A General Three-Dimensional Tensor FDTD-Formulation for Electrically Inhomogeneous Lossy Media Using the Z-Transform

A general three-dimensional tensor finite-difference time-domain (TFDTD) formulation is derived to model electrically inhomogeneous lossy media of arbitrary shapes. The time domain representation of electric losses is achieved using Z-transforms. The regular cubical grid structure is maintained everywhere in the calculation domain by defining a 3-D face-fraction based 3 $\times$ 3 permittivity tensor on the interfaces that describes the relationship between the (known) average flux density vector and the (unknown) local electric field vector. For electrically lossy media, this tensor is complex in the frequency domain. However, it can be modified for use with the Z-transform. Only this modified real form is inverted, then transformed from the frequency into the Z-domain, and finally into the time domain. Furthermore, a local interface matrix is used to describe the relationship between the local electric field in the grid node and its counterpart on the other side of the interface. This matrix is complex in the frequency domain for lossy media. By applying the Z-transform, this matrix can also be transformed into the time domain using only real modified matrix elements. The accuracy of the method is confirmed by comparisons with analytical solutions.