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

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 μ_rd(H), it was shown that the peak value of the electric field transient can be expressed in terms of an effective permeability μ_E(β) and the time at which the peak occurs can be expressed in terms of effective permeability μ_τ(β), where the applied pulse parameter β≡Q₀/σd² is a fundamental combination of the nonmagnetic problem parameters (the charge per unit circumference, Q₀, transported along the cylinder during the pulse; the electrical conductivity σ, of the material; and the wall thickness, d, of the cylinder). For a simple exponential relative differential permeability model μ_rd(H)=μ_riexp(-H/H₁), the analytical procedure can be extended significantly. An applied pulse parameter ζ≡Q₀/σd²B_s has been identified that includes the saturation magnetization B_s , which is a magnetic problem parameter. Furthermore, the effective permeabilities μ_E(ζ) can be expressed as μ_ζ=μ_riΩ_E(ζ), where Ω_E(ζ) is a fundamental quantity that depends only on ζ. Similarly, μ_τ(ζ) can be expressed as μ_τ(ζ)=μ_riΩ_τ (ζ), where Ω_E(ζ) is a fundamental quantity that depends only on ζ. Results of numerical calculations for Ω_E(ζ) and Ω_τ(ζ) are presented