Inverse problem of three-dimensional current density reconstruction from its magnetic field: theory and application to electric arcs

Electric arc plasma is an essential and very complex phenomenon in circuit breakers during the fault current interruption. The current density in the electric arc gives very important information to understand the physics of the electric arc plasma and the characteristics of the current interruption process, such as the distributions of temperature, pressure and energy, the electrode erosion, the interactions between the arc plasma with nearby materials. At present, available experimental diagnostic methods to investigate the electric arc behavior include electrical measurements, optical and magnetic diagnostic techniques. However, noninvasive experimental methods able to obtain the three-dimensional electric arc current density cannot be found in literature. This work proposes a noninvasive magnetic diagnostic technique able to reconstruct the three-dimensional electric arc current density from the magnetic field measurements by solving a magnetic inverse problem. This thesis attempts to investigate this specific inverse problem with respect to the theoretical considerations, the numerical implementations, and the experimental tests, in order to develop an effective and noninvasive diagnostic approach to study the electric arc dynamics during the short-current interruption transients lasting few milliseconds. This inverse problem is known to be mathematically and numerically very challenging due to its inherent ill-posed nature in the sense that a small perturbation of the magnetic field data may cause a large deviation of the solution. In this thesis, the mathematical formulations of the inverse problem is established complemented with a theoretical analysis. The key ingredient of this inverse problem is to solve a Biot-Savart integral equation via the Biot-Savart operator, which maps a current density onto its corresponding magnetic field. By calculating the curl and the divergence of the Biot-Savart operator, it is shown that the Biot-Savart generally does not satisfy the magneto-quasi-static Maxwell’s equations. To this end, a common mode approach is proposed by splitting the original current density into a known common mode component and an unknown component, which enables us to restrict the unknown current density to a Sobolev space in which the Biot-Savart operator satisfies the Maxwell’s equations and acts as an inverse of the curl operator. The magnetic inverse problem is known to be ill-posed. In order to solve this ill-posed inverse problem, the mapping properties of the Biot-Savart operator have been analyzed. In particular, the nullspace of the Biot-Savart operator as well as its orthogonal complement are characterized explicitly with respect to L2 scalar product in Sobolev spaces, showing the classes of functions that do and do not generate a magnetic field outside the solution domain. The nullspace is found to be non-trivial, indicating the ill-posedness of the problem in terms of the uniqueness issue. The Moorse-Penrose generalized inverse of the Biot-Savart operator is established and a least-squares solution in the sense of minimum norm is then derived via the singular value decomposition, which indicates the ill-posedness of the inverse problem in terms of the stability issue. In order to address the ill-posedness of the inverse problem, the Tikhonov regularization is used leading to an unique and stable best-approximate solution in the sense of minimal norm. The numerical implementation of both forward and inverse problems is then studied using Whitney elements in Whitney spaces W^p (p = 0, …, 3) with both scalar basis functions associated with node and volume elements and vector basis functions associated with edge and face elements. The idea of using scalar basis functions is to represent the current density in terms of its x-, y-, and z-component separately. The current density in general is a zero-divergence vector field complemented by some prescribed boundary conditions. Although scalar basis functions can be easily defined in comparision to vector basis functions, the disadvantage is also apparent: the current density represented with node basis function has “too much continuity” and that with volume basis function has “too little continuity” leading to non-physical solution spaces. However, the numerical implementations using vector basis functions can provide a physically reasonable solution space. Noting the divergence of the current density is zero, a current vector potential is introduced which can be appropriately represented using edge elements while the current density can be appropriately represented using face elements. Then the appropriate incorporations of the boundary conditions and the zero-divergence constraint are introduced when solving the problem in terms of the electric current density and the current vector potential. Numerical studies are carried out using the simulated magnetic field data considering the simplified geometries of a low-voltage air circuit breaker (LVCB). The simulation results show the capability and feasibility of the proposed approach to reconstruct the time-dependent three-dimensional electric arc current density from the magnetic fields. Moreover, several factors that affect the reconstructions are investigated numerically, including the sensor number and arrangement, the meshing resolution of the domain, the ideal and non-ideal modeling of the magnetic sensors, and random variations in sensor orientation and position. The numerical results show that an eight-by-eight or sixteen-by-sixteen planar array of magnetic sensors would be the optimal arrangement considering the complexity of the sensor array setup, the time consumption of the solution process, and the accuracy of the reconstruction. The application of the proposed approach to realistic circuit breakers is investigated considering a miniature circuit breaker (MCB) with ferromagnetic regions. The numerical analysis is carried out in time domain taking into account eddy currents in conductive regions and magnetization currents in ferromagnetic regions. The simulation results indicate the capability and applicability of the proposed approach to reconstruct the arc current density distributions in real circuit breakers. The complexity and dynamic behaviour of the arc plasma in circuit breakers require measuring the magnetic field at high spatial and temporal sampling rates. The ill-posedness of the inverse problem leads to strong sensitivity of the solution to signal noise. Therefore, high quality and accurate magnetic measurements are required. This is a demanding challenge due to the reduced space available to locate magnetic sensors and the transient arcing process. For the experimental validation of the proposed approach, a simplified arc chamber of the low-voltage circuit breaker is considered. An experimental system has been developed including the electric circuits of a pulsed current source, an arc-breaking device, current and voltage transducers to measure the total current and the arc voltage, a magnetic field measurement system, and a complementary optical diagnostic method based on a charge-coupled device (CCD) camera. The magnetic field measurement system is developed based on a Hall-effect magnetic sensor array and a data acquisition system. The bandwidth of the magnetic sensor is 20 kHz with a maximum measurable magnetic flux density of 9 mT covering the full range of the magnetic field outside the arc chamber. The data acquisition system consists of eight analog-to-digital converters (ADCs) with a sampling rate of 200 kHz and a 16-bit resolution, which is enough considering the bandwidth of the magnetic signals. Experimental tests are carried out using the proposed approach. The reconstructions from the experimental magnetic field data are compared with the optical images by the CCD camera. Moreover, a Monte-Carlo analysis is carried out in order to analyse the effect of the magnetic field measurement uncertainty on the current density reconstructions. The experimental validation is conducted in an indirect manner based on the errors in the reconstructed magnetic field which show a good agreement between the reconstructed and the measured magnetic fields with a relative error in magnetic field smaller than 5%.