An empirical correlation for the calculated effective permeability for a nonlinear impulse problem with a simple exponential model for the relative differential permeability

A nonlinear impulse problem is considered in which a simple exponential representation for the relative differential magnetic permeability is used to model the effects of afield dependent permeability (including magnetic saturation) on the performance of an electromagnetic shield subjected to intense pulsed electromagnetic fields. Previously, an analytical procedure was developed that characterized the maximum value of the transient electric field induced at the inner surface of the shield in terms of a dimensionless effective permeability that depends on a fundamental dimensionless combination of problem parameters. A series of finite difference time domain (FDTD) calculations was used to characterize the dimensionless effective permeability numerically for selected values of the dimensionless applied pulse parameter. In this paper, a simple empirical relationship is presented that represents to very good accuracy the calculated effective permeability for the simple exponential permeability model. The relationship has only a single free parameter, yet it fits the numerical results to remarkable accuracy for values of the applied pulse parameter from zero to saturation. This relationship conveniently summarizes the nonlinearity of the problem for the maximum electric field.