Numerical computation of transient quasistatic electric fields

In many problems involving applications ranging from microwave substrates to high voltage isolators, capacitive and conductive effects have to be simultaneously taken into account, whereas inductive phenomena can be neglected. This means that the quasistatic electric field intensity is curl free and can hence be described by an electric scalar potential, but both the conduction and displacement current density have to be considered. In case of time harmonic problems, the use of the complex notation allows the constant permittivity and conductivity of linear materials to be described by a complex permittivity or, equivalently, by a complex conductivity. In problems with general time variation or ones involving nonlinear material properties, a transient treatment is necessary. The aim of this paper is to summarise the differential equations and boundary conditions for the electric scalar potential in this case and to describe its solution by the finite element method (FEM). The FEM discretisation is shown to lead to a set of ordinary differential equations which can be solved by time stepping. A numerical example is also presented.