Determination of the effective permittivity of dielectric mixtures with the transmission line matrix method

In this article, the effective permittivity of two-phase dielectric mixtures is calculated by applying the transmission line matrix (TLM) method. Two slightly different TLM algorithms are considered: a hybrid approach, which combines the TLM method with a subgriding technique based on dual capacitor circuits, to allow a refined description of the material, and a standard or pure TLM approach, which uses a mesh size smaller than the typical dimension of insertions in order to appropriately describe details of the geometry. A study of the statistical distribution of permittivity for insertions in random positions is also presented, showing that the effective permittivity of the mixture tends to concentrate around the mean value as insertions reduce in size. Both TLM techniques are applied to dielectric mixtures in two-dimensional situations. When the concentration of insertions is small, the results are in close agreement with prediction formulae while for higher concentration values, deviations are observed, although basically the results fall within the range predicted by theoretical bounds. Numerical results obtained using the two TLM approaches present a similar qualitative behavior; nevertheless, a clear difference is observed between them. The study of special periodic situations with coated insertions allows us to identify the pure TLM results as more accurate than those of the hybrid approach and also explains why homogeneous distributions provide numerical values close to the theoretical limits. The effects of shape on permittivity are also modeled and deviations to the Wiener bounds are discussed in detail, using two- and three-dimensional examples in practical situations.