On energy consistent vector hysteresis operators

Abstract

Incremental models for magnetic vector hysteresis have been developed in previous works in accordance with basic principles of thermodynamics. In this paper, we present an equivalent representation of the associated hysteresis operator in terms of a co-energy functional which is useful for magnetic field computations based on a scalar potential. Using convex duality, we further define the corresponding energy functional and the associated inverse hysteresis operator which is required for computations based on the vector potential. The equivalence of the two representations with the energy-based hysteresis models proposed in earlier works is demonstrated and numerical results for some typical test problems are presented obtained by finite element simulation of corresponding scalar and vector potential formulations.

Figures (6)

• Figure 1: Hysteresis loop for one pinning force $(\chi_1 = 71, w_1 = 1)$ with $\mathbf{H}^i = (H_m\,\text{sin}(t^i),0)$, where $H_m = \{180,600\}$.
• Figure 2: Hysteresis loop for one pinning force $(\chi_1 = 71, w_1 = 1)$ with $\mathbf{H}^i = (H_m^i\,\text{sin}(t^i),\text{cos}(t^i))$, where $H_m^i = 110\min(t^i/6\pi,1)$.
• Figure 3: Hysteresis loop for multiple pinning forces ($N_{\chi} = 20$) with $\mathbf{H}^i = (H_m\,\text{sin}(t^i),0)$, where $H_m = \{100,200,300,400,500\}$.
• Figure 4: Sketch of the geometry of the TEAM 32 problem with iron (grey), coil (green and red), air (cyan). Evaluations of the magnetic field and flux at the points $C_6$, $C_{1,2}$ and $C_{3,4}$ are considered later on.
• Figure 5: Vertical component $B_y(t)$ of magnetic flux at the points $C_6$, $C_{1,2}$ and $C_{3,4}$ for the scalar potential and vector potential formulation.