Through cartesian discretization of the differential operator
in (4.13), [66] obtained a finite difference method
on a rectangular grid. With interpretation according to the TLM method [67],
[68], a method with lumped inductances, capacitances, resistances and
admittances on a rectangular mesh was presented by [65] and
[69]. The value for the lumped resistance is obtained
from (4.11) and that for the lumped admittance from (4.12). The
value of the lumped inductance is calculated directly from (4.9), while
the lumped capacitance is obtained by multiplication of the capacitance
from (4.9) with DxDy. Dx and Dy denote the
rectangular grid cell lengths in x-direction and in y-direction, respectively. Other
circuit elements can be introduced easily at every point of the grid. The circuit
structure of the lumped elements on the grid leads to a sparse matrix. However, a very
fine grid has to be used to obtain reasonably good accuracy. Figure 4.4
depicts a comparison of the analytic results from (4.18) to
the LCR grid method results for the impedance Z_{in} of a port at position (x=10mm, y=10mm) on a rectangular cavity with dimensions (L=160mm, W=120mm) and
h=7mm. A grid spaciny of Dx = Dy = 2mm was used for the LCR grid method
simulation. Even with that fine grid the comparison shows some slight deviations of the
impedance magnitude minima, which indicate a small inaccuracy of simulated inductance.
This is a disadvantage of the method, because the necessity of the fine grid leads to
higher simulation costs compared to the FEM method in Section 4.4.

Rectangular planes have been used for the comparison in Figure 4.4 to
enable the comparison with the analytical solution. When the method is used for planes
with edges that are not parallel to one of the cartesian directions
or
, a dense grid will be necessary to obtain an accurate discretization of the
geometry at that edge. Although sub-gridding at the edge is an opportunity to reduce the
overall mesh size, the effort for geometry discretization is much higher compared to the
finite element method in Section 4.4, which uses triangular meshing.

(a) Magnitude comparison. | (b) Phase angle comparison. |