Jinlong Dong, Luca Di Rienzo, Olivier Chadebec and Jianhua Wang
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…
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.
Details
Keywords
Carlo de Falco, Luca Di Rienzo, Nathan Ida and Sergey Yuferev
The purpose of this paper is the derivation and efficient implementation of surface impedance boundary conditions (SIBCs) for nonlinear magnetic conductors.
Abstract
Purpose
The purpose of this paper is the derivation and efficient implementation of surface impedance boundary conditions (SIBCs) for nonlinear magnetic conductors.
Design/methodology/approach
An approach based on perturbation theory is proposed, which expands to nonlinear problems the methods already developed by the authors for linear problems. Differently from the linear case, for which the analytical solution of the diffusion equation in the semi-infinite space for the magnetic field is available, in the nonlinear case the corresponding nonlinear diffusion equation must be solved numerically. To this aim, a suitable smooth map is defined to reduce the semi-infinite computational domain to a finite one; then the diffusion equation is solved by a Galerkin method relying on basis functions constructed via the push-forward of a Lagrangian polynomial basis whose degrees of freedom are collocated at Gauss–Lobatto nodes. The use of such basis in connection with a suitable under-integration naturally leads to mass-lumping without impacting the order of the method. The solution of the diffusion equation is coupled with a boundary element method formulation for the case of parallel magnetic conductors in terms of E and B fields.
Findings
The results are validated by comparison with full nonlinear finite element method simulations showing very good accordance at a much lower computational cost.
Research limitations/implications
Limitations of the method are those arising from perturbation theory: the introduced small parameter must be much less than one. This implies that the penetration depth of the magnetic field into the magnetic and conductive media must be much smaller than the characteristic size of the conductor.
Originality/value
The efficient implementation of a nonlinear SIBC based on a perturbation approach is proposed for an electric and magnetic field formulation of the two-dimensional problem of current driven parallel solid conductors.
Details
Keywords
Luca Di Rienzo and Jens Haueisen
To define a methodology for comparing sensor arrays for solving magnetostatic linear inverse problems.
Abstract
Purpose
To define a methodology for comparing sensor arrays for solving magnetostatic linear inverse problems.
Design/methodology/approach
A singular value decomposition related projection method is used for comparing sensor arrays and we applied it to a biomagnetic inverse problem, as an example. Furthermore, a theoretical reference sensor system is introduced and used as a benchmark for the analysed sensor arrays.
Findings
The method has turned out to be effective in comparing three different theoretical sensor arrays, showing the superiority of the two arrays constituted by three‐axial sensors.
Research limitations/implications
The method has been applied only to the case of over‐determined problems. The underdetermined case will be considered in future work.
Practical implications
From the applicative point of view, the illustrated methodology is useful when one has to choose between existing sensor arrays or in the design phase of a new sensor array.
Originality/value
A new methodology is proposed for comparing sensor arrays. The advantage of the methodology are to take into account the regularization in the solution of the inverse problem and to be general, not depending on a particular source configuration.
Details
Keywords
Youpeng Huangfu, Song Wang, Shuang Wang, Hailin Li, Dongsheng Yuan, Shuhong Wang and Luca Di Rienzo
The purpose of this paper is to implement the macro-modeling and passivity enforcement for the equivalent high frequency circuit model of a single-phase winding for an alternating…
Abstract
Purpose
The purpose of this paper is to implement the macro-modeling and passivity enforcement for the equivalent high frequency circuit model of a single-phase winding for an alternating current (AC) three-phase motor. It provides a stable and strictly passive Foster-type circuit macro-model for the winding. Consequently, a stable circuit network is guaranteed when it is connected with an external passive circuit. The equivalent circuit is validated on a three-phase permanent magnet synchronous motor. Furthermore, the corresponding three-phase windings macro-model could be obtained accordingly.
Design/methodology/approach
The following techniques are used: the least square method, vector fitting method, the fast residue perturbation method, circuit synthesis, sequence quadratic programming method and simulated annealing method.
Findings
This work presents an effective approach to model an equivalent high frequency circuit macro-model for a single-phase winding. Simultaneously, both the characteristics of port passivity and component passivity are guaranteed.
Originality/value
This paper carries out both the port passivity and the component passivity enforcement for a single-phase winding of a motor during the macro-modeling procedure. This equivalent motor winding model can be applied to obtain the conducted electromagnetic interference and the overvoltage performance analysis for an adjustable speed motor drive system.
Details
Keywords
Luca Di Rienzo, Sergey Yuferev, Nathan Ida and Cesare Mario Arturi
To provide a time domain formulation for reconstruction of transient currents flowing in massive parallel conductors from magnetic data collected in the dielectric space…
Abstract
Purpose
To provide a time domain formulation for reconstruction of transient currents flowing in massive parallel conductors from magnetic data collected in the dielectric space surrounding the conductors.
Design/methodology/approach
A boundary integral equation (BIE) formulation involving Mitzner's and Rytov's high order surface impedance boundary conditions (SIBCs) is used. Input data of the inverse problem are the magnetic fields at given locations near the conductors. In order to validate the inversion algorithm, the magnetic field data are computed solving the direct problem with FEM for given current waveforms.
Findings
The improvement in reconstruction accuracy of the new time domain BIE formulation employing high order SIBCs has been demonstrated numerically in a simple test case. The range of validity of the technique has been extended to current pulses of longer duration and the computational burden has shown to increase only by a factor of 4.
Research limitations/implications
The proposed formulation can be compared with other possible formulations, both in the time and in the frequency domain.
Practical implications
Based on this formulation a new current sensing technique is proposed for realization of low cost current sensors based on magnetic sensor arrays.
Originality/value
The inverse problem addressed in the paper has been solved for the first time.
Details
Keywords
Luca Di Rienzo, Jens Haueisen and Cesare Mario Arturi
To demonstrate the improvement of three‐axial magnetic sensors systems for magnetocardiography when using minimum norm solutions (MNS).
Abstract
Purpose
To demonstrate the improvement of three‐axial magnetic sensors systems for magnetocardiography when using minimum norm solutions (MNS).
Design/methodology/approach
The thesis is proved by using figures of merit and by means of repeated numerical simulations, starting from a BEM model for forward calculations.
Findings
We found out that both for under‐determined and over‐determined problems two figures of merit show better performance of a three‐axial sensor system when compared with two mono‐axial sensor systems. We also showed the positive impact of three component magnetic field data on MNS by means of repeated simulations.
Research limitations/implications
The analysis is limited to theoretical sensor systems and can be applied also to realistic measurement set‐ups. Noise is considered uncorrelated. The analysis could be carried out with the help of other figures of merit. More refined models for the human body could be adopted.
Practical implications
The use of three axial sensor systems is encouraged in the field of magnetocardiography.
Originality/value
Numerical analysis of inversion algorithms using three‐dimensional magnetic field data in magnetocardiography have been never carried out.
Details
Keywords
Roland Eichardt, Daniel Baumgarten, Luca Di Rienzo, Sven Linzen, Volkmar Schultze and Jens Haueisen
The purpose of this paper is to examine the localisation of ferromagnetic objects buried in the underground. More specifically, it deals with the reconstruction of the…
Abstract
Purpose
The purpose of this paper is to examine the localisation of ferromagnetic objects buried in the underground. More specifically, it deals with the reconstruction of the XY‐positions, the depths (Z‐positions), the number, and the extension of the objects based on geomagnetic measurements. This paper introduces a minimum‐norm reconstruction approach and evaluates its performance in a simulation study.
Design/methodology/approach
A minimum‐L2‐norm estimation based on the truncated singular value decomposition method with lead field weighting is proposed in order to localise geomagnetic sources. The sensor setup and positions are taken from real measurements. The source space is formed by an automatically generated grid. At each grid point, a magneto‐static dipole is assumed.
Findings
Sources with different depths and XY‐positions could be successfully reconstructed. The proposed approach is not overly sensitive to errors/noise in measurement values and sensor positions.
Originality/value
The approach described in this paper can be used for applications like geoprospection, archaeology, mine clearing, and the clean‐up of former waste deposits.
Details
Keywords
Óscar Lahuerta, José Ortega, Claudio Carretero, Juan Pablo Martínez and Jesús Acero
The purpose of this paper is the identification of the magnetic characteristics of the induction load by means of the B–H curve proposed by Fröhlich.
Abstract
Purpose
The purpose of this paper is the identification of the magnetic characteristics of the induction load by means of the B–H curve proposed by Fröhlich.
Design/methodology/approach
An electromagnetic description of the inductor system is performed to substitute the effects of the induction load, for a mathematical condition, the so-called impedance boundary condition (IBC).
Findings
A significant reduction in the computational cost of electromagnetic simulation has been achieved through the use of the IBC, resulting in a computation time approximately 400 times faster than time domain simulation. Moreover, an alternative method has been developed to experimentally identify the parameters that determine the magnetic behavior of the induction load. Finally, further research has been conducted to understand the relationship between the equivalent impedance of an induction load and the excitation current level.
Practical implications
This work is performed to achieve a better understanding of the fundamentals involved in the electromagnetic modeling of an induction heating system.
Originality/value
This paper introduces the dependence on the excitation level based on a first harmonic approximation and extends the IBC to nonlinear magnetic materials which allows the identification of the magnetic characteristics of the induction load.