Non ohmic heterogeneous insulation: the single layer with a temperature gradient and the double layer

Two special types of heterogeneity are considered. The first is produced in a non-isothermal sample by the combined effects of the field and the temperature on the local field. The second concerns the isothermal superposition of two layers obeying an improved model for the field dependence of σ, compared to the exponential or Eⁿ approximation used in previous calculations. In the first part, we calculate the field distribution in a stressed semi-insulating sample submitted to a temperature gradient, using the Wagner´s approximation for the temperature dependence of σ and a polynomial expansion of Poole-Frenkel´s formula for its field dependence. The methods used for a flat sample can be applied to the configuration of a cable. In the second part, the classical results concerning the Maxwell-Wagner model for a superposition of two ohmic layers are first recalled. Then, the case of non-ohmic layers is treated, using for the field dependence of the conductivity the empirical relation σ(E)=[1-E/E*]^-1 σ(0), σ(0) being the low-field limit of σ and E* the breakdown field. The conditions under which an increase of the applied voltage tends to level off the fields in the bilayer are derived. Finally, the stored energy due to the non-ohmic character of materials is shown to obey the same criteria as the uniformization of the field