Application of Whitney elements for the reconstruction of electric arc current density in low-voltage circuit breakers

Abstract

Purpose This paper aims to present the mathematical formulations of a magnetic inverse problem for the electric arc current density reconstruction in a simplified arc chamber of a low-voltage circuit breaker. Design/methodology/approach Considering that electric arc current density is a zero divergence vector field, the inverse problem can be solved in Whitney space W2 in terms of electric current density J with the zero divergence condition as a constraint or can be solved in Whitney space W1 in terms of electric vector potential T where the zero divergence condition naturally holds. Moreover, the tree gauging condition is applied to ensure a unique solution when solving for the vector potential in space W1. Tikhonov regularization is used to treat the ill-posedness of the inverse problem complemented with L-curve method for the selection of regularization parameters. A common mode approach is proposed, which solves for the reduced electric vector potential representing the internal current loops instead of solving for the total electric vector potential. The proposed inversion approaches are numerically tested starting from simulated magnetic field values. Findings With the common mode approach, the reconstruction of current density is significantly improved for both formulations using face elements in space W2 and using edge elements in space W1. When solving the inverse problem in space W1, the choice of the regularization operator has a key role to obtain a good reconstruction, where the discrete curl operator is a good option. The standard Tikhonov regularization obtains a good reconstruction with J-formulation, but fails in the case of T-formulation. The use of edge elements requires a tree-cotree gauging to ensure the uniqueness of T. Moreover, additional efforts have to be taken to find an optimal regularization operator and an optimal tree when using edge elements. In conclusion, the J-formulation is to be preferred. Originality/value The proposed approaches are able to reconstruct the three-dimensional electric arc current density from its magnetic field in a non-intrusive manner. The formulations enable us to incorporate a priori knowledge of the unknown current density into the solution of the inverse problem, including the zero divergence condition and the boundary conditions. A common mode approach is proposed, which can significantly improve the current density reconstruction.