A nonlinear magnetic circuit model for periodic eddy current problems using T,ϕ-ϕ formulation

2017 ◽  
Vol 36 (3) ◽  
pp. 649-664 ◽  
Author(s):  
Rene Plasser ◽  
Gergely Koczka ◽  
Oszkár Bíró
Keyword(s):  
Eddy Current ◽  
Magnetic Circuit ◽  
Iteration Process ◽  
Equation System ◽  
Circuit Model ◽  
Computational Time ◽  
Combined Method ◽  
3D Fem ◽  
Content Type ◽  
Transformer Model

Purpose A transformer model is used as a benchmark for testing various methods to solve 3D nonlinear periodic eddy current problems. This paper aims to set up a nonlinear magnetic circuit problem to assess the solving procedure of the nonlinear equation system for determining the influence of various special techniques on the convergence of nonlinear iterations and hence the computational time. Design/methodology/approach Using the T,ϕ-ϕ formulation and the harmonic balance fixed-point approach, two techniques are investigated: the so-called “separate method” and the “combined method” for solving the equation system. When using the finite element method (FEM), the elapsed time for solving a problem is dominated by the conjugate gradient (CG) iteration process. The motivation for treating the equations of the voltage excitations separately from the rest of the equation system is to achieve a better-conditioned matrix system to determine the field quantities and hence a faster convergence of the CG process. Findings In fact, both methods are suitable for nonlinear computation, and for comparing the final results, the methods are equally good. Applying the combined method, the number of iterations to be executed to achieve a meaningful result is considerably less than using the separated method. Originality/value To facilitate a quick analysis, a simplified magnetic circuit model of the 3D problem was generated to assess how the different ways of solutions will affect the full 3D solving process. This investigation of a simple magnetic circuit problem to evaluate the benefits of computational methods provides the basis for considering this formulation in a 3D-FEM code for further investigation.

A novel saturated open-core fault current limiter and its nonlinear magnetic circuit model

2017 ◽  
Vol 36 (6) ◽  
pp. 1739-1749 ◽  
Author(s):  
Lei Li ◽  
Lin Li
Keyword(s):  
Nonlinear Equations ◽  
Magnetic Circuit ◽  
Physical Experiment ◽  
Circuit Model ◽  
Fault Current ◽  
Fault Current Limiter ◽  
Content Type ◽  
Dc Power ◽  
Current Limiter ◽  
Open Core

Purpose This paper aims to present a novel energy-efficient saturated open-core fault current limiter (FCL) with special permanent magnet (PM) modules. Design/methodology/approach The special PM modules are used to drive the cores of FCL into a saturated state from different directions in the normal operation condition, reducing the DC current of the saturated open-core FCL. An equivalent magnetic circuit model of the saturated open-core FCL with PM modules is built to calculate the magnetic flux density in the cores of FCL. By applying the modified nodal approach on the circuit, the nonlinear equations of the magnetic circuit can be achieved. The Newton – Raphson method is used to solve the nonlinear equations. The model shows good accuracy verified by finite element simulation and a physical experiment. Findings Compared with the original saturated open-core FCL structure with PMs, the novel saturated open-core FCL structure can save 84% DC power. The physical experiment results show that the saturated open-core FCL has a good performance on limiting the fault current. Originality/value A novel saturated open-core FCL structure with PM modules is proposed in this paper. A physical model of the saturated open-core FCL structure with PM modules is manufactured and tested. About 84% DC power can be reduced by using the PM modules in this saturated open-core FCL, and it can save most of the cost of the saturated open-core FCL.

3D-FEM computation and experimental study of eddy currents for characterization of surface cracks

2017 ◽  
Vol 8 (5) ◽  
pp. 603-610 ◽  
Author(s):  
Salaheddine Harzallah ◽  
Mohamed Chabaat
Keyword(s):  
Finite Element ◽  
Eddy Current ◽  
Eddy Currents ◽  
Research Work ◽  
Main Idea ◽  
Shape Reconstruction ◽  
Finite Element Discretization ◽  
3D Fem ◽  
Element Discretization ◽  
Content Type

Purpose The purpose of this paper is to present a new approach for computing by measuring and testing related 3D Eddy currents. In the process, a magnetic vector is formulated from the theoretical setup and obtained results from relevant applications are checked for the consistency of the theory. Besides, cracks detection as well as its propagation is studied through the two parameters: SIF and J-integral. A simulation by a numerical approach using finite-element discretization of 3D governing equations is employed to detect damaged zones and cracks. This approach has been used in the aircraft industry to control cracks. Besides, it makes it possible to highlight the defects of parts while preserving the integrity of the controlled products. Obtained results are compared and agreed with those of other researchers. Design/methodology/approach Finite-element discretization of 3D for solving problem in eddy current testing is presented in this paper. The main idea is the introduction of categorization for the shape reconstruction using the non-destructive testing by 3D-EC. The results are presented for a simple eddy current problem using the finite-element method as an experimental support. Findings In this research work, results of the various cases of simulation have been obtained. From these results of various boxes of simulation, one can conclude that the calculation of the impedance in only one point is not enough to confirm the presence or the absence of a defect for materials. Then, this confirmation leads us to the calculation of the impedance along the plate. The detection of an external defect requires the energy of the sensor by high frequencies .The position of defect (internal, in the middle, external) has a large effect on the impedance. The use of this sensor type in industrial application is frequent because of its precision (minimal error) and its low costs. The major disadvantage of this type of sensor lies in the fact that it is unable to detect a defect. Originality/value This paper fulfills an identified need to detect cracks in materials and eventually to study their propagation.

Explicit time integration of eddy current problems using a selective matrix update strategy

2017 ◽  
Vol 36 (5) ◽  
pp. 1364-1371 ◽  
Author(s):  
Jennifer Susanne Dutiné ◽  
Markus Clemens ◽  
Sebastian Schöps
Keyword(s):  
Eddy Current ◽  
Time Integration ◽  
Schur Complement ◽  
Equation System ◽  
Differential Algebraic Equation ◽  
Time Step ◽  
Explicit Time Integration ◽  
Content Type ◽  
Ode System ◽  
Update Strategy

Purpose Discretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems. Design/methodology/approach The resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible. Findings Numerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation. Originality/value The explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.

Calculation of eddy current and hysteresis losses during transient states in laminated magnetic circuits

2017 ◽  
Vol 36 (3) ◽  
pp. 665-682 ◽  
Author(s):  
Marek Golebiowski
Keyword(s):  
Eddy Current ◽  
Magnetic Permeability ◽  
Magnetic Circuit ◽  
A Priori ◽  
Content Type ◽  
Hysteresis Losses ◽  
Transient States ◽  
Iir Filter ◽  
Magnetic Circuits ◽  
Nonlinear Magnetization

Purpose The purpose of this paper is to develop the method of taking the eddy current losses in the laminated magnetic circuits into account during implicit transient calculations. The nonlinear magnetization characteristic of iron and the hysteresis losses can also be considered in the simulations done with the developed method. Design/methodology/approach The paper presents complex equivalent magnetic permeability derived from the presumed angular frequency in a laminated magnetic circuit. On this basis, the synthesis of a magnetic permeability as a function of the Laplace variable “s” is presented. After transformation of the variable “s” to a variable “z” of the Z transformation, it is possible to conduct discrete time calculation of transient states of magnetic circuits including the eddy current losses. An iterative process is developed to take the saturation of the magnetic circuit in these calculations into account. As regards hysteresis losses, the scalar model of magnetic hysteresis by Juhani Tellinen was implemented. The new method is validated by calculations of a two-coil transformer. Findings It is important to take into account the losses in sheet metal directly in the implicit transient calculations. This possibility is provided by the presented method based on the synthesis of the equivalent magnetic permeability μ^(s). The presented method was proved to be correct and efficient. The calculated sheet metal losses were compared with the results presented in literature. Good conformance of results was attained. Practical implications The method enables calculation of eddy current and hysteresis losses in laminated magnetic circuits during calculations of transient states. It does not need, unlike the previous methods, previously provided information (“a priori”) about the content of higher harmonics in waveforms. The method takes into account mutual dependence of transient waveforms of currents in the analysed system and losses of laminated magnetic circuit, expressed by eddy currents and hysteresis losses. Its implementation comes down to using in calculations a filter of the IIR type and corresponds to its calculation complexity. The author plans to use the presented method in the finite elements method transient calculations. Originality/value A new approach is a synthesis of the equivalent magnetic permeability in Laplace domain, which creates an equivalent RC circuit for permeability. Analytic equations for parameters of this equivalent circuit are original. A method for considering nonlinear magnetization characteristic and hysteresis losses was presented. In calculations of transient states of systems with magnetic circuits, one can use the developed equivalent circuit of magnetic permeability in a form of the IIR filter. Operator magnetic permeability includes fractional derivative of Laplace’s variable “vs”. Therefore, the equivalent IIR filter includes “history” of the processes that take place in the laminated magnetic circuit to the current, calculated time moment. This “history” in terms of its content is limited only by the degree of the applied IIR filter. It enables to calculate “step by step”, without previous (“a priori”) knowledge about harmonic components of the whole waveforms. It was necessary in the previously used methods, when determining parameters of magnetic permeability. The method proposed in the paper allows for calculations with taking into account direct dependence of an electric part of the system on its magnetic part.

A new analytical approach to determine slotting based eddy current losses in permanent magnets of PMSM taking into account axial and circumferential segmentation

2015 ◽  
Vol 34 (2) ◽  
pp. 398-412 ◽  
Author(s):  
Cornelius Bode ◽  
Wolf-Rüdiger Canders ◽  
Markus Henke
Keyword(s):  
Eddy Current ◽  
Analytical Approach ◽  
Permanent Magnets ◽  
Eddy Currents ◽  
3D Fem ◽  
Eddy Current Losses ◽  
Content Type ◽  
High Frequencies ◽  
Reaction Field ◽  
3D Fea

Purpose – The purpose of this paper is to calculate slotting-based eddy currents in permanent magnet excited synchronous machine (PMSM) taking into account axial and circumferential segmentation of magnets. Design/methodology/approach – An analytical approach to calculate eddy current losses in PM caused by slotting harmonics of PMSM is presented. The eddy current reaction field is taken into account as well as axial and circumferential segmentation of the magnets. Findings – The analytical model provides results comparable to 3D-FEM calculations even at high frequencies at reduced computation costs. To generalize the results the magnetic Reynold’s number is introduced. Originality/value – Taking into account the axial and circumferential segmentation in the PDE; the approach is much more accurate compared to known approaches; accuracy is comparable to 3D-FEA.

A 3-D analytic eddy current model for a finite width conductive plate

2013 ◽  
Vol 33 (1/2) ◽  
pp. 688-706 ◽  
Author(s):  
Subhra Paul ◽  
Jonathan Z. Bird
Keyword(s):  
Eddy Current ◽  
Finite Thickness ◽  
Translational Motion ◽  
Plate Thickness ◽  
Computational Time ◽  
Finite Width ◽  
Magnetic Vector Potential ◽  
Content Type ◽  
Conductive Plate ◽  
Conducting Plate

Purpose – A 3-D analytic modeling technique for calculating the eddy current distribution, force and power loss in a conductive plate of finite width and thickness is presented. The derived equations are expressed in a general form so that any magnetic source can be utilized. The model assumes the length of the conductive plate is large and the thickness of the plate is thin but not negligible. The paper aims to discuss these issues. Design/methodology/approach – The conducting and non-conducting regions are formulated in terms of decoupled magnetic vector potential components. In order to accurately compute the eddy current fields and forces the source field only needs to be applied on the surface of the conducting plate. The primary focus is on reducing the eddy current computational time. Findings – The accuracy of the presented approach is verified by utilizing a magnetic rotor that has both a rotational and translational motion. The proposed method is computationally efficient and its accuracy is validated using the finite element method. Research limitations/implications – The conducting plate thickness is assumed to be thin (but not negligible), and this enables the field interaction through the edge of the plate to be neglected. The lateral force is not calculated in the proposed approach. Practical implications – The calculation procedure presented is computationally fast and therefore this can enable the 3-D eddy current forces to be computed in near real-time. Originality/value – This paper presents a fully 3-D analytic based eddy current formlation for computing the eddy current fields and forces in a conducting plate of finite thickness and finite width. The modeling approach is shown to be computationally accurate and relatively fast.

Locating complex eigenvalues for analytical eddy-current models used to detect flaws

2019 ◽  
Vol 38 (6) ◽  
pp. 1800-1809
Author(s):  
Grzegorz Tytko ◽  
Łukasz Dawidowski
Keyword(s):  
Eddy Current ◽  
Design Methodology ◽  
Complex Function ◽  
Precise Determination ◽  
Argument Principle ◽  
Content Type ◽  
Conductive Material ◽  
Complex Roots ◽  
Complex Eigenvalues

Purpose Discrete eigenvalues occur in eddy current problems in which the solution domain was truncated on its edge. In case of conductive material with a hole, the eigenvalues are complex numbers. Their computation consists of finding complex roots of a complex function that satisfies the electromagnetic interface conditions. The purpose of this paper is to present a method of computing complex eigenvalues that are roots of such a function. Design/methodology/approach The proposed approach involves precise determination of regions in which the roots are found and applying sets of initial points, as well as the Cauchy argument principle to calculate them. Findings The elaborated algorithm was implemented in Matlab and the obtained results were verified using Newton’s method and the fsolve procedure. Both in the case of magnetic and nonmagnetic materials, such a solution was the only one that did not skip any of the eigenvalues, obtaining the results in the shortest time. Originality/value The paper presents a new effective method of locating complex eigenvalues for analytical solutions of eddy current problems containing a conductive material with a hole.

Electromagnetic analysis of an AC electric arc furnace including the modeling of an AC arc

2010 ◽  
Vol 29 (3) ◽  
pp. 667-685 ◽  
Author(s):  
Arash Kiyoumarsi ◽  
Abolfazl Nazari ◽  
Mohammad Ataei ◽  
Hamid Khademhosseini Beheshti ◽  
Rahmat‐Allah Hooshmand
Keyword(s):  
Electromagnetic Field ◽  
Electric Arc ◽  
Electromagnetic Analysis ◽  
Arc Furnace ◽  
Molten Bath ◽  
3D Fem ◽  
Content Type ◽  
Three Phase ◽  
Ac Arc ◽  
Different Parts

PurposeThe purpose of this paper is to present a 3D finite element model of the electromagnetic fields in an AC three‐phase electric arc furnace (EAF). The model includes the electrodes, arcs, and molten bath.Design/methodology/approachThe electromagnetic field in terms of time in AC arc is also modeled, utilizing a 3D finite element method (3D FEM). The arc is supposed to be an electro‐thermal unit with electrical power as input and thermal power as output. The average Joule power, calculated during the transient electromagnetic analysis of the AC arc furnace, can be used as a thermal source for the thermal analysis of the inner part of furnace. Then, by attention to different mechanisms of heat transfer in the furnace (convection and radiation from arc to bath, radiation from arc to the inner part of furnace and radiation from the bath to the sidewall and roof panel of the furnace), the temperature distribution in different parts of the furnace is calculated. The thermal model consists of the roof and sidewall panels, electrodes, bath, refractory, and arc. The thermal problem is solved in the steady state for the furnace without slag and with different depths of slag.FindingsCurrent density, voltage and magnetic field intensity in the arcs, molten bath and electrodes are predicted as a result of applying the three‐phase AC voltages to the EAF. The temperature distribution in different parts of the furnace is also evaluated as a result of the electromagnetic field analysis.Research limitations/implicationsThis paper considers an ideal condition for the AC arc. Non‐linearity of the arc during the melting, which leads to power quality disturbances, is not considered. In most prior researches on the electrical arc furnace, a non‐linear circuit model is usually used for calculation of power quality phenomena distributions. In this paper, the FEM is used instead of non‐linear circuits, and calculated voltage and current densities in the linear arc model. The FEM results directly depend on the physical properties considered for the arc.Originality/valueSteady‐state arc shapes, based on the Bowman model, are used to calculate and evaluate the geometry of the arc in a real and practical three‐phase AC arc furnace. A new approach to modeling AC arcs is developed, assuming that the instantaneous geometry of the AC arc at any time is constant and is similar to the geometry of a DC arc with the root mean square value of the current waveform of the AC arc. A time‐stepping 3D FEM is utilized to calculate the electromagnetic field in the AC arc as a function of time.

Efficient evaluation of the earth return mutual impedance of overhead conductors over a horizontally multilayered soil

2014 ◽  
Vol 33 (4) ◽  
pp. 1379-1395 ◽  
Author(s):  
Jae-bok Lee ◽  
Jun Zou ◽  
Benliang Li ◽  
Munno Ju
Keyword(s):  
Unit Length ◽  
Oscillatory Integral ◽  
Electromagnetic Transients ◽  
Closed Form Expression ◽  
Computational Time ◽  
Horizontal Distance ◽  
Content Type ◽  
Mutual Impedance ◽  
The Earth ◽  
Highly Oscillatory

Purpose – The per-unit-length earth return mutual impedance of the overhead conductors plays an important role for analyzing electromagnetic transients or couplings of multi-conductor systems. It is impossible to have a closed-form expression to evaluate this kind of impedance. The purpose of this paper is to propose an efficient numerical approach to evaluate the earth return mutual impedance of the overhead conductors above horizontally multi-layered soils. Design/methodology/approach – The expression of the earth return mutual impedance, which contains a complex highly oscillatory semi-infinite integral, is divided into two parts intentionally, i.e. the definite and the tail integral, respectively. The definite integral is calculated using the proposed moment functions after fitting the integrand into the piecewise cubic spline functions, and the tail integral is replaced by exponential integrals with newly developed asymptotic integrands. Findings – The numerical examples show the proposed approach has a satisfactory accuracy for different parameter combinations. Compared to the direct quadrature approach, the computational time of the proposed approach is very competitive, especially, for the large horizontal distance and the low height of the conductors. Originality/value – The advantage of the proposed approach is that the calculation of the highly oscillatory integral is completely avoided due to the fact that the moment function can be evaluated analytically. The contribution of the tail integral is well included by means of the exponential integral, though in an asymptotic way. The proposed approach is completely general, and can be applied to calculate the earth return mutual impedance of overhead conductors above a soil structure with an arbitrary number of horizontal layers.

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