Tensor Definition
Mathematical models represent the frequency dependent response of insulating materials through a set of complex numbers. A complex permittivity matrix accounts for the three dimensional nature of electric displacement in anisotropic printed circuit substrates. This tensor maps the electric field vector to the displacement field across the x, y, and z axes of the material.
Material Interaction
Laminates often exhibit different dielectric properties along the glass fibers compared to the resin rich areas between them. Within a complex permittivity matrix, the diagonal elements represent the permittivity in the primary directions while off-diagonal elements describe the coupling between those axes. Designers use these matrices to simulate signal propagation in millimetre-wave circuits where direction-dependent loss and speed become evident.
Advanced software tools require this level of detail to predict how electromagnetic waves behave in non-isotropic environments. Calculations involving the weave of the glass fabric and the density of the resin provide the necessary inputs for the matrix coefficients.
Frequency Dependence
Real and imaginary components of the matrix vary across the operational spectrum of the circuit. Because the complex permittivity matrix includes the loss tangent as an imaginary term, it tracks both the energy storage and the energy dissipation within the substrate. At higher frequencies, the molecular dipoles in the polymer cannot keep pace with the alternating field, leading to changes in the matrix values.
Accurate modeling of these shifts prevents timing errors in high speed digital buses.
