letter, the iterative FEM is applied to the characterisation of a radiating slot, which is used widely as an element of waveguide slot array antennas. This radiating slot is approximately half a wavelength in size, which means that internal resonance is liable to occur. Therefore, the radiation-type boundary condition is incorporated with this method to prevent such internal resonance. Theory: Fig. 1 shows the geometry of a slot on a rectangular waveguide and the definition of surfaces used in the proposed method. As shown in Fig. 1, the volume enclosed by A 2in , A 2out , A c will be discretised into finite elements, and so A 2in , A 2out denote the boundary surfaces where the finite element meshes are terminated. On these boundary surfaces A 2in and A 2out , each boundary condition is imposed as follows: With the given boundary conditions, the electric field in the volume and on the boundary surfaces can be calculated by seeking the stationary point of the functional given by Initially, it is assumed that only the TE 10 mode exists inside the rec- tangular waveguide and there is no field outside. Thus, out in eqn. 1 is equal to zero and in in eqn. 1 can be obtained easily from solving the TE 10 mode inside the rectangular waveguide. Using the typical finite element procedure, we can calculate the electric field inside the volume. However, this field is not the accurate final solution we are seeking since the field on the boundary is not a real solution but an assumed one. To obtain a more accurate solution, we update the boundary field using the equivalent magnetic current, which is given by the field calculated on slot surfaces A 1in and A 1out . In this procedure, the appropriate Green function should be used in the updating integral equation. Outside the waveguide, the simple free-space Green function is employed and so the updated out can be easily calculated. However, inside the rectangular waveguide, the waveguide Green function, represented by the infinite series, which converges very slowly, should be used. Since it is almost impossible to calculate this waveguide Green function as it is, its series form is accelerated so that it converges fast enough to calculate it numerically using the Poisson summation formula and the Kummer transform [6]. Although the infinite series is accelerated so that it itself rapidly converges, the numerical integration for updating the boundary fields is still a time-consuming procedure. However, this numerical inte- gration is performed only once at the first iteration since its value is stored in a matrix form and this matrix is invariant during the iterations. Fig. 1 Geometry of radiating slot in rectangular waveguide Fig. 2 Resonant lengths of longitudinal slot on broad wall of rectangular waveguide (WR-90) against offset from centreline w = 1.59 mm, f = 9.375 GHz a t = 1.27 mm —■— proposed method ——— result from [8]  result from [9] b t = 0.38 mm —■— proposed method ——— result from [8] × measured [8] U → U → U →