**Lucia Cascio**, **Giampaolo Tardioli** and **Wolfgang J.R. Hoefer**, in collaboration with **Tullio E. Rozzi** from the University of Ancona, Ancona, Italy

The TLM method is widely regarded as an efficient and flexible technique for the analysis of a large class of electromagnetic problems. One of the main limitations of this and other numerical techniques is that the spatial discretization fails to accurately describe the singularities of the electromagnetic field, which occur for example close to sharp edges. Unless a very fine discretization is used, the singular behavior around the corner is poorly represented and the frequency domain characteristics of the structure will typically be shifted. This error is very often unacceptable when we are dealing with narrowband structures such as filters.

The accuracy of the discretized model can be improved by introducing a better description of the field singularity, through local modification of the algorithm. We have taken a new approach based on the quasi-static approximation of the Green's functions for an infinite conductive wedge. The field distribution around a corner is represented in terms of an equivalent circuit, expressed by a Z matrix, which can be implemented easily and efficiently in TLM.

Due to the quasi-static approximation, the voltages at the ports of the equivalent circuit depend only linearly on the frequency. Therefore, using a bilinear discretization scheme to approximate the frequency dependance, we obtain a recursive formulation to describe the corner condition in the TLM process:

In this expression *Y_0* is the TLM link line admittance, *V_k_r* and *V_k_i* are the vectors of the voltages incident and reflected at the terminals of the equivalent circuit at the time step *k*.

The new method has been applied to analyze discontinuities in the transverse section of a WR(28) rectangular waveguide (*a*=7.112 mm). To validate the model of a knife edge, a symmetrical inductive iris with aperture *d*=*a*/2 has been analyzed both with the corner modification and the regular TLM algorithm, and the results have been compared with Marcuvitz's formulae. The scattering parameters obtained for different discretizations are shown in Fig. 1-a. Note that the corner modification improves considerably the accuracy of the TLM algorithm (Fig. 1-b) even when a very coarse mesh is used. To further test the efficiency of the proposed method, an iris-coupled waveguide bandpass filter, with center frequency of 33.18 GHz and bandwidth of 0.94 GHz, has been analyzed. Also in this case the corner correction results in a much faster convergence to Marcuvitz's curves as compared with the standard TLM algorithm (Fig. 2).

To verify the model of the 90³ wedge, a symmetrical iris, of thickness *t*=*a*/6 and aperture *d*=*a*/2 has been examined. Comparison with the uncorrected TLM algorithm and other techniques has shown that in this case the correction is less effective since, for this kind of discontinuities, the standard TLM method provides good accuracy even with relatively coarse discretizations (Fig. 3).

Further details can be found in:

L. Cascio, G. Tardioli, T. E. Rozzi, and W. J. R. Hoefer, "A Quasi-Static Modification of TLM at Knife Edge and 90 Wedge Singularities" to be presented at the 1996 IEEE MTT Symposium, San Francisco, CA.

Fig. 1: S-parameters for the thin iris in WR(28) waveguide: a) TLM with corner correction, b) TLM without corner correction

Fig. 2: Iris coupled bandpass filter in WR(28) waveguide: a) TLM with corner correction, b) TLM without corner correction

Fig. 3: S-parameters for the thick iris in WR(28) waveguide: a) TLM with corner correction, b) TLM without corner correction