Mathematical Models of Hysteresis

alternating series astroid classical Preisach model computed congruency property corresponding decreased derive discussion double integral equal vertical chords experimental data experimentally measured expression FIELD Oe first-order transition curves formula fu(t hysteresis loops hysteresis nonlinearities hysteresis transducer hysteretic energy losses hysteretic losses identification problem input extrema input u(t input variations integral equation irreversible M₁ magnetic field MAGNETIZATION emu magnetostrictive mathematical models of hysteresis monotonically increased negative saturation nonlinear Preisach model notation numerical implementation output f(t output increments output value Preisach-type models Q(u_ restricted Preisach model Ru(t S-W model scalar hysteresis scalar Preisach models second-order reversal curves shown in Fig solution spherical harmonics ẞ diagram SS v(a ẞ)dadẞ Su(t superconducting symmetry theorem thermodynamics triangle u₁ u₂ values of input vector hysteresis vector Preisach models wiping-out property αλβ θα λα μα дадв