that a very coarse grid was used with EF equations, with only two cells between the microstrip and the ground plane and four cells across the microstrip. The maximum error in the 20᎐60 GHz frequency band is 1.5, 6.7, and 13.5% using the enhanced EF, split-cell, and staircase techniques, respectively.
Any diagonal microstrip line at an arbitrary angle to the grid can be modeled accurately using singularity-enhanced EF equations by properly selecting the aspect ratio of the FDTD grid. Currently available enhanced equations are limited to diagonal edges, and therefore, one aspect ratio of the Ž grid is determined by the angle of the microstrip line accord-. ing to ⌬ yr⌬ x s tan . Ongoing research hopefully will eliminate this limitation in the future.
IV. CONCLUSIONS
Using the singularity-enhanced EF equations for diagonal metal edges, microstrip lines which are at an angle to the grid can be modeled very accurately and efficiently. A coarse grid may be used, saving computer memory and time, but not sacrificing accuracy. For example, the error in the computed effective dielectric constant of a microstrip line at 40 GHz is 0.8, 6.1, and 12.8% using enhanced EF, split-cell, and staircase techniques, respectively.
The enhanced-equation technique avoids the need to use a fine subgrid near the microstrip line, and has the advantage of negligible computing overhead when compared with the subgrid approach. The singularity-enhanced EF equations, which differ from the standard Yee equations by only a few additional coefficients, can be easily implemented in a standard FDTD code. Tests have shown that the new equations are stable even when the time step is increased to the maximum allowed value.