A nonlinear analytical procedure for electromagnetic transients in ferromagnetic shields with an exponential permeability model

Abstract

Electrically conductive, ferromagnetic shielding materials exhibit nonlinear behavior (including saturation) under intense pulsed electromagnetic field conditions such as those associated with lightning, electromagnetic pulse (EMP), and electrostatic discharge (ESD). Previously, an analytical procedure was developed to characterize the nonlinear electric field transients induced at the inner surface of long, thin-walled, cylindrical, electrically conductive, ferromagnetic shields by axially-directed short-duration surface current pulses on the outer surface. For a general relative differential permeability function /spl mu//sub rd/(H), it was shown that the peak value of the electric field transient can be expressed in terms of an effective permeability /spl mu//sub E/(/spl beta/) and the time at which the peak occurs can be expressed in terms of effective permeability /spl mu//sub /spl tau//(/spl beta/), where the applied pulse parameter /spl beta//spl equiv/Q/sub 0///spl sigma/d/sup 2/ is a fundamental combination of the nonmagnetic problem parameters (the charge per unit circumference, Q/sub 0/, transported along the cylinder during the pulse; the electrical conductivity /spl sigma/, of the material; and the wall thickness, d, of the cylinder). For a simple exponential relative differential permeability model /spl mu//sub rd/(H)=/spl mu//sub ri/exp(-H/H/sub 1/), the analytical procedure can be extended significantly. An applied pulse parameter /spl zeta//spl equiv/Q/sub 0///spl sigma/d/sup 2/B/sub s/ has been identified that includes the saturation magnetization B/sub s/, which is a magnetic problem parameter. Furthermore, the effective permeabilities /spl mu//sub E/(/spl zeta/) can be expressed as /spl mu//sub /spl zeta//=/spl mu//sub ri//spl Omega//sub E/(/spl zeta/), where /spl Omega//sub E/(/spl zeta/) is a fundamental quantity that depends only on /spl zeta/. Similarly, /spl mu//sub /spl tau//(/spl zeta/) can be expressed as /spl mu//sub /spl tau//(/spl zeta/)=/spl mu//sub ri//spl Omega//sub /spl tau//(/spl zeta/), where /spl Omega//sub E/(/spl zeta/) is a fundamental quantity that depends only on /spl zeta/. Results of numerical calculations for /spl Omega//sub E/(/spl zeta/) and /spl Omega//sub /spl tau//(/spl zeta/) are presented.