Compel (Vol. 23, 2004)

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Volume 23 Number 3 2004

ISBN 0-86176-978-3

ISSN 0332-1649

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Selected papers from the 11th International Symposium on Electromagnetics Fields in Electrical Engineering ISEF 2003 Guest Editor: Professor S. Wiak Co-editors: Professor M. Trlep and Professor A. Krawczyk

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COMPEL

ISSN 0332-1649

The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

Volume 23 Number 3 2004

Selected papers from the 11th International Symposium on Electromagnetics Fields in Electrical Engineering ISEF 2003 Guest Editor Professor S. Wiak Co-editors Professor M. Trlep and Professor A. Krawczyk

Access this journal online __________________________ 596 Editorial advisory board ___________________________ 597 Abstracts and keywords ___________________________ 598 Editorial __________________________________________ 605 Special issue section Application of Haar’s wavelets in the method of moments to solve electrostatic problems Aldo Artur Belardi, Jose´ Roberto Cardoso and Carlos Antonio Franc¸ a Sartori ____________________________________

606

A 3D multimodal FDTD algorithm for electromagnetic and acoustic propagation in curved waveguides and bent ducts of varying cross-section Nikolaos V. Kantartzis, Theodoros K. Katsibas, Christos S. Antonopoulos and Theodoros D. Tsiboukis ______________________________________

613

The highly efficient three-phase small induction motors with stator cores made from amorphous iron M. Dems, K. Kome˛za, S. Wiak and T. Stec __________________________

Access this journal electronically The current and past volumes of this journal are available at:

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625

CONTENTS

CONTENTS continued

Optimal shape design of a high-voltage test arrangement P. Di Barba, R. Galdi, U. Piovan, A. Savini and G. Consogno____________

633

Cogging torque calculation considering magnetic anisotropy for permanent magnet synchronous motors Shinichi Yamaguchi, Akihiro Daikoku and Norio Takahashi_____________

639

Magnetoelastic coupling and Rayleigh damping A. Belahcen ____________________________________________________

647

Modelling of temperature-dependent effective impedance of non-ferromagnetic massive conductor Ivo Dolezˇel, Ladislav Musil and Bohusˇ Ulrych ________________________

655

Field strength computation at edges in nonlinear magnetostatics Friedemann Groh, Wolfgang Hafla, Andre´ Buchau and Wolfgang M. Rucker_____________________________________________

662

Genetic algorithm coupled with FEM-3D for metrological optimal design of combined current-voltage instrument transformer Marija Cundeva, Ljupco Arsov and Goga Cvetkovski___________________

670

Adaptive meshing algorithm for recognition of material cracks Konstanty M. Gawrylczyk and Piotr Putek ___________________________

677

Incorporation of a Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations – application of the Newton-Raphson method J. Gyselinck, P. Dular, N. Sadowski, J. Leite and J.P.A. Bastos____________

685

The modelling of the FDTD method based on graph theory Andrzej Jordan and Carsten Maple _________________________________

694

Inverse problem – determining unknown distribution of charge density using the dual reciprocity method Dean Ogrizek and Mladen Trlep ___________________________________

701

Finite element modelling of stacked thin regions with non-zero global currents P. Dular, J. Gyselinck, T. Zeidan and L. Kra¨henbu¨hl ___________________

707

Reliability-based topology optimization for electromagnetic systems Jenam Kang, Chwail Kim and Semyung Wang _______________________

715

A ‘‘quasi-genetic’’ algorithm for searching the dangerous areas generated by a grounding system Marcello Sylos Labini, Arturo Covitti, Giuseppe Delvecchio and Ferrante Neri __________________________________________________

724

CONTENTS continued

Development of optimizing method using quality engineering and multivariate analysis based on finite element method Yukihiro Okada, Yoshihiro Kawase and Shinya Sano __________________

733

An improved fast method for computing capacitance L. Song and A. Konrad __________________________________________

740

Power losses analysis in the windings of electromagnetic gear Andrzej Patecki, Sławomir Ste˛pien´ and Grzegorz Szyman´ski ____________

748

Finite element analysis of the magnetorheological fluid brake transients Wojciech Szela˛g_________________________________________________

758

Magnetic stimulation of knee – mathematical model Bartosz Sawicki, Jacek Starzyn´ski, Stanisław Wincenciak, Andrzej Krawczyk and Mladen Trlep _______________________________________________ 767

2D harmonic analysis of the cogging torque in synchronous permanent magnet machines M. Łukaniszyn, M. Jagiela, R. Wro´bel and K. Latawiec ________________

774

Determination of a dynamic radial active magnetic bearing model using the finite element method Bosˇtjan Polajzˇer, Gorazd Sˇtumberger, Drago Dolinar and Kay Hameyer __

783

Electromagnetic forming: a coupled numerical electromagnetic-mechanical-electrical approach compared to measurements A. Giannoglou, A. Kladas, J. Tegopoulos, A. Koumoutsos, D. Manolakos and A. Mamalis ________________________________________________

789

Regular section 2D harmonic balance FE modelling of electromagnetic devices coupled to nonlinear circuits J. Gyselinck, P. Dular, C. Geuzaine and W. Legros _____________________

800

Finite element analysis of coupled phenomena in magnetorheological fluid devices Wojciech Szela˛g_________________________________________________

813

Comparison of the Preisach and Jiles-Atherton models to take hysteresis phenomenon into account in finite element analysis Abdelkader Benabou, Ste´phane Cle´net and Francis Piriou _______________

825

Error bounds for the FEM numerical solution of non-linear field problems Ioan R. Ciric, Theodor Maghiar, Florea Hantila and Costin Ifrim ________

835

EDITORIAL ADVISORY BOARD

Professor O. Biro Graz University of Technology, Graz, Austria Professor J.R. Cardoso University of Sao Paulo, Sao Paulo, Brazil Professor C. Christopoulos University of Nottingham, Nottingham, UK Professor J.-L. Coulomb Laboratoire d’Electrotechnique de Grenoble, Grenoble, France Professor X. Cui North China Electric Power University, Baoding, Hebei, China Professor A. Demenko Poznan´ University of Technology, Poznan´, Poland Professor E. Freeman Imperial College of Science, London, UK Professor Song-yop Hahn Seoul National University, Seoul, Korea Professor Dr.-Ing K. Hameyer Katholieke Universiteit Leuven, Leuven-Heverlee, Belgium Professor N. Ida University of Akron, Akron, USA Professor A. Jack The University, Newcastle Upon Tyne, UK

Professor D. Lowther McGill University, Ville Saint Laurent, Quebec, Canada

Editorial advisory board

Professor O. Mohammed Florida International University, Florida, USA Professor G. Molinari University of Genoa, Genoa, Italy

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Professor A. Razek Laboratorie de Genie Electrique de Paris - CNRS, Gif sur Yvette, France Professor G. Rubinacci Universita di Cassino, Cassino, Italy Professor M. Rudan University of Bologna, Bologna, Italy Professor M. Sever The Hebrew University, Jerusalem, Israel Professor J. Tegopoulos National Tech University of Athens, Athens, Greece Professor W. Trowbridge Vector Fields Ltd, Oxford, UK Professor T. Tsiboukis Aristotle University of Thessaloniki, Thessaloniki, Greece Dr L.R. Turner Argonne National Laboratory, Argonne, USA

Professor A. Kost Technische Universitat Berlin, Berlin, Germany

Professor Dr.-Ing T. Weiland Technische Universitat Darmstadt, Darmstadt, Germany

Professor T.S. Low National University of Singapore, Singapore

Professor K. Zakrzewski Politechnika Lodzka, Lodz, Poland

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 p. 597 # Emerald Group Publishing Limited 0332-1649

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COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 Abstracts and keywords # Emerald Group Publishing Limited 0332-1649

Application of Haar’s wavelets in the method of moments to solve electrostatic problems Aldo Artur Belardi, Jose´ Roberto Cardoso and Carlos Antonio Franc¸ a Sartori Keywords Electrostatics, Density measurement, Optimization techniques Presents the mathematical basis and some results, concerning the application of Haar’s wavelets, as an expansion function, in the method of moments to solve electrostatic problems. Two applications regarding the evaluation of linear and surface charge densities were carried out: the first one on a finite straight wire, and the second one on a thin square plate. Some optimization techniques were used, whose main computational performance aspects are emphasized. Presents comparative results related to the use of Haar’s wavelets and the conventional expansion functions. A 3D multimodal FDTD algorithm for electromagnetic and acoustic propagation in curved waveguides and bent ducts of varying cross-section Nikolaos V. Kantartzis, Theodoros K. Katsibas, Christos S. Antonopoulos and Theodoros D. Tsiboukis Keywords Electromagnetic fields, Acoustic waves, Wave physics, Finite difference time-domain analysis This paper presents a curvilinearlyestablished finite-difference time-domain methodology for the enhanced 3D analysis of electromagnetic and acoustic propagation in generalised electromagnetic compatibility devices, junctions or bent ducts. Based on an exact multimodal decomposition and a higherorder differencing topology, the new technique successfully treats complex systems of varying cross-section and guarantees the consistent evaluation of their scattering parameters or resonance frequencies. To subdue the non-separable modes at the structures’ interfaces, a convergent grid approach is developed, while the tough case of abrupt excitations is also studied. Thus, the proposed algorithm attains significant accuracy and savings, as numerically verified by various practical problems.

The highly efficient three-phase small induction motors with stator cores made from amorphous iron M. Dems, K. Kome˛za, S. Wiak and T. Stec Keywords Inductance, Design, Iron Applies the field/circuit two-dimensional method and improved circuit method to engineering designs of the induction motor with stator cores made of amorphous iron. Exploiting of these methods makes possible computation of many different specific parameters and working curves in steady states for the ‘‘high efficiency’’ three-phase small induction motor. Compares the results of this calculation with the results obtained for the classical induction motor with identical geometric structure. Optimal shape design of a high-voltage test arrangement P. Di Barba, R. Galdi, U. Piovan, A. Savini and G. Consogno Keywords Finite element analysis, Electrostatics, High voltage Discusses the automated shape design of the electrodes supplying an arrangement for highvoltage test. Obtains results that are feasible for industrial applications by means of an optimisation algorithm able to process discrete-valued design variables. Cogging torque calculation considering magnetic anisotropy for permanent magnet synchronous motors Shinichi Yamaguchi, Akihiro Daikoku and Norio Takahashi Keywords Magnetic fields, Torque, Laminates This paper describes the cogging torque of the permanent magnet synchronous (PM) motors due to the magnetic anisotropy of motor core. The cogging torque due to the magnetic anisotropy is calculated by the finite element method using two kinds of modeling methods: one is the 2D magnetization property method, and the other is the conventional method. As a result, the PM motors with parallel laminated core show different cogging torque waveform from the PM motors with the rotational laminated core due to the influence of the magnetic anisotropy. The amplitudes of the cogging torque are different depending on

the modeling methods in the region of high flux density. Magnetoelastic coupling and Rayleigh damping A. Belahcen Keywords Rayleigh-Ritz methods, Finite element analysis, Vibration measurement This paper presents a magnetoelastic dynamic FE model. As first approach, the effect of magnetostriction and strong coupling is not considered. The effect of Rayleigh damping factors on the vibrational behaviour of the stator core of a synchronous generator is studied using the presented model. It shows that the static approach is not accurate enough and the difference between calculations with damped and undamped cases is too important to be ignored. However, the difference between damped cases with reasonable damping is not very important. Modelling of temperature-dependent effective impedance of non-ferromagnetic massive conductor Ivo Dolez˘el, Ladislav Musil and Bohus˘ Ulrych Keywords Modelling, Numerical analysis, Inductance Impedance of long direct massive conductors carrying time-variable currents is a complex function of time. Its evolution is affected not only by the skin effect but also by the temperature rise. This paper presents a numerical method that allows one to compute the resistance and internal inductance of a non-ferromagnetic conductor of any cross-section from values of the total Joule losses and magnetic energy within the conductor, and also illustrates the theoretical analysis based on the field approach on a typical example and discusses the results. Field strength computation at edges in nonlinear magnetostatics Friedemann Groh, Wolfgang Hafla, Andre´ Buchau and Wolfgang M. Rucker Keywords Magnetic fields, Integral equations, Nonlinear control systems, Vectors Magnetostatic problems including iron components can be solved by a nonlinear

indirect volume integral equation. Its unknowns are scalar field sources. They are evaluated iteratively. In doing so the integral representation of fields has to be calculated. At edges singularities occur. Following a method to calculate the field strength on charged surfaces a way out is presented.

Abstracts and keywords

599 Genetic algorithm coupled with FEM-3D for metrological optimal design of combined current-voltage instrument transformer Marija Cundeva, Ljupco Arsov and Goga Cvetkovski Keywords Transformers, Genetic algorithms, Magnetic fields The combined current-voltage instrument transformer (CCVIT) is a complex non-linear electromagnetic system with increased voltage, current and phase displacement errors. Genetic algorithm (GA) coupled with finite element method (FEM-3D) is applied for CCVIT optimal design. The optimal design objective function is the metrological parameters minimum. The magnetic field analysis made by FEM-3D enables exact estimation of the four CCVIT windings leakage reactances. The initial CCVIT design is made according to analytical transformer theory. The FEM-3D results are a basis for the further GA optimal design. Compares the initial and GA optimal output CCVIT parameters. The GA coupled with FEM-3D derives metrologically positive design results, which leads to higher CCVIT accuracy class. Adaptive meshing algorithm for recognition of material cracks Konstanty M. Gawrylczyk and Piotr Putek Keywords Sensitivity analysis, Mesh generation, Optimization techniques Describes the algorithm allowing recognition of cracks and flaws placed on the surface of conducting plate. The algorithm is based on sensitivity analysis in finite elements, which determines the influence of geometrical parameters on some local quantities, used as objective function. The methods are similar to that of circuit analysis, based on differentiation of stiffness matrix. The algorithm works iteratively using gradient method. The information on the gradient of the goal

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function provides the sensitivity analysis. The sensitivity algorithm allows us to calculate the sensitivity versus x and y, so the nodes can be properly displaced, modeling complicated shapes of defects. The examples show that sensitivity analysis applied for recognition of cracks and flaws provides very good results, even for complicated shape of the flaw. Incorporation of a Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations: application of the Newton-Raphson method J. Gyselinck, P. Dular, N. Sadowski, J. Leite and J.P.A. Bastos Keywords Finite element analysis, Vector hysteresis, Magnetic fields, Newton-Raphson method This paper deals with the incorporation of a vector hysteresis model in 2D finite-element (FE) magnetic field calculations. A previously proposed vector extension of the well-known scalar Jiles-Atherton model is considered. The vectorised hysteresis model is shown to have the same advantages as the scalar one: a limited number of parameters (which have the same value in both models) and ease of implementation. The classical magnetic vector potential FE formulation is adopted. Particular attention is paid to the resolution of the nonlinear equations by means of the Newton-Raphson method. It is shown that the application of the latter method naturally leads to the use of the differential reluctivity tensor, i.e. the derivative of the magnetic field vector with respect to the magnetic induction vector. This second rank tensor can be straightforwardly calculated for the considered hysteresis model. By way of example, the vector Jiles-Atherton is applied to two simple 2D FE models exhibiting rotational flux. The excellent convergence of the Newton-Raphson method is demonstrated. The modelling of the FDTD method based on graph theory Andrzej Jordan and Carsten Maple Keywords Modelling, Finite difference time-domain analysis, Magnetic fields, Graph theory Discusses a parallel algorithm for the finitedifference time domain method. In particular, investigates electromagnetic field propagation

in two and three dimensions. The computational intensity of such problems necessitates the use of multiple processors to realise solutions to interesting problems in a reasonable time. Presents the parallel algorithm with examples, and uses aspects of graph theory to examine the communication overhead of the algorithm in practice. This is achieved by observing the dynamically changing adjacency matrix of the communications graph. Inverse problem – determining unknown distribution of charge density using the dual reciprocity method Dean Ogrizek and Mladen Trlep Keywords Density measurement, Reciprocating engines, Algorithmic languages Presents the use of the dual reciprocity method (DRM) for solving inverse problems described by Poisson’s equation. DRM provides a technique for taking the domain integrals associated with the inhomogeneous term to the boundary. For that reason, the DRM is supposed to be ideal for solving inverse problems. Solving inverse problems, a linear system is produced which is usually predetermined and ill-posed. To solve that kind of problem, implements the Tikhonov algorithm and compares it with the analytical solution. In the end, tests the whole algorithm on different problems with analytical solutions. Finite element modelling of stacked thin regions with non-zero global currents P. Dular, J. Gyselinck, T. Zeidan and L. Kra¨henbu¨hl Keywords Laminates, Finite element analysis, Eddy currents Develops a method to take the eddy currents in stacked thin regions, in particular lamination stacks, into account with the finite element method using the 3D magnetic vector potential magnetodynamic formulation. It consists in converting the stacked laminations into continuums with which terms are associated for considering the eddy current loops produced by both parallel and perpendicular fluxes. Non-zero global currents can be considered in the

laminations, in particular for studying the effect of imperfect insulation between their ends. The method is based on an analytical expression of eddy currents and is adapted to a wide frequency range. Reliability-based topology optimization for electromagnetic systems Jenam Kang, Chwail Kim and Semyung Wang Keywords Design, Optimization techniques, Topology, Sensitivity analysis This paper presents a probabilistic optimal design for electromagnetic systems. A 2D magnetostatic finite element model is constructed for a reliability-based topology optimization (RBTO). Permeability, coercive force, and applied current density are considered as uncertain variables. The uncertain variable means that the variable has a variance on a certain design point. In order to compute reliability constraints, a performance measure approach is widely used. To find reliability index easily, the limit-state function is linearly approximated at each iteration. This approximation method is called the first-order reliability method, which is widely used in reliability-based design optimizations. To show the effectiveness of the proposed method, RBTO for the electromagnetic systems is applied to magnetostatic problems. A ‘‘quasi-genetic’’ algorithm for searching the dangerous areas generated by a grounding system Marcello Sylos Labini, Arturo Covitti, Giuseppe Delvecchio and Ferrante Neri Keywords Programming, Algorithmic languages, Soil testing Sets out a method for determining the dangerous areas on the soil surface. The touch voltages are calculated by a Maxwell’s subareas program. The search for the areas in which the touch voltages are dangerous is performed by a suitably modified genetic algorithm. The fitness is redefined so that the genetic algorithm does not lead directly to the only optimum solution, but to a certain number of solutions having pre-arranged ‘‘goodness’’ characteristics. The algorithm has been called ‘‘quasi-genetic’’ algorithm

and has been successfully applied to various grounding systems. Development of optimizing method using quality engineering and multivariate analysis based on finite element method Yukihiro Okada, Yoshihiro Kawase and Shinya Sano Keywords Multivariate analysis, Finite element analysis, Torque, Optimization techniques Describes the method of optimization based on the finite element method. The quality engineering and the multivariable analysis are used as the optimization technique. In addition, this method is applied to a design of IPM motor to reduce the torque ripple. An improved fast method for computing capacitance L. Song and A. Konrad Keywords Capacitance, Computer applications, Production cycle In the design of chip carriers, appropriate analysis tools can shorten the overall production cycle and reduce costs. Among the functions to be performed by such computer-aided engineering software tools are self and mutual capacitance calculations. Since the method of moments is slow when applied to large multi-conductors systems, a fast approximate method, the average potential method (APM), can be employed for capacitance calculations. This paper describes the improved average potential method, which can further reduce the computational complexity and achieve more accuracy than the APM. Power losses analysis in the windings of electromagnetic gear Andrzej Patecki, Sławomir Ste˛pien´ and Grzegorz Szyman´ski Keywords Power measurement, Electromagnetic fields, Eddy currents Presents 3D method for the computation of the winding current distribution and power losses of the electromagnetic gear. For a prescribed current obtained from measurement, the transient eddy current

Abstracts and keywords

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field is defined in terms of a magnetic vector potential and an electric scalar potential. From numerically obtained potentials the power losses are determined. The winding power losses calculation of an electromagnetic gear shows that a given course of the current generates skin effect and significantly changes the windings resistances. Also presents the designing method for reducing power losses.

! potential T and magnetic scalar potential V . Since the problem is of low frequency and the electric conductivity of biological tissues is very small, consideration of electric vector potential only is quite satisfactory.

Finite element analysis of the magnetorheological fluid brake transients Wojciech Szela˛g Keywords Newton-Raphson method, Fluid dynamics, Finite element anaylsis Deals with coupled electromagnetic, hydrodynamic, thermodynamic and mechanical motion phenomena in magnetorheological fluid brake. Presents the governing equations of these phenomena. The numerical implementation of the mathematical model is based on the finite element method and a step-by-step algorithm. In order to include non-linearity, the NewtonRaphson process has been adopted. The method has been successfully adapted to the analysis of the coupled phenomena in the magnetorheological fluid brake. Present the results of the analysis and measurements.

2D harmonic analysis of the cogging torque in synchronous permanent magnet machines M. Łukaniszyn, M. Jagiela, R. Wro´bel and K. Latawiec Keywords Magnetic devices, Torque, Flux density, Fourier transforms Presents an approach to determine sources of cogging torque harmonics in permanent magnet electrical machines on the basis of variations of air-gap magnetic flux density with time and space. The magnetic flux density is determined from the twodimensional (2D) finite element model and decomposed into the double Fourier series through the 2D fast Fourier transform (FFT). The real trigonometric form of the Fourier series is used for the purpose to identify those space and time harmonics of magnetic flux density whose involvement in the cogging torque is the greatest relative contribution. Carries out calculations for a symmetric permanent magnet brushless machine for several rotor eccentricities and imbalances.

Magnetic stimulation of knee – mathematical model Bartosz Sawicki, Jacek Starzyn´ski, Stanisław Wincenciak, Andrzej Krawczyk and Mladen Trlep Keywords Finite element analysis, Bones, Vectors, Mathematical modelling Arthritis, the illness of the bones, is one of the diseases which especially attack the knee joint. Magnetic stimulation is a very promising treatment, although not very clear as to its physical background. Deals with the mathematical simulation of the therapeutical technique, i.e. the magnetic stimulation method. Considers the low-frequency magnetic field. To consider eddy currents one uses the pair of potentials: electric vector

Determination of a dynamic radial active magnetic bearing model using the finite element method ˘ tumberger, Bos˘tjan Polajz˘er, Gorazd S Drago Dolinar and Kay Hameyer Keywords Magnetic fields, Modelling, Nonlinear control systems The dynamic model of radial active magnetic bearings, which is based on the current and position dependent partial derivatives of flux linkages and radial force characteristics, is determined using the finite element method. In this way, magnetic nonlinearities and cross-coupling effects are considered more completely than in similar dynamic models. The presented results show that magnetic nonlinearities and cross-coupling effects can

change the electromotive forces considerably. These disturbing effects have been determined and can be incorporated into the real-time realization of nonlinear control in order to achieve cross-coupling compensations.

Electromagnetic forming: a coupled numerical electromagnetic-mechanicalelectrical approach compared to measurements A. Giannoglou, A. Kladas, J. Tegopoulos, A. Koumoutsos, D. Manolakos and A. Mamalis Keywords Electromagnetic fields, Finite element analysis, Manufacturing systems, Numerical analysis Undertakes an analysis of electromagnetic forming process. Despite the fact that it is an old process, it is able to treat current problems of advanced manufacturing technology. Primary emphasis is placed on presentation of the physical phenomena, which govern the process, as well as their numerical representation by means of simplified electrical equivalent circuits and fully coupled fields approach of the electromagneticmechanical-electric phenomena involved. Compares the numerical results with measurements. Finally, draws conclusions and perspectives for future work.

2D harmonic balance FE modelling of electromagnetic devices coupled to nonlinear circuits J. Gyselinck, P. Dular, C. Geuzaine and W. Legros Keywords Finite element analysis, Nonlinear control systems, Harmonics, Frequency multipliers This paper deals with the two-dimensional finite element analysis in the frequency domain of saturated electromagnetic devices coupled to electrical circuits comprising nonlinear resistive and inductive components. The resulting system of nonlinear algebraic equations is solved straightforwardly by means of the NewtonRaphson method. As an application example we consider a three-phase transformer feeding

a nonlinear RL load through a six-pulse diode rectifier. The harmonic balance results are compared to those obtained with timestepping and the computational cost is briefly discussed.

Finite element analysis of coupled phenomena in magnetorheological fluid devices Wojciech Szela˛g Keywords Couplers, Electromagnetism, Fluids, Finite element analysis This paper deals with coupled electromagnetic, hydrodynamic and mechanical motion phenomena in magnetorheological fluid devices. The governing equations of these phenomena are presented. The numerical implementation of the mathematical model is based on the finite element method and a step-by-step algorithm. In order to include non-linearity, the NewtonRaphson process has been adopted. A prototype of an electromagnetic brake has been built at the Poznan´ University of Technology. The method has been successfully adapted to the analysis of this brake. The results of the analysis are presented.

Comparison of the Preisach and Jiles-Atherton models to take hysteresis phenomenon into account in finite element analysis Abdelkader Benabou, Ste´phane Cle´net and Francis Piriou Keywords Finite element analysis, Electromagnetism, Energy In this communication, the Preisach and JilesAtherton models are studied to take hysteresis phenomenon into account in finite element analysis. First, the models and their identification procedure are briefly developed. Then, their implementation in the finite element code is presented. Finally, their performances are compared with an electromagnetic system made of soft magnetic composite. Current and iron losses are calculated and compared with the experimental results.

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Error bounds for the FEM numerical solution of non-linear field problems Ioan R. Ciric, Theodor Maghiar, Florea Hantila and Costin Ifrim Keywords Error analysis, Magnetic fields, Field testing A bound for a norm of the difference between the computed and exact solution vectors for static, stationary or quasistationary

non-linear magnetic fields is derived by employing the polarization fixed point iterative method. At each iteration step, the linearized field is computed by using the finite element method. The error introduced in the iterative procedure is controlled by the number of iterations, while the error due to the chosen discretization mesh is evaluated on the basis of the hypercircle principle.

Editorial

Editorial This special issue is devoted to the papers that were presented at the International Symposium on Electromagnetic Fields in Electrical Engineering ISEF’03. The symposium was held in Maribor, Slovenia on 18-20 September 2003. The city of Maribor is known for its beauty, charm and academic flavour as well. Therefore, the participants of ISEF’03 found there very good atmosphere to present their papers and debate on them. After the selection process, 159 papers have been accepted for the presentation at the symposium and almost 90 per cent were presented at the conference both orally and in the poster sessions. The papers have been divided into the following groups: . Computational Electromagnetics; . Electromagnetic Engineering; . Coupled Field and Special Applications; . Bioelectromagnetics and Electromagnetic Hazards; . Magnetic Material Modelling.

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It is the tradition of the ISEF meetings that they comprise quite a vast area of computational and applied electromagnetics. Moreover, the ISEF symposia aim at joining theory and practice, thus the majority of papers are deeply rooted in engineering problems, being simultaneously of high theoretical level. Bearing this tradition, we hope to touch the heart of the matter in electromagnetism. The present issue of COMPEL contains 27 papers which have been selected by the editors on the basis of the reviewing process done by the chairmen of the sessions. This selection, however, gave the number of papers much bigger than the number imposed by the COMPEL. Thus, going to the number required we also considered differentia specifica of COMPEL – the selected papers are of more computational aspect than the remaining part of high-qualified papers. The latter ones are expected to be published elsewhere. We, the Editors of the special issue, would like to express our thanks to COMPEL for giving us the opportunity to present, at least, the flavour of the ISEF meeting. We also thank our colleagues for their help in reviewing the papers. Finally, we would like to wish the prospective readers of the issue to find within many subjects of interest. Mladen Trlep Chairman of the Organising Committee Andrzej Krawczyk Scientific Secretary Sławomir Wiak Chairman of the ISEF symposium

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 p. 605 q Emerald Group Publishing Limited 0332-1649

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The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Application of Haar’s wavelets in the method of moments to solve electrostatic problems

606

Aldo Artur Belardi Centro Universita´rio de FEI, Sa˜o Paulo, Brazil

Jose´ Roberto Cardoso and Carlos Antonio Franc¸a Sartori Escola Polite´cnica, Universidade de Sa˜o Paulo, Sa˜o Paulo, Brazil Keywords Electrostatics, Density measurement, Optimization techniques Abstract Presents the mathematical basis and some results, concerning the application of Haar’s wavelets, as an expansion function, in the method of moments to solve electrostatic problems. Two applications regarding the evaluation of linear and surface charge densities were carried out: the first one on a finite straight wire, and the second one on a thin square plate. Some optimization techniques were used, whose main computational performance aspects are emphasized. Presents comparative results related to the use of Haar’s wavelets and the conventional expansion functions.

1. Formulation In order to illustrate the proposed methodology, the main theoretical aspects of the method of moments and of the Haar’s wavelets, concerning one- and two-dimensional configurations, are presented in this paper. 1.1 Method of moments Although the method of moments is a well known numerical technique, and the complete description and details of this method have already been presented in many papers, in order to guide the reader through the overall method explanation, a brief summary of this method is given. In a simplified way, it can be mentioned that the method of moments basis is the application of approximation functions, such as the one represented by the following expression (Harrington, 1968): X X an Lg n ¼ an kLg n ; W m l ¼ k f ; W m l for m ¼ 1; 2. . .; N ð1Þ f ðxÞ ¼ n

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 606–612 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540511

n

In the aforementioned expression, an represents the unknown coefficients; gn is the expansion function, e.g. the pulse or the Haar’s wavelets, “L” a mathematical operator, and “Wm” is a weighting function. Expression (1) can also be represented in a matrix form by ½A
*½a
¼ ½B
; where [a ] is the unknown coefficients column matrix, and the matrixes [A] and [B]: 2 3 2 3 kLg 1 ; W 1 l . . . kLgn ; W 1 l k f ; W 1l 6 7 6 7 ½A
¼ 4 kLg 1 ; W 2 l . . . kLgn ; W 2 l 5 and ½B
¼ 4 k f ; W 2 l 5 ð2Þ kLg 1 ; W n l . . . kLgn ; W n l k f ; W nl As a first application, the potential distribution on a finite and straight wire that can be calculated using the next equation is taken into consideration (Balanis, 1990):

Z

rðr 0 Þ dl 0 Rðx; x 0 Þ

ð3Þ

Application of Haar’s wavelets

Thus, making use of the method of moments, knowing the approximated solution function f(x), the expansion function g(x) and the weighting function W(x), the potential on a finite straight wire can be estimated by the inner product of these functions: Z a 1 gðxÞW ðxÞf ðxÞ dx ð4Þ V ðxÞ ¼ k g; W ; f l ¼ R RðxÞ 2a

607

1 V ðx; y ¼ 0; z ¼ 0Þ ¼ 4p1

Consequently, the surface density r(r 0 ) can be approximated by the N term expansion. If the wire is divided into uniform segments D ¼ L=N ; after applying the weight delta function of Dirac W m ¼ dðxm 2 x 0 Þ ¼ 1; the inner product will become: V ðxÞ ¼ kW m ; f ; Lgl ¼ dðx 2 xm Þ £

Z L N 1 X g n ðx 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 0 an 4p1 n¼1 0 ðxm 2 x 0 Þ2 þ a 2

ð5Þ

Assuming the charges placed in the center of each subdivision in relation to the axis, substituting the values of x by the distance of the charge position to the point P(xm), we will have an integral that is the only function of the x 0 . For a fixed potential V, the equation can be represented, using the matrix notation, ½V m
¼ ½Z mn
½a n
; in which Zmn is defined by: Z mn ¼

Z 0

L

g n ðx 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 0 ðxm 2 x 0 Þ2 þ a 2

ð6Þ

The same approach can be used, if a two-dimensional application is considered. 1.2 The Haar’s wavelets Different types of functions can be used as expansion functions. Among them are the pulse function, the truncated cosine as well as the wavelets. In this paper, the Haar’s wavelets are used as the expansion function. Thus, considering a two-dimensional application, after applying the method of moments and considering the Haar’s wavelets, a function f(x, y) can be approximated by (Aboufadel and Schlicker, 1999): f ðx; yÞ ¼

1 X k¼21

ck fðx; yÞ þ

1 1 X X

dj;k f P ðx; yÞcj;k ðx; yÞ

ð7Þ

j¼21 k¼21

where “j”, and “k” are, respectively, the resolution and the translation levels. Moreover, once the Haar’s wavelets, and the so-called mother function (8) and the scale or father function (9) are applied, the formulation will result in a product combination of (10) and (11) given by (12): Þ j=2 j cðH j;k ðxÞ ¼ 2 cð2 x 2 kÞ j; k [ Z

ð8Þ

(

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f ðH Þ ðxÞ ¼

608

n

1

0 # x , 1; and

0

for other intervals

ð9Þ

Þ j cðH j;k ðxÞ ¼ ½fðxÞ cðxÞcð2xÞcð2x 2 1Þ. . .cð2 x 2 kÞ


ð10Þ

Þ j cðH j;k ð yÞ ¼ bfðyÞcðyÞcð2yÞcð2y 2 1Þ. . .cð2 y 2 kÞc

ð11Þ

o Þ ðH Þ cðH j;k ðxÞ; cj;k ðyÞ ¼ fðxÞfð yÞ; fðxÞcð yÞ; . . .; cð2x 2 1Þcð2y 2 1Þ

ð12Þ

As an illustration, Figure 1 shows the Haar’s function regarding two dimensions and one level of resolution, for a point P(xm, ym). On the other hand, if the potential in a finite and very thin plane plate is considered as an application, it can be evaluated by (Newland, 1993):

Figure 1. Representation of the Haar’s function for two-dimensional and one level of resolution

V ðx; yÞ4p1 ¼ aj bj

Z

a

2a

þ

Z

b

2b

Application of Haar’s wavelets

fðx; yÞ dx dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxm 2 x 0 Þ2 þ ð ym 2 y 0 Þ2

1 1 X X

aj;k bj;k

j¼21 k¼21

Z

a

2a

Z

b

2b

Þ cðH j;k ðx; yÞ dx dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxm 2 x 0 Þ2 þ ð ym 2 y 0 Þ2

ð13Þ

609

In a very similar matrix notation that was used for the one-dimensional application, the previous equation can be described as ½V m
¼ ½Z mn
½an
in which Zmn is defined by: Z b Z a gn ðx 0 ; y 0 Þ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy 0 dx ð14Þ Z mn ¼ 2 2 2a 2b 4p1 ðxm 2 x 0 Þ þ ð ym 2 y 0 Þ It should be observed that the previous formulation is indexed by two parameters, “j” and “k”, allowing us to vary the precision of the results through these levels. Concerning the characteristic of the method and the application of the Haar’s wavelets, the main aspects are related to the resulting scattered matrices and null coefficients, an interesting property to be considered regarding the computational aspects. If one remembers that the equation to determine the coefficients of the approximation function can be written as in equation (15), those aspects can be realized based on the following approach: ½Z mn
½r
¼ ½V


ð15Þ

where Zmn is a square matrix that is not necessarily a sparse one, since it depends on the expansion function that was chosen. Moreover, taking advantages of the fact that the Haar’s matrix [ H ] is a sparse matrix, applying the matrix algebra, it will result (Wagner and Chew, 1995): ð16Þ ½Z 0mn
½r 0
¼ ½V 0
else, ½Z 0mn
¼ ½H
½Z mn
½H T
;

½r 0
¼ ½H T
21 ½r
; and ½V 0
¼ ½H
½V


ð17Þ

Consequently, we will obtain: ½H
½Z mn
½H T
½H T
21 ½r
¼ ½H
½V


ð18Þ

Thus, after applying such an approach, we obtained a symmetric matrix and, due to the properties of Haar’s function, a number of “near” zero matrix elements. Additionally, the assumption of threshold levels, a percentage of the difference between the maximum positive value and the minimum negative one, will help us to obtain an additional computing time reduction. It is based on the fact that, once it is adopted, the matrix elements that are smaller than this number will be assumed to be zero. 2. Applications and discussion Applying the aforementioned formulation, we obtained some results related to two applications: the first one related to a finite and straight wire, and another one regarding a thin plane plate. In the two applications it is assumed that a constant potential distribution is equal to 1 V.

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Table I presents the results regarding the charge surface density on a 1.0 m straight wire, when it is divided into 16 equal segments, as a function of the resolution ( j) and the translation (k) levels. Those results can be considered as the ones suitable to validate this approach. Figure 2 shows the surface charge density on a 1.0 m straight wire, and diameter equal to 0.0001 m, when the level 4 of resolution is applied for the wavelets, and 32 subdivisions are used. The wire is at a potential of 1.0 V.

Expansion function Point

Table I. Charge surface density ( pC/m) on a straight finite wire as a function of the resolution levels

Figure 2. The surface charge (pC/m) on a 1.0 m straight wire for 32 subdivisions

1 2 3 4 5 ... 12 13 14 15 16

Haar wavelet (level) 2 8.835 8.835 8.835 8.835 7.970 ... 7.970 8.835 8.835 8.835 8.835

3 9.376 9.376 8.274 8.274 8.059 ... 8.059 8.274 8.274 9.376 9.376

Pulse 4 9.957 8.764 8.411 8.219 8.102 ... 8.102 8.219 8.411 8.764 9.957

9.957 8.764 8.411 8.219 8.102 ... 8.102 8.219 8.411 8.764 9.957

Figure 3 shows the surface charge density on a square plate ð1:0 m £ 1:0 mÞ; when 16 subdivisions are considered, and the level 5 of resolution is applied for the wavelets. The plate is at a potential equal to 1.0 V. Table II shows the results obtained after comparing the values of the computing time (Patterson and Hennessy, 2001), function of the number of divisions in each axe of the plate, with and without applying the null value detection (NVD) approach. Moreover, we can compare the results obtained by the proposed methodology considering the expansion function as being the pulse with the other ones using the wavelets. The validation of the aforementioned approach was carried out based on the application of the statistical indexes regarding the paired data and the corresponding average correlation (Papoulis, 1991). For the straight wire, and 32 subdivisions, when the results related to the use of wavelets are compared with the pulse function ones, statistically, we could see no statistical difference between the two approaches (tcalc . 95 per cent). Moreover, it was verified that average comparative values related to the charge density value is less than 0.025 per cent, for the straight finite wire, and for square plane plate applications.

Application of Haar’s wavelets

611

Figure 3. The surface charge ( pC/m) on a 1.0 m £ 1.0 m plate, for 16 subdivisions

Divisions 4£4 8£8 16 £ 16 32 £ 32

Computing time (s) Without NVD With NVD 0.321 7.931 451.960 27,273.738

0.25 5.488 222.60 11,994.487

Difference ( per cent) 22.12 30.80 50.75 56.02

Table II. Computing time as a function of the number of the subdivisions and of the use of the NVD approach

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3. Conclusion This paper presented the main features of wavelets as an expansion function in the method of the moments. Although the proposed methodology can be applied to more complex problems, some applications in electrostatics were demonstrated. Based on the theoretical features and on the statistical indexes applied to comparative results regarding the Haar’s wavelets and the pulse functions, the proposed methodology was validated. The main advantages concerning the matrix arrangements and its numerical treatment, as well as the related computing time were discussed in the paper. References Aboufadel, E. and Schlicker, S. (1999), Discovering Wavelets, Wiley, New York, NY, pp. 1-42. Balanis, C. (1990), Advanced Engineering Electromagnetics, Wiley, New York, NY, pp. 670-95. Harrington, R.F. (1968), “Field computation by moment methods”, Electrical Science, pp. 1-40. Newland, D.E. (1993), Random Vibrations Spectral and Wavelet Analysis, Addison Wesley, Reading, MA, pp. 315-33. Papoulis, A. (1991), Probability Random Variables and Stochastic Processes, McGraw-Hill, New York, pp. 265-78. Patterson, D.A. and Hennessy, J.L. (2001), Computer Organization and Design the Hardware/Software Interface, 1st ed., Morgan Kaufmann, Los Altos, CA, pp. 26-51. Wagner, R.L. and Chew, W.C. (1995), “A study of wavelets for the solution of electromagnetic integral equations”, IEEE Transactions on Antennas and Propagation, Vol. 43 No. 8, pp. 802-10.

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The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

3D multimodal A 3D multimodal FDTD algorithm AFDTD algorithm for electromagnetic and acoustic propagation in curved waveguides 613 and bent ducts of varying cross-section Nikolaos V. Kantartzis, Theodoros K. Katsibas, Christos S. Antonopoulos and Theodoros D. Tsiboukis Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece Keywords Electromagnetic fields, Acoustic waves, Wave physics, Finite difference time-domain analysis Abstract This paper presents a curvilinearly-established finite-difference time-domain methodology for the enhanced 3D analysis of electromagnetic and acoustic propagation in generalised electromagnetic compatibility devices, junctions or bent ducts. Based on an exact multimodal decomposition and a higher-order differencing topology, the new technique successfully treats complex systems of varying cross-section and guarantees the consistent evaluation of their scattering parameters or resonance frequencies. To subdue the non-separable modes at the structures’ interfaces, a convergent grid approach is developed, while the tough case of abrupt excitations is also studied. Thus, the proposed algorithm attains significant accuracy and savings, as numerically verified by various practical problems.

Introduction The systematic modelling of arbitrary electromagnetic compatibility (EMC) applications or bent ducts with irregular cross-sections remains a fairly demanding area of contemporary research, since curvilinear coordinates complicate the separation of the wave equation and therefore, the extraction of a viable analytical solution. Furthermore, the involved fabrication details of such structures in both electromagnetics and acoustics, having a critical influence on the overall frequency response, enforce regular numerical realisations to utilise extremely fine meshes with heavy computational overheads. Soon after the detection of these shortcomings, several effective techniques have been presented for their mitigation (Farina and Sykulski, 2001; Przybyszewski and Mrozowski, 1998; Sikora et al., 2000) or the evolution of flexible discretisation perspectives (Felix and Pagneux, 2002; Rong et al., 2001) and robust lattice ensembles (Bossavit and Kettunen, 2001; Podebrad et al., 2003; Zagorodov et al., 2003). Among them, the finite-difference time-domain (FDTD) renditions (Taflove and Hagness, 2000) in accordance with the highly-absorptive perfectly matched layers (PMLs) (Berenger, 2003) constitute trustworthy simulation tools, especially in Cartesian grids. In this paper, a novel multimodal FDTD formulation, founded on a 3D curvilinear regime, is introduced for the precise analysis of electromagnetic and acoustic waves inside curved arrangements of varying cross-section. Owing to the essential role of

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 613-624 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540520

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614

the excitation scheme, the framework so developed, projects each component on the basis of a transverse mode series to derive an ordinary differential equation which performs reliable vector representations and launches the initial fields in close proximity to the discontinuity. To annihilate the erroneous oscillations near the curvature or bend, an impedance matrix that can be quantitatively integrated up to a sufficient number of modal counterparts is defined. In this context, the new FDTD concepts for electromagnetics are easily extended to acoustics in a completely dual manner, whereas a higher-order differencing tessellation suppresses the dispersion error mechanisms and evaluates the scattering properties irrespective of geometrical peculiarities or frequency spectrums. Conversely, dissimilar interface media distributions that do not follow the grid lines are handled via a convergent transformation. Numerical results, addressing diverse realistic EMC configurations, waveguides, junctions and ducts – terminated by appropriately constructed curvilinear PMLs – demonstrate the considerable accuracy, the stability as well as the drastic computational savings of the proposed approach. The 3D multimodal FDTD method in general coordinate systems Let us consider the general waveguide of Figure 1(a), including a toroidal-like section of length ut with inner and outer mean radii R1(u) and R2(u), respectively. The two straight parts can have arbitrary cross-sections, while their walls may be flexible or rigid. Actually, the most difficult part of this simulation is the bent waveguide sector that generates non-separable modes not easy to determine because of the cumbersome calculations required at every frequency and the incomplete fulfilment of the suitable continuity conditions regarding the straight parts. Comparable observations may be performed from Figure 1(b), where the more complex three-port Y junction is depicted. Herein, the discontinuity and the two ports prohibit any variable separation of Maxwell’s or the linearised Euler’s equations. To circumvent the prior defects, our methodology introduces a multimodal decomposition for the propagating quantities that is applied to prefixed planes in the bend. Thus, each component f can be written in terms of infinite series as X jk ðr; uÞFk ðr; uÞ with jk ðr; uÞ ¼ Bk cos½kpðr 2 R1 ðuÞÞ=RðuÞ
; ð1Þ f ðr; uÞ ¼ k

and Bk ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 2 dk0 Þ=RðuÞ

for

Z

R1 þsl

jk ðr; uÞjl ðr; uÞ dr ¼ dkl ;

ð2Þ

R1

where RðuÞ ¼ R2 ðuÞ 2 R1 ðuÞ; Fk are scalar coefficients and jk are eigenfunctions complying with the corresponding transverse electromagnetic or acoustic eigenproblem. Two characteristic functions for R1(u) and R2(u), with au ¼ u=ut ; s ¼ sr 2 sl ; sr ¼ 1:25sl and t ¼ v=c0 ; run into several practical applications, are R1 ðuÞ ¼ st 2 ðau 2 1:5Þ þ sl 2 0:5sr ;

ð3Þ

R2 ðuÞ ¼ 2st 2 ðau 2 1:5Þ þ sl þ 0:5sr :

ð4Þ

The key issue in such systems is the initial mode coupling, occurring in two distinct ways: one due to the curvature of the waveguide or duct and the other due to its

A 3D multimodal FDTD algorithm

615

Figure 1. (a) Geometry of an arbitrarily-curved waveguide with a varying cross-section, and (b) transverse cut of a three-port Y junction comprising two ducts of elliptical cross-section and different dimensions

varying cross-section. The former normally contributes to the generation of higher-order modes and the latter induces the symmetric ones. It is stressed that the existing schemes cannot simulate this intricate situation contaminating so, the final outcomes. The proposed algorithm overcomes this artificial hindrance by projecting the appropriate governing laws to jk in order to extract an equivalent set of ordinary differential equations which combine electric and magnetic fields in electromagnetics or velocity and pressure in acoustics. Their solutions lead to very accurate excitation models and the most substantial; they preserve lattice duality, even when the source is placed quite close to the discontinuity. For illustration and without loss of generality, we concentrate on Maxwell’s curl analogues at a certain plane in the interior of the bend. Then, by means of the respective matrix terminology, one obtains the modified forms of Ampere’s and Faraday’s laws, 1›t ½E
¼ t 21 ðW A þ W B W C ÞH 2 W D E

ð5Þ

m›t ½H
¼ 2tW C E þ ðW E 2 W D ÞH;

ð6Þ

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with E and H the electric and magnetic field intensities defined at the general coordinates (u, v, w) of g(u, v, w) metrics. The elements of matrices W i ði ¼ A; . . .; EÞ describe the fundamental details of the curvature and should be carefully calculated. Hence, after enforcing the boundary constraints, we obtain 8 for k ¼ l <0 A W kl ¼ : Ckl k 2 =ðk 2 2 l 2 Þ for k – l   W Bkl ¼ t 2 2 b2k dkl W Ckl

¼

W Dkl ¼

for bk ¼ kp=s

8 < R1 ðuÞ þ s=2

for k ¼ l

: Ckl ðk 2 þ l 2 Þ=ðk 2 2 l 2 Þ2

for k – l

8 < dk0 ½R2 ðuÞ 2 R1 ðuÞ
=RðuÞ

for k ¼ l

: W E k 2 =ðk 2 2 l 2 Þ2 kl

for k – l

ð7Þ

and W Ekl ¼ Bk Bl ½ð21Þkþl R2 ðuÞ 2 R1 ðuÞ


with Ckl ¼ Bk Bl ½ð21Þkþl 2 1
:

ð8Þ

In order to guarantee the sufficient annihilation of vector parasites and avoid undesired instabilities, owing to arbitrary cross-sections, equations (5) and (6) will not be directly advanced in the time-domain. A closer inspection to their structure reveals that they can be efficiently combined by means of an impedance matrix K and satisfy the general relation E ¼ KH: Substitution of electric/magnetic field quantities and their temporal derivatives in this last expression leads to the ordinary differential equation of

›t ½K
¼ 2tW C 2 t 21 KðW A þ W B W C ÞK þ KW D 2 W D K þ W E K;

ð9Þ

whose solution can be promptly acquired. The previous procedure – being fully dual for the acoustic case – provides the required E and H values at specific planes inside the bend. These values are next inserted as excitation terms in the FDTD formulae to proceed with the update mechanism in the usual manner. Having determined the correct variation of the propagating waves at the aforementioned problematic areas, the rest of the computational space could be discretised through the traditional FDTD approach. Nonetheless, when the second-order Yee’s method is implemented for the treatment of complicated EMC or acoustic devices, large dispersion errors are induced that degrade its total performance, not to mention the discrepancies due to the partial imposition of continuity conditions at curvilinear grids (Figure 2). Therefore, it becomes apparent that a more orderly discretisation strategy should be pursued. In the light of these abstractions, we develop a parametric family of 3D higher-order non-standard FDTD operators

A 3D multimodal FDTD algorithm

617

Figure 2. A dual generalised curvilinear FDTD mesh with two kinds of cells

( ) 3 h  i X t rA t M Qz;L f u;v;w þ f z^hDz=2 ; ¼ u;v;w 4Dz h¼1 h  i h  t2Dt=2 i. t t tþDt=2 T f u;v;w ¼ f u;v;w 2 f u;v;w C T ðDtÞ 2 r B ›ttt f u;v;w ; h  t Dz f 

i

ð10Þ ð11Þ

which achieve very improved and robust approximations of the spatial and temporal derivatives appearing in Maxwell’s or Euler’s laws. Coefficients rA, rB are real numbers, z [ ðu; v; wÞ and M is the accuracy order. Moreover, factor L denotes the size of the stencil, hDz, along every axis with a customary value of L ¼ 3; while Q [·] is the non-standard difference operator, given by ( ) M L h  i gðu; v; wÞ X h  i X t t M z ðmÞ z   V Y U f Qz;L f u;v;w ¼ : ð12Þ C S ðkLDzÞ m¼1 m l¼1 m;l z;lDz u;v;w Vm and Ym according to the calligraphic style appearing in equation (12). Algorithmic consistency and control the assignment of fields to space-time entities by obeying the subsequent gauges M X

Vzm ¼ 1=2;

ð13Þ

m¼1 L X

Yzm;l ¼ 1

;m:

ð14Þ

l¼1

On the other hand, functions CS (kLDz), CT (Dt) enable the smooth transition from the continuous to discrete state. To pick an acceptable argument for distinct wavenumbers k, the Fourier transform of the already computed components at certain locations, is utilised. After the frequency spectrum has been specified, we take its maximum value for the optimal selection. Operators U (m)[·], in equation (12), yield coarse topologies without compromising the accuracy. So, unlike the Yee’s technique with the two mesh points for derivative evaluation, our scheme concerns a whole set of nodes. A typical v-directed U (m)[·] is defined as

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þ1 h  h  i i t ðDvÞm X t t UðmÞ f 2Du;rlDv=2;rDw 2 f Du;2rlDv=2;2rDw : v;lDv f u;v;w ¼ 3m 2 2 r¼21

ð15Þ

For handling areas near boundary walls in terms of the enlarged stencils, we establish a filtering process with an additional degree of freedom L. For instance, the manipulation of a v-axis stencil, results in L        1X aj f i; jþl;k þ f i; j2l;k ; f 0 i; j21;k þ f 0 i; j;k þ f 0 i; jþ1;k ¼ ð16Þ 2 l¼0 with the primes indicating the modified quantities. The gradually calculated filtering coefficients aj take into account the boundary curvature and the nodes that extend outside the higher-order FDTD grid. A convergent treatment for complex material interfaces The staircasing regime and the inability of the simple FDTD method to satisfy the necessary continuity conditions on both sides of arbitrarily-embedded media interfaces, have a crucial impact on its stability. Actually, when a component is discontinuous along a 3D grid line, Yee’s algorithm loses the global convergence. To alleviate this drawback, a new higher-order formulation is discussed, which alters the stencils around these boundaries and correctly represents their physical location. For this objective, assume a curvilinear interface between two media ðmat ¼ A; BÞ with fairly different properties and the propagation of an acoustic wave through them. The basic premise is to appoint a spatial parameter to each region, i.e. sA and sB, respectively, as a guideline of the distance from the first/last cell to the physical position of the wall relative to cell dimensions. Evidently, s mat fulfils the constraint s A ¼ 0:5 2 s B ; and is computed only once. For example, the corrected value of the acoustic pressure derivative ›w p at the boundary becomes  21 h  i. nþ1 nþ1 ›w pi;j;k ¼ 2 2sBi;j;k þ 1 piþ1=2;k;j 2 p B;nþ1 du: ð17Þ The essential material term on the right hand side of equation (17) becomes p B;nþ1 ¼ p~ A;nþ1 þ ðr A 2 r B Þ

p~ A;nþ1 þ p~ B;nþ1 ; r A n2w þ r B n2u þ r B n2v

ð18Þ

where r mat is the mass density of the medium and n ¼ ½nu ; nv ; nw
T a unit vector normal to the interface. The two transitional variables in the nominator of equation (18) are recovered directly by extrapolation. So,    nþ1   A  nþ1 ~ B;nþ1 ¼ p~ nþ1 ~ nþ1 þ p ; ð19Þ p~ A;nþ1 ¼ 1 þ sA i;j;k p i21=2;j;k þ si;j;k p i23=2;j;k and 2p i;jþ1=2;k i;j21=2;k with a similar process holding for the electromagnetic case, as well. The realisation of equations (17)-(19) entails a predictor-corrector scheme in which the higher-order non-standard FDTD concepts serve as the predictor stage for extracting all field distributions in the entire domain, while a corrector stage changes the solutions locally via the above technique. Finally, problems involving open ends are treated by a curvilinear version of the PML. The appropriate expressions are obtained in

such a way that can be easily applied to both lossless and lossy media. Consequently, the optimised absorber is built using a consistent variable scaling and accomplishes considerable attenuation rates, even for small depths or lattice resolutions. Numerical results The efficiency of the novel multimodal FDTD algorithm with its higher-order implementation is validated by various modern problems in both physical fields where conventional methods lack to offer adequate solutions. Particular attention has been paid to the construction of PMLs and non-Cartesian grids. Let us, first, explore the propagation of electromagnetic waves and specifically the inclined-slot coupled elliptical cavity of Figure 3. Its analysis is deemed difficult due to its curvilinear shape and the varying cross-section. Here, we have chosen b1 ¼ 6:76; b2 ¼ 8:56; b3 ¼ 19:21; c1 ¼ 18:37; c2 ¼ 9:63; c3 ¼ 25:82; l ¼ 57:46 mm while the ratio of the two cavity bases is 2.91. The domain has 26 £ 18 £ 32 cells with Dx ¼ 2:581; Dy ¼ 1:925; Dz ¼ 0:817 mm and Dt ¼ 1:432 ps: Also, the slot inclination u ¼ 608 and the open ends are backed by a six-cell PML. Figures 4 and 5 show the magnitude of S-parameters

A 3D multimodal FDTD algorithm

619

Figure 3. A sidewall inclined-slot coupled elliptical cavity

Figure 4. Magnitude of S-parameters for the sidewall elliptical cavity

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Figure 5. Shielding effectiveness for the sidewall elliptical cavity

(excitation is at port 2) and the shielding effectiveness, respectively, compared with the reference solution (Rong et al., 2001). As observed, the higher-order non-standard forms are very accurate and the most important is that they need almost 91 per cent lower resources than the Yee’s scheme. Analogously, Figure 6 shows the shielding effectiveness of the conducting aperture of Figure 7 that is fed by two rectangular horn waveguides. Its dimensions are a ¼ 9:27; b ¼ 12:78; c ¼ 24:53; l ¼ 35:64; l 1 ¼ 3:92; l 2 ¼ 1:56 and w ¼ 1:68 mm: Note the satisfactory agreement between the reference and the computed plots as well as the inability of the second-order FDTD technique to give acceptable results. The next application studies the case of a three-port Y junction (Figure 1(b)) whose straight parts have different cross-sections and curvatures given by equations (3) and (4). A typical configuration is a ¼ 2:97; b ¼ 4:56; d ¼ 31:84; sl ¼ 20:14; sr ¼ 15:29 mm and a PML depth of eight cells. The magnitude of S-parameters is shown in Figure 8. Again our

Figure 6. Shielding effectiveness for the thin iris-coupled aperture

A 3D multimodal FDTD algorithm

621

Figure 7. A thin aperture with two elliptical irises

Figure 8. Magnitude of S-parameters for the three-port Y junction

approach overwhelms the classical one, an issue also confirmed by the values of Table I which furnishes the first five resonance frequencies of the structure. Numerical verification now proceeds to the analysis of acoustic waves in bent ducts. In this context, Figure 9 shows the reflection coefficient for a two-port system of varying cross-section bounded by the hard-to-model flexible walls. The ratio of the radii is R2 =R1 ¼ 2:69; ut ¼ 1208 and the open ends are truncated by a four-cell PML. From the results, it is deduced that the higher-order solutions – even for coarse meshes – are very close to those of Felix and Pagneux (2002), without arousing any artificial oscillations. The second example examines the Y junction of Figure 1(b) with a ¼ 10:21; b ¼ 15:83; d ¼ 47:69; sl ¼ 18:11 and sr ¼ 21:84 cm: Figure 10 shows the transmission coefficient T13 versus normalised frequency for a six-cell PML. The accuracy of the proposed strategy is, indeed, promising despite the presence of flexible boundaries. Finally, in Figure 11, several indicative snapshots of acoustic pressure,

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Table I. Resonance frequencies for the three-port Y junction

Figure 9. Magnitude of the reflection coefficient for a 3D bent duct

Figure 10. Magnitude of T13 for an acoustic three-port Y junction

addressing diverse types of ducts, are demonstrated. In particular, we investigate a straight flexible duct (grid: 42 £ 126 £ 40 cells), a 908 bend with rigid walls (grid: 34 £ 124 £ 56 cells, R2 =R1 ¼ 2Þ; a bent duct of varying cross-section (grid: 118£100 £ 58 cells, R2 =R1 ¼ 2:95; ut ¼ 1808Þ and a three-port Y junction Second-order FDTD method Reference solution (GHz) (Taflove and Hagness, 2000) 1.86534 4.62379 5.34762 9.29582 11.64217

Proposed method

Simulation 86 £ 60 £ 142

Relative error (per cent)

Simulation 28 £ 24 £ 40

Relative error (per cent)

1.82406 4.46450 5.01628 8.49526 10.45071

2.213 3.445 6.196 8.612 10.234

1.86511 4.62203 5.34467 9.28754 11.62948

0.012 0.038 0.055 0.089 0.109

A 3D multimodal FDTD algorithm

623

Figure 11. Snapshots of acoustic pressure for various bent ducts and a Y junction with flexible or rigid walls

(grid: 102 £ 102 £ 62 cells). Herein, lattice resolution has been increased in order to attain the optimal propagation of the demanding acoustic phenomena. As can be deduced, our algorithm is fairly effective, since it leads to sufficiently smooth variations without any artificial reflections or material oscillations. Conclusions A 3D curvilinear FDTD technique implementing a rigorous multimodal decomposition has been introduced in this paper. The proposed formulation incorporates

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a higher-order topological perspective to manipulate challenging realistic structures in the areas of electromagnetics and acoustics. Extensive numerical verification reveals the advantages and the universality of the method which is capable of conducting remarkably accurate simulations independent of the source frequency and the curvature of the device.

624

References Berenger, J-P. (2003), “Making use of the PML absorbing boundary condition in coupling and scattering FDTD codes”, IEEE Transactions on Electromagnetic Compatibility, Vol. 15 No. 2, pp. 189-97. Bossavit, A. and Kettunen, L. (2001), “Yee-like schemes on staggered cellular grids: a synthesis between FIT and FEM approaches”, IEEE Transactions on Magnetics, Vol. 37 No. 5, pp. 861-7. Farina, M. and Sykulski, J. (2001), “Comparative study of evolution strategies combined with approximation techniques for practical electromagnetic optimisation problems”, IEEE Transactions on Magnetics, Vol. 37 No. 5, pp. 3216-20. Felix, S. and Pagneux, V. (2002), “Multimodal analysis of acoustic propagation in three-dimensional bends”, Wave Motion, Vol. 36, pp. 157-68. Podebrad, O., Clemens, O. and Weiland, T. (2003), “New flexible subgridding scheme for the finite integration technique”, IEEE Transactions on Magnetics, Vol. 39 No. 3, pp. 1662-5. Przybyszewski, P. and Mrozowski, M. (1998), “A conductive wedge in Yee’s mesh”, IEEE Microwave Guided Wave Letters, Vol. 8 No. 2, pp. 66-8. Rong, A., Yang, H., Chen, X. and Cangellaris, A. (2001), “Efficient FDTD modeling of irises/slots in microwave structures and its application to the design of combline filters”, IEEE Transactions on Microwave Theory and Techniques, Vol. 49 No. 12, pp. 2266-75. Sikora, R., Gratkowski, S. and Komorowski, M. (2000), “Eddy-current based determining of conductivity and permittivity”, COMPEL, Vol. 19 No. 2, pp. 352-6. Taflove, A. and Hagness, S. (2000), Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House, Boston, MA. Zagorodov, I., Schuhmann, R. and Weiland, T. (2003), “A uniformly stable conformal FDTD-method in Cartesian grids”, International Journal of Numerical Modelling, Vol. 16, pp. 127-41.

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The highly efficient three-phase small induction motors with stator cores made from amorphous iron

Three-phase small induction motors 625

M. Dems, K. Kome˛za and S. Wiak Institute of Mechatronics and Information Systems, Technical University of Lodz, Lodz, Poland

T. Stec AMM Technologies, Huntfield Heights, Australia Keywords Inductance, Design, Iron Abstract Applies the field/circuit two-dimensional method and improved circuit method to engineering designs of the induction motor with stator cores made of amorphous iron. Exploiting of these methods makes possible computation of many different specific parameters and working curves in steady states for the “high efficiency” three-phase small induction motor. Compares the results of this calculation with the results obtained for the classical induction motor with identical geometric structure.

Introduction The US, Energy Policy Act of 1992 (EPACT) requires minimum efficiency of electric motors. Similar requirements were introduced in Australia, New Zealand, Europe and other countries all over the world. Similarly, all three-phase induction motors from 0.73 to 185 kW must meet minimum energy performance and “high efficiency” standards. Silicon steel is currently used in electric motors. In this steel, the core losses cannot be reduced. Superconductors can reduce copper losses increasing efficiency of electric motors. However, superconductors that can work generally in low temperatures are not suitable in wide range applications, especially for small electric motors. Therefore, efficiency of electric motors and generators (especially in the range of fractional horsepower and small motors) may rather be increased by using new and more effective magnetic materials. For decades the Electrical and Electronic industries have taken advantage of high performance amorphous metal materials in the design and manufacture of components and products. Amorphous metals are man-made ribbons of powered metal that are generally impregnated into glass substrates and due to their purity and molecular consistency, they offer far lower energy losses in magnetic field and electron flow physics. They are widely used in transformers and other electronic components. Because electric motors represent the single largest application for the consumption of electric power, it has long been hoped that some day, these materials could be used in the manufacture of electric motors (Jordan and Woods, 1984). Amorphous iron used in electric motors saturate and they cannot transport an infinite volume of the magnetic field. Therefore, efficiency of smaller motors is 60-75 per cent whilst the bigger motors achieve efficiency of up to 95 per cent (Lu and Kokernak, 2000; Mischler et al., 1981; Stec, 1995).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 625-632 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540539

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The increase is achievable by almost total reduction of core losses in the stator of the induction motors and subsequent reduction of copper losses. Therefore, even a very small motor achieves efficiency and power factor on the same level, as it is currently achievable only in big and the biggest motors. It is particularly important for the high-speed induction motors, supplied by inverters. The process of modern design of electromagnetic devices requires the use of high accuracy modelling methods such as finite element or accuracy circuit methods. The main aim of this paper will be the application of these methods in professional designs of electric induction motors with stator cores made of amorphous iron. The circuit and field-circuit analyses make it possible to compute many different specific parameters and working curves for the “high efficiency” three-phase small induction motor with stator cores made of amorphous iron. The results of this calculation will be compared with the result obtained for the classical induction motor. Amorphous iron characteristics The magnetic properties of the amorphous iron are different from that of the silicon iron. The authors have conducted investigations for the stator core made of non-oriented steel sheet V 400-50 A (DIN) with its thickness 0.5 mm and maximal core loss p1:0=50 ¼ 1:6 W=kg; and the amorphous steel METGLAS 2605SA1 (annealed) with thickness 0.0254 mm and maximal core loss of about p1:0=50 ¼ 0:44 W=kg (measured results). Figure 1 shows B/H curves for both materials and the curve of core losses for amorphous steel and frequency of 50 Hz. The saturation flux density of the amorphous iron is lower than that of the silicon steel. The right characteristic relates to annealed material Metglas produced by Honeywell. Materials’ producers are mainly focused on implementation of these materials in transformers and do publish data related to no annealed (as cast) materials suitable for electric motors. The results of measurements show that losses generated in the no annealed materials are little bit higher, however several times lower in comparison with currently used silicon steel (Stec, 1994). Description of investigated motor The examined motor was designed on the basis of the classical structure of two-pole induction motor Sg71-2B, whose output power is 0.55 kW, stator windings are delta connected, and supply voltage is 220 V for the frequency of 50 Hz. The additional data: 24 stator slot, external stator diameter Dse ¼ 106 mm; internal stator diameter Dsi ¼ 58 mm; stack length l s ¼ 62 mm; width of air-gap delta ¼ 0:25 mm:

Figure 1. Magnetisation curves for both materials and curves of core loss for amorphous steel

Owing to predefined constant and relatively small stator diameter and lower saturation density for amorphous materials and lower stacking factor of 82.5 per cent, compared with the 96 per cent for the silicon steel longer stack length for amorphous motor has been assumed. As the main goal of the comparative analysis we select (and design as well) two motors whose effective stack lengths are l su ¼ 66 mm; which give the nominal stack length l s ¼ 80 mm for amorphous stator core, and l s ¼ 68:75 mm for conventional stator. Additionally an induction motor with stator core made of silicon iron (with the same nominal stack length as for amorphous stator core of l s ¼ 80 mm) giving the effective stack length l su ¼ 76:8 mm was also investigated. The circuit and field-circuit analyses have been done for 2D structure of the motor and for all its constructions with frequency of 50 Hz. Distribution of the magnetic field in the motor Magnetic field distributions have been calculated for two selected motor structures with stator core made of silicon and amorphous iron, and with the same effective stack lengths (Vector fields, 2004). The exemplary flux density distributions are shown in Figure 2. The radial flux density distributions in the stator tooth and the stator yoke with stator core made of amorphous iron are shown in Figure 3. In Figure 4, distributions of the flux density in the stator tooth versus angle, and selected two values of the radius: r¼ 35 and 29.1 mm are shown. These curves correspond to the average value of flux density in the stator tooth and the minimum value of the flux density in the stator tooth, respectively. The maximum value of the flux density in the stator tooth, calculated by means of circuit method, for the given motor is Bsd ¼ 1:559 T: In Figure 5, the flux density distributions in the stator yoke versus angle (for two values of the radius: the average radius of the stator yoke r ¼ 45 mm; and the minimal radius of the stator yoke r ¼ 41 mm) are shown. The maximal value of the flux density in the stator yoke in the described motor, calculated using circuit method, is Bsy ¼ 1:363 T: The “typical-classical” air-gap flux density distribution in this motor is shown in Figure 6. Figure 6 also shows a distribution of the flux density in the rotor yoke versus angle, while the radius is selected as r ¼ 16 mm: Similar calculations have been carried out for the motor with stator core made of silicon iron with the same effective stack lengths (l su ¼ const), and the same nominal stack length (l s ¼ const). The results of comparative analysis are presented in Table I. Owing to the comparative study, by use of both the circuit method and FEM, performed for the maximum flux density of each part of the core of the motor, and average value of the flux density in the air-gap we conclude that: values of the flux density in the middle of the air-gap and values of magnetising current are smaller for the motor with stator core made of amorphous iron than for the conventional one. This is consistent with the results of calculations obtained from circuit method (Table I). Generally, we conclude that values of the flux density obtained from field/circuit method are almost the same as obtained by means of classical circuit method. The values of magnetising current obtained from field/circuit method are lower than

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Figure 2. Distribution of the flux density in the induction motor with stator core made of silicon (a) and amorphous (b) iron (effective stack lengths lsu ¼ 66 mm)

those obtained from circuit method. The discrepancies between analysed values are acceptable. Moreover, the correlation between flux density values for different stator cores are the same. The working curves of the motor The circuit and field-circuit methods make possible computation of the parameters and working curves for induction motors with stator cores made of amorphous and silicon iron. Figure 7 shows the efficiency and power factor for the motors with amorphous and conventional stator cores for the same effective (l su ¼ const) and nominal (l s ¼ const) stack lengths. We obtain higher efficiency for the motor with amorphous stator core than for the motor with conventional stator. It results from the lower core losses and lower

Three-phase small induction motors 629

Figure 3. Distribution of flux density in the stator tooth (a) and the stator yoke (b) versus radial position

Figure 4. Distribution of flux density in the stator tooth versus angle for two values of the radius

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magnetising current and in effect the lower total stator current. The stator current of the analysed motors is shown in Figure 8. Table II presents the total losses in these motors. Table II shows that core losses in the motor with stator core made of amorphous iron are lower than for the conventional motor, even for frequency of 50 Hz. The results

630

Figure 5. Distribution of flux density in the stator yoke versus angle for two values of the radius

Figure 6. Distribution of flux density in the air-gap and in the rotor yoke versus angle

Table I. The maximum values of flux density in the motor core calculated by use of circuit method and FEM

Figure 7. Efficiency (a) and power factor (b) for the motors with stator cores made of amorphous and silicon iron

Stator iron Amorphous Silicon, lsu ¼ const Silicon, ls ¼ const

Bry0 (T) Brd0 (T) Bp0 (T) Bsd0 (T) Bsy0 (T) Im (A) Circuit FEM Circuit FEM Circuit FEM Circuit FEM Circuit FEM Circuit FEM 1.016 1.178 1.015

1.365 1.374 1.245

1.068 1.247 1.086

1.191 1.183 1.016

0.523 0.610 0.532

0.401 0.402 0.403

1.559 1.564 1.363

1.532 1.520 1.331

1.363 1.365 1.168

1.370 0.6893 0.571 1.365 0.8536 0.770 1.171 0.7104 0.580

Three-phase small induction motors 631 Figure 8. Stator current of motors with stator cores made of amorphous and silicon iron

Stator iron Amorphous Silicon, lsu¼const Silicon, ls¼ const

Core losses (W)

Stator winding losses (W)

Rotor winding losses (W)

Total losses (W)

2.43 20.37 17.40

67.62 66.50 68.05

49.70 41.34 48.09

143.4 151.8 158.0

of simulations show high winding losses in the amorphous motor, with the condition of constant and low stator diameter causing increasing length of the machine. It is expected that future research work will be focussed on increasing outside stator diameter and reduction of the machine length, which would give significant efficiency increasing of this motor with amorphous core. Conclusion This paper presents the results of comparative analysis of the induction motors: classical structure and new structure made from the amorphous steel stator core. The obtained results show that, even for 50 Hz, induction motors made from the amorphous steel stator core gives the better efficiency and power factor than the classical motor structure. More efficient motor, with proposed amorphous stator core, is expected for 100 Hz or higher frequency of the supply voltage. References Jordan, H.E. and Woods, E.J. (1984), “Fractional horsepower motor constructed with amorphous iron”, Proceedings of Motor-Con, April, pp. 37-42. Lu, N. and Kokernak, J.M. (2000), “Amorphous metals for radial airgap electric machines”, Proceedings of ICEM 2000, 28-30 August 2000, Espoo, Finland, pp. 618-22. Mischler, W.R., Rosenberg, G.M., Freschmann, P.C. and Tompkins, R.E. (1981), “Test results on a low loss amorphous iron induction motor”, IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100 No. 6, pp. 2907-11. Stec, T. (1994), “Materials metglas 2605S-2 and 2605TCA in application to electric rotating machines”, International Conference on Modern Electrical Drivers, January 1994, NATO Advanced Institute, Turkey.

Table II. Total losses in motors with amorphous and conventional stator cores

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Stec, T. (1995), “Electric motors from amorphous magnetic materials”, Proceedings of International Symposium on Non Linear Electromagnetic Systems, 17-20 September 1995, The University of Cardiff. Vector Fields (2004), PC OPERA-2D – version 1.5, Software for electromagnetic design. Further reading Wiak, S., Kome˛za, K. and Dems, M. (1998), “Electromagnetic field and parameters modeling of induction motors by means of FEM”, Proceedings 32 Spring. International Conference MOSIS’98, 5-7 May, Ostrava, Czech Republic, Vol. 3, pp. 275-81.

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Optimal shape design of a high-voltage test arrangement

Optimal shape design

P. Di Barba and R. Galdi Department of Electrical Engineering, University of Pavia, Pavia, Italy

633

U. Piovan Weidmann Transformerboard Systems AG, Rapperswil, Switzerland

A. Savini Department of Electrical Engineering, University of Pavia, Pavia, Italy

G. Consogno Weidmann Transformerboard Systems AG, Rapperswil, Switzerland Keywords Finite element analysis, Electrostatics, High voltage Abstract Discusses the automated shape design of the electrodes supplying an arrangement for high-voltage test. Obtains results that are feasible for industrial applications by means of an optimisation algorithm able to process discrete-valued design variables.

1. Introduction Safe and long-lasting operation of transformer insulation systems strongly depends on the quality of components like insulating barriers and angle rings; they are characterized by materials with different dielectric properties, arranged in complicate geometries and subject to electric field ( Moser et al., 1979). In this frame, the adoption of effective techniques of automated optimal design makes the virtual prototyping of components an attractive alternative to time-consuming trial-and-error design cycles ( Bramanti et al., 2002). 2. The device The problem considered here is to design an arrangement to test the dielectric surface strength of cylindrical pressboard samples when subject to applied AC stress in the order of 2-6 kV/mm. The proposed test arrangement exhibits two pairs of toroidal electrodes immersed in oil and coaxially located with respect to the sample under test as shown in Figure 1. To prevent discharge along field lines located in the oil bulk, two barriers have been placed in the axial direction between electrodes. 3. Design problem The shape of the electrode cross-section is to be synthesised in order to obtain a trapezoidal distribution of tangential electric field along the cylindrical surface. Field distribution should originate the inception of discharge path just along the surface of transformerboard sample. A vector of nine design variables (A1, A2, H1, R2, R3, R4, R8, W1, W2) defines the geometry of the electrode pair as pointed out in Figure 2; R7 is a constant parameter while B1 is a dependent variable. In particular, each design variable is characterized by a discrete variation, according to prescribed steps, to take into account technological bounds connected with fabrication of components.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 633-638 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410543482

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Figure 1. Test arrangement

Figure 2. Cross-section of electrode pair with design variables

In order that a controlled discharge takes place along the sample surface, a trapezoidal profile of tangential field should be synthesised, flat in the middle and sharply decreasing to zero near electrodes. The controlled region is the external surface of transformerboard sample. The objective function can then be represented by the average deviation between actual and prescribed field profile of tangential field along the sample. The shaded area in Figure 3 gives the geometrical interpretation of the objective function; the field deviation is computed whenever the field profile gets out of the tolerance shown by dotted lines. The finite-element simulation of electric field is based on triangular elements with quadratic approximation of potential. The grid is particularly dense in the so-called triple point of test arrangement, namely the subregion where transformerboard, oil and electrode are close to each other; a detail is shown in Figure 4.

Optimal shape design

635

4. Optimisation method An evolutionary strategy was adopted for optimisation; it is worth recalling that an evolutionary strategy is based on the random walk of an individual, representing the current solution, through the design space. In the authors experience, evolutionary algorithms have proven to be successful when several design variables are managed and no information about convexity and smoothness of the objective function is given a priori (Neittaanmaki et al., 1996). At each iteration, the current individual generates a new one, according to a modification probability function which is updated during the optimisation. To this aim, a normally distributed displacement, with variable standard deviation, is added to the coordinates of the current individual in the design space. The deviation-updating

Figure 3. Prescribed tangential field and objective function

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Figure 4. Finite-element mesh near the triple-point of test arrangement

criterion takes into account the history of the individual itself, trying to better approximate the found solution without neglecting the possible existence of local minima. Owing to the technological processes involved, the design variables are discrete in variation, i.e. only a discrete set of values is feasible for them. For these reasons, a direct discrete approach was chosen, by setting up a Gaussian discrete sample generator. The algorithm is based on the generation formula xi ¼ mi þ ki Dxi

ð1Þ

where x and m are new and current value of ith variable, respectively, Dx is the associated step of variation and k is a random integer number with uniform probability density. In other words, kDx represents the discrete displacement allowed by technological limitations that should be added to the coordinates of the current individual to obtain the new one. 5. Optimisation results Two different optimisations have been run, starting from the same initial geometry: first, design variables have been processed as continuous-valued variables. After convergence, the objective function has been evaluated in two ways: either considering results as real numbers or rounded a posteriori to the nearest discrete value, respectively. Next, the optimiser has been restarted, this time processing the design variables as discrete-valued variables, each of which has its own step within the corresponding range of variation. From Table I, it can be noted that the improvement of the objective function depends on the arithmetic representation of design variables in a remarkable way. In particular, results from continuous-valued optimisation appear to be the best ones in principle; however, if they would be accepted, an infinitely-precise technology should be assumed against any reasonable evidence. Only results stemming from discrete-valued optimisation can be considered realistic and compatible with technological bounds; moreover, the improvement of the objective

Discrete optimization results Design variables A1 (mm) A2 (mm) H1 (mm) R2 (mm) R3 (mm) R4 (mm) R8 (mm) W1 (deg) W2 (deg) Relative improvement (per cent)

Starting point

Continuous optimisation results

Valuea

Stepa

Valueb

Stepb

Rounded results of continuous optimisationc

10.000 7.000 20.000 2.000 5.000 0.200 5.000 7.500 2.000

13.770 5.971 19.533 1.816 7.841 0.157 3.533 22.761 20.579

14.000 3.000 20.000 1.500 7.500 0.140 11.000 2.000 2 2.100

1.000 14.000 0.500 3.000 Const. 20.000 0.500 1.400 0.500 5.500 0.010 0.140 1.000 5.000 0.500 2 2.500 0.100 2 2.700

1.000 0.500 Const. 0.200 0.500 0.010 1.000 0.500 0.100

14.000 6.000 20.000 2.000 8.000 0.160 4.000 2 3.000 2 0.600

0

95.6

83

88.1

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637

70.1

Notes: aR2 interval of variation has eight subdivisions; bR2 interval of variation has 20 subdivisions; c rounding is made according to the nearest step (the step of R2 was 0.5 mm)

Table I. Optimisation results

function corresponding to discrete-valued optimisation is better than the improvement obtainable by rounding off continuous-valued results. Finally, the optimal shape of electrodes corresponding to discrete-valued optimisation is shown in Figure 5 along with equipotential lines. In order to compare the rate of convergence, the history of the objective function is shown in Figure 6 for both discrete and continuous cases, respectively. Data refer to Table I (in particular, variable R2 varies in a range characterised by eight subdivisions; objective function is expressed in arbitrary units). It can be noted that the overall computational cost is higher in the case of continuous-valued variables; the search tolerance was the same in both cases.

Figure 5. Field map of optimal geometry

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Figure 6. Objective function history: discrete case vs continuous case

6. Conclusion In this paper, the optimal shape design of a test arrangement for high-voltage operation has been developed. A routine generating discrete samples with normal distribution has been implemented, governing the movement of discontinuous variables in the design space. The case study is an example of virtual prototyping that gives rise to a realistic design. References Bramanti, A., Di Barba, P., Piovan, U. and Savini, A. (2002), “Design optimisation of transformer insulation”, in Krawczyk, A. and Wiak, S. (Eds), Electromagnetic Fields in Electrical Engineering – ISEF’01, IOS Press, Amsterdam, pp. 170-3. Moser, H.P. et al.(1979), “Transformerboard”, Scientia Electrica. Neittaanmaki, P., Rudnicki, M. and Savini, A. (1996), Inverse Problem and Optimal Design in Electricity and Magnetism, Oxford Science Publication, Oxford University Press, Oxford. Further reading Infolytica Corporation (1998-1999), ElecNet Version 6.7, Getting Started Guide, available at: www.infolytica.com/

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Cogging torque calculation considering magnetic anisotropy for permanent magnet synchronous motors

Cogging torque calculation

639

Shinichi Yamaguchi and Akihiro Daikoku Advanced Technology R&D Center, Mitsubishi Electric Corporation, Amagasaki, Hyogo, Japan

Norio Takahashi Department of Electrical and Electronic Engineering, Faculty of Engineering, Okayama University, Tsushima, Okayama, Japan Keywords Magnetic fields, Torque, Laminates Abstract This paper describes the cogging torque of the permanent magnet synchronous (PM) motors due to the magnetic anisotropy of motor core. The cogging torque due to the magnetic anisotropy is calculated by the finite element method using two kinds of modeling methods: one is the 2D magnetization property method, and the other is the conventional method. As a result, the PM motors with parallel laminated core show different cogging torque waveform from the PM motors with the rotational laminated core due to the influence of the magnetic anisotropy. The amplitudes of the cogging torque are different depending on the modeling methods in the region of high flux density.

Introduction Permanent magnet synchronous (PM) motors, which are widely used in industry or automotive applications etc., are often required to be miniaturized, to reduce the cogging torque and the torque ripple, to improve the positioning accuracy or to reduce the noise and vibrations. To miniaturize the PM motors, the tooth-wound motors and non-oriented soft magnetic material lamination with low core loss are often used (Daikoku and Yamaguchi, 2002). On the other hand, it is well known that the soft magnetic material laminations, even the non-oriented soft magnetic material laminations used for motors, have the magnetic anisotropy. The magnetic anisotropy of the non-oriented soft magnetic material lamination with low core loss (high-grade core) is greater than that with high core loss (low-grade core). The magnetic anisotropy causes the non-uniform permeance distribution of stator core, which often produces the cogging torque, in a certain kind of motor design. Especially on the tooth-wound motors, low order components of cogging torque are produced due to the magnetic anisotropy, because the number of poles is close to the number of stator slots. So it is very important to consider the magnetic anisotropy for designing miniaturized and low cogging torque motors, but the report concerning this is few. In this paper, we calculate and discuss about the cogging torque due to the magnetic anisotropy, using the 2D magnetization property method (Fujiwara et al., 2002). Also the results using the 2D magnetization property method are compared with those using the conventional method.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 639-646 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540548

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Analysis model and analysis methods Analysis model The magnetic characteristic in the transverse direction of the non-oriented soft magnetic material lamination is inferior to that in the rolling direction. When the motor is manufactured using a circle-like core and laminated for the rolling direction in parallel, the permeance oscillates twice per one revolution of rotor due to the magnetic anisotropy. It is expected that the influence of the magnetic anisotropy on cogging torque becomes large, if the difference of the number of poles and the number of stator slots is two. As an example that is easy to be influenced by the magnetic anisotropy, we examine the PM motor having ten poles and 12 stator slots. The analysis model is shown in Figure 1 and the specifications are shown in Table I. We calculate the cogging torque of the motor in the two cases: one is the case considering the magnetic anisotropy, which corresponds to the parallel laminated core in the stator, and the other is the case neglecting the magnetic anisotropy, which corresponds to the rotational laminated core in the stator. When the cogging torque oscillates n times per one rotation of the rotor, we call this “(n/p)th-order component of the cogging torque”, where p is the number of pole pairs. If the PM motor is manufactured ideally without the magnetic anisotropy, etc., then the 12th(¼ 60/5)-order component is the lowest one of cogging torque, which corresponds to the least common multiple (LCM) of the number of poles and the number of stator slots. However, when the motor is manufactured using a circle-like core and

Figure 1. Analysis model

Table I. Specifications of analysis model

Number of poles Number of stator slots Material of rotor yoke Material of stator core Rotor outer diameter Stator outer diameter Core length Air gap Residual induction of permanent magnet

10 12 S45C ( JIS C 4051) 50A290, 50A1300 ( JIS C 2552) 35 mm 80 mm 39 mm 0.6 mm 0.4-1.4 T

laminated for the rolling direction in parallel, the second(¼ 10/5)-order component of the cogging torque is produced by the asymmetry of the stator core. Two kinds of materials of stator core are examined: one is 50A290(JIS C 2552) belonging to the category with the lowest iron loss (high-grade core), and the other is 50A1300 belonging to the low-grade core. Also, we investigate the influence of the residual induction of the permanent magnet on the cogging torque. Analysis methods The cogging torque of the motor is calculated by the 2D finite element method (FEM). Table II shows the discretization data of FEM. To investigate the modeling methods for the magnetic anisotropy, we used two kinds of methods: one is the 2D magnetization property method, and the other is the conventional method. In the anisotropic materials, the direction of H vector is different from that of B vector, as shown in Figure 2. The conventional method uses the magnetic characteristics in only rolling and transverse directions. The permeability tensor mconv of the conventional method is expressed as follows. 2

3 Bx " # 0 H x
u ¼u ¼08 mx_conv 0 6 7 H B 6 7
mconv ¼ 4 ð1Þ 5¼ By
my_conv 0 0 H y


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uH ¼uB ¼908

It is reported that the magnetic easy axis of the conventional method is different from the actual magnetic easy axis, which coincides with the rolling direction (Nakata et al., 1994). This is because the components mx_conv and my_conv of the conventional method are independent of each other. The 2D magnetization property model is proposed in order to avoid the inconvenience of the conventional method. The permeability tensor m2D mag of 2D magnetization property method is expressed as follows (Fujiwara et al., 2002). Here, uHB is the angle between H and B vectors as shown in Figure 2. Unknown variables Number of elements Number of nodes Number of elements in the air gap Order of elements Calculation model

14,137 14,454 14,830 300 (Rotational direction) £ 6 (Radial direction) First order quadrate elements 1/2

Table II. Discretization data of FEM

Figure 2. Relationship between B and H in anisotropic material

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2

m2D mag

B 6 H cos uHB 6 ¼6 4 B 2 sin uHB H "

642 ¼

3 2 B cos uB B sin uHB 7 6 H 7 6 H cos uH 7¼6 B 5 4 cos uHB 0 H

mx_2D mag

0

0

my_2D mag

#

0

3

7 7 B sin uB 7 5 H sin uH ð2Þ

From these formulas, it is clear that the components of mx and my of the conventional method are different from the components of those of the 2D magnetization property method. That is, both of mx_2D mag and my_2D mag of the 2D magnetization property method consist of (B, uB, H, uH) to consider the mutual permeability between mx and my. As opposed to this, mx_conv of the conventional method consists of (Bx, Hx) and my_conv consists of (By, Hy). In this paper, we measured the 2D magnetization property of the material to use the 2D magnetization property method. The 2D magnetization properties of 50A290 and 50A1300 are shown in Figures 3 and 4. The magnetic characteristic over 2.0 T is extrapolated, because it is difficult to measure in high flux density regions (Fujiwara et al., 2002). From these figures, it is observed that the angle uHB ð¼ uH 2 uB Þ of low-grade core (50A1300) is smaller than that of high-grade core (50A290).

Figure 3. 2D magnetization property of high-grade core (50A290)

Figure 4. 2D magnetization property of low-grade core (50A1300)

It is expected that the influence of the magnetic anisotropy is larger in the motors with high-grade core than in the motors with low-grade core. Calculation results Influence of magnetic anisotropy on cogging torque Figure 5(a) shows the calculated result of the cogging torque waveform using the 2D magnetization property method to consider the magnetic anisotropy, which corresponds to the parallel laminated core. The residual induction of the permanent magnet is 1.1 T and the grade of stator core is 50A290. Also, the calculated result neglecting the magnetic anisotropy is shown in Figure 5(b); this corresponds to the rotational laminated core. These figures show only 360 electrical degrees in consideration of symmetry. These figures indicate that the cogging torque waveform is very different due to the influence of the magnetic anisotropy. When the magnetic anisotropy is neglected (Figure 5(b)), the cogging torque oscillates 12 times per 360 electrical degrees, which is determined by the LCM of the number of poles and number of stator slots; this is the 12th-order component of cogging torque. On the other hand, when the magnetic anisotropy is considered (Figure 5(a)), the second-order component of cogging torque is the main component of the waveform. The amplitude of the cogging torque considering the magnetic anisotropy is about 40 times larger than the amplitude of the cogging torque neglecting the magnetic anisotropy.

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Difference arising from the modeling method The calculated results of the cogging torque using the 2D magnetization property method and the conventional method are shown in Figure 6, when the grade of stator core is 50A290 and the residual induction of permanent magnet is made into a parameter to change the magnetic flux density of stator core. We performed the frequency analysis for each cogging torque waveform, and only the second-order components of the cogging torque waveform are shown in this figure. These calculated results using both models show that the cogging torque, produced due to the magnetic anisotropy, increases when the residual induction of permanent magnet becomes large.

Figure 5. Calculated results of cogging torque waveform (when the residual induction of permanent magnet is 1.1 T and the grade of stator core is 50A290)

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Figure 6. Calculated results using the 2D magnetization property method and conventional method (when the grade of stator core is 50A290)

Figure 7. Comparison of flux-line by modeling methods (when the residual induction of permanent magnet is 1.4 T and the grade of stator core is 50A290)

This is because the influence of the magnetic anisotropy becomes remarkable when the flux density in stator core is high in the case of the large residual induction of permanent magnet. Also, the amplitudes of the cogging torque calculated by both models are almost the same in low flux density region. But, the calculated result using the 2D magnetization property method differs from that using the conventional method in high flux density region. Especially, when the residual induction of the permanent magnet is 1.4 T, the amplitude using the conventional method is about half of the amplitude using the 2D magnetization property method. The flux distribution calculated by both methods are shown in Figure 7, when the residual induction of permanent magnet is 1.4 T. The flux distribution at the region enclosed with the dashed circle is considerably different in both models. The flux distribution in this area by the 2D magnetization property method is reasonable compared with the flux distribution by the conventional method.

Difference arising from the core grade Figure 8 shows the calculated results of the second-order component of the cogging torque, when the grade of the stator core is 50A1300 (low-grade core). The second-order component of cogging torque shown in Figure 8, using the low-grade core, is about half of the high-grade core shown in Figure 6, regardless of the magnitude of the residual induction of the permanent magnet. The influence of the magnetic anisotropy on the cogging torque becomes remarkable when the high-grade core is used.

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Conclusions In this paper, we examined the influence of the magnetic anisotropy on the cogging torque of the permanent magnet synchronous motor. We used the 2D magnetization property method and conventional method in FEM to consider the magnetic anisotropy. Also, we measured the 2D magnetization properties of the two materials. As an example, we examined the PM motor having ten poles and 12 stator slots; that is easy to be influenced by the magnetic anisotropy. The obtained results are as follows. (1) The second-order component of cogging torque is produced, when the PM motor has the magnetic anisotropy. The amplitude of the cogging torque considering the magnetic anisotropy is about 40 times larger than that neglecting the magnetic anisotropy. (2) The cogging torque produced by the magnetic anisotropy increases as the residual induction of the permanent magnet increases. (3) The amplitudes of the second-order component in the cogging torque are different depending on the modeling methods of the magnetic anisotropy. When the residual induction of the permanent magnet is 1.4 T, the amplitudes of the second-order component using the conventional method is about half of the amplitude of second-order component using the 2D magnetization property method. (4) The amplitude of the second-order component of cogging torque using 50A1300 is about half of that using 50A290. The influence of magnetic anisotropy on cogging torque becomes remarkable when the high-grade core is used.

Figure 8. Calculated results when the grade of stator core is 50A1300 (low-grade core)

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References Daikoku, A. and Yamaguchi, S. (2002), “Design of tooth-wound permanent magnet synchronous motors for reducing torque ripples”, ICEM2002, No. 359. Fujiwara, K., Adachi, T. and Takahashi, N. (2002), “A proposal of finite-element analysis considering two-dimensional magnetic properties”, IEEE Trans. Magn., Vol. 38 No. 2, pp. 889-92. Nakata, T., Fujiwara, K., Takahashi, N., Nakano, N. and Okamoto, N. (1994), “An improved numerical analysis of flux distributions in anisotropic material”, IEEE Trans. Magn., Vol. 30 No. 5, pp. 3395-8.

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Magnetoelastic coupling and Rayleigh damping

Magnetoelastic coupling

A. Belahcen Laboratory of Electromechanics, Helsinki University of Technology, Finland

647

Keywords Rayleigh-Ritz methods, Finite element analysis, Vibration measurement Abstract This paper presents a magnetoelastic dynamic FE model. As first approach, the effect of magnetostriction and strong coupling is not considered. The effect of Rayleigh damping factors on the vibrational behaviour of the stator core of a synchronous generator is studied using the presented model. It shows that the static approach is not accurate enough and the difference between calculations with damped and undamped cases is too important to be ignored. However, the difference between damped cases with reasonable damping is not very important.

Introduction The problem of acoustic noise in electrical machines seems to be an eternal subject. In the beginning of the last century, analytical methods were developed to analyse and predict the noise of electrical machines. Later, in the same century, the development of computers and numerical methods allowed for more accurate analysis of the phenomenon. At this point more complicated magnetic phenomena such as saturation has been taken into account. The specific topology and construction of the machines also could be accounted for. Nowadays, with powerful computers, both elastic and the magnetic problems are solved numerically. Moreover, these two phenomena are coupled and need to be solved simultaneously to take into account the effects of one model on the other. Magnetostriction and the effect of mechanical stress on the magnetic properties of the material are the most important aspect of coupling between magnetic and elastic fields. Different models of magnetoelastic coupling were presented. Ren et al. (1995) presented a model that takes into account the effect of stress on both magnetisation and magnetostriction. In this model, the magnetic and magnetostrictive nodal forces are calculated directly from the magnetic energy by means of the local Jacobian derivative. However, in the presented calculations, the effect of stress on magnetisation could not be taken into account due to lack of data. Later, Mohammed et al. (1999) presented a similar model and used a hypothetical data derived from analytical model. Delaere (2002) presented also a model of magnetoelastic coupling in which the effect of stress on magnetisation is modelled with additional magnetising current. In these models, the mechanical system is considered static and the effect of moving FE-mesh is not considered except in the work of Ren et al. (1995). When dealing with the noise problem, either a harmonic analysis or a dynamic analysis is needed. The elastic problem can be solved with harmonic analysis, but the magnetic problem is better solved with dynamic time-stepping analysis. The coupling between these problems requires the same kind of solution procedure. Hence, the need for a coupled dynamic time-stepping magnetoelastic model. The elastic problem needs knowledge of material properties that depend on the structure and construction of the machine. One of these properties is the mechanical damping. In dynamic time-domain analysis of elastic structures, the damping can be taken into account in the form

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 647-654 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540557

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of Rayleigh damping. The effect of the choice of Rayleigh damping factors is the main goal of this paper. This work also develops and uses the model presented by Mohammed et al. (1999) by adding damping and inertia. In some publications (Ishibashi et al., 1998; Jang and Lieu, 1991; Neves et al., 1999), the modal damping is set to approximate values. In this work, the Rayleigh damping factors are varied and the modal damping corresponding to a given pair of Rayleigh damping factors is calculated. The goal is to show the effect of these choices on the vibrational behaviour of electrical machines. Methods The model presented here consists of the magnetic field and circuit equations and the elastic field equations as well as the nodal force equations including magnetostrictive forces. Magnetic field and circuit equations In a voltage fed electrical machine, the magnetic field equations need to be solved with the circuit equations of the magnetising windings. The set of equations needed has been presented by Arkkio (1987): 2 6 6 4

SðAkþ1 Þ ½D r
T ½KD s
T Dr

Cr

0

KD s

0

Gs

32

Akþ1

3

2

S 0 ðAk ÞAk þ ½D r
T u rk þ ½D s
T K T i sk

3

76 r 7 6 7 76 u kþ1 7 ¼ 6 7 ð1Þ D r Ak 2 C r u rk 2 G r i rk 54 5 4
s 5 s s s s s s i kþ1 KD Ak 2 H i k 2 C V kþ1 þ V k

where A is the magnetic vector potential, C, D, G and H are coupling matrices between vector potential, stator currents and voltages and rotor voltages, K the connection matrix of stator windings, i, u and V current and voltages, S the magnetic stiffness matrix, superscripts s and r stand, respectively, for stator and rotor, subscripts k and k þ 1 stand for step numbers. Elastic field The equations of the dynamic elastic field in time domain are presented by Zienkiewicz (1983): ½M þ gDtC þ bDt 2 K
U kþ1     1 2 2 2b þ g Dt K U k þ 22M þ ð1 2 2gÞDtC þ 2     1 þ b 2 g Dt 2 K U k21 þ M 2 ð1 2 gÞDtC þ 2       1 1 2 2b þ g F k þ þ b 2 g F k21 Dt 2 ¼ bF kþ1 þ 2 2

ð2Þ

where C, K and M are, respectively, damping stiffness and masse matrices, F and U are, respectively, nodal force and displacement vectors, b and g constants defining

the time integration scheme, Dt length of the time step, k 2 1; k and k þ 1 stand, respectively, for the previous present and next time steps. Equation (2) can be written in a compact form: ~ kþ1 ¼ F~ kþ1 KU

ð3Þ

Matrices K and M can be calculated from the finite element mesh and the material properties such as Poisson ratio, Young’s modulus and specific masse. Calculation of the damping matrix C requires the knowledge of a damping factor, which cannot be determined. The usual procedure is to calculate C using the so-called Rayleigh damping: C ¼ aM þ bK

ð4Þ

The factors a and b can be calculated from measured modal damping but their meaning is only approximate. The effect of the choice of these factors is studied here. Coupled model and forces Combining the equations for the magnetic field, winding circuits and elastic field and applying the Newton-Raphson iteration leads to a system of coupled magnetoelastic equations: 32 3 2 ;U kþ1 Þ DAnkþ1 Akþ1 ½D r
T ½KD s
T ››US Akþ1 SðAkþ1 ;U kþ1 Þþ ›SðAkþ1 ›A 76 7 6 76 Du rn 6 Dr Cr 0 0  kþ1 7 76 7  6 76 sn 7 ¼ r nkþ1 ð5Þ 6 76 Di kþ1 7 6 0 Gs 0 KD s 54 5 4 ~ ~ ›F~ kþ1 DU nkþ1 0 0 K2 2 ›Fkþ1 ›A

›U

where r is the residuals vector, r contains the magnetic residuals and the elastic ones. The first ones are formed from the currents, voltages and residuals of previous iterations while the second ones are formed from forces and previous residuals. The force vector is assembled from the element force vectors in a standard fashion. The element force vector is calculated from the solution of the magnetic field using the Jacobian derivative method: ›W e e F ¼2 ð6Þ ›U f¼cte where the magnetic energy is given by: Z Z We ¼ V

B

 H T · dB dV

ð7Þ

0

so that Fe ¼2

Magnetoelastic coupling

 Z Z B Z  1 ›B 2 ›detðJ Þ B ›s ›y ^ detðJ Þ þ detðJ Þ y H · dB þ B · dB dV ^ 2 ›U ›U ›U V 0 0 ›s ð8Þ

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^ the surface of the reference element and where the integration is carried out on V det(J ) stands for the determinate of the Jacobian matrix of the transformation from local to reference element. The last term in the force equation (8) stands for magnetostrictive forces, which are calculated from measured data of reluctivity as function of applied mechanical stress and magnetic flux density B.

650 Results In this part of the work, calculations were made with static and dynamic elastic approach and without strong coupling. The magnetostrictive forces are not taken into account. Different values of the damping factors a and b were used in these calculations to show the effect of damping on the vibrational behaviour of rotating electrical machines. Table I gives the used values as well as the corresponding relative modal damping at 60 Hz and 1 kHz. The modelled machine is a 3.4 MVA synchronous generator connected to a sinusoidal voltage source of 6,600 V 60 Hz. The speed and the magnetising current are kept constant. Figure 1 shows the original and the deformed geometry of the machine calculated with static approach at a given time step. The calculated forces at the same time step are shown in Figure 2. The deformations and forces calculated with dynamic approach and different damping are similar to these calculated with the static one. The stator core shrinks under the effect of magnetic forces. Figure 3 shows the calculated radial displacement of a node on the outer edge of the stator. The corresponding frequency spectrum is shown in Figure 4. The result of Figure 3 was obtained using undamped dynamic approach. Figure 5 shows a plot of the magnetic flux density at the last time step. The difference in the harmonic contents of Figure 3 compared with the same result obtained for static and dynamic approach is shown in Figure 6. The difference for undamped dynamic approach and the damped one with a ¼ 0 and b ¼ 1:1075E 2 5 is shown in Figure 7. Analysis and discussion As a first approximation, the static elastic calculation can be enough to predict the vibrations of the calculated electrical machines at low frequencies up to 200 Hz. a

Table I. Calculated cases and corresponding modal damping

0.00E + 00 4.00E + 01 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 0.00E + 00 5.00E + 00 5.00E + 00

b

Modal damping at 60 Hz

Modal damping at 1,000 Hz

0.00E + 00 0.00E + 00 1.11E2 01 1.11E2 02 1.11E2 03 1.11E2 04 1.11E2 05 1.11E2 08 1.11E2 09 1.11E2 09 1.11E2 10

0.00E + 00 5.31E2 02 2.09E + 01 2.09E + 00 2.09E2 01 2.09E2 02 2.09E2 03 2.09E2 06 2.09E2 07 6.63E2 03 6.63E2 03

0.00E + 00 3.18E2 03 3.48E + 02 3.48E + 01 3.48E + 00 3.48E2 01 3.48E2 02 3.48E2 05 3.48E2 06 4.01E2 04 3.98E2 04

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Figure 1. Deformation of the stator core

Figure 2. Magnetic nodal forces

However, these are not the frequencies of interest when dealing with the noise problem. The most annoying noise frequencies are around 1,000 Hz. For these frequencies, the relative difference between the static and dynamic approach is almost 100 per cent. The effect of different damping factors is less pronounced the relative difference varies around 10 per cent. Although the difference in calculated vibrations with different damping is quit small, its effect on the noise can be considerable. This is due to the frame of the machine that in some cases can vibrate with larger amplitudes. Hence, accurate

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Figure 3. Radial displacement of a node on the outer edge of the stator

Figure 4. Frequency contents of the radial displacement of the same node on the outer edge of the stator

calculation of the vibrations of the machine is needed to correctly predict and analyse their noise. Conclusions A magnetoelastic dynamic FE model has been presented. As a first approach, the effect of magnetostriction and strong coupling is not considered. The effect of Rayleigh

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Figure 5. Plots of the field lines in the machines

Figure 6. Differences in frequency contents between static and dynamic

damping factors on the vibrational behaviour of the stator core of a synchronous generator has been studied. It is shown that the static approach is not enough accurate. The differences between calculations of damped and undamped cases are too important to be ignored. The problem is how to choose these damping factors. However, the difference between damped cases with reasonable damping is not very important.

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Figure 7. Differences in frequency contents between undamped and damped with a ¼ 0 and b ¼ 1.1075E 2 5

References Arkkio, A. (1987), “Analysis of induction motors based on the numerical solution of the magnetic field and circuit equations”, Doctoral thesis, Helsinki, Finland, available at: http://lib.hut.fi/ Diss/198X/isbn951226076X/ Delaere, K. (2002), “Computational and experimental analysis of electrical machine vibrations caused by magnetic forces and magnetostriction” Doctoral thesis, Leuven, Belgium. Ishibashi, F., Noda, S. and Mochizuki, M. (1998), “Numerical simulation of electromagnetic vibration of small induction motor”, IEE Proceedings Electrical Power Applications, Vol. 145 No. 6, pp. 528-34. Jang, G.H. and Lieu, D.K. (1991), “The effect of magnetic geometry on electric motor vibration”, IEEE Transactions on Magnetics, Vol. 27 No. 6, pp. 5202-4. Mohammed, O., Calvert, T. and McConnell, R. (1999), “A model for magnetostriction in coupled nonlinear finite element magneto-elastic problems in electrical machines”, Proceedings of the International Conference on Electric Machines and Drives, IEMD’99, pp. 728-35. Neves, C.G., Carlson, R., Sadowski, N., Bastos, J.P.A. and Soeiro, N.S. (1999), “Forced vibrations calculation in switched reluctance motor taking into account viscous damping”, Proceedings of the International Conference on Electric Machines and Drives, IEMD’99, pp. 110-2. Ren, Z., Ionescu, B., Besbes, M. and Razek, A. (1995), “Calculation of mechanical deformation of magnetic material in electromagnetic devices”, IEEE Transactions on Magnetics, Vol. 31 No. 3, pp. 1873-6. Zienkiewicz, O.C. (1983), The Finite Element Method, McGraw-Hill, London.

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Modelling of temperature-dependent effective impedance of non-ferromagnetic massive conductor

Nonferromagnetic conductor 655

Ivo Dolezˇel and Ladislav Musil Institute of Electrical Engineering, Czech Academy of Sciences, Praha, Czech Republic

Bohusˇ Ulrych Faculty of Electrical Engineering, University of West Bohemia, Plzenˇ, Czech Republic Keywords Modelling, Numerical analysis, Inductance Abstract Impedance of long direct massive conductors carrying time-variable currents is a complex function of time. Its evolution is affected not only by the skin effect but also by the temperature rise. This paper presents a numerical method that allows one to compute the resistance and internal inductance of a non-ferromagnetic conductor of any cross-section from values of the total Joule losses and magnetic energy within the conductor, and also illustrates the theoretical analysis based on the field approach on a typical example and discusses the results.

Introduction Simulations of fast phenomena in various electrical systems (grounding devices, windings of electrical machines, etc.) often do not respect changes of the resistance and internal inductance due to skin effect. The reason consists in a widespread opinion that the increase of the resistance leads to reduction of the surge effects while its neglection provides results that are worse than the physical reality, and when the device is designed for less favourable parameters, its safety is higher. In case of steep pulses or waves, however, the effective resistance of the conductor may reach values by an order higher than the DC resistance. In such a case, neglecting the skin effect can lead to quite incorrect ideas about the voltage and current phenomena in the device. The situation (particularly in fault regimes) is also affected by the temperature rise of the current-carrying parts, which brings about an additional increase of their resistances. The complete analysis of the effect represents a coupled electromagnetic-thermal problem and the paper offers a methodology on how to cope with it. We consider a surge skin effect in a direct massive non-ferromagnetic conductor of a general cross-section placed in a linear medium from which the produced heat is transferred by convection (which is usual, for example, at various kinds of grounders, or overhead and cable lines in power distribution systems). The financial support of the Research Plan MSM 212300016 is gratefully acknowledged.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 655-661 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540566

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Formulation of the problem A sufficiently long direct massive conductor of constant cross-section is supplied from the current source by current i(t) of general time dependence. Its material is non-ferromagnetic ðm ¼ m0 Þ and temperature dependencies of its physical parameters (electrical conductivity g, thermal conductivity l, specific mass r and specific heat c) are known. The aim of the analysis is to find the time evolution of various quantities characterising the process. Of great importance are particularly the effective resistance Reff(t) and internal inductance Leff(t) of the conductor that are generally influenced by the: . time dependence of the passing current i(t), . corresponding time-dependent distribution of the current density within its cross-section, and . consequent temperature rise due to the Joule losses in the conductor. The task is handled as a quasi or hard-coupled problem. This means that all required quantities are calculated simultaneously. Mathematical model The continuous mathematical model of the problem consists of two partial differential equations describing the: (1) non-stationary electromagnetic field expressed by distribution of the vector potential A, and (2) non-stationary temperature field in the conductor produced by the Joule losses. The non-stationary electromagnetic field produced by current i(t) within the conductor reads (Ida, 2000) rot rot A þ m0 g

›A ¼ m0 J 0 ›t

ð1Þ

where J0 denotes the vector of the uniform current density within the conductor corresponding to the external current i(t) from the source. The term J e ¼ 2g

›A ›t

expresses the eddy current density. When accepting the Cartesian co-ordinate system according to Figure 1 (conductor of cross-section Vc with boundary Gc, Ga being the artificial boundary whose significance is dealt with later on), J0 has only one non-zero position-independent component J0z(t), Je and A also one non-zero, but position-dependent component J ez ðx; y; tÞ; and Az ðx; y; tÞ: Now, equation (1) may be rewritten as

›2 A z ›2 A z ›A z ¼ 2m0 J 0z þ 2 2 m0 g ›x 2 ›y ›t

ð2Þ

(other co-ordinate systems may be used, of course, as well). Uniqueness of the solution may be secured, for example, by imposing indirect boundary condition

Nonferromagnetic conductor 657

Figure 1. The investigated arrangement

Z
J 0z 2 g Vc

 ›A z dS ¼ iðtÞ ›t

ð3Þ

(parameter g is not constant, because it is a function of temperature T that is variable within Vc). Practically, circular artificial boundary Ga placed at a sufficient distance from the conductor is characterised by a constant value of component Az of the vector potential. Distribution of this quantity on the definition area V ¼ Va < Vc follows from equations (2) and (4). The initial condition reads Az ðx; y; 0Þ ¼ 0; because at the beginning of the process no magnetic field is supposed to affect the arrangement. 0 (t) per unit length of the conductor calculated from The effective resistance Reff distribution of the specific Joule losses is R ð J 0z þ J ez Þ2 V wJ dS 0 ; wJ ¼ : ð4Þ R eff ¼ c 2 g i 0 The effective internal inductance Leff (t) per unit length may be calculated from the magnetic field energy by the formula

! R ›A z 2 ›A z 2 þ dS Vc ›x ›y 0 Leff ¼ : ð5Þ m0 i 2 The non-stationary temperature field is described by equation (Holman, 2002) ›T divðl grad TÞ ¼ rc 2 wJ ð6Þ ›t where the temperature T ¼ Tðx; y; tÞ: In our case, this equation within the conductor may be written as

 › ›T › ›T ›T 2 wJ : l l ð7Þ þ ¼ rc ›x ›y ›x ›y ›t The boundary condition along the surface of the conductor reads 2l

›T ¼ ac ðT 2 T 0 Þ for t $ 0 ›n

ð8Þ

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where n denotes the outward normal. Radiation is (but not necessarily) neglected due to lower temperatures usually not exceeding 100-2008C. Here ac denotes the coefficient of the convective heat transfer from the conductor surface into ambient medium of temperature T0. Finally, the initial condition for the temperature reads Tðx; y; 0Þ ¼ T 0 : The algorithm of calculation of the discretised model at the (k+1)th time level on a grid with N nodes consists of the following steps. (1) Iterative computation of values Azi;kþ1 from values Azi;k ; where i ¼ 1; . . .; N by means of discretised equations (2) and (3) with respect to the dependence gi;k ¼ gðT i;k Þ: (2) Explicit determination of values T i;kþ1 from values T i;k ; where i ¼ 1; . . .; N by means of discretised equation (7) with respect to dependencies li;k ¼ lðT i;k Þ; rci;k ¼ rcðT i;k Þ: (3) Computation of R 0eff kþ1 by means of numerical approximation of equation (4). (4) Computation of L0eff kþ1 by means of numerical approximation of equation (5). For very short processes such as fast current pulses the heating process is practically adiabatic (convection and radiation can be neglected) and the temperature rise of the conductor may sufficiently accurately be calculated from the simple calorimetric equation. Computation of particular tasks was realised by the combination of professional codes PC OPERA 7.0 and Femlab 2.3 with single-purpose user programs developed and written by the authors in Matlab 6.5. Illustrative example A hollow copper water-cooled conductor (Figure 2) whose quarter is depicted in common with the discretisation mesh carries a current pulse according to Figure 3. It is necessary to determine the time evolution of its effective resistance and internal inductance per unit length ðR0eff ; L0eff Þ as well as the temperature T (its starting value T 0 ¼ 208C). The temperature dependencies of the specific electrical and thermal conductivities g ¼ gðTÞ and l ¼ lðTÞ for copper are shown in Figure 4. An analogous dependence for rc(T) may with a sufficient accuracy be expressed as a linear function in the form

Figure 2. The investigated conductor and mesh of the area

Nonferromagnetic conductor 659 Figure 3. Current pulse through the conductor

Figure 4. Temperature dependencies g ¼ g (T) and l ¼ l(T ) for copper

rcðTÞ ¼ 3; 631; 200 þ 934:5ðT 2 20Þ J=K m3 : These parameters are adjusted automatically during the calculation. Computations were realised on a grid with about 6,700 triangular elements (Figure 2), with good geometrical convergence of the results. Stability of the numerical process was secured by accepting the time step Dt # 5 £ 1026 s: Basic solution to the task took (according to the selected time step) several tens of minutes (PC 2.4 GHz). The most important results follow. Figure 5 shows the time dependence of the total Joule losses DP 0 per unit length that (to some extent) copies the current pulse in Figure 3. Its growth at the beginning of the process is, however, substantially faster due to highly expressed skin effect. This results in the time dependence of the effective 0 (Figure 6) whose maximum value is about five times higher than the resistance R eff direct-current value.

Figure 5. Time dependence of the Joule losses DP 0 per unit length

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Figure 6. Time dependence of the 0 effective resistance Reff per unit length

Figure 7. Time dependence of the effective internal 0 inductance Leff per unit length

Figure 8. Time evolution of the average temperature T of the conductor

0 An analogous shape as DP 0 exhibits the magnetic field energy per unit length W m ; the 0 corresponding time evolution of the effective internal inductance Leff per unit length is shown in Figure 7. Its minimum value is, on the other hand, about five times lower than its direct-current value. Finally, Figure 8 shows the time evolution of the temperature. As mentioned above, the process of heating is practically adiabatic, which was validated by solution and comparison of the corresponding equations. Cooling of the conductor can be observed only in several seconds.

Conclusion The presented methodology allows solving this hard-coupled electromagnetic-thermal problem with good accuracy and reasonable time of computation. Nevertheless, problems may appear with increasing steepness of the time variations of the feeding current due to the necessity to reduce the time step in order to keep the stability of computations. Next work in the field will be aimed at testing the integral model of the skin effect, extending the methodology by conductors fed from the voltage source (an ordinary differential circuit equation has to be added to the system), mutual influence of several near conductors (proximity effect), influence of ferromagnetic material, etc. References Holman, J.P. (2002), Heat Transfer, McGraw-Hill, New York, NY. Ida, N. (2000), Engineering Electromagnetics, Springer, Berlin.

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Field strength computation at edges in nonlinear magnetostatics Friedemann Groh, Wolfgang Hafla, Andre´ Buchau and Wolfgang M. Rucker Institute for Theory of Electrical Engineering, University of Stuttgart, Stuttgart, Germany Keywords Magnetic fields, Integral equations, Nonlinear control systems, Vectors Abstract Magnetostatic problems including iron components can be solved by a nonlinear indirect volume integral equation. Its unknowns are scalar field sources. They are evaluated iteratively. In doing so the integral representation of fields has to be calculated. At edges singularities occur. Following a method to calculate the field strength on charged surfaces a way out is presented.

1. Introduction Magnetostatic field problems that include iron components can be solved with integral equations. The resulting nonlinear problem is tackled iteratively with a sequence of linear problems. At each linear step one uses a spatially dependent permeability function that is interpolated by a volume mesh. Then the field strengths of the preceding iteration steps are to be calculated at each node of the volume elements. At the edges of the iron component field strength singularities usually occur. Assuming that temporarily all edges are uniformly chamfered with an edge radius 1. With small values of 1 the appropriate calculated field strength H1 should approximate the initially wanted solution sufficiently. However, generating such meshes is rather laborious. Therefore, an alternative method is proposed. 2. Singularity vectors To regularise a problem means to slightly modify it to keep initially divergent values finite. The deviation from the original problem shall be controlled by a parameter 1. Edge singularities occurring with the evaluation of integral representation formulas of fields will be regularised by omitting all sources within a sphere of radius 1. Exploiting the methods (Guiggiani and Gigante, 1990; Huber et al., 1997) for computation of strongly singular integrals, which arise when calculating the field strength at a charged surface, leads to a practical procedure. First one applies the suggested regularisation method to the field H1 at nodes of a single second-order element E which carries a surface charge s Z r0 2 r H 1; E ðr 0 Þ ¼ sðrÞ dA: ð1Þ 3 jr 0 2 rj AE ðjr 0 2rj.1Þ COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 662-669 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540575

To compute this integral expression one defines polar coordinates at the reference element

j ¼ j0 þ r cos w h ¼ h0 þ r sin w: This work was partially supported by DFG, grant: RU 720/3 – 1.

ð2Þ

In Figure 1, this polar coordinate system is shown. As explained later, the tangential plane is introduced to define a kind of first-order approximation to the complicated intersection line of the curved element surface and a sphere of radius 1. The kernel of the H1 integral representation is expanded into the Laurent series with respect to radius r later 1 sðrÞJ ðrÞ ¼ b 21 ðwÞ þ b H ðr; wÞ: r jr 0 2 rj r0 2 r

3

ð3Þ

Field strength computation

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Therein bH is the nonsingular part that contains only positive powers of r. The Jacobian of the element parameterisation is denoted by J. The coefficient b2 1 is b 21 ðwÞ ¼

aðwÞ jaðwÞj

3

sðr 0 ÞJ ðr 0 Þ:

ð4Þ

It includes the first-order approximation of the distance vector between evaluation and source points ð5Þ r 0 2 r ¼ raðwÞ þ Oðr 2 Þ: A proper parameterisation of the sector where we do not integrate the surface charge is essential. For this purpose the curved element surface is substituted with its tangential plane at the point r0. The intersection of the tangential plane with a sphere of radius 1 around the center r0 is just a circle. It has to be parameterised by using the polar coordinates defined at the reference element (Figure 1). r 1 ðwÞ ¼ r 0 þ r1 ðwÞaðwÞ:

ð6Þ

Figure 1. Polar coordinates at the reference element. Transformation to the curved element surface. The tangential plane at r0 and its intersection with a sphere of radius 1

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Since the distance from r 1 ðwÞ to r0 is equal to 1 the angle dependent radius r1 ðwÞ must satisfy ð7Þ 1 ¼ r1 ðwÞjaðwÞj: Thus, the domain of integration is transformed to the polar coordinates at the reference element (Figure 2). Integration of the representation formula Z b Z rc ðwÞ 1 b 21 ðwÞ þ b H ðr; wÞ dr dw ð8Þ H 1;E ðr 0 Þ ¼ a r 1 ðw Þ r yields a sum of a log 1 dependent term and another one which has been described by Guiggiani and Gigante (1990) and Huber et al. (1997). As 1 tends to 0 the log 1 dependent term is the only divergent one. H 1;E ðr 0 Þ ¼ 2sðr 0 Þlog 1 S E ðr 0 Þ þ H E ðr 0 Þ ð9Þ Hence it is reasonable to call S E ðr 0 Þ the singularity vector of the considered element at the point r0. With the notation of Figures 1 and 2, Z b aðwÞ J ðr 0 Þ dw: ð10Þ S E ðr 0 Þ ¼ 3 a jaðwÞj Z b Z b Z rc ðwÞ H E ðr 0 Þ ¼ logðjaðwÞjrc ðwÞÞb 21 ðwÞ dw þ dw b H ð r ; wÞ d r : ð11Þ a

a

0

The singularity vector turns out to be a parameterisation invariant expression. It depends only on unit vectors tangential to the element boundary line as shown in Figure 3. ð12Þ S E ðrÞ ¼ n £ ðe þ 2 e 2 Þ n ¼ e 2 £ e þ

Figure 2. The domain of integration transformed to polar coordinates

Figure 3. Edge singularity vector

To obtain the H1 field at a junction with m joining elements one has to sum up each element contribution. Consequently, all singularity vectors joining a node have to be added to define the total singularity vector there. The case of a three element junction results in (Figure 4): Sðr 0 Þ ¼ n 1 £ ðe 1 2 e 3 Þ þ n 2 £ ðe 2 2 e 1 Þ þ n 3 £ ðe 3 2 e 2 Þ:

Field strength computation

ð13Þ

It is worth, mentioning the case when all normal vectors nj are equal and the singularity vector vanishes. However, in discretising smooth surfaces with C 0 elements small unmeant edges at element junctions cannot be avoided. To overcome this difficulty one leaves out the singularity vector as soon as its length falls below a sensible fixed treshold.

665

3. Numerical results To calculate the regularised field strength at edges a small ball of material was left out. Therefore, the resulting field vector is attached to a point outside the material border. Since the inside field strengths are to be calculated, it is recommended to use the singular log 1-dependent part in equation (9) with the opposite sign. Instead of log 1; a constant L is introduced. In the following its values are chosen between zero and one. The regularised field strength at a m-element junction becomes m X H E j ðr 0 Þ: ð14Þ H 1 ðr 0 Þ ¼ Lsðr 0 ÞSðr 0 Þ þ j¼1

Actually, the situation is more complicated. With finite surface charges at edges, as it is always the case when using numerical methods, a saddle point of the field potential usually occurs very close to the edge (Figure 5). Thus, the field vectors change their signs not immediately at the singularity. 3.1 Singularity vector at post processing The first example is an electrostatic calculation (Figure 6). Two electrodes, both at a potential of 10 V, are placed in free space. First, an indirect integral equation was solved to obtain the surface charge distribution. Since the Galerkin method was applied to assemble the system matrix no singularity vectors were needed. They come in when the field strength on the charged surfaces is to be calculated. This might happen when using the collocation method. Here the electric field strength is evaluated simply to demonstrate that the singularity vector concept works.

Figure 4. Singularity vector of node at three-element junction

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666 Figure 5. Saddle point of the field strength potential is very close to the edge singularity. It usually occurs if the surface charge keeps finite at the edge

Figure 6. Result of an electrostatic calculation: two conductors with potential 10 V. The electric field strengths on the front surfaces of both electrodes are plotted

3.2 Influence on a nonlinear calculation The material law that relates the magnetic field strength H inside iron to the flux density B leads to nonlinear magnetostatic equations. Such problems can be solved by using an indirect integral equation whose unknowns are scalar field sources (Hafla et al., 2002). To evaluate them we proceed iteratively. At each iteration step one solves a linear equation which contains a spatially dependent permeability. Interpolating this function on a volume mesh requires to calculate a field strength at the element nodes. Therefore, the field strength integral formula is evaluated with

respect to the sources calculated at the preceding iteration step. This is where the singularity vector method comes in. The calculations presented subsequently use the measured B(H) characteristic of an AlNiCo-steel alloy (Figure 7). To demonstrate the influence of the singularity vectors on a nonlinear calculation, results with different regularisation constants, L, are compared. A simple iteration scheme was applied. The iron core shown in Figure 8 was exited by the magnetic field of cylindrical coil around the drawn in evaluation point axis. This excitation field had maximum field strength values of 5,000 A/m. The coil current Jc was homogeneous. Figures 9 and 10 at last show the dependence of the calculated solution from the regularisation constant L. Roughly speaking, one can say that the treatment of edge singularities affect a solution calculated by a simple iteration scheme at least up to 1 percent (Figure 10).

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Figure 7. B(H) is the characteristic of an AlNiCo-steel alloy

Figure 8. Half of the iron core of a contactor. The magnetic field strength will be calculated along the drawn in evaluation point axis

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Figure 9. Magnetic field strength z-component along the evaluation axis inside the iron core (Figure 8) for different regularisation constants L

Figure 10. Deviation of the magnetic field strength z-component evaluated with two nonzero regularisation constants referred to a calculation without singularity vectors

Figure 11. The component Hz of the magnetic field strength calculated inside a iron cube by using four different meshes and an alternative method based on BEM-FEM coupling (Edyson). The z-axis passes the center of the cube

3.3 Comparison to a different method A cube having side lengths L ¼ 6:25 m was immersed in the homogeneous field H c ¼ 90; 000 ez A/m (Figure 11).

4. Conclusions As the singularity vector method is easy to manage, we regard it to be a reasonable procedure whenever one has to deal with edge singularities. However, it has to prove its value in practice. References Guiggiani, M. and Gigante, P. (1990), “A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method”, ASME Journal of Applied Mechanics, Vol. 57, pp. 906-19. Huber, Ch.J., Rucker, W.M., Hoschek, R. and Richter, K.R. (1997), “A new method for the numerical calculation of Cauchy principal value integrals in BEM applied to electromagnetics”, IEEE Transactions on Magnetics, Vol. 33 No. 2, pp. 1386-9. Hafla, W., Groh, F., Buchau, A. and Rucker, W.M. (2002), “Magnetostatic field computations by a integral equation method using a difference field concept and the fast multipole method”, Proceedings of the 10th IGTE Symposium on Numerical Field Calculation in Electrical Engineering, pp. 262-6.

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Genetic algorithm coupled with FEM-3D for metrological optimal design of combined current-voltage instrument transformer Marija Cundeva, Ljupco Arsov and Goga Cvetkovski Ss. Cyril and Methodius University, Faculty of Electrical Engineering, Skopje, R. Macedonia Keywords Transformers, Genetic algorithms, Magnetic fields Abstract The combined current-voltage instrument transformer (CCVIT) is a complex non-linear electromagnetic system with increased voltage, current and phase displacement errors. Genetic algorithm (GA) coupled with finite element method (FEM-3D) is applied for CCVIT optimal design. The optimal design objective function is the metrological parameters minimum. The magnetic field analysis made by FEM-3D enables exact estimation of the four CCVIT windings leakage reactances. The initial CCVIT design is made according to analytical transformer theory. The FEM-3D results are a basis for the further GA optimal design. Compares the initial and GA optimal output CCVIT parameters. The GA coupled with FEM-3D derives metrologically positive design results, which leads to higher CCVIT accuracy class.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 670-676 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540584

Introduction The instrument transformers are one of the main components of the power measurement systems. Therefore their metrological characteristics are of high importance and should comply with the IEC standards specifications (IEC 60044-3, 1980). The 20 pffiffikV ffi combined pffiffiffi current-voltage instrument transformer (VT ratio : 20; 000 V= 3 : 100 V= 3 and CT ratio : 100 A : 5 A) is a non-linear electromagnetic system consisting of two tape-wounded ring magnetic measurement cores with mutual magnetic influence and two electrical winding systems which increase the voltage, current and phase displacement errors (Cundeva and Arsov, 2002). In order to achieve metrologically optimal design of the combined current-voltage instrument transformer (CCVIT) the magnetic field distribution has to be determined. The analytical theory can be applied on simple electromagnetic structures. However, in the case of the CCVIT the numerical methods are indispensible (Lesniewska and Chojnacki, 2002). The magnetic field analysis in the three-dimensional domain is achieved by using the finite element method (FEM) and the original program package FEM-3D (Cundev et al., 2000). The genetic algorithm (GA) is a powerful optimisation tool convenient for practical optimal design application of complex electromagnetic devices (Cvetkovski et al., 1997). In this paper, the GA coupled with the FEM-3D results is implemented in the procedure of optimal design of CCVIT. The objective function in the process of optimal design of this complex system (Figure 1) is the minimum of the metrological parameters.

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Figure 1. CCVIT electromagnetic system

FEM-3D magnetic field analysis of CCVIT The CCVIT is a closed and bounded system and the magnetic field distribution is described by the system of Maxwell equations. After the introduction of magnetic ~ as an auxiliary quantity the magnetic field distribution is expressed vector potential A by the Poisson non-linear partial differential equation as follows: ! ! ! ~ ~ ~ › › › › › › A A A ~ ~ ~ ~vðBÞ ~vðBÞ ~vðBÞ þ þ ¼ 2~jðx; y; zÞ ð1Þ ›x ›y ›z ›x ›y ›z ~ is the magnetic flux density, ~v is the magnetic reluctivity and ~jðx; y; zÞ the where B volume current density. Equation (1) can be solved by numerical methods only. The magnetic field analysis is done by an original and universal program package FEM-3D developed at the Faculty of Electrical Engineering, Skopje (Cundev et al., 2000). The magnetic anisotropy, the different reluctivities along the co-ordinate axis and the lamination of the magnetic cores are considered. The complex FEM-3D analysis is made for various regimes of input voltage Uv of the VT core and current values Ic of the CT core (from plug out regime to 120 per cent of the rated values Uvn and Icn) at rated loads of both measurement cores (S nv ¼ 50 VA for the VT and S nc ¼ 15 VA for the CT and cos w ¼ 0:8 for both cores). The magnetic field distribution is derived by non-linear iterative calculation. The main flux and the leakage fluxes via the input VT voltage and the input CT current are derived by FEM-3D enabling exact calculation of the leakage reactances of the four windings of the CCVIT. The CCVIT winding leakage fluxes are given in Tables I-IV. Ic/Icn Uv/Uvn 0.2 0.4 0.6 0.8 1.0 1.2

only VT

0.0

0.2

0.146 0.293 0.435 0.581 0.728 0.875

0.145 0.293 0.435 0.581 0.728 0.875

0.133 0.275 0.418 0.564 0.710 0.857

0.4 0.6 Csv1 [V s] 0.110 0.257 0.400 0.546 0.701 0.840

0.093 0.240 0.382 0.529 0.675 0.822

0.8

1.0

1.2

0.075 0.222 0.364 0.511 0.657 0.804

0.057 0.205 0.347 0.493 0.640 0.787

0.040 0.187 0.330 0.476 0.623 0.770

Table I. FEM-3D calculated leakage fluxes of the VT primary winding with 24,000 turns

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CCVIT optimal design by using GA The initial design of CCVIT is made according to the analytical transformer theory. The FEM-3D results are the basis for further GA optimal design. The specifications and restrictions of the instrument transformer standards (IEC 60044-3, 1980) must be implemented into the objective function fopt. The original and universal GA optimisation program developed at the Faculty of Electrical Engineering (Cvetkovski et al., 1997) performs maximisation of the objective function. The objective function in the optimization process is the minimum of the mutually coupled voltage error pv and the current error pc, as well as the minimum of the VT voltage error pv0.25 by 25 per cent of the VT rated input power Snv and the minimal difference of the VT voltage error at 25 per cent of the rated input power and by rated input power, as requested in the IEC 60044-3 (1980). Therefore, the objective function is defined as follows: f opt ¼

Ic/Icn Uv/Uvn Table II. FEM-3D calculated leakage fluxes of the VT secondary winding with 120 turns

0.2 0.4 0.6 0.8 1.0 1.2

Uv/Uvn Ic/Icn Table III. FEM-3D calculated leakage fluxes of the CT primary winding with six turns

0.2 0.4 0.6 0.8 1.0 1.2

Uv/Uvn Ic/Icn Table IV. FEM-3D calculated leakage fluxes of the CT secondary winding with 120 turns

0.2 0.4 0.6 0.8 1.0 1.2

1 1 1 1 þ þ þ 1 þ jpc j 1 þ jpv j 1 þ jpv0:25 j 1 þ jpv þ pv0:25 j

only VT

0.0

0.2

0.505 1.016 1.510 2.018 2.525 3.037

0.503 1.015 1.510 2.017 2.526 3.036

0.460 0.954 1.449 1.956 2.464 2.975

only CT

0.0

0.2

64.4 128.9 193.3 257.8 322.2 385.0

63.6 127.3 190.9 254.6 318.1 380.5

63.0 126.6 190.3 254.0 317.5 379.6

only CT

0.0

0.2

0.409 0.817 1.226 1.635 2.044 2.443

0.404 0.808 1.211 1.615 2.018 2.414

0.400 0.803 1.207 1.612 2.014 2.408

0.4 0.6 Csv2 [mV s] 0.383 0.893 1.387 1.896 2.431 2.914

0.321 0.832 1.326 1.834 2.342 2.853

0.4 0.6 Csc1 [mV s] 62.4 126.1 189.7 253.4 316.9 379.4

61.7 125.4 189.0 252.8 316.2 378.5

0.4 0.6 Csc2 [mV s] 0.396 0.800 1.203 1.608 2.010 2.407

0.391 0.796 1.199 1.603 2.006 2.401

ð2Þ

0.8

1.0

1.2

0.260 0.770 1.265 1.773 2.280 2.792

0.199 0.710 1.204 1.712 2.220 2.730

0.139 0.651 1.145 1.653 2.161 2.672

0.8

1.0

1.2

61.2 124.8 188.4 252.2 315.5 377.6

60.5 124.2 187.8 251.6 315.0 377.4

60.0 123.6 187.2 251.0 314.4 376.7

0.8

1.0

0.388 0.792 1.195 1.600 2.002 2.395

0.384 0.788 1.192 1.596 1.998 2.394

1.2 0.381 0.784 1.188 1.592 1.995 2.390

The rated regime for both measurement cores has been selected for metrological optimal design of the CCVIT. In the mathematical model of the CCVIT all quantities which affect the objective function were made to be dependent on the 11 optimisation variables: number of turns of the primary VT and secondary CT winding, current densities in all four coils and geometrical parameters of both magnetic cores. In Figures 2-5 the changes of the number of turns and the current densities of the primary VT and secondary CT winding throughout the GA generations are displayed in logarithmic scale. The objective function changes throughout the GA generations are shown in Figure 6.

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Figure 2. Changes throughout the GA generations of the VT primary winding number of turns

Figure 3. Changes throughout the GA generations of the VT primary winding current density

Figure 4. Changes throughout the GA generations of the CT secondary winding number of turns

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The results derived by the GA optimisation process must be adjusted to practical values for further realisation of the instrument transformer prototype (optimised design model). The comparison of the initial and GA optimal output and the optimised parameters of CCVIT is given in Table V. The designed combined current-voltage instrument transformer from the initial accuracy class 3 of the voltage transformation core and accuracy class 1 of the current transformation core has been optimized to achieve higher accuracy class: 1 of the VT core and 0.5 of the CT core. In Figures 7-10, the CCVIT metrological characteristics of the three CCVIT designs (initial, optimal and optimised) are compared.

Figure 5. Changes throughout the GA generations of the CT secondary winding current density

Figure 6. Changes throughout the GA generations of the objective function

Table V. Initial, optimal and optimized transformer metrological parameters

Parameter

Initial

Optimal

Voltage error at 0.25 Snv (per cent) Voltage error (per cent) (at rated regime for both cores) Current error (per cent) (at rated regime for both cores) Voltage phase displacement error (min) Current phase displacement error (min)

2 0.752

0.775

0.771

20.775

2 0.786

2 2.42 2 0.68 2 74.23 0.57

20.0000135 272.574 2.593

Optimized

0.3154 2 72.402 1.097

Genetic algorithm

675 Figure 7. VT voltage error via the VT load at rated power factor cos f ¼ 0.8

Figure 8. VT phase displacement error via the VT load at rated power factor cos f ¼ 0.8

Figure 9. CT current error via the CT load at rated power factor cos f ¼ 0.8

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676 Figure 10. CT phase displacement error via the CT load at rated power factor cos f ¼ 0.8

Conclusions The GA optimisation coupled with FEM-3D methodology presented in this paper derived positive design results from metrological aspect. The optimal design of the CCVIT is one accuracy class higher than the initial analytical design. References Cundeva, M. and Arsov, L. (2002), “Metrologically improved design of combined current-voltage instrument transformer by using FEM-3D”, 12th IMEKO TC4 International Symposium on Electrical Measurement and Instrumentation, Zagreb, Croatia, pp. 282-5. Cundev, M., Petkovska, L., Stoilkov, V. and Cvetkovski, G. (2000), “From macroelements to finite elements for 3D magnetic field analysis”, Int. Conf. Finite Element Methods for Three-Dimensional Problems FEM-3D, Jyvaskyla, Finland, pp. 5-6. Cvetkovski, G., Petkovska, L., Cundev, M. and Gair, S. (1997), “Genetic algorithms for design optimization of electrical machines”, 9th Int. Symp. ISTET’97, Palermo, Italy, pp. 381-4. IEC 60044-3 (1980), Instrument Transformers, Part 3: Combined Transformers, Geneve. Lesniewska, E. and Chojnacki, J. (2002), “Influence of the correlated localisation of cores and windings on measurement properties of current transformers”, Studies in Applied Electromagnetics and Mechanics, IOS Press, Vol. 22, pp. 236-41.

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Adaptive meshing algorithm for recognition of material cracks

Adaptive meshing algorithm

Konstanty M. Gawrylczyk and Piotr Putek Electrical Engineering Department, Technical University of Szczecin, Szczecin, Sikorskiego, Poland

677

Keywords Sensitivity analysis, Mesh generation, Optimization techniques Abstract Describes the algorithm allowing recognition of cracks and flaws placed on the surface of conducting plate. The algorithm is based on sensitivity analysis in finite elements, which determines the influence of geometrical parameters on some local quantities, used as objective function. The methods are similar to that of circuit analysis, based on differentiation of stiffness matrix. The algorithm works iteratively using gradient method. The information on the gradient of the goal function provides the sensitivity analysis. The sensitivity algorithm allows us to calculate the sensitivity versus x and y, so the nodes can be properly displaced, modeling complicated shapes of defects. The examples show that sensitivity analysis applied for recognition of cracks and flaws provides very good results, even for complicated shape of the flaw.

Introduction Using finite element method for the aim of surface shape recognition requires very exact discretization of area under consideration. The most effective is flexible discretization, which adapts to shape of identified area. The solution of inverse problem consisting in recognition of the conductivity distribution was shown by Gawrylczyk (1998, 2002). This paper proposes the method for sensitivity analysis of the magnetic vector potential in selected measurement points versus the position of nodes in area of identification. The proposed method is based on differentiation of stiffness matrix of finite elements. The sensitivity analysis versus node position was analyzed by Gawrylczyk (1996) for the purpose of establishing the optimal nodes distribution, which minimized the energetic functional. However, such approach is not appropriate in this case, due to the fact that the sensitivity of only energetic functional provides too few information for optimization process. So it is impossible to solve inverse problem based on this quantity. Therefore, in this paper, sensitivity of the magnetic vector potential in chosen nodes was calculated, amount of which should match up the solved problem. The obtained information allows us to modify the nodes positions in such a manner that the distribution of measured field and that calculated by FEM-algorithm is closer. On the basis of equality of both fields, one could assume that the shape of conducting or ferromagnetic surface was identified correctly. The goal function As the goal function for optimization task, the quadratic error value was assumed: X ðRk 2 R k Þ2 ð1Þ J¼ k

So the mth component of the goal function gradient becomes:

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 677-684 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540593

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gradm J ¼ 2

X ðRk 2 R k Þ ›Rm =›jm

ð2Þ

k

where Rk is the referenced potential in measure points. Owing to the fact that the described method delivers only the sensitivity of vector potential R, the sensitivity of other quantities, e.g. the induced voltage in the probe coil, will require integrating this quantity. Sensitivity of the magnetic vector potential on nodal positions Providing gradient information for optimization procedure according to equation (2) requires evaluation of the vector potential sensitivity ›Rm =›jm : A similar method was described in Gawrylczyk (1998). Method of stiffness matrix derivative The analysis with FEM leads to solving the following linear equations: ½A
½R
¼ ½b


ð3Þ

where the stiffness matrix [A ] includes the geometrical properties of the analyzed model and its physical parameters, [R ] is the magnetic vector potential to be determined, and [b ] defines the known excitation vector. The variation of any geometrical or physical parameter in the matrix [A ] implies adequate change of the magnetic vector [R ], so that the excitation vector remains unchanged. In this way, evaluation of the sensitivity ›R=›j has been reduced to solve equations system:

› › ½A
½R
þ ½A
½R
¼ ½0
›j ›j

ð4Þ

From equation (4), we can conclude that the calculation of nodal sensitivity requires differentiation of stiffness matrix versus parameter j. In the case of 2D analysis, which is the object of this paper, differentiation of the matrix [A ] should be carried out versus coordinates xm and ym of the mth node of finite elements. Analytical calculation of stiffness matrix derivative Let us consider the stiffness matrix for Helmholtz’s equation:

› 2 R z ›2 R z þ 2 ¼ jvmgRz ›x 2 ›y The stiffness matrix of first-order elements is given by equation: 2 2 3 2 bi þ c2i bi bj þ ci cj bi bk þ ci ck 2 6 7 6 2 1 6bb þcc j vm g D 2 0e e e 6 b j þ cj bj bk þ cj ck 7 i j Ae ¼ 6 i j 7þ 41 5 2De mre 4 6 2 2 1 b i b k þ ci ck b j b k þ cj ck bk þ ck

ð5Þ

3

1

1

2

7 17 5 ð6Þ

1

2

The derivative of stiffness matrix [Ae] versus the coordinates of ith node takes the form:

2

b2i þ c2i

b i b j þ ci cj

› 2bi 6 6 bi bj þ ci cj ½Ae
¼ 6 ›x i 4mre ðDe Þ2 4 b i b k þ ci ck 2

0

6 6 2ci þ 2mre De 4 ci 2

b j b k þ cj ck ci

22cj cj 2 ck b2i þ c2i

› þci 6 6 bi bj þ ci cj ½Ae
¼ 6 ›yi 4mre ðDe Þ2 4 bi bk þ ci ck 2 þ

0

6 6 2bi 2mre De 4 bi 1

3

Adaptive meshing algorithm

7 b j b k þ c j ck 7 7 5 2 bk þ c2k

b2j þ c2j

2ci

1

b i b k þ c i ck

3

2

2

7 6 cj 2 ck 7 þ jvm0e ge bi 6 1 5 4 12 2ck 1 bi bj þ ci cj b2j þ c2j bj bk þ cj ck 2bi 22bj

bj 2 bk

bi

bi bk þ ci ck

3

1

1

2

7 17 5 2

1

679

ð7Þ

3

7 bj bk þ cj ck 7 7 5 2 2 b k þ ck 3

2

3

2

1

1

7 6 bj 2 bk 7 2 jvm0e ge ci 6 1 5 4 12 2bk 1

2

7 17 5

1

2

The derivative versus the remaining nodes: (xj, yj) and (xk, yk) may be obtained in a cyclic way. Recognition of surface shape After calculation of derivative matrix ›½Ae
for specified nodes according to equation (7) and solving equations system (3) with respect to [R ], one can obtain the sensitivity matrix [S ]. In the event of the same number of measurement nodes and moving points, the determined matrix is squared. Solving the inverse problem The relationship between position of the nodes and field distribution over conducting or ferromagnetic plate is given by the equation: 3 2 32 3 2 Dy1 s11 s12 . . . s1m DR1 7 6 76 7 6 6 DR2 7 6 s21 s22 . . . s2m 7 6 Dy2 7 7 6 76 7 6 7 6 . 7¼6 . 6 ð8Þ .. .. 7 7 6 .. 7 6 . 7 6 . . ... . 76 . 7 6 . 7 6 . 5 4 54 5 4 DRk sk1 sk2 . . . skm Dym This formula allows us to proceed with inverse job in the y-direction. Similarly, the change of nodes positions in the x-direction can be calculated. However, large corrections of nodes positions may cause discontinuity of mesh, so it should be limited to a proper range. For this aim, it is necessary to check the elements for positive area and displace the neighboring elements too, if required.

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Algorithm for cracks identification First, the measurement data of flux density distribution over the conducting surface should be carried forward. In the test case, the data obtained from computer modeling are introduced. Then, the user would determine the search area (Figure 1). The simulating algorithm starts iterations with the basic configuration, in this case with the shape without cracks. Then, it follows the modification of the nodal position because of the difference between the distribution of measured and modeled fields in selected points. This task requires iterative approach due to the differential definition of sensitivity and discretization errors of finite element method. Numerical examples The model of probe for nondestructive testing The construction of a probe for an eddy-current apparatus is axial-symmetric and can be described by means of two-dimensional model (Gawrylczyk, 1998). It consists of an exciting coil, which induces field inside tested material, as well as of measurement coils. The field distribution over the plate can be measured in different ways, e.g. with

Figure 1. Algorithm for cracks identification

biasing coils of large diameter or with pancake coil. For harmonic excitations in 2D space the magnetic vector potential R in a given point is proportional to the voltage induced in circular measurement coil. First example shows the recognition of surface cracks on the plate. It is assumed that the probe as well as the plate with crack has the axial symmetry (Figures 2 and 3). The identification process was carried out with the following parameters: . conductivity and relative permeability of the plate: g ¼ 2; 000 S=m and mrel ¼ 100; . relative permeability of core: mrel ¼ 100; and . current density and frequency: J ¼ 106 A=m2 and f ¼ 1 kHz:

Adaptive meshing algorithm 681

In the test case, the position of only eight nodes was modified. The algorithm converged very well. The crack shape was reconstructed in three iterations, as shown in Figure 4.

Figure 2. The model of probe

Figure 3. Two-dimensional finite elements for probe and crack

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Figure 4. Results of identification in consecutive iterations

Based on the fact that the shape of the crack for the test case was known, two indicators of error can be defined: . field indicator: Np ðRi 2 Ri Þ · ðRi 2 Ri Þ* 1 X DN ¼ N p i¼1 Ri · R i *

.

where Ri is the magnetic vector potential for ith iteration, and Ri is the reference potential. nodes indicator: Kp ð yi 2 yi Þ2 1 X DK ¼ K p i¼1 y_ 2 i

where yi is the coordinate of ith displaced node, yi is the reference coordinate of the crack. Figures 5 and 6 show these two errors for different material parameters. In the second test problem the crack was detected on the inner and outer sides of the tube (Figures 7 and 8). While identified area was located at the same side as measurement coil, the algorithm revealed very good convergence. In Figure 7, the crack on the inner tube side is shown. In this case, the measurement coils were placed directly over detected crack in seven positions, and the parameters of simulation were assumed as follows: . conductivity and relative permeability of tube: g ¼ 2; 000 S=m and mrel ¼ 100; and . current density and frequency: J ¼ 106 A=m2 and f ¼ 1 kHz: The recognition of cracks positioned on the other side of tube wall causes considerable difficulties. In Gawrylczyk (1996) one can find conclusion, that in this case multi-frequency method and Tikhonov regularization should be used. Other solution is to locate the secondary coils on the outer side of the tube, but it may not be always possible.

Adaptive meshing algorithm 683

Figure 5. Errors DN and DK for mre ¼ 100 and g ¼ 2 £ 103 S/m after six iterations

Figure 6. Errors DN and DK for mre ¼ 1 and g ¼ 2 £ 107 S/m after five iterations

Figure 7. The field distribution in the tube

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Conclusions The success of numerical recognition of material shape strongly depends on exact measurement of magnetic flux density. In the examples shown in this paper, the situation was comfortable, because the field distribution was obtained by means of computer modeling. The precision of calculated sensitivity has secondary meaning. For recognition of real cracks the application of data filtering and regularization of equations system will be necessary. A very good choice is the solution of over-determined equation systems for the case of excess of measurement data. References Gawrylczyk, K.M. (1996), “Sensitivity analysis applied to relocation of nodes in 3D tetrahedral meshes”, Proc.7th Int. IGTE Symp., 23-25 September, Graz, pp. 198-201. Gawrylczyk, K.M. (1998), “Sensitivity evaluation methods of electromagnetic quantities with FE-analysis”, COMPEL, Vol. 17 Nos 1-3, pp. 78-84. Gawrylczyk, K.M. (2002), “Sensitivity analysis of electromagnetic quantities by means of FEM”, Journal of Technical Physics, Polish Academy of Sciences, Vol. 43 No. 3, pp. 323-34.

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Incorporation of a Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations Application of the Newton-Raphson method

A Jiles-Atherton vector hysteresis model 685

J. Gyselinck and P. Dular Department of Electrical Engineering (ELAP), University of Lie`ge, Lie`ge, Belgium

N. Sadowski, J. Leite and J.P.A. Bastos Department of Electrical Engineering (GRUCAD), Federal University of Santa Catarina, Floripanopolis, Brazil Keywords Finite element analysis, Vector hysteresis, Magnetic fields, Newton-Raphson method Abstract This paper deals with the incorporation of a vector hysteresis model in 2D finite-element (FE) magnetic field calculations. A previously proposed vector extension of the well-known scalar Jiles-Atherton model is considered. The vectorised hysteresis model is shown to have the same advantages as the scalar one: a limited number of parameters (which have the same value in both models) and ease of implementation. The classical magnetic vector potential FE formulation is adopted. Particular attention is paid to the resolution of the nonlinear equations by means of the Newton-Raphson method. It is shown that the application of the latter method naturally leads to the use of the differential reluctivity tensor, i.e. the derivative of the magnetic field vector with respect to the magnetic induction vector. This second rank tensor can be straightforwardly calculated for the considered hysteresis model. By way of example, the vector Jiles-Atherton is applied to two simple 2D FE models exhibiting rotational flux. The excellent convergence of the Newton-Raphson method is demonstrated.

1. Introduction In the domain of numerical electromagnetism, an increasing number of publications is devoted to the inclusion of hysteresis models in finite-element (FE) field computations. Mostly the well-known Preisach (1935) model and Jiles-Atherton model (Jiles et al., 1992) are used. A comparison of these two scalar models is presented by Benabou et al. (2003). They are applicable to 1D, 2D and 3D FE models displaying unidirectional flux (Chiampi et al., 1995; Sadowski et al., 2002; Saitz, 1999). In applications having rotational flux in part of the computation domain, a vector hysteresis model should be used (Dupre´ et al., 1998; Gyselinck et al., 2000). To date the inclusion of a scalar or vector hysteresis model in FE computations remains challenging. One possible difficulty may reside in the fact that the basic variable of the FE formulation does not coincide with the (main) input variable of The research was carried out in the frame of the Inter-University Attraction Pole IAP P5/34 for fundamental research funded by the Belgian federal government. P. Dular is a Research Associate with the Belgian National Fund for Scientific Research (FNRS).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 685-693 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540601

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the hysteresis model. This problem may be solved by inverting the hysteresis model (Gyselinck et al., 2000; Sadowski et al., 2002). Further, a suitable iterative method to deal with the nonlinearity of the hysteretic media has to be found. For nonlinear problems with isotropic nonhysteretic materials, the NR method is the obvious choice as it offers a quadratic convergence rate near the solution and is easy to implement (Bastos and Sadowski, 2003). In the presence of hysteretic media, the implementation of the NR method is somewhat less self-evident (Gyselinck et al., 2000; Sadowski et al., 2002), and the fixed-point method is often adopted (Chiampi et al., 1995; Saitz, 1999). In Section 2, it is shown that a straightforward elaboration of the NR method naturally leads to the explicit use of the differential reluctivity tensor when considering, e.g. the 2D magnetic vector potential formulation. In the third section, an orginal method to calculate this tensor for a vector generalisation of the Jiles-Atherton model (Bergqvist, 1996) is presented. This vector model is then applied to a simple 2D FE model exhibiting rotational flux. 2. Newton-Raphson method applied to 2D FE model We consider a 2D magnetic field problem in a domain V in the xy-plane (Bastos and Sadowski, 2003). In a subdomain Vs, the current density j s ðx; y; tÞ ¼ js ðx; y; tÞ1 z ; along the z-axis, is given. The magnetic field vector h ðx; y; tÞ and the induction vector b ðx; y; tÞ both have a zero z-component and are related by a hysteretic model in (part of) V. For any continuous vector potential a ¼ aðx; y; tÞ1 z ; the induction b ¼ curl a satisfies div b ¼ 0: The FE discretisation of V leads to the definition of basis functions a l ¼ aðx; yÞ1 z for the potential a : aðx; y; tÞ ¼

n X l¼1

a l ðx; yÞal ðtÞ or aðx; y; tÞ ¼

n X

al ðx; yÞal ðtÞ:

ð1Þ

l¼1

Commonly triangular elements and piecewise linear nodal basis functions are adopted. The weak form of Ampe`re’s law curl h ¼ j s reads ðcurl h; a 0k ÞV ¼ ð j s ; a 0k ÞVs ) ð h; curl a 0k ÞV þ k · ; · l›V ¼ ð j s ; a 0k ÞVs ;

ð2Þ

where a 0k is a continuous test function and ð · ; · ÞV denotes the integral of the scalar product of the two vector arguments over the domain V. For the sake of brevity, the contour integral k · ; · l›V ; originating from the partial integration, is disregarded in the following. By using the n basis functions a k as test functions as well, we obtain a system of n nonlinear algebraic equation (2). Because of the hysteretic material behaviour in V, magnetostatic equation (2), with given excitation js ðx; y; tÞ; has to be solved in the time domain (time stepping). Starting from the known solution a 2 ðx; yÞ and known material state ðb 2 ¼ curl a2 ; h 2 ; . . .Þ at a time instant t 2 , the solution at the next instant t þ ¼ t 2 þ Dt can be obtained by means of an iterative NR scheme. The ith NR þ þ iteration, i ¼ 1; 2; . . .; produces the ith approximation aþ ðiÞ ¼ aði21Þ þ DaðiÞ ; where þ the increment DaðiÞ follows from linearisation of equation (2) around the ði 2 1Þth þ 2 solution aþ ði21Þ : The iterative scheme is initialised with að0Þ ¼ a : The linearisation of equation (2) requires the evaluation of its derivatives with respect to the nodal values al. Given that

›h ›h curl a l ; ¼ ›a l ›b

ð3Þ

the differential reluctivity tensor ›h=›b emerges quite naturally and linearised equation (2) can be written as  n
X ›h curl a l ; curl a k DalðiÞ ¼ ð j s ; a k ÞVs 2 ð h ði21Þ ; curl a k ÞV : ð4Þ ›b V l¼1 For nonhysteretic nonlinear isotropic materials this amounts to the well-known expression of the Jacobian matrix. Indeed, the differential reluctivity tensor can be written as

›h dn ¼ n1 þ 2 2 b b; ›b db

ð5Þ

where the scalar reluctivity n(b) is a single-valued function of the magnitude of b; b b is the dyadic square of b and 1 is the unit tensor. In the xy coordinate system, the matrix representation of reluctivity tensor (5) thus is 3 2 ›hx ›hx " # " # " # 6 7 bx bx bx by 1 0 ›h 6 ›bx ›by 7 d n 7 6 ð6Þ ›b ¼ 6 ›hy ›hy 7 ¼ n 0 1 þ 2 db 2 by bx by by : 5 4 ›bx ›by The integrand




 ›h curl a l · curl a k ›b

in the lefthand-side of equation (4) can then be written as dn ðgrad al · grad aÞðgrad ak · grad aÞ: db 2 For hysteretic material models, the differential reluctivity tensor also depends on the history of the material. For instance, for the vector Preisach model considered by Dupre´ et al. (1998) and Gyselinck et al. (2000), the history consists of extreme values of the magnetic field projected on a number of spatial directions. The vectorised Jiles-Atherton model (Bergqvist, 1996) is dealt with in Section 3.

ngrad al · grad ak þ 2

3. Scalar and vectorised Jiles-Atherton model In the scalar Jiles-Atherton model, the material is characterised by five (scalar) parameters. The determination of these parameters, commonly denoted by a; a; ms ; c and k, is discussed by Benabou et al. (2003) and Jiles et al. (1992). The equations relevant to its vectorisation and FE implementation are given hereafter. The scalar magnetisation m ¼ b=m0 2 h consists of a reversible part mr and an irreversible part mi: m ¼ mr þ mi with mi ¼ ðm 2 cman Þ=ð1 2 cÞ and mr ¼ cðman 2 mi Þ;

ð7Þ

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where the anhysteretic magnetisation man is a single-valued function of the effective field he:

 he a man ðhe Þ ¼ ms coth ð8Þ with he ¼ h þ am: 2 he a The irreversible character of the material is given by


 dmi 1 dh ¼ ðman 2 mi Þ with d ¼ sign : dt dhe dk

ð9Þ

An alternative definition, based on Bergqvist (1996), may be adopted in order to prevent dmi =dhe (and db=dh) from becoming negative: dmi jman 2 mi j dmi if dh · ðman 2 mi Þ . 0; else ¼ ¼ 0: k dhe dhe

ð10Þ

The differential susceptibility dm=dh and the differential permeability db=dh can then be calculated for given b, h and sign(dh):
 db dm dm ¼ m0 1 þ ðb; h; signðdhÞÞ ¼ and dh dh dh

dman dmi þ ð1 2 cÞ dhe dhe : ð11Þ dman dmi 1 2 ac 2 að1 2 cÞ dhe dhe c

For a given state ðh 2 ; b 2 Þ and a given b +, the corresponding h + can be obtained by numerically integrating dh=db : Z bþ dh ðb; h; signðh þ 2 h 2 ÞÞ db: ð12Þ h þ ¼ h þ ðh 2 ; b 2 ; b þ Þ ¼ h 2 þ b 2 db We now outline the vector extension as proposed by Bergqvist (1996), but limit the analysis to the isotropic case. In the vector generalisation of equations (7)-(12), the scalar fields are replaced by vector fields, e.g. b becomes b, while the scalar differential quantities are replaced by tensors, e.g. dh=db becomes ›h=›b: The division in equation (11) is replaced by the multiplication of the nominator by the inverse of the denominator. The scalar 1 is replaced by the unit tensor 1 where necessary. The vector extension of equations (8) and (10) needs special attention. m an and ›m an =›h e are single-valued functions of h e : ! he ›m an man h eh e dman h e h e and m an ¼ man ðjh e jÞ ¼ 12 2 : ð13Þ þ jh e j ›h e he dhe h2e he According to Bergqvist (1996), the vector extension of equation (10) consists in assuming that the increment dm i is parallel to m an 2 m i ; proportional to jm an 2 m i j=k and nonzero only if dh · ðm an 2 m i ). Considering a local coordinate system x 0 y 0 , with the x 0 -axis along the vector m an 2 m i (Figure 1), we thus have " # " # ›m i jm an 2 m i j 1 0 ›m i ¼ ¼ 0: ð14Þ if dh · ðm an 2 m i Þ . 0 else ›h e k ›h e 0 0 0 0 x y

The matrix representation of ›m i =›h e in a coordinate system xy is then " #     cos u sin u ›m i ›m i T : ¼R R with R ¼ ›h e xy ›h e x 0 y 0 2sin u cos u

ð15Þ

Using all the above equations (or their vector extension), ›b =›h can be calculated for given b and h; and given direction of dh: By inverting (the matrix representation of) ›b=›h; ›h=›b is obtained.

A Jiles-Atherton vector hysteresis model 689

4. Application examples By way of example, the vector Jiles-Atherton model presented in Section 3 will now be applied to two simple 2D FE models. The five parameter values of the hysteresis model are those of the electrical steel (FeSi) considered in Benabou et al. (2003): ms ¼ 1; 145; 500 A=m; a ¼ 59 A=m; k ¼ 99 A=m; c ¼ 0:55 and a ¼ 1:3 £ 1024 : 4.1 FE model with spatially uniform field In a calculation domain in the xy-plane, in which the same material model is adopted throughout, a spatially uniform magnetic field h ðtÞ can be obtained by imposing a current layer k ðx; y; tÞ1 z ¼ hðtÞ · ð1 n £ 1 z Þ on the outer boundary of the domain, where 1 n ðx; yÞ is the outward unit vector (perpendicular to 1 z ). Note that the magnetic field and induction are uniform regardless of the FE discretisation. The unknowns of the FE problem are all the nodal values of the magnetic vector potential, except for one, which is, e.g. set to zero. We consider a square domain comprising 68 first-order triangular elements (Figure 2). (A FE mesh having only two elements could do the job as well.) A magnetic field with sinusoidally varying x and y components: h ðtÞ ¼ hx ðtÞ1 x þ hy ðtÞ1 y ¼ h^ x cosð2pftÞ1 x þ h^ y cosð2pft þ fÞ1 y ;

ð16Þ

is imposed as described above. As a static hysteresis model is considered, the frequency f is arbitrary. We take, e.g. f ¼ 1 Hz: If the phase angle f is nonzero (and h^ y is nonzero as well), the excitation is rotational; if f is zero, the flux is unidirectional. Calculations have been carried out with a unidirectional magnetic field ðf ¼ 0Þ of maximum value rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 hmax ¼ h^ þ h^ x

y

equal to 200, 400 and 800 A/m. It has been verified that the FE results do not depend on the spatial direction of the field, as can be expected for an isotropic material.

Figure 1. Local coordinate system x 0 y 0 with x 0 -axis along m an 2 m i

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Figure 2. Model with uniform field and FE discretisation – magnetic field vectors in the case of rotating excitation at t ¼ 0.65 (left) and 0.755 s (right)

Figure 3. bh-loci at alternating excitation (left) and bxhx-loci at rotational excitation (right), with hmax¼ 200 (dashed line) and 400 A/m (full line)

Calculations with a purely rotational flux, f ¼ p=2 and hmax ¼ h^ x ¼ h^ y ; have also been done, considering the same values for hmax (see also Figure 2). Two periods, from t ¼ 0 to 2 s, are time stepped using the implicit Euler scheme. The time step Dt is taken to be either 1/200, 1/400 or 1/800 s. During the initial interval [0,trelax], with, e.g. trelax ¼ 0:25 s; the imposed current layer kðx; y; tÞ1 z (or magnetic field h ðtÞ) is multiplied by a smooth step function f relax ðtÞ ¼ ð1 2 cosðpt=t relax Þ=2; in order to step smoothly through the first magnetisation curve of the hysteretic material. It has been verified that in steady-state and in the case of purely rotational flux, the h and b loci are circular, as, again, can be expected for an isotropic material. Two bh-loci at alternating excitation and two bxhx-loci at rotational excitation are shown in Figure 3. The nonlinear FE equations are solved by means of the NR method, as explained above. A relative tolerance of 102 4 is adopted. The average number of NR iterations per time step is listed in Tables I and II for the different calculations. One observes an excellent convergence of the NR scheme, requiring only three or four iterations in the case of alternating and rotational flux, respectively. In the considered hmax range, the number of iterations depends little on hmax. The number of iterations decreases as the time step is decreased, as could be expected.

4.2 T-joint of three-phase transformer The same vector Jiles-Atherton model is next applied to a 2D FE model of a T-joint of a three-phase transformer, which is shown in Figure 4. The Dirichlet condition a ¼ 0 is imposed on the upper boundary. On the other boundaries, the Neumann condition (zero tangential magnetic field) is implicitly considered. In the lower left and right square corners (of side 1 m) of the model, a uniform current density corresponding with a magnetomotive force of 400 cos (2pft) and 400 cos (2pft+2p/3) A t, respectively, is imposed. As in the first example, two periods are time stepped (with f ¼ 1 Hz; t relax ¼ 0:25s). A FE mesh having 458 first-order triangular elements is used. Figure 4 shows the flux pattern obtained at four equidistant instants in the first half of the second period. The b-loci and h-loci obtained in the six points indicated in Figure 4(a) are shown in Figure 5.

A Jiles-Atherton vector hysteresis model 691

Dt (s) hmax (A/m)

1/200

1/400

1/800

200 400 800

3.31 3.26 3.26

3.14 3.02 2.91

2.96 2.84 2.61

Table I. Number of NR iterations per time step in the case of alternating flux

Dt (s) hmax (A/m)

1/200

1/400

1/800

200 400 800

4.01 4.09 4.21

3.78 4.06 4.10

3.58 3.72 4.00

Table II. Number of NR iterations per time step in case of rotating flux

Figure 4. 2D FE model of T-joint of transformer - location of six points in the model – flux lines at t ¼ 1, 1.125, 1.25 and 1.375 s (a to d)

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Figure 5. The b-loci (left) and h-loci (right) in the six points indicated in Figure 4a

A fast convergence of the NR scheme is again observed: the average number of iterations per time step is 4.6, 3.89 and 3.38 with Dt equal to 1/200, 1/400 and 1/800 s, respectively. 5. Conclusions The implementation of a vector Jiles-Atherton model in 2D FE magnetic field computations has been studied. When solving the nonlinear equations by means of the NR method, the differential reluctivity tensor naturally emerges. This tensor is also a natural output of the considered vector hysteresis model. The vector Jiles-Atherton model has been successfully applied to two simple 2D FE models with rotational flux. The Newton-Raphson scheme has been shown to converge very well. References Bastos, J.P.A. and Sadowski, N. (2003), “Electromagnetic modeling by finite element methods”, Electrical Engineering and Electronic Series, 117, Marcel Dekker, New York, NY. Benabou, A., Cle´net, S. and Piriou, F. (2003), “Comparison of Preisach and Jiles-Atherton models to take into account hysteresis phenomenon for finite element analysis”, Journal of Magnetism and Magnetic Materials, Vol. 261, pp. 305-10. Bergqvist, A. (1996), “A simple vector generalisation of the Jiles-Atherton model of hysteresis”, IEEE Trans. Magn., Vol. 32 No. 5, pp. 4213-5. Chiampi, M., Chiarabaglio, D. and Repetto, M. (1995), “A Jiles-Atherton and fixed-point combined technique for time periodic magnetic field problems with hysteresis”, IEEE Trans. Magn., Vol. 31 No. 6, pp. 4306-11. Dupre´, L., Gyselinck, J. and Melkebeek, J. (1998), “Complementary finite element methods in 2D magnetics taking into account a vector Preisach model”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3048-51. Gyselinck, J., Vandevelde, L. and Melkebeek, J. (2000), “Calculation of noload lsses in an induction motor using an inverse vector Preisach model and an eddy current loss model”, IEEE Trans. Magn., Vol. 36 No. 4, pp. 856-60.

Jiles, D., Thoelke, B. and Devine, M. (1992), “Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis”, IEEE Trans. Magn., Vol. 28, pp. 27-35. Preisach, F. (1935), “Uber die magnetische nachwirkung”, Zeitschrift ¨fr Physik, Vol. 94, pp. 277-302. Sadowski, N., Batistela, N.J., Bastos, J.P.A. and Lajoie-Mazenc, M. (2002), “An inverse Jiles–Atherton model to take into account hysteresis in time-stepping finite-element calculations”, IEEE Trans. Magn., Vol. 32 No. 2, pp. 797-800. Saitz, J. (1999), “Newton-Raphson method and fixed-point technique in finite element computation of magnetic field problems in media with hysteresis”, IEEE Trans. Magn., Vol. 35 No. 3, pp. 1398-401.

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The modelling of the FDTD method based on graph theory

694

Polish Japanese Institute of Information Technology, Warsaw, Poland

Andrzej Jordan Carsten Maple Department of Computing and Information Systems, University of Luton, Luton, UK Keywords Modelling, Finite difference time-domain analysis, Magnetic fields, Graph theory Abstract Discusses a parallel algorithm for the finite-difference time domain method. In particular, investigates electromagnetic field propagation in two and three dimensions. The computational intensity of such problems necessitates the use of multiple processors to realise solutions to interesting problems in a reasonable time. Presents the parallel algorithm with examples, and uses aspects of graph theory to examine the communication overhead of the algorithm in practice. This is achieved by observing the dynamically changing adjacency matrix of the communications graph.

Introduction The finite-difference time domain (FDTD) method has been widely used for the analysis of high-frequency electromagnetic field propagation. The method was proposed by Yee (1966) (Figure 1), and has since been developed by many others (Kunz and Luebbers, 1993; Taflove and Hagness, 2000). Applications of the FDTD method include the analysis and optimisation of antennae for radio, television and radar. The method has also been applied to fibreoptics, the design of microwave circuits and the influence of electromagnetic fields on the human body (Hsing-Yi and Hou-Hwa, 1994). The theoretical basis of the FDTD method (Hsing-Yi and Hou Hwa, 1994; Kunz and Luebbers, 1993; Taflove and Hagness, 2000; Yee, 1966) is supplied by the standard vector Maxwell’s equations. These vector equations can be transformed into the following system of scalar equations in Cartesian coordinates.

 ›E x 1 ›H z ›H y ›H x 1 ›E y ›E z ¼ 2 2 gE x ¼ 2 ð1aÞ 1 ›y ›t ›z ›t m ›z ›y

 ›E y 1 ›H x ›H z ›H y 1 ›E z ›E x ¼ 2 2 gE y ¼ 2 ð1bÞ 1 ›z ›t ›x ›t m ›x ›z

 ›E z 1 ›H y ›H x ›H z 1 ›E x ›E y ¼ 2 2 gE z ¼ 2 ð1cÞ 1 ›x ›t ›y ›t m ›y ›x COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 694-700 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540610

where s, m, and 1 are the conductivity, permeability and permittivity, respectively, of the medium in which the propagation occurs. Following the notation of Yee, This work was supported by a grant of the Polish Scientific Committee no.: 4T11C 007 22 and internal grants of PJIIT and the University of Luton.

Modelling of the FDTD method

695

Figure 1. Yee cell

we consider the field at a point in space (i, j, k) with coordinates (idx, jdy, kdz) where dx ¼ dy ¼ dz are the dimensions of the cubic cell, and dt is the time-step. Taking
1=2 10 ~ E¼ E; m0 we can obtain a set of difference equations. For (example, taking only E and H components) the two-dimensional problem, these equations are as follows.



 1 1 ~ n22 1 1 n ~ i; j; k þ ¼ CAz i; j; k þ Ez þ CBz i; j; k þ Ez i; j; k þ 2 2 2 2 

 1 1 1 1
H yn21 i þ ; j; k þ ð2aÞ 2 H yn21 i 2 ; j; k þ 2 2 2 2

 1 1 1 1 þH xn21 i; j 2 ; k þ 2 H xn21 i; j þ ; k þ 2 2 2 2 


 1 1 1 1 1 n21 H xn i; jþ ; kþ ¼ H xn22 i; jþ ; kþ þRC E~ y i; jþ ; kþ1 2 2 2 2 2


ð2bÞ 1 1 1 n21 n21 n21 2E~ y i; jþ ; k þ E~ z i; j; kþ 2 E~ z i; jþ1; kþ 2 2 2

 
 1 1 1 1 1 n21 H yn i þ ; j; kþ ¼ H yn22 i þ ; j; kþ þRC E~ z i þ1; j; kþ 2 2 2 2 2 ð2cÞ


 1 1 1 n21 n21 n21 2E~ z i; j; kþ þ E~ x i þ ; j; k 2 E~ x i þ ; j; kþ1 2 2 2 It is necessary to note that Ex, Ey, Ez, are computed for time-steps of even index and Hx, Hy, Hz, are computed for time-steps of odd index. Numerical implementations of

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the Maxwell equations above are used in educational and professional programs (Users manual of QuickWave-3D v. 1.8 by QUED) which compute the electromagnetic field propagation in different media (Kunz and Luebbers, 1993; Taflove and Hagness, 2000). The computational expense of solving such problems necessitates the use of parallel algorithms for finding solutions within a reasonable time in cases of large domains. Such computations may be performed on MPPs or networked clusters. A parallel algorithm for FDTD Let us consider the two-dimensional electromagnetic wave propagation problem. In this case we compute E~ z , Hx and Hy only. The parallel algorithm involves dividing the geometric space into a number of subdomains, with one processor given the responsibility of calculating the field in that subdomain at each time interval (Figure 2) (Smyk et al., 2001). The subdomains are further divided into discrete elements using a regular mesh. Each processor computes E~ z ; Hx and Hy, using the values of the vectors in each element in that subdomain at the previous time interval. Since each subdomain is associated with a different processor, calculating values adjacent to the boundary between two subdomains requires communication between two processors. The cycle of computation and data exchange in the 2D problem is as follows. . Compute the value of E~ z : . Send the values of E~ z computed on the edges to neighbouring processors. . Compute the field on the domain boundary using Mur’s boundary conditions. . Compute the component of Hx and Hy. . Send to neighbouring processors the components Hx and Hy which are computed on the edges. Figure 2 shows the computations in the interior of the space and on the edges.

Figure 2. Division of the domain in the parallel version of the FDTD method

The parallel computation of the electromagnetic field is performed using a master-slave principle. The master processor sends data to the slave processors and receives the result of computations from them. Most of the data are sent between the neighbouring processors in the iterative computation of electromagnetic field distribution (sending the data with the result of computations on the edges of the subregions). It should be noted that when considering the three-dimensional system the communications are concerning the planes between the subregions.

Modelling of the FDTD method

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Application of graph theory to the optimisation and command of processors in parallel computations Analyses of data exchange, total time for computation and communication are all important for the consideration of the effectiveness of a parallel algorithm. A problem that must be considered when implementing parallel algorithm is the automatic command and control of processors that can be integrated into a system on which the problem is being solved. Graph theoretical methods can be used in both cases. Before presenting how graphs may be applied to the analysis of parallel computations we require some basic definitions (Cormen et al., 1994). Definition: Graph. A graph G ¼ kV ; El is defined by a non-empty finite set of vertices, V(G), and a (possibly empty) collection of edges, EðGÞ # V £ V : Each edge is identified with two, not necessarily distinct, vertices in V. If the edges are associated with an ordered pair of vertices, then G is called a directed graph. In general, more than one edge may be associated with the same pair of vertices, in such cases G is said to have multiple or parallel edges. If the two vertices associated with an edge are coincident, that edge is said to be a self-loop. In cases where each element e [ E is distinct and none are self-loops, the graph is known as a simple graph. We define the order of G, ord(G) to be the number of vertices in the graph, that is ordðGÞ ¼ jV j: In the scenario of parallel computation, the vertices represent the computation nodes (processors) and the edges represent the communications between the processors. An example of a graph and its associated matrix is shown in Figure 3. The vertices v1 ; v2 ; . . .; vordðGÞ are numbered arbitrarily. Any graph can be represented by a ordðGÞ £ ordðGÞ matrix A ¼ ðai:j Þ such that ( 1 if ðvi ; vj Þ [ E ai;j ¼ 0 otherwise This matrix is known as the adjacency matrix of G. In the FDTD method an adjacency matrix is calculated for each time step. This matrix is dynamic in the sense that it is

Figure 3. Graph representation G ¼ ðV ; EÞ: (a) G with six vertices and eight edges, and (b) the adjacency matrix of G

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Figure 4. 2D propagation problem: P1-P9 slave processors; p5 is the source of the electromagnetic wave; and Mc denotes Mur’s boundaries

Figure 5. (a) Graph representation of the electromagnetic field computations. P1-P9 are slave processors, P10 is the master, and (b) adjacency matrix representation of Figure 5(a) at a time well-advanced of the inclusion of the source

possible for this matrix to change at each time step. The element of A at a particular time depends on the communication between the slave processor at that time. Therefore, we produce a sequence of adjacency matrices AðN Þ; ¼ A 1 ; A 2 ; . . .; A N }; where the index of the matrix denotes the time interval. That is, the matrix A n is a representation of the communication graph at time interval n. This matrix also depends on the position of the electromagnetic source. An example of the method presented above is shown in Figures 4-6. Figure 4 shows the position of the source of the electromagnetic wave, ps, the distribution of Ez, Hx, Hy

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Figure 6. (a) Graph visualisation, and (b) the block matrix representation for the graph shown in Figure 5(a)

at an arbitrary point in space and the communication edges between the processors P1 and P9. Figure 5(a) shows the graph for these nine processors and the master node, node 10. Figure 5(b) shows the adjacency matrix in the case where all slave processors are in the communicative state. This occurs at a time well advanced from the inclusion of the source into the domain. Figure 6 shows the graph representations for the 3D case. In this case, the adjacency matrix represents the communication between the block matrices shown in Figure 6(a). It is necessary to stress that this is a logical representation. For example, matrix A1, which has dimension 9 £ 9 is the same as the figure shown in Figure 5(b). It is necessary to point out that the adjacency matrix from Figure 5(a) is dynamically modified from the moment the source is introduced to the domain. The dynamic modifications result from the physical properties of the electromagnetic wave propagation. At the start, the only computational processor is P4 since ps, the point source, is in the subdomain of P4. As the computation progresses we will introduce other processors and the order in which these processors are introduced depends on the geometry of the domain in which we are computing the electromagnetic wave propagation. In practical implementations we used the parallel program Antenna 26, developed at PJWSTK, running on the Linux operating system. A sinusoidal wave source was placed in the domain at position (10,8). The mesh used was of size 41 £ 41: The frequency was 426 MHz, dt ¼ 0:5 £ 10211 s; dx ¼ dy ¼ dz ¼ 0:02 m: The dynamics of the adjacency matrix is shown in Figure 7(a)-(c) for the time step n ¼ 1 ðt ¼ 0:5 £ 10211 sÞ;

Figure 7. The dynamic changes of the adjacency matrix using Antenna 26 program

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n ¼ 300 ðt ¼ 1:5 £ 1029 sÞ; n ¼ 500 ðt ¼ 2:5 £ 1029 sÞ: In the case where n ¼ 1 the adjacency matrix has all elements set to zero since during this period the wave does not arrive at the communication edges. For the time which corresponds to n ¼ 300 we have communications between processor P1 and processors P0, P2 and P4, between processor P2 and processors P1, and P5 and so forth. Note that the adjacency matrix is created with processors labelled P0, P1,. . ., P8, which differs from that of Figure 4. Conclusions and further work We have presented a method for the analysis of parallel computations of electromagnetic field propagation using FDTD. The direct implementation of the graph theory to the distributed algorithms enables: . to observe and analyze dynamic relations between slave processors, . to analyze task structure and to optimize the speed up of the algorithms. Using directed weighted graphs in a manner similar to that presented herein will allow simple analysis of computations and communications. This is the subject of future work. References Cormen, T.H., Leiserson, C.E. and Rivest, R.L. (1994), Introduction to Algorithms, The Massachusetts Institute of Technology. Hsing-Yi, C. and Hou-Hwa, W. (1994), “Current and SAR induced in a human head model by the electromagnetic fields irradiated from a cellular phone”, IEEE Trans. Microwave Theory Techniq., Vol. 42 No. 12, pp. 2249-54. Kunz, K.S. and Luebbers, R.J. (1993), The Finite Difference Time Domain Method for Electromagnetics, CRC Press, Boca Raton, FL. Smyk, A., Cieslak, P., Tudruj, M., Jordan, A. and Butrylo, B. (2001), “Optimization of FDTD parallel software based on the use of vampir monitor”, WMSCI’01, July 2001, Orlando-Florida Vol. XV, pp. 343-8. Taflove, A. and Hagness, S.C. (2000), “Computational electrodynamics”, The Finite-Difference Time-Domain Method, Artech House, Inc., Boston, MA. Yee, K.S. (1966), “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media”, IEEE Trans. Antennas Propagat., Vol. AP-14 No. 5, pp. 302-57.

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Inverse problem – determining unknown distribution of charge density using the dual reciprocity method

Inverse problem

701

Dean Ogrizek Elektro Maribor d.d. Vetrinjska 2, 2000 Maribor, Slovenija

Mladen Trlep Faculty of Electrical Engineering and Computer Science, University of Maribor, Smetanova 17, 2000 Maribor, Slovenija Keywords Density measurement, Reciprocating engines, Algorithmic languages Abstract Presents the use of the dual reciprocity method (DRM) for solving inverse problems described by Poisson’s equation. DRM provides a technique for taking the domain integrals associated with the inhomogeneous term to the boundary. For that reason, the DRM is supposed to be ideal for solving inverse problems. Solving inverse problems, a linear system is produced which is usually predetermined and ill-posed. To solve that kind of problem, implements the Tikhonov algorithm and compares it with the analytical solution. In the end, tests the whole algorithm on different problems with analytical solutions.

1. Introduction The scalar field as the electrostatic field (ES) in space is described by Poisson’s equation (1) in V and by boundary conditions on G (equation (2)): Dw ¼ b;

b ¼ bðx; y; zÞ ¼ 2

w ¼ w


›w ¼ q
›n

rðx; y; zÞ 1

ð1Þ ð2Þ

The solution of Poisson’s equation can be determined, in case the information about the field type (in our case ES field) and boundary conditions is known. The information about the material and source distribution is considered to be known as well. In most cases, the unknown values of the potentials and the normal derivatives on part of the boundary and within the problem are determined. In that case we can say that we are dealing with a direct problem. Example of direct problem is shown in Figure 1. Many times the values of the potential and the normal derivative on the boundary are known and we wish to determine the unknown distribution of the space charge density bðx; yÞ: In that case we are dealing with an inverse problem. In the case of inverse problems the values of the potential and the normal derivative are usually determined by measurement. This is the reason why we are dealing with an inverse problem any time we want to determine the unknown distribution of space charge density bðx; yÞ or the unknown distribution of material within a problem by the known information on the boundary. Example of an inverse problem is shown in Figure 2.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 701-706 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540629

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Figure 1. Example of direct problem

Figure 2. Inverse problem – determination of the unknown distribution of the space charge density b(x, y)

2. Description of Poisson’s equation with the DRM The boundary integral equation for the ith point on G is expressed by equation (3): Z Z bw* dV ð3Þ ci wi þ ðq* w 2 w* qÞ dG ¼ G

V

where w* is the fundamental solution, q* is its normal derivative and b is the space charge density. When DRM is used, b is approximated by the following expression: Nb X b¼ a j fj ð4Þ j¼1

where aj are a set of initially unknown coefficients and fj are approximating functions in the Nb observed point in the domain V and on the boundary G. Equation (3) can be rewritten for the ith boundary node Partridge et al., 1992 in the final form (equation (5)):
 Z Z Nb X ci wi þ ðq* w 2 w* qÞ dG ¼ aj ci w^ij þ ðq* w^j 2 w* q^ j Þ dG ð5Þ G

j¼1

G

Equation (5) can be expressed in matrix form by (Partridge et al., 1992). ^ ^ 2 GQ; a ¼ F 21 b H w 2 Gq ¼ C a; C ¼ H F

ð6Þ

where w and q are node value vectors of the potential and of the normal derivative on the boundary, while a is the vector of the unknown coefficients and b is the vector of the charge density values in the Nb observed points, mark “^” stands for particular solution (Partridge et al., 1992). When we wish to determine the unknown charge density b(x, y) in the case of an inverse problem, the problem can be translated to the solution of equation (6) which represents the usually over determined system. In this case, we have two possibilities: (a) First calculate a and then b from equation (7) Aa ¼ d ) b ¼ F a A ¼ C d ¼ H w 2 Gq ð7Þ (b) We calculate b directly from Ab ¼ d A ¼ CF 21 d ¼ H w 2 Gq ð8Þ

Inverse problem

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3. Test cases The whole algorithm of determination of the unknown distribution of space charge density using the DRM and Tikhonov algorithm (Hansen, n.d.; Hansen and O’Leary, 1993) was tested on a square 6 £ 6 model (Figure 3), for three different distributions of the space charge density (constant, linear and quadratic – Table I) and for three

Figure 3. Test problem with the presentation of observed points

Constant

w(x, y) ›w/›x ¼ ›w/›y b (x, y)

(x + y + 1)2 2(x + y + 1) 4

Distribution of charge density b(x, y) Linear (x + y + 1)3 3(x + y + 1)2 12(x + y + 1)

Quadratic (x + y + 1)4 4(x + y + 1)2 24(x + y + 1)2

Table I. Different distribution of charge density

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different numbers of boundary elements (40, 80 and 120 boundary elements). For testing we chose problems for which the analytical solutions are known. The analytical solutions were used as a reference to the solutions obtained by algorithm. 3.1 Constant charge distribution Figure 4 shows calculated b(x,y) and the deviations from the exact equivalent charge density for the case of 80 boundary elements. Maximum deviations with 40 and 120 boundary elements are shown in Table II. 3.2 Linear charge distribution Figure 5 shows calculated b(x,y) and the deviations from the exact equivalent charge density for the case of 80 boundary elements. For the case of 40 and 120 boundary elements, maximum deviations are shown in Table III. 3.3 Quadratic charge distribution Figure 6 shows calculated b(x,y) and the deviations from the exact equivalent charge density for the case of 80 boundary elements. For the case of 40 and 120 boundary elements, maximum deviations are shown in Table IV. 4. Conclusion This paper presents the determination of unknown distribution of charge density by the use of DRM and Tikhonov algorithm. The results show that in all test cases good results can be obtained. The worst but still good results were obtained in the case of

Figure 4.

Table II. Maximum deviations with 40 and 120 boundary elements

Elements Maximum deviation from the exact solution (percent)

40

120

3.7

2

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Figure 5.

Elements Maximum deviation from the exact solution (percent)

40

120

6.5

6

Table III. Maximum deviations for the case of 40 and 120 boundary elements

Figure 6.

Elements Maximum deviation from the exact solution (percent)

40

120

7.8

6.6

Table IV. Maximum deviations for the case of 40 and 120 boundary elements

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quadratic distribution of charge density (Table IV). To improve the accuracy of the algorithm, we should spend more time on the discretization of such problems. References Hansen, P.C. (n.d.), “The L-curve and its use in the numerical treatment of inverse problems”, Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark, available at: www.imm.dtu.dk/~pch/TR/Lcurve.ps Hansen, P.C. and O’Leary, D.P. (1993), “The use of the L-curve in the regularization of discrete ill-posed problems”, SIAM J. Sci. Comput., Vol. 14, pp. 1487-503. Partridge, P.W., Brebbia, C.A. and Wrobel, L.C. (1992), The Dual Reciprocity Boundary Element Method, Computational Mechanics Publications, Southampton, Boston, MA. Further reading Demmel, P.W. (2000), Prevod in priredba: Egon Zakrajsˇek, Uporabna numericˇna linearna algebra, Ljubljana. Hensel, E. (1991), Inverse Theory and Applications for Engineers, Prentice Hall, Englewood Cliffs, NJ. Sun, Y. (1996), “Boundary element models for Laplace, Poisson and Helmholz field computation and application to inverse analysis”, Computational Mechanics Publications, Southampton, Boston, MA, Okayama University. Trlep, M., Hamler, A. and Hribernik, B. (2000), “The use of DRM for inverse problems of Poisson’s equation”, IEEE Transactions on Magnetics, Vol. 36 No. 4, pp. 1649-52.

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Finite element modelling of stacked thin regions with non-zero global currents P. Dular and J. Gyselinck

Finite element modelling

707

Department of Electrical Engineering, Institut Montefiore, University of Lie`ge, Lie`ge, Belgium

T. Zeidan and L. Kra¨henbu¨hl

CEGELY (UMR CNRS 5005), E´cole Centrale de Lyon, E´cully, France Keywords Laminates, Finite element analysis, Eddy currents Abstract Develops a method to take the eddy currents in stacked thin regions, in particular lamination stacks, into account with the finite element method using the 3D magnetic vector potential magnetodynamic formulation. It consists in converting the stacked laminations into continuums with which terms are associated for considering the eddy current loops produced by both parallel and perpendicular fluxes. Non-zero global currents can be considered in the laminations, in particular for studying the effect of imperfect insulation between their ends. The method is based on an analytical expression of eddy currents and is adapted to a wide frequency range.

Introduction The consideration of thin conducting regions in a 3D finite element analysis is an important source of difficulty regarding the mesh as well as the numerical solving. An isolated thin volumic region can be efficiently reduced to surface elements satisfying the actual distributions or interface conditions of the fields (Ren, 1998). Nevertheless, when numerous thin regions are stacked and separated with insulating layers, e.g. lamination stacks and foil windings, the whole resulting region must remain volumic and its homogenization is usually the only feasible solution for a finite element analysis (De Gersem and Hameyer, 2001; Dular and Geuzaine, 2002; Dular et al., 2003a). The aim of this paper is to present a homgenization technique for stacked thin conducting regions, in particular lamination stacks, separated by insulating layers and having a non-zero global current in each thin region, which extends the method proposed by Dular et al. (2003a). The eddy currents produced by both parallel and perpendicular fluxes are considered. The global currents can be coupled between regions, which opens the method to the study of short-circuited or imperfectly insulated lamination stacks. The method makes use of the magnetic vector potential magnetodynamic finite element formulation with edge elements and is developed in the frequency domain. It consists in considering the formulation terms of the analytical solution of the equations governing the eddy current distribution in the section of each thin region. The associated mean current and magnetic flux densities along each lamination The research was carried out in the frame of the Inter-University Attraction Pole IAP P5/34 for fundamental research funded by the Belgian federal government. P. Dular is a Research Associate with the Belgian National Fund for Scientific Research (F.N.R.S.).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 707-714 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540638

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thickness are converted into a continuum for the whole stack, which is then homogenized, and remain the primary unknowns, from which their actual distribution can be determined. Magnetodynamic a-v formulation A bounded domain V of the three-dimensional Euclidean space is considered, in which the magnetodynamic problem is defined. The eddy current conducting part of V is denoted as Vc and the non-conducting one as VCc ; with V ¼ Vc < VCc : Massive conductors belong to Vc and source conductors, with a given current density js, are regions of Vs, with Vs , VCc : The general expression of the electric field e via a magnetic vector potential a involves the gradient of an electric scalar potential v in Vc, i.e. e ¼ 2›t a 2 grad v

in Vc ;

with b ¼ curl a

in V;

ð1-2Þ

so that the Faraday equation is satisfied; b is the magnetic flux density and v can usually be fixed to zero in passive conductors. With these two potentials, the a-v magnetodynamic formulation is obtained from the weak form of the Ampere equation, i.e. (Dular et al., 1999a; Golovanov et al., 1998) ðm 21 curl a;curl a0 ÞV þ ðs›t a;a0 ÞVc þ ðs grad v;a0 ÞVc 2 ðj s ;a0 ÞVs ¼ 0; ;a0 [ F a ðVÞ; ð3Þ where m is the magnetic permeability, s the electric conductivity, Fa(V) the function space defined on V and containing the basis functions for a as well as for the test function a0 , ( , )V denotes a volume integral in V of the product of its vector field arguments. The direct way to consider the eddy currents in a lamination stack is to model this stack as a set of massive conductors (i.e. a subset of Vc) separated by insulating layers. This is nevertheless generally unfeasible in view of the large number of laminations encountered in iron cores. Another method consists in considering the lamination stack as a source conductor (i.e. a subset of Vs) through a precalculated current density js corresponding to the eddy currents in the laminations. The actual distribution of the magnetic flux density has to be considered as well. The expression of such current and magnetic flux densities and their use in the a-v magnetodynamic formulation is developed in the following. Current density in a lamination stack A lamination stack region Vls is considered (Figure 1) as a subset of the source conductor domain Vs. Each lamination has a thickness d, an electric conductivity s

Figure 1. Lamination stack with its local coordinate system associated with each lamination

and a magnetic permeability m, which can vary from one lamination to another, and has a local coordinate system (ia, ib, ig). The directions ia and ib are parallel to the associated lamination, while ig is along its perpendicular direction. The direction ia is considered as the a priori unknown direction of the magnetic flux density ba parallel to the lamination, and consequently, ib is the main direction of the eddy current loops generated by variations of ba, with the associated current density jb. In addition, a magnetic flux density perpendicular to the lamination generates a current density denoted as jab. The current density in one lamination is then expressed as j ¼ j ab þ j b ;

ð4Þ

i.e. as the superposition of eddy current densities generated by time-varying flux perpendicular and parallel to the lamination, respectively. The current density jab can be considered through an anisotropic conductivity tensor sab with non-zero components only in the directions of ia and ib, while jb should undergo a pretreatment for avoiding, at the discrete level, the discretization of each lamination separately. Actually, the current density jab adds a non-zero component to the current loops described by jb and will thus, have to be considered jointly with jb in the final formulation. A non-zero component of jb can also be obtained in the case of imperfectly insulated laminations, which will be considered first. Model for eddy currents Expressions of eddy current density and magnetic field The actual distributions of ba (or the associated magnetic field h a ¼ m 21 b a ) and jb have to be considered first. From the Maxwell equations, the second-order partial differential equations for jb and ha written for one lamination (Ida and Bastos, 1997), neglecting the fringing effects for jb, lead to their analytical expressions, i.e. for their modulus in complex notation, j b ðgÞ ¼ J 1 sinhðkgÞ þ J 2 coshðkgÞ;

ð5Þ

h a ðgÞ ¼ H 1 coshðkgÞ þ H 2 sinhðkgÞ;

ð6Þ

k ¼ ð1 þ j Þd 21 ;

ð7Þ

with

where J1, J2, H1 and H2 are constants depending on the exterior constraints, g is the the lamination, position along the g-direction (equal to zero at the mid-thickness ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pof Figure 1) and d is the skin depth in the lamination, i.e. d ¼ 2=vsm; with the pulsation v ¼ 2pf ; f is the frequency. From the Ampere equation curl h a ¼ j b ; one has ›g h a ¼ j b ; which, once applied to equations (5) and (6), implies relations between J1 and H1, and J2 and H2, i.e. H 1 ¼ J 1 =k;

H 2 ¼ J 2 =k:

ð8-9Þ

Consequently, there remain only two constants in equations (5) and (6) for which no expression can generally be obtained a priori (i.e. before computing the finite element solution). The key point is rather to express these constants in terms of the mean

Finite element modelling

709

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values of the considered quantities along the thickness of each lamination, which will actually be the mean fields to be considered in the homogenized lamination stack. The mean magnetic field and current density are, respectively, Z d=2 h a ¼ 1 h a ðgÞ dg ¼ H 1 sinhðkd=2Þ=ðkd=2Þ; ð10Þ d 2d=2 Z d=2 j b ¼ 1 j b ðgÞ dg ¼ J 2 sinhðkd=2Þ=ðkd=2Þ; ð11Þ d 2d=2 from which H1 and J2 can be expressed in terms of the mean magnetic flux density b a ¼ mh a and current density ¯j b, i.e. H 1 ¼ b a m 21 ðkd=2Þ sinh21 ðkd=2Þ;

ð12Þ

J 2 ¼ j b ðkd=2Þ sinh21 ðkd=2Þ:

ð13Þ

with equations (8-9), (12) and (13), expressions (5) and (6) can finally be written in terms of b a and j b : In the case of zero global currents, constants J2 and H2 are equal to zero. Magnetodynamic a-v formulation with homogenized lamination stacks The term associated with the current density jb in the weak formulation (3) is ( j s, a0 )Vs and has to be developed using the expression (5) of j b. Its contribution in equation (3) has to be similar to the one obtained in the case of a massive conductor region, i.e. Vls. The contribution of the actual distribution of the magnetic flux density b ¼ curl a, involved in (m 2 1curl a, curl a0 )V in equation (3), has to be considered in Vls as well. From equation (1), with v ¼ 0; and the Ohm law j b ¼ s e b ; one has for the b-component of the magnetic vector potential a, a b ¼ 2j b =jvs:

ð14Þ

This expression is also used for the test function a0 , which consists in using some basis functions satisfying the properties of the field in the laminations as test functions. The terms associated with the magnetic flux density ba and the current density jb in the weak formulation (3) can then be written as ðm 21 curl a; curl a0 ÞVls 2 ðj b ; a0 ÞVls ¼ ðm 21 b a ; b0a ÞVls þ ðð jvsÞ21 j b ; j0b ÞVls ;

ð15Þ

in which all the field quantities are defined in the actual lamination stack before homogenization, i.e. in Vls. Using expressions (5) and (6), equation (15) becomes ðm 21 b a ; b0a ÞVls þ ðð jvsÞ21 j b ; j0b ÞVls ¼ ðmH 1 coshðkgÞ; H 01 coshðkgÞÞVls þ ðð jvsÞ21 J 1 sinhðkgÞ; J 01 sinhðkgÞÞVls þ ðmH 2 sinhðkgÞ; H 02 sinhðkgÞÞVls þ ðð jvsÞ21 J 2 coshðkgÞ; J 02 coshðkgÞÞVls ;

ð16Þ

in which the effect of zero and non-zero global currents, i.e. the odd and even components of jb in expression (5), respectively, are decoupled in order to be

considered separately in the following. The products of odd and even components of expressions (5) and (6) give no contribution to equation (16). The terms in equation (16) associated with zero global currents, or with the even component of ba in expression (6), gives ðmH 1 cosh2 ðkgÞ; H 01 ÞVls þ ðð jvsÞ21 J 1 sinh2 ðkgÞ; J 01 ÞVls

ð17Þ

¼ ðmðcosh2 ðkgÞ þ sinh2 ðkgÞÞH 1 ; H 01 ÞVls ¼ ðm coshð2kgÞH 1 ; H 01 ÞVls :  ls ; i.e. after averaging the g-dependent function in After homogenizing Vls to V equation (17), this last term in equation (17) can be expressed in terms of the mean magnetic flux density ba, i.e. also using equation (12), ðm coshð2kgÞH 1 ; H 01 ÞVls ¼ ðmðkdÞ21 sinhðkdÞH 1 ; H 01 ÞV ls ¼ ðmðkdÞ21 sinhðkdÞm 22 ðkd=2Þ2 sinh22 ðkd=2Þb a ; b 0a ÞV ls ¼ ðm 21 ðkd=4Þ2 sinhðkd=2Þ coshðkd=2Þ sinh22 ðkd=2Þb a ; b 0a ÞV ls

ð18Þ

¼ ðm 21 ðkd=2Þ tanh21 ðkd=2Þb a ; b 0a ÞV ls ¼ ðm 21 F R21 b a ; b 0a ÞV ls ; with ð19Þ F R ¼ tanhðkd=2Þ=ðkd=2Þ: In the same way, the terms in equation (16) associated with non-zero global currents, or with the odd component of ba in expression (6), give ðmH 2 sinh2 ðkgÞ; H 02 ÞVls þ ðð jvsÞ21 J 2 cosh2 ðkgÞ; J 02 ÞVls

ð20Þ

¼ ðð jvsÞ21 ðsinh2 ðkgÞ þ cosh2 ðkgÞÞJ 2 ; J 02 ÞVls ¼ ðð jvsÞ21 coshð2kgÞJ 2 ; J 02 ÞVls :  ls ; equation (19) can be expressed in terms of the mean After homogenizing Vls to V current density j b ; i.e.     ð jvsÞ21 coshð2kgÞJ 2 ; J 02 V ¼ ð jvsÞ21 ðkdÞ21 sinhðkdÞ J 2 ; J 02 V ls ls   ¼ ð jvsÞ21 ðkdÞ21 sinhðkdÞðkd=2Þ2 sinh22 ðkd=2Þj b ; j0b V ls   ð21Þ 21 22  ¼ ð jvsÞ ðkd=4Þ2 sinhðkd=2Þ coshðkd=2Þ sinh ðkd=2Þj b ; j0b V ls     ¼ ð jvsÞ21 ðkd=2Þ tanh21 ðkd=2Þj b ; j0b V ¼ ð jvsÞ21 F R21 j b ; j0b  : ls

Vls

The mean magnetic vector potential a b can be shown to be related to the mean current density j b through (Kra¨henbu¨hl et al., 2003) j b ¼ 2jvsF R a b :

ð22Þ

Term (21) can then be written in terms of the mean magnetic vector potential a b ; i.e.     ð jvsÞ21 F R21 j b ; j0b V ¼ jvsF R a b ; a 0b V : ð23Þ ls

ls

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711

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Using equations (18) and (23), the magnetodynamic a-v formulation, in complex notation, can finally be written as ðm 21 curl a;curl a0 ÞV2V ls þðsjva;a0 ÞVc þðs gradv;a0 ÞVc

712

ð24Þ

  a 0 ÞV ls þððmF R Þ21 curl a;curl þðsab F R jva; a 0 ÞV ls 2ðj s ;a0 ÞVs 2V ls ¼0; ;a0 [Fa ðVÞ; where curl a is the mean magnetic flux density defined in the homogenized lamination  ls : As for the current density jab, it is considered jointly with ¯jb using the stack V conductivity tensor sab.  ls ; through b a and j b ; enables us to The solution of this formulation (24) in V directly calculate the actual distributions of jb and ba in each separate lamination of Vls using expressions (5) and (6) with equations (8-9), (12) and (13).

Application Two 3D finite element models have been developed and applied to the study of lamination stacks with non-zero global currents. Both models are based on the magnetic vector potential magnetodynamic formulation. The first model directly considers the stack as a set of massive conductors separated by insulating layers (Dular et al., 2003b). In view of the large number of laminations encountered in iron cores, such a model is generally unfeasible and is applied here only to simple local problems for validation. The second model considers the lamination stack as a continuum, using the developed homogenization method. In order to lighten the computational work for the test problem and to focus on the main part of interest, the studied domain is reduced to the lamination stack. The consideration of resistances connecting the lamination ends is done through conductive volume regions in contact with the lamination ends. The direct model needs to mesh each lamination separately, to take the small skin depth into account for high frequencies, while the mesh can be coarser for the homogenized or continuum model for any frequency. The stack contains nine laminations. The laminations are characterized by a thickness d ¼ 1 mm, a relative permeability mr ¼ 1; 000 and a conductivity s ¼ 107 S=m: The stack is excited by an enforced magnetomotive force, giving a flux parallel to the laminations, at frequency 1,000 Hz. A non-uniform even distribution of the resistance along the lamination ends is considered (higher value at stack ends). The magnetomotive force is fixed as a weak global quantity associated with the global magnetic flux, of which the distribution is obtained through the finite element analysis (Dular et al., 1999b). A non-zero flux crossing the lateral boundary of the stack is defined through the use of a surface scalar potential associated with the surface magnetic vector potential (Dular et al., 1999b). This scalar potential is multivalued and has to undergo discontinuities along lines making the lateral surface simply connected. Such constraints defines the boundary conditions. The mean magnetic field h a computed with the continuum model is compared to the field ha obtained with the direct method in the massive conducting regions (Figure 2 ). It is also used to post-compute the actual distribution of ha

through expression (6), with equations (12), (9) and (13), which appears accurate. The same comparison is done for the current density jb (Figure 3), of which the actual distribution is post-computed through expression (5), with equations (8), (12) and (13).

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713

Figure 2. Magnetic field along the stack thickness with the direct method (all laminations considered as massive conductors) and the continuum model (mean value and post-computed actual distribution)

Figure 3. Current density modulus along the stack half-thickness with the direct method and the continuum model

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Conclusions A method has been developed for taking the eddy currents in lamination stacks into account with the finite element method using the 3D magnetic vector potential magnetodynamic formulation. In order to avoid the explicit definition of all laminations, the laminated region is converted into a continuum in which the eddy currents produced by both parallel and perpendicular fluxes are considered, due to adapted terms in the weak formulation, where the mean fields are the primary unknowns. The method is adapted to a wide frequency range, i.e. for low skin depths in the laminations. It appears attractive for directly considering the eddy current effects which are particularly significant for high frequency components. Moreover, the consideration of non-zero global currents in the laminations enables the study of imperfect insulations between the lamination ends. References De Gersem, H. and Hameyer, K. (2001), “A finite element model for foil winding simulation”, IEEE Trans. Magn., Vol. 37 No. 5, pp. 3427-32. Dular, P. and Geuzaine, C. (2002), “Spatially dependent global quantities associated with 2D and 3D magnetic vector potential formulations for foil winding modeling”, IEEE Trans. Magn., Vol. 38 No. 2, pp. 633-6. Dular, P., Henrotte, F. and Legros, W. (1999a), “A general and natural method to define circuit relations associated with magnetic vector potential formulations”, IEEE Trans. Magn., Vol. 35 No. 3, pp. 1630-3. Dular, P., Gyselinck, J., Geuzaine, C., Sadowski, N. and Bastos, J.P.A. (2003a), “A 3D magnetic vector potential formulation taking eddy currents in lamination stacks into account”, IEEE Trans. Magn., Vol. 39 No. 3, pp. 1147-50. Dular, P., Gyselinck, J., Geuzaine, C., Sadowski, N. and Bastos, J.P.A. (2003b), “Modeling of thin insulating layers with dual 3D magnetodynamic formulations”, IEEE Trans. Magn., Vol. 39 No. 3, pp. 1139-42. Dular, P., Gyselinck, J., Henrotte, F., Legros, W. and Melkebeek, J. (1999b), “Complementary finite element magnetodynamic formulations with enforced magnetic fluxes”, COMPEL, Vol. 18 No. 4, pp. 656-67. Golovanov, C., Mare´chal, Y. and Meunier, G. (1998), “3D edge element based formulation coupled to electric circuits”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3162-5. Ida, N. and Bastos, J.P.A. (1997), Electromagnetics and Calculation of Fields, Springer-Verlag, New York, NY. Kra¨henbu¨hl, L., Dular, P., Zeidan, T. and Buret, F (2003), “Homogenization of lamination stacks in linear magnetodynamics”, Proceedings of Compumag 2003, Saratoga Srings, USA. Ren, Z. (1998), “Degenerated Whitney prisms elements – general nodal and edge shell elements for field computation in thin structures”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 2547-50.

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Reliability-based topology optimization for electromagnetic systems Jenam Kang, Chwail Kim and Semyung Wang

Reliability-based topology optimization 715

Department of Mechatronics, Gwangju Institute of Science and Technology (K-JIST), Gwangju, South Korea Keywords Design, Optimization techniques, Topology, Sensitivity analysis Abstract This paper presents a probabilistic optimal design for electromagnetic systems. A 2D magnetostatic finite element model is constructed for a reliability-based topology optimization (RBTO). Permeability, coercive force, and applied current density are considered as uncertain variables. The uncertain variable means that the variable has a variance on a certain design point. In order to compute reliability constraints, a performance measure approach is widely used. To find reliability index easily, the limit-state function is linearly approximated at each iteration. This approximation method is called the first-order reliability method, which is widely used in reliability-based design optimizations. To show the effectiveness of the proposed method, RBTO for the electromagnetic systems is applied to magnetostatic problems.

Introduction The goal of probabilistic optimization is to obtain a reliable design by considering uncertainties. The uncertain variables have variances on a certain design point. In deterministic optimization, these uncertainties are not considered. Thus, deterministic optimum designs can be unreliable for failures. In probabilistic optimization, minimizing a performance and satisfying probabilistic constraints on target should be done simultaneously (Belegundu, 1988). Reliability-based design optimization (RBDO) has the same objective as deterministic optimization. The main difference between deterministic optimization and RBDO is constraints. In RBDO, probabilistic constraints are formulated to construct approximated linear (or quadratic) functions to closely represent the nonlinear limit-state functions for reliability index (or safety index) calculation and optimization using appropriate transformations. The research on the topology optimization of electromagnetic systems began few years back (Dyck and Lowther, 1996; Wang and Kang, 2002; Wang and Kim, 2000). The principle of the topology optimization on electromagnetic systems is the same as that of structural systems (Bendsoe and Kikuchi, 1988). Topology optimization is mostly used for a conceptual design of products while other conventional methods, sizing or shape optimization, are focused on improving the current design. In this paper, a reliability-based topology optimization (RBTO) for electromagnetic systems is proposed. RBTO is based on probabilistic (or reliability) constraints. Permeability, coercive force, and applied current density are considered as uncertain variables. This work was supported by the Center of Innovative Design Optimization Technology (iDOT), Korea Science and Engineering Foundation.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 715-723 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540647

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In order to compute probability constraints, a reliability index approach (RIA) and a performance measure approach (PMA) had been used. Most researchers (Nicolaidis and Burdisso, 1988; Reddy et al., 1994; Wang et al., 1995) have used reliability index as probabilistic constraints in RBDO. This approach is called as RIA. Recently, a PMA (Tu et al., 1999) is proposed for RBDO to evaluate probabilistic constraints in an inverse reliability analysis which is consistent with the conventional RIA. However, in this paper, the PMA is only used to compute probability constraints for electromagnetic systems because the PMA is inherently robust for RBDO and is more efficient in evaluating the violated probabilistic constraint (Tu et al., 1999). To show the effectiveness of the proposed method, RBTO for the electromagnetic systems is applied to magnetostatic problems. To evaluate probabilistic constraints, the limit-state function is linearly approximated at each iteration. This approximation method is called as the first-order reliability method (FORM), which is widely used in RBDO. For reliability and sensitivity evaluation (Haug et al., 1986), subroutines are developed by Visual C++ language. Design optimization tools (DOT) (Vanderplaats Research & Development, Inc., 1999) is used as an optimizer, and ANSYS Inc. (2002) as an analyzer. Then, to show the efficiency of the developed program, the RBTO of a C-core actuator is performed to reduce the volume while maintaining an energy constraint. Topology optimization for electromagnetic systems Topology optimization and design sensitivity analysis An objective of topology optimization is to find an optimum material distribution that maximizes or minimizes an objective function while satisfying the given constraints. A general topology optimization problem takes the form maximize or minimize f ðA; mÞ ~
¼ l V ðAÞ

[A for all A subject to aV ðA; AÞ

ð1Þ

where f ðA; mÞ is an objective function, such as magnetic energy, magnetic force or ~ is
is the virtual vector potential; A torque, or uniform flux; A is the vector potential; A
the energy the space of the virtual vector potential; m is the permeability; and aV ðA; AÞ
the load linear form, are functions of permeability, m, coercive bilinear form; l V ðAÞ force, Hcf, and system output, A. An integral objective function form in electromagnetic systems may be written as ZZZ gðA; 7A; uÞ dV ð2Þ C¼ V

where u is the design vector of permeability and 7A is the gradient of vector potential. The adjoint equation for the adjoint variable l is ZZZ
au ðl; lÞ ¼ ½g A l
þ g 7A 7l

dV ð3Þ V

which must hold for all admissible virtual vector potentials or electric field densities ~ l
[ A:

Using the variational form of the objective function of equation (2) and the direct differentiation result, the sizing design sensitivity equation is (Haug et al., 1986; Wang and Kang, 2002; Wang and Kim, 2000), ZZZ ½g A A0 þ g7A 7A0 þ gu du
dV C0 ¼ V

¼

ZZZ

g u du dV þ

¼

V

717

ZZZ

V

ZZZ

½g A A0 þ g7A 7A0
dV

ð4Þ

V

g u du dV þ l 0du ðlÞ 2 a0du ðA; lÞ

Maxwell and variational equations for electromagnetic systems The magnetostatic field can be described using the set of Maxwell’s equations. 7 £ H ¼ J s;

H¼

1 ðB 2 m0 M Þ; m

7·B ¼ 0

ð5Þ

where H, B, and m0 are the magnetic field intensity, the magnetic flux density, and the permeability of free space, respectively. The vector M represents the magnetization vector (A/m) in the permanent magnet. It is the zero-vector outside the permanent magnet region. The magnetization vector is related to the coercive force and the residual flux density (equation (6)). Hc ¼

m0 1 M ¼ Br m m

ð6Þ

where Hc and Br are the coercive force and the residual magnetic flux density, respectively. By introducing a vector potential B ¼ 7 £ A and considering equation (6), we have a single governing equation
 1 7 £ A ¼ Js þ 7 £ Hc ð7Þ 7£ m
integrating Multiplying both sides of equation (7) with the virtual vector potential A; over the domain, and applying boundary conditions leads to the variational equation
¼ l V ðAÞ
aV ðA; AÞ

~
[A for all A

ð8Þ

~ is the space of the admissible vector potential, the energy bilinear form is where A
 ZZZ  1
¼
aV ðA; AÞ ð7 £ AÞ · 7£A dV ð9Þ m V and the load linear form is

Reliability-based topology optimization

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718


¼ l V ðAÞ

ZZZ


þ H c · ð7 £ AÞ

dV ½ Js · A

ð10Þ

V

Density method As in the study of Wang and Kang (2002) and Wang and Kim (2000), for topology optimization, the density method is used to represent a fictitious material with properties like permeability m as a p-powered function of r, that is,

m ¼ ðm0 mr 2 m0 Þr P þ m0 ; p . 1 Z rðxÞ dV # V ; 0 # rðxÞ # 1; x [ V

ð11Þ

V

where P is the penalization factor, r is the density function, and mr is the relative permeability.

Formulation of RBTO for electromagnetic systems RBTO for magnetostatic field RBTO for electromagnetic systems has the same objective as the deterministic topology optimization. Before the formulation, objective, constraints, design variables and uncertain variables should be selected. In RBTO for electromagnetic systems, the objective function is to minimize the volume. To estimate the failure probability, the magnetic energy is considered as the limit-state function for static problems. As mentioned earlier, design variables are density functions ri in each finite element and the permeability, applied current density, and coercive force are uncertain variables for RBTO. In this paper, all uncertain variables are assumed to be normally distributed. The general form of RBTO for static problems is described as follows. Find the design variable vector r ¼ ðr1 ; r2 ; . . .; rn Þ such that Minimize

Total volume V ðri Þ

Subject to

P f ðXÞ ¼ P½Gðri ; xj Þ , 0
# P t ð12Þ

0 # ri # 1 i ¼ 1; . . .; n and j ¼ 1; 2; 3 where

ð13Þ

G ¼ C 2 Cmin Z

1 gðA; uÞ dV ¼ C¼ 2 V

Z

B · H dV

ð14Þ

V

X ¼ ½X 1 ; X 2 ; X 3
T ¼ ½m; J s ; H c
T

ð15Þ

In the above equations, Pt is the failure probability limit, and C is the magnetic energy. The limit-state means that if the magnetic energy C is smaller than the limit value Cmin, then the system fails.

Reliability-based topology optimization

Design sensitivity of RBTO for magnetostatic field In the magnetostatic field, the topology sensitivity equation can be derived from equation (4). The sensitivity equation is ZZZ g u du dV þ l 0du ðlÞ 2 a0du ðA; lÞ C0 ¼

719

V

¼

ZZZ



½g J s ; g m ; g H c
½dJ s ; dm; dH c


T



dV þ

ZZZ

V

½ldJ s þ ð7 £ lÞdH c
dV V

ZZZ 


 1 2 ð7 £ AÞ · 2 2 dm ð7 £ lÞ dV m V Z Z Z  Z Z Z  ¼ ½g J s þ l
dV dJ s þ ½gH c þ ð7 £ lÞ
dV dH c V

þ

ð16Þ

V

Z Z Z  V

gm þ

  1 ð7 £ AÞ · ð7 £ l Þ dV dm m2

Therefore, the derivatives of G with respect to three uncertain variables are derived as  ZZZ  ›G ›G ›C 1 ¼ ¼ ¼ ð7 £ AÞ · ð7 £ lÞ dV ð17Þ 2 ›X 1 ›m ›m V m

›G ›G ›C ¼ ¼ ¼ ›X 2 ›J s ›J s ›G ›G ›C ¼ ¼ ¼ ›X 3 ›H c ›H c

ZZZ

l dV

ð18Þ

ð7 £ lÞ dV

ð19Þ

V

ZZZ V

Numerical example Magnetostatic problem with one uncertain variable A numerical example is a C-core as shown in Figure 1. The C-core actuator has three parts, which consists of an armature, core, and coil. The width of both armature and core is 20 mm. The length of the core and blade are 60 and 50 mm, respectively. The relative permeability is 1,000 at both core and armature. The current density of the coil is 2.0 A/mm2. The limit-state function is the magnetic energy that should be larger than 140 J/m, which is the target magnetic energy (Ct) in this example. The target magnetic energy can be obtained by FEA of initial FE model that uses all elements of design domain.

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The C-core is modeled as two-dimensional elements (PLANE13) in ANSYS. In this paper, the design domain is meshed into 1,140 elements, and the penalization factor is 3. An uncertain variable is the permeability. The uncertain variable has 10 percent variance of the initial value and is assumed to be normally distributed. The deterministic topology optimization (DTO) is written as Minimize

Total volume of design domain

Subject to

G ¼ C 2 Ct $ 0;

ð20Þ

Ct ¼ 140

where Ct is the target magnetic energy and the optimal result is shown in Figure 2. If the PMA is used for the probabilistic constraint, the RBTO problem is written as Minimize

Total Volume

Subject to

G ¼ C 2 Ct $ 0

when bs ¼ 3;

Ct ¼ 140

ð21Þ

and the optimal result is shown in Figure 3. Reliability reanalysis can be obtained by HL-RF method. A summary for RBTO of the magnetostatic problem with one uncertain variable is shown in Table I. The objective, the used volume of the C-core, in DTO is smaller than results of RBTO. But DTO has low reliability, b ¼ 20:0703: It means that the optimum in DTO has about 50 percent of failure probability. When reliability is considered, more volume is used to satisfy the probabilistic constraint. It is because

Figure 1. C-core actuator

Figure 2. Optimal result using DTO

the feasible design region becomes smaller because of the uncertain variable. RBTO results show that the proposed method achieves the target reliability index. Magnetostatic problem with two uncertain variables The RBTO of the same C-core model is performed with two uncertain variables, permeability and current density. The permeability and current density have 10 and 5 percent variance of the initial values, respectively. Both uncertain variables have assumed to be normally distributed. The optimum results of RBTO are shown in Figure 4. As seen in Figure 4, the optimum shapes are different from the earlier results. Because of increased uncertain variables, more volume is used to satisfy the reliability target in comparison to the RBTO with one uncertain variable. From Table II, we find that the results of RBTO have a much larger volume than DTO. It is possible that a DTO model with a large volume like using a safety

Reliability-based topology optimization 721

Figure 3. Optimal result using RBTO with one uncertain variable

DTO RBTO with PMA

Objective (volume) at mean value (percent)

Energy at mean value (J/m)

Force (Fx) (N m)

Reliability

59.03 73.63

139.96 140.25

26048.5 26053.2

2 0.070274 2.96038

Table I. Comparison between DTO and RBTO with one uncertain variable

Figure 4. Optimal result using RBTO with two uncertain variables

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factor can have a high-reliability design. To check this possibility, a new DTO problem is formulated as Maximize

C ¼ Energy

Subject to

V # 0:864V0

ð22Þ

where V is the volume used in optimal design, V0 is the initial volume, and the optimal result is shown in Figure 5. Table II shows the comparison of these three optimum designs. DTO result with more volume has a better reliability than the initial DTO result. But it is still lower than the reliability target. Also, the magnetic energy of the DTO model is smaller than RBTO models.

Conclusion In this paper, the RBTO for electromagnetic systems using the finite element method is proposed. RBTO for electromagnetic systems is topology optimization based on probabilistic (or reliability) constraints. Permeability, applied current, and coercive force are considered as uncertain variables. Since the PMA is inherently robust for RBDO and is more efficient in evaluating the violated probabilistic constraint, the PMA is used in order to compute the probability constraints for electromagnetic systems. To show the effectiveness of the proposed method, RBTO is applied to magnetostatic problems.

Table II. Comparison between DTO and RBTO with two uncertain variables

Figure 5. Optimal result using DTO with the volume as much as RBTO result

DTO RBTO with PMA DTO (volume used the same as RBTO result)

Objective (volume) at mean value

Energy at mean value

Force (Fx) (N m)

Reliability

59.03 86.40 86.40

139.96 140.81 140.73

2 6048.5 2 6110.6 2 6102.9

2 0.070274 3.00241 2.40125

References ANSYS, Inc. (2002), ANSYS Users Manual. Belegundu, A.D. (1988), “Probabilistic optimal design using second moment criteria”, J. Mech. Transmissions Automat. Design ASME, Vol. 110, pp. 324-9. Bendsoe, M.P. and Kikuchi, N. (1988), “Generating optimal topologies in structural design using a homogenization method”, Computer Method in Applied Mechanics and Engineering, Vol. 71, pp. 197-224. Dyck, D.N. and Lowther, D.A. (1996), “Automated design of magnetic devices by optimizing material distribution”, IEEE Transactions on Magnetics, Vol. 32 No. 3. Haug, E.J., Choi, K.K. and Komkov, V. (1986), Design Sensitivity Analysis of Structural Systems, Academic Press, London. Nicolaidis, E. and Burdisso, R. (1988), “Reliability based optimization: a safety index approach”, J. Eng. Mech. Div. ASCE, Vol. 100, pp. 111-21. Reddy, M.V., Grandhi, R.V. and Hopkins, D.A. (1994), “Reliability based structural optimization: a simplified safety index approach”, Computers and Structures, Vol. 53 No. 6, pp. 1407-18. Tu, J., Choi, K.K. and Park, Y.H. (1999), “A new study on reliability-based design optimization”, Journal of Mechanical Design, Vol. 121, pp. 557-64. Vanderplaats Research & Development, Inc. (1999), Design Optimization Tools User Manual. Wang, S. and Kang, J. (2002), “Topology optimization of nonlinear magnetostatics”, IEEE Transactions on Magnetics, Vol. 38 No. 2, pp. 1029-32. Wang, S. and Kim, Y. (2000), “A study on the topology optimization of electromagnetic systems”, CEFC 2000, p. 336. Wang, L.P., Grandhi, R.V. and Hopokins, D.A. (1995), “Structural reliability optimization using an efficient safety index calculation procedure”, International Journal of Numerical and Mathematical Engineering, Vol. 38 No. 10, pp. 1721-38.

Reliability-based topology optimization 723

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724

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A “quasi-genetic” algorithm for searching the dangerous areas generated by a grounding system Marcello Sylos Labini and Arturo Covitti Department of Electrotechnics and Electronics, Polytechnic of Bari, Bari, Italy

Giuseppe Delvecchio University of Bari, Italy

Ferrante Neri Department of Electrotechnics and Electronics, Polytechnic of Bari, Bari, Italy Keywords Programming, Algorithmic languages, Soil testing Abstract Sets out a method for determining the dangerous areas on the soil surface. The touch voltages are calculated by a Maxwell’s subareas program. The search for the areas in which the touch voltages are dangerous is performed by a suitably modified genetic algorithm. The fitness is redefined so that the genetic algorithm does not lead directly to the only optimum solution, but to a certain number of solutions having pre-arranged “goodness” characteristics. The algorithm has been called “quasi-genetic” algorithm and has been successfully applied to various grounding systems.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 724-732 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540656

Introduction As is well-known, a grounding system leaking a fault current IF generates touch voltages UT( P ) in all the points P of the soil surface. These voltages can be dangerous. It is therefore very important to search for the areas of the soil surface where the touch voltages are greater than the maximum permissible touch voltage UTp, this value being fixed by Standards on the basis of the time tF during which the fault persists. In previous papers, the authors have defined two optimization methods for determining the point of the soil surface in which the touch voltage UT( P ) is maximum: the first one consists of a stochastic method followed, in cascade, by a deterministic method (Covitti et al., 2002a, b); the second method is based on the genetic algorithms (Delvecchio et al., 2003). In this paper, the same authors do not suggest the search for the point of the soil surface in which the touch voltage is maximum but they suggest the search for all the soil surface areas in which the touch voltages UT( P ) are higher than the permissible touch voltage UTp. In this way, the designing of the grounding system is less hard since it is no more necessary to design a completely safe grounding system; on the contrary, we need to determine, during the designing, only the dangerous areas and then make these areas beyond the reach of people. The determination of the dangerous areas is carried out by a “quasi-genetic” algorithm. It is still based on the genetic algorithms (De Jong et al., 1997; Mitchell, 1996; Rudnicki, 2000), but a new formulation of the fitness has been worked out, so that there is no more the need to define an individual as being “the best” but the individuals are classified as “good” and “bad”. The individuals must have some pre-arranged “goodness” requirements so as “to survive”.

The calculation of the touch voltage UT( P ) by the Maxwell’s subareas method A good method for calculating the touch voltages UT( P ) generated by a grounding electrode leaking a known fault current IF is the Maxwell’s subareas method (Sylos Labini et al., 2003). It, briefly, consists in subdividing the electrode into a suitable number N of elementary parts (subareas), in calculating the voltages U 1 ; U 2 ; . . .; U N taken by each subarea and produced by all the leaking subareas, and in imposing the equipotentiality of the grounding electrode. The voltage taken by the generic subarea i is given by: U i ¼ Ri;1 I 1 þ Ri;2 I 2 þ · · · þ Ri;N I N

ði ¼ 1; 2; . . .; N Þ

ð1Þ

where Ii are the currents (or subcurrents) leaked by each subarea; Ri,j is the voltage coefficient between the subareas i and j. Each Ri,j can be calculated by the formulas given by Sylos Labini et al. (2003) on the basis of the voltage generated by the inducing subarea j and by its electrical image j0 in the barycenter of the induced subarea i. If we impose the equipotentiality of the whole electrode (i.e. if we impose U 1 ¼ U 2 ¼ · · · ¼ U N ¼ U E ; U E being the earthing voltage) the Maxwell’s method leads us to write the following system of ðN þ 1Þ linear equations: ( Ri;1 I 1 þ Ri;2 I 2 þ · · · þ Ri;N I N ¼ U E ði ¼ 1; 2; . . .; N Þ ð2Þ I1 þ I2 þ · · · þ IN ¼ IF The solution of this system gives the earthing voltage UE and also the N subcurrents Ii. These subcurrents are needed for calculating the voltage U( P ) in any point of the soil surface. In fact, the contribution, in terms of voltage, given by each subarea j and by its electrical image j0 in a point P of the soil surface can be calculated by the equation: U j ðPÞ ¼ RP; j I j

ð3Þ

where RP, j is the voltage coefficient existing between the inducing subarea j and the induced point P (RP, j can be calculated by the formulas given by Sylos Labini et al. (2003)). Consequently, the total voltage U( P ) due to all the subareas can be calculated as the sum of the voltages generated in P by each subarea and by its electrical image by the formula: U ðPÞ ¼

N X

U j ðPÞ:

ð4Þ

j¼1

Once the voltage U( P ) is known, it is possible to calculate the touch voltage in P by the well-known formula: U T ðPÞ ¼ U E 2 U ðPÞ:

ð5Þ

The “quasi-genetic” algorithm From what we have seen in the previous section, for a given grounding system that injects in the soil a known fault current IF the touch voltage in a generic point P of the soil surface is a function UT( P ) of the point P, that is to say, it is a function of the

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x and y co-ordinates of this point. In a similar way, the permissible touch voltage UTp can be considered as being a function of the point P of the soil surface, this function taking the constant value UTp according to the x and y co-ordinates of P. Consequently, it is possible to draw attention to the probable danger of the leaking grounding system on the basis of the relative position, in the space, of both the surface representing the function U T ðx; yÞ and the plane UTp. The following three cases can occur: (1) The plane UTp is above the surface of U T ðx; yÞ (Figure 1); (2) The plane UTp is below the surface of U T ðx; yÞ (Figure 2); (3) The plane UTp intersects the surface of U T ðx; yÞ (Figure 3). In case (1) all touch voltages are smaller than UTp, that is no point of the soil surface is dangerous. Consequently, the grounding system designed on the basis of the value UTp is not dangerous. In case (2) all the points of the soil surface are dangerous. The grounding system under examination is dangerous, and so a new grounding system must be designed. In case (3), there are only some areas of the soil surface in which the touch voltage exceeds the permissible value UTp. In this case there is no need to design another

Figure 1. A well-designed grounding system: there is no dangerous point on the soil surface

Figure 2. A badly-designed grounding system: all the points on the soil surface are dangerous

A “quasi-genetic” algorithm

727 Figure 3. Four dangerous areas generated by the grounding system

grounding system, provided that the dangerous areas existing on the soil surface are carefully found. In other words, it is necessary to determine on the soil surface the points obtained as orthogonal projections of the intersection points between the surface UT( P ) and the plane UTp (Figure 3). The determination of the dangerous areas is carried out by a “quasi-genetic” algorithm. As is well-known, the genetic algorithms are methods based on the analogy with the natural systems. These genetic algorithms have often been implemented in order to search for the optimum solution of a given problem (De Jong et al., 1997). The method suggested in this paper, i.e. the “quasi-genetic” algorithm, is partly identical to the genetic algorithm methods but it does not search for the only individual having a better fitness compared to other individuals, because it searches for various sets of individuals, each set having a touch voltage which is higher than a pre-fixed value. Briefly, the “quasi-genetic” algorithm consists of the following. A first sampling of the function UT( P ) is done at random. This sampling concerns a small number Np of points P of the soil surface. These points represent the population to which we apply the genetic algorithm method. Each point Pðx; yÞ; in terms of genetic algorithms, is an individual having x and y as chromosomes. The x and y co-ordinates of each point P are expressed in binary numbers. The binary numbers are considered to be string variables, so they undergo, in each iteration, cross-over and mutation. The authors have chosen the “one-point crossover” technique and established that each individual thus generated has a 4 percent probability of undergoing a mutation. Regarding cross-over, the probability that cross-over will occur is equal to 100 percent. In other terms, in each iteration, N cross-over always occur, where N is a number chosen by us a priori. Therefore, the choice of the pairs of chromosomes which undergo cross-over occurs, in a stochastic way, N times each iteration. This way of choosing the pairs of chromosomes implies that the same chromosome can generate individuals several times in the same iteration. At each iteration, the population of individuals is made up of parents as well as of newly generated individuals. For each iteration, the touch voltage of each individual (or point P) is calculated. This value is then compared with the permissible touch voltage UTp. The individuals whose UT( P ) is lower than the value UTp are “killed”,

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which means that they are removed from the memory location, while the individuals (dangerous points) whose UT( P ) exceeds the value UTp “survive”. Moreover, in the same iteration, the data are further processed by means of a “virtual filter” that removes all the “duplicate” individuals, that is all the newly generated points Pðx; yÞ having the same genetic makeup (i.e. the same co-ordinates x and y, respectively) of the individuals which already belong to the population of dangerous points. If the method now explained is iterated an adequate number of times it allows the determination of all the dangerous areas existing in the domain under examination. The procedure allows the population “to grow” or “to contract” according to need: the final population can have a number of individuals different from the initial one, this number being smaller or greater than the initial number according to the number of dangerous areas that really exist on the soil surface. In short, it is worth noting that in this paper the fitness function has been defined by the authors in a different way compared to Delvecchio et al. (2003). In fact, in Delvecchio et al. (2003) the fitness of each individual (or point P) was the touch voltage in P; now the fitness takes only the values 0 and 1, according to whether UT( P ) is smaller or greater than UTp, respectively. Numerical results Some results concerning two rectangular meshed grounding grids are given as an example in this paper. The first grounding grid is a classic grid, that is made up of regular meshes; it is 60 £ 40 m in size and has six regular meshes. The second grid is the same grid studied by Huang et al. (1995); it is 80 £ 60 m in size and is made up of five horizontal conductors and seven vertical conductors; this grid has 24 unequally spaced meshes which are wider at the center and narrower at the grounding system’s edges. We will call the first grounding grid “classic” and the second one “unequally spaced” later on; this way of indicating the latter has been taken from Huang et al. (1995). The “unequally spaced” grounding system has the following advantages compared to the “classic” one (Otero et al., 1998). If we consider two meshed rectangular grounding grids (one is “classic” and the other is “unequally spaced”) having both the same perimeter and the same number of conductors according to the two directions, burying to the same depth in the same soil and leaking the same fault current IF, the “unequally spaced” grounding grid generates touch voltages which are smaller compared to those generated by the “classic” grounding grid; moreover, the surface of the touch voltages UT( P ) generated by the “unequally spaced” grounding grid is flatter than that generated by the “classic” grounding grid. In other words, the highest values of touch voltages generated by the “unequally spaced” grounding grid are very similar to each other, which does not occur in the case of the “classic” one (compare Figures 4 and 5). The two grounding grids under examination are both made up of cylindrical conductors having a section S ¼ 50 mm2 ; are buried to a depth h ¼ 0:5 m in a homogeneous soil having resistivity r ¼ 100 V m: Moreover, I F ¼ 100 and 400 A are the fault currents leaked by the “classic” and “unequally spaced” grounding grids, respectively. U Tp ¼ 37 and 58 V are the permissible touch voltages for the “classic” and “unequally spaced” grounding grids, respectively. Regarding the soil surface areas under study, we have considered the following areas:

.

.

in respect of the “classic” meshed grid, a rectangular area whose sides are 61 £ 41 m (Figure 4); in respect of the “unequally spaced” grid, a rectangular area whose sides are 80 £ 60 m (Figure 5).

As for the number of initial points to take into account (that is, the number of individuals constituting the initial population) N p ¼ 100 points has been chosen. With regard to the number of cross-over to carry out in each iteration, N ¼ 100 cross-over has been chosen. Moreover, a 4 percent probability of mutation has been set. The results obtained by an Intel CELERON processor having a clock frequency equal to 1,000 MHz, relating to the two cases, are given in Table I and Figures 6 and 7, and in Table II and Figures 8 and 9, respectively. For each grounding system five simulations have been carried out and for each of them the number of iterations has been fixed equal to N iter ¼ 10: Figures 7 and 9 and Tables I and II show that, for both grounding electrodes, the “quasi-genetic” algorithm converges quickly towards the dangerous areas, with a very small number of iterations and extremely low calculation times.

A “quasi-genetic” algorithm

729

Figure 4. Voltages on the soil surface (in volt) generated by a “classic” grid (60 £ 40 m in size, with six regular meshes)

Figure 5. Voltages on the soil surface (in volt) generated by an “unequally spaced” grid (80 £ 60 m in size, with 24 unequally spaced meshes)

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It is worth noting that a further increase in the number of iterations does not involve advantages over the determination of dangerous areas and calculation times. Conclusions The algorithm carried out allows to foresee, in the designing stage, all the probable dangerous areas that the grounding system can generate on the soil surface. It follows

Simulations carried out Table I. “Classic” meshed grounding grid (Niter ¼10)

Figure 6. “Classic” grid: population at the third iteration

Figure 7. “Classic” grid: population at the tenth iteration

1 2 3 4 5

Number of points found

Calculation times (s)

752 747 786 767 764

8.34 7.90 9.00 8.54 8.23

that this algorithm turns out to be very useful since the forecast of the dangerous areas allows a considerable saving in the costs needed to make the same grounding system. On the basis of the various designing tests, it may be inferred that the “quasi-genetic” algorithm is particularly effective not only for the low calculation times but also for the reliability of the results. From a scientific point of view, the authors have given a further contribution compared to the genetic algorithm method previously defined by the same authors. In fact, the new algorithm is still based on the characteristics peculiar to the Simulations carried out 1 2 3 4 5

Number of points found

Calculation times (s)

558 549 576 584 539

5.71 5.72 6.13 6.19 5.64

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Table II. “Unequally spaced” meshed grounding grid (Niter ¼ 10)

Figure 8. “Unequally spaced” grid: population at the third iteration

Figure 9. “Unequally spaced” grid: population at the tenth iteration

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evolutionary programming; however it does not aim at optimizing an objective function but at searching for sets of dangerous points. References Covitti, A., Delvecchio, G., Marzano, C. and Sylos Labini, M. (2002a), “A global optimization method for designing meshed grounding grids”, Proceedings CEFC 2002, Tenth Biennial IEEE Conference on Electromagnetic Field Computation, 16-19 June, Perugia, Italy, p. 321. Covitti, A., Delvecchio, G., Sylos Labini, M. and Verde, D. (2002b), “A global optimization method for determining the maximum touch voltage generated by grounding systems”, Studies in Applied Electromagnetic and Mechanics, IOS Press, Amsterdam, Vol. 22, pp. 373-8. De Jong, K., Fogel, D.B. and Schwefel, H.P. (1997), “A history of evolutionary computation”, in Ba¨ck, Th., Fogel, D.B. and Michalewicz, Z. (Eds), Handbook of Evolutionary Computation, Oxford University Press, New York, NY, Institute of Physics Publishing, Bristol, pp. A2.3:1-A2.3:6. Delvecchio, G., Sylos Labini, M. and Neri, F. (2003), “A genetic algorithm method for determining the maximum touch voltage generated by a grounding system”, in Rudnicki, M. and Wiak, S. (Eds), Optimization and Inverse Problems in Electromagnetism, Kluwer Academic Publisher, Dordrecht, pp. 85-92. Huang, L., Chen, X. and Yan, H. (1995), “Study of unequally spaced grounding grids”, IEEE Power Delivery, Vol. 10 No. 2, pp. 716-22. Mitchell, M. (1996), Introduction to Genetic Algorithms, MIT Press, Boston, MA. Otero, A.F., Cidras, J. and Garrido, C. (1998), “Genetic algorithm based method for grounding grid design”, Evolutionary Computation Proceedings, IEEE World Congress on Computational Intelligence, 4-9 May. Rudnicki, M. (2000), “Evolutionary and genetic tools in optimization”, Jyvaskyla 10th Summer School, Lecture Notes. Sylos Labini, M., Covitti, A., Delvecchio, G. and Marzano, C. (2003), “A study for optimizing the number of subareas in the Maxwell’s method”, IEEE Transactions on Magnetics, Vol. 39 No. 3, pp. 1159-62. Further reading Amoruso, V., De Nisi, S., Negro, G. and Sylos Labini, M. (1995), “A complete computer program for the analysis and design of grounding grids”, International Journal of Power and Energy Systems, Vol. 15 No. 3, pp. 122-7.

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Development of optimizing method using quality engineering and multivariate analysis based on finite element method

Development of optimizing method 733

Yukihiro Okada Matsushita Electric Industrial Co., Ltd, Morofuku, Daito, Osaka, Japan

Yoshihiro Kawase and Shinya Sano Department of Information Science, Gifu University, Yanagido, Gifu, Japan Keywords Multivariate analysis, Finite element analysis, Torque, Optimization techniques Abstract Describes the method of optimization based on the finite element method. The quality engineering and the multivariable analysis are used as the optimization technique. In addition, this method is applied to a design of IPM motor to reduce the torque ripple.

1. Introduction The finite element analysis is widely used for a design of many devices and equipments. It is combined with an optimization technique and is useful in shortening the development period. Optimization technique has various methods and many applications. Since an interior permanent magnet (IPM) motor can use both reluctance torque and magnet torque, it can realize a more efficient motor. However, the IPM motor has many studies such as arrangement of the permanent magnet or a current phase and so on. Especially when using in a household-electric-appliances field, low vibration is also required with high efficiency. From this reason, the reduction of torque ripple, which cause the vibration, becomes very important. So, we have developed an optimizing method based on the finite element method (FEM), and the quality engineering (QE) (Taguchi, 1976; Wang et al., 1999) and the multivariable analysis (MA) (Haga and Hashimoto, 1980) are used as the optimization technique. In addition, this optimizing method is applied to reduce the torque ripple of an IPM motor. 2. Optimizing method Our optimizing method consists of two steps. The flow chart of this method is shown in Figure 1. 2.1 Step 1 An influence of design parameters on an output is investigated by the QE. In the QE, an ideal function as shown in Figure 2 is decided as the basic characteristic. Using the basic characteristics, both restraint of variation to the output Y and adjustment of inclination are performed. As a factor to deal with, there are some factors shown in Figure 3. Here, the signal factor M is input data. The noise factor is the parameter that cannot be changed by a designer (for example, temperature, dispersion of material and so on).

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 733-739 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540665

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Figure 1. Flowchart

Figure 2. Basic characteristics

Figure 3. Relationship of some factors

The control factor is the design parameter that can be changed by a designer (for example, size, a kind of material and so on). It is shown that output Y changes a value by the control factor or the noise factor even if the signal factor is same. Next, the procedure of the QE is shown briefly. First, design parameters, signal factor, and noise factor are determined. Next, the levels of the design parameters and each factor are decided and an orthogonal array is decided from the number and the level of design parameters. All the models obtained from the orthogonal array are analyzed. Finally, it asks for the rate of contribution and the factorial effect figure, and the influence on the output is investigated. Here, if the influence of the parameter is small, its parameter

can be excluded from the object for examination. By reducing the number of parameters to be examined, this optimizing method carries out a shorter development period. 2.2 Step 2 An approximation formula that is called the multiple regression equation is generated from the MA using the results of the QE at Step 1. From this approximate formula, the optimal combination of design parameters is searched. The approximation formula is a second-order equation like the following formula. n n n X n X X X ai X i þ aii X 2i þ aij X i X j ð1Þ Output ¼ a0 þ i

i

Development of optimizing method 735

i–j i–j

where ai ði ¼ 0; . . .; nÞ is a coefficient, X i ði ¼ 1; . . .; nÞ is the design parameter and n is the number of design parameters. When there are few data than (n+1)(n+2)/2, equation (2), not equation (1), is used. n n X X ai X i þ aii X 2i ð2Þ Output ¼ a0 þ i

i

The important point in generation of the approximation formula is which term to take into consideration. Because, the accuracy of approximation formula may become bad if all terms are taken into consideration. Here, the prediction sum of squares (PSS) (Haga and Hashimoto, 1980) is used to select which term is considered. The accurate approximation formula is obtained using the combination that has the smallest PSS. The optimal combination of the design parameter is searched for using the approximation formula, and it is checked by the FEM. If the required characteristic is not satisfied, the approximation formula is modified and the same process is repeated. At Step 2, most calculation used the approximation formula (1) or (2). Since the FEM analysis that needs much calculating time is used only for the check, it can reduce the calculation time. Thus, at Step 1, the number of parameters is cut down; at Step 2, the number of calculation by the FEM is cut down. That is, the optimal combination of the design parameters, which satisfies the required characteristic, can be searched efficiently. 3. Application to design of IPM motor We applied this optimizing method for the design of a rotor structure of IPM motor (Honda et al., 1997). The aim is to reduce the torque ripple while maintaining the present average torque. The analysis model of initial shape is shown in Figure 4. There are four poles and 24 slots. The permanent magnet is ferrite magnet. The coil is distributed winding, and the current waveform is sine waveform. The two-dimensional

Figure 4. IPM motor of initial shape

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Figure 5. Mesh of initial shape

Figure 6. Distribution of magnetic flux density

Figure 7. Design parameters

magnetic field analysis is performed. The analysis region is 1/4 region from symmetry. The mesh of initial shape is shown in Figure 5. The number of elements and nodes is about 9,000 and 4,600, respectively. In the gap, it is divided into six in the radial direction and 90 in the circumferential direction. For the calculation of torque, Maxwell’s stress is used, the integrated plane is made into the center of the gap. The magnetic flux density distribution is shown in Figure 6. In Figure 6, the portion shown with a circle causes magnetic saturation. The design parameters A-D examined by the QE are shown in Figure 7. The parameter A is the distance from the center of circle, parameter B is the distance between the outside and inside permanent magnet, parameter C is the thickness of outside permanent magnet, and parameter D is the thickness of inside permanent magnet. These parameters are changed with the three levels. As the noise factor, the value of magnetization Mr is varied, and has two levels (N1 : Mr and N2 : 0.9*Mr).

As the signal factor, the value of current I0 is changed, and has three levels (M1 : I0/3, M2 : 2/3*I0, and M3 : I0). So, the orthogonal array L9 shown in Table I is used because the number of design parameter is four and the number of level is three. Nine models shown in Table I are analyzed. The factorial effects for average torque and torque ripple are shown in Figure 8. The rate of contribution is shown in Table II. From Figure 8 and Table II, the influence of each design parameter is obtained quantitatively. The parameter A has the largest influence in all parameters. Especially, the torque ripple is decided by the parameter A. Since the inclination of the average torque and torque ripple for the parameter A is reverse, it has the relation of a trade-off. Next, although the parameters B and D affect Model no.

A

B

C

D

1 2 3 4 5 6 7 8 9

1 1 1 2 2 2 3 3 3

1 2 3 1 2 3 1 2 3

1 2 3 2 3 1 3 1 2

1 2 3 3 1 2 2 3 1

Development of optimizing method 737

Table I. Orthogonal array L9

Figure 8. Factorial effects

Design parameters Average torque Torque ripple

A

B

C

D

41 87

25 5

7 4

27 4

Table II. Rate of contribution (percent)

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Figure 9. Error of the approximation formula

Figure 10. Comparison of the presumed value and FEM

Figure 11. Comparison of torque waveforms

average torque, they do not affect the torque ripple. Finally, the parameter C has little influence. Here, although the parameter C can be excluded for examination, the approximation formula is created using all parameters to reduce the amount of magnet. The error of the average torque and torque ripple obtained from the approximation formula is shown in Figure 9. In Figure 9, the model number is shown in Table I. The average error is 0.5 percent for average torque, 2.8 percent for torque ripple, and the approximation formula is in good agreement. The comparison of the presumed value and the FEM is shown in Figure 10 at the optimal combination obtained from the approximation formula for the average torque and torque ripple. Although the error of the torque ripple is large, the average torque is maintained and the torque ripple can be reduced. Using the previous data and the new data shown in Figure 10, all coefficients are modified and the new approximation formula is created. Thus, by searching the optimal combination of design parameter from the approximation formula, checking analysis by FEM is performed. The torque waveforms for the initial shape and final one are shown in Figure 11. From Figure 11, the average torque is the same and

the torque ripple is reduced to 65 percent. Furthermore, the amount of permanent magnets can be reduced to 14 percent with change of arrangement and thickness. 4. Conclusions The optimizing method, which used the QE and the MA based on the FEM, is described. This method is applied to the design of the IPM motor. The influence of four parameters on average torque and torque ripple is obtained quantitatively. Comparing with the initial shape, the final one can maintain average torque and cut down the torque ripple by 65 percent. Furthermore, the amount of permanent magnets can be reduced. From these results, the usefulness of this optimizing method is found. References Haga, T. and Hashimoto, T. (1980), Regression Analysis and Principal Component Analysis, JUSE Press Ltd.. Honda, Y., Yokote, S., Higaki, T., Morimoto, S. and Takeda, Y. (1997), “Magnet design and motor performances of a double layer interior permanent magnet synchronous motor”, Trans. IEE Japan, Vol. 117-D No. 10, pp. 1221-6. Taguchi, G. (1976), Design of Experiments, MARUZEN Co. Wang, H.T., Liu, Z.J., Chen, S.X. and Yang, J.P. (1999), “Application of Taguchi method to robust design of BLDC motor performance”, IEEE Trans. Magn., Vol. 35 No. 5, pp. 3700-2.

Development of optimizing method 739

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An improved fast method for computing capacitance L. Song and A. Konrad

740

Department of Electrical and Computer Engineering, University of Toronto, Toronto, Ontario, Canada Keywords Capacitance, Computer applications, Production cycle Abstract In the design of chip carriers, appropriate analysis tools can shorten the overall production cycle and reduce costs. Among the functions to be performed by such computer-aided engineering software tools are self and mutual capacitance calculations. Since the method of moments is slow when applied to large multi-conductors systems, a fast approximate method, the average potential method (APM), can be employed for capacitance calculations. This paper describes the improved average potential method, which can further reduce the computational complexity and achieve more accuracy than the APM.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 740-747 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540674

Introduction A ceramic chip carrier is the support structure for integrated circuits, where the connections between chips and pins of chips constitute a complicated 3D network of conductors. The design of this network would have a significant influence on the performance of integrated circuits. An appropriate computer-aided engineering (CAE) tool is therefore desirable for predicting the performance of chip carrier networks at the design stage. The focus of this paper is a fast method for the computation of chip carrier lead capacitances. Self capacitance can be calculated as the ratio of total charge on the conductor to the potential of conductor relative to ground potential. Mutual capacitance is similarly obtained from total charge and potential difference. With the knowledge of the conductor geometry, the so-called potential coefficient matrix can be calculated, and then the potentials can be expressed as the product of the potential coefficient matrix and the charge densities. In the method of moments (MoM) (Harrington, 1983) conductors are assigned constant potential values. The MoM yields the charge densities by inverting the potential coefficient matrix. The method works well, but is extremely slow for complex situations since it involves the inversion of a full matrix. An approximate but very fast method, the average potential method (APM) (Konrad and Sober, 1986), is based on the assumptions of constant charge density distribution and varying potential on a conductor. Although the assumptions of APM are physically absurd, it has been shown that numerical results obtained by the APM are actually quite good (Konrad and Sober, 1986). APM does not require matrix inversion for the computation of self capacitances and requires the inversion of only a 2 £ 2 matrix for the computation of mutual capacitances. Motivated by the physical fact that charge density on a conductor is relatively high along edges and low around the center, the improved average potential method (IAPM) is proposed. Instead of assuming constant charge density distribution, IAPM assumes that charge densities are zero around the center of a conductor, but non-zero constant along the edges. For 3D conductors, edges can be defined as regions on surfaces where curvatures are relatively large. The dimension of the potential coefficient matrix and hence the complexity are further reduced under the assumptions of IAPM. Finally, it is

shown that IAPM can achieve a more accurate result than APM, since it has a more physically reasonable assumption than APM. The IAPM Computation of self capacitance Consider a conductor with its surface divided into N subsections {Dsn }: The charge density as a function of position can be represented by

sðx; yÞ <

N X

an f n

ð1Þ

n¼1

where

( fn ¼

1

on Dsn

0

on Dsm ; m – n

ð2Þ

The representation of potential at the midpoint {xm ; ym ; zm } of each Dsm can be derived from Poisson’s equation (Harrington, 1983) as Vm ¼

N X

m ¼ 1; 2; . . .; N

lmn an

ð3Þ

n¼1

where lmn ¼

Z

dx0 Dxn

Z Dyn

dy0

Z Dzn

dz0

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 2 0 4p1 ðxm 2 x Þ þ ðym 2 y0 Þ2 þ ðzm 2 z0 Þ2

ð4Þ

Equation (3) can be written in matrix form as V ¼ La

ð5Þ

The self capacitance of the conductor can be computed with the knowledge of V and a as follows N X

Cg ¼

an · Dsn

n¼1 N X

:

ð6Þ

V n =N

n¼1

L is the potential coefficient matrix mentioned in the Introduction and can be computed from equation (4) or its approximate equations (Harrington, 1983) with the knowledge of the conductor geometry. The MoM assumes that V is a constant column matrix and calculates a from a ¼ L 21 V

ð7Þ

The APM assumes that a is a constant column matrix, so one can calculate V directly from equation (5).

An improved fast method

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IAPM differs from APM in that an is not assumed to be constant for all n. Without loss of generality, one can assume that ( 1; n [ E an ¼ 0; n [ C ð8Þ

742

where E denotes the set of subsections along the edges of the conductor, and C denotes the set of subsections around the center of the conductor. One has E < C ¼ {1; 2; . . .; N } E >C ¼f

ð9Þ

If Ne and Nc denote the size of the sets E and C, respectively, one can reduce the N £ N matrix L to a N £ N e matrix Le, and rewrite equation (5) as V ¼ Leae

a e ¼ ½an jn [ E
T

ð10Þ

Hence the complexity is further reduced by N c =N : The key point in implementing the IAPM is how to separate the subsections in the sets E and C. The example given below demonstrates it for a specific case where the conductor in consideration is a square plate. In this case, the square plate is subdivided into N small squares. Suppose the side length of the square conductor is a, and the side length of the small subsections is b, then the relationship between a and b is given by a ¼ N s b; where Ns is an integer related to N by the equation N ¼ N 2s : The IAPM forces the charge densities at the center region of the conducting plate to be zero. The center region is defined as a square region with sides having a length equivalent to Wz subdivisions. An example with N s ¼ 7 and W z ¼ 3 is shown in Figure 1. Note that Ns and Wz must either be both odd, or both even. Also, they should satisfy 0 # W z # ðN s 2 2Þ: It is easy to verify that for a square conducting plate, N c ¼ W 2z

Figure 1. Subsections and assigned charge densities for a square, conducting plate

and

N e ¼ N 2s 2 W 2z

ð11Þ

As a rule of thumb one could set W z < N s =2; which would reduce the computational complexity by approximately 25 per cent. Note that the IAPM becomes the APM when W z ¼ 0; which means that APM is a special case of IAPM. Computation of mutual capacitance To illustrate the computation of mutual capacitance, the method described above is applied to the symmetric, two-conductor problem of two square, parallel, conducting plates. If both conductors are subdivided into N subsections, one obtains a 2N £ 2N potential coefficient matrix: " t# " t# " tt # ½l
½ltb
V a ; a¼ ð12Þ L¼ and V ¼ bt bb b ½l
½l
V ab Owing to symmetry, ½ltt
¼ ½lbb
;

½ltb
¼ ½lbt
;

V t ¼ 2V b ;

Thus, from equation (5), one obtains (Harrington, 1983),  tt  t t lmn 2 ltb mn a ¼ V

a t ¼ 2a b

ð13Þ

ð14Þ

Following the same IAPM procedure as in the case of the single, square, conducting plate, one can create a column matrix of a t and compute V t from equation (14). Then, the mutual capacitance is given by N P

Cm ¼

n¼1 N P

atn Dsn

2

n¼1

:

ð15Þ

V tn =N

Simulations and comparisons Comparisons are made between the MoM, APM and IAPM by evaluating the self and mutual capacitances. The self capacitance of a single, square, conducting plate, of side length a, is computed. Similarly, the mutual capacitance of two square, parallel, conducting plates is found. Each plate has a side length of a, and the distance between the plates is d. Since Wz is an important parameter in both cases, first the accuracy of the IAPM as a function of Wz is shown in Figure 2. In this example, a ¼ 10 m and N s ¼ 16: For the mutual capacitance calculations, d ¼ 1 m is used. The results obtained by the MoM can be considered as accurate for N s ¼ 16: One can see from Figure 2 that by properly choosing Wz, the IAPM can perform better than the APM. Note that when W z ¼ 0; the IAPM becomes identical to the APM. In the following simulations, Wz is set to an integer nearest to N s =2: Figure 3 compares the convergence of the MoM, APM and IAPM for a ¼ 10 m and N s ¼ 16: For the mutual capacitance calculations d ¼ 1 m is used. Ns varies from 2 to 16, which means the number of subsections varies from 4 to 256. Again one can see that the IAPM

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Figure 2. Capacitance calculations by the IAPM when Wz varies

results are closer to the MoM results than the APM results. Note that the IAPM also achieves lower complexity. Finally, the performances of the MoM, APM, and IAPM are compared for variations in the side length a. The variation of self capacitance is shown in Figure 4(a). The normalized mutual capacitance given by C m d=ð1a 2 Þ is shown in Figure 4(b) as a function of the normalized distance, d=a: In these simulations N s ¼ 16:

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Figure 3. Comparison of the convergence of MoM, APM and IAPM when Ns varies

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Figure 4. Comparisons for variation of geometrical parameters a and d/a

Conclusions The MoM, although accurate, is too slow for problems where hundreds of conducting paths are involved. While the MoM assumes a constant potential for conductors and calculates the charge densities by inversion of a full matrix, an approximate alternative is to assume that charge densities are given and calculate the average potential by simple matrix multiplication. While the APM assumes that the charge densities are

constant, in the IAPM presented above the charge density distribution becomes a {0, 1} function on subsections. Since this IAPM assumption is closer to reality, the calculated capacitances are more accurate than the ones obtained by the APM. References Harrington, R.F. (1983), Field Computation by Moment Methods (also IEEE Press, Piscataway, NJ, 1994), Robert E. Krieger Publishing Co., Malabar, FL, Chapter 2. Konrad, A. and Sober, T.J. (1986), “A fast method for computing the capacitance of ceramic chip carrier conductors”, Proc. IEEE Workshop on Electromagnetic Field Comput. (Cat. No. 87-TH0192-5), 20-21 October 1986, Schenectady, New York, NY, pp. C41-7.

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Power losses analysis in the windings of electromagnetic gear

748

Institute of Control and Systems Engineering, Poznan´ University of Technology, Poznan´, Poland

Andrzej Patecki, Sławomir Ste˛pien´ and Grzegorz Szyman´ski Keywords Power measurement, Electromagnetic fields, Eddy currents Abstract Presents 3D method for the computation of the winding current distribution and power losses of the electromagnetic gear. For a prescribed current obtained from measurement, the transient eddy current field is defined in terms of a magnetic vector potential and an electric scalar potential. From numerically obtained potentials the power losses are determined. The winding power losses calculation of an electromagnetic gear shows that a given course of the current generates skin effect and significantly changes the windings resistances. Also presents the designing method for reducing power losses.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 748-757 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540683

Introduction In analysing the electromagnetic field of electrical machines such as induction motor or generator, it is necessary to take the eddy current of the conductors into account. The eddy current problem especially concerns the windings when time varying current flows in a few conductors and generates skin effect. The skin effect is a special class eddy current problem (Biro et al., 2000) and has an impact on the current distribution in the conductors and their power losses. Moreover for some current variations, the windings resistance increases significantly. For solving the problem, the windings should be considered as massive conductors. As an excitation two sources: voltage or current can be applied. The voltage excitation is natural and in this case the current variation should be found. In prescribed current case the problem is inverse and consists of finding the voltage course. Unfortunately, the most useful for electrical machines modelling is the prescribed current case because of current measurement possibilities in the windings. As a typical application a part of electromagnetic gear is presented. The view of analysed system is shown in Figure 1. The movable permanent magnets generate current in the windings. Owing to electronic switching elements are connected to the stator circuit, the current variation in each phase of presented generator is a reason of skin effect. The windings current was measured and shown in Figure 2. Some works have been done recently for solving the skin effect problem with prescribed voltage or current in massive conductors (Badics, 1992; Biro et al., 2000; Gaier and Haas, 1996; Mayergoyz, 1993). Most of the models treat the problems in two ways. If the voltage excitation is given the field quantities are derived from a magnetic vector potential and an electric scalar potential (Badics, 1992; Biro et al., 2000). In case, when the current is given the field is described in terms of a current vector potential and a magnetic scalar potential (Badics, 1992; Biro et al., 2000). But these models use different potentials and it is difficult to realise them in one integrated CAD system. Therefore, if the voltage forced model corresponds with a magnetic vector potential

Power losses analysis

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Figure 1. A part of electromagnetic gear

Figure 2. Measured current variation

and an electric scalar potential then a current forced model should be expressed using the same potentials. For the simulation, the example is free of movement and permanent magnets. The windings as a massive conductor are excited from current source with given current variation. The simulations presented in this paper are illustrated by the power losses analysis in massive windings. Additionally, the windings division to reduce power losses is proposed.

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Formulation Using the magnetic vector potential A and electric scalar potential V as electromagnetic field variables, the electric field intensity E in conducting region (VC) and magnetic flux density B in conducting and non-conducting region ðVC < VN Þ are defined as (Biro and Preis, 1989; Nakata et al., 1988; Patecki et al., 1998): B¼7£A

750

E¼2

in VC < VN

›A 2 7V ›t

in VC

ð1Þ ð2Þ

The total current i(t) is connected to the boundary surface P. The situation is shown in Figure 3. In this case, the boundary value problem in terms of potentials is expressed as follows (Badics, 1992; Biro et al., 2000)

 1 ›A þ 7V ¼ 0 in VC < VN 7£A þs ð3Þ 7£ m ›t
 ›A 27 · s þ 7V ¼ 0 ›t
7£

1 7£A m

in VC

ð4Þ

 ¼0

 1 7£A £n¼0 m

in VC < VN

ð5Þ

on P1 and P2

ð6Þ







 ›A þ 7V dP ¼ iðtÞ ›t P1
 Z ›A þ 7V dP ¼ 2iðtÞ s ›t P2 Z

s

Figure 3. A current forced conductor

on P1 and P2

ð7Þ

The system of equations (3)-(5) is well known for eddy current problem (Biro and Preis, 1989; Nakata et al., 1988; Patecki et al., 1998), which defines magnetic field strength inside and around a conductor. Obviously, induced current and current divergence in conducting region are considered. In the conducting region conductivity s and permeability m everywhere are given. Next equation (6) imposes only normal component of magnetic vector potentials on boundary surface and adjacent magnetic field strength to the same surface. The normal components of flowing current are prescribed on the boundary surfaces. The above boundary value problem can be written in integral form (Nakata et al., 1988; Patecki et al., 1998):
 Z Z Z 1 ›A dV þ 7£ 7 £ A dV þ s s7V dV ¼ 0 ð8Þ m ›t VC
s P

›A dP þ ›t

I

s7V dP ¼ 0

ð9Þ

P

Equation (8) defines a volume current density in conducting region. The current continuity law is expressed in equation (9) and defines the current flowing through the closed surface in conducting region. Naturally, equation (7) which defines the current prescribed on two electrodes P1 and P2 must be considered in equation (9). The total power can be suitably derived from Poynting’s vector and defined as a sum of the power loss and the time derivative of the magnetic energy: I

½E £ H
dP ¼

2 PðVC
Z

2

jJðtÞj › dV þ ›t s VC

Z

2

ðmjHj Þ dV

ð10Þ

VC
Obviously in this case, the most important and interesting is the power loss for heat. Z

2

jJðtÞj dV ¼ pðtÞ ¼ s VC

Z

 2s · 7V 2 s s VC



› A2 ›t

ð11Þ

dV

In the presented part of electromagnetic gear, the windings should be considered as a multiconducting system. Then iterative procedure for global matrix of multiconducting system becomes: 2

D C þ Dt i

6 6 FT 6 6 6 HT 6 1 6 6 . 6 .. 6 4 HTj

F

H1

···

G · Dti

K 1 · Dt i

···

KT1 · Dti

m1 · Dti

···

.. .

.. .

..

· Dti

0

···

KTj

.

Hj

3

2

A iþ1

3

2

D 2 Dt Ai i

3

7 6 7 7 6 7 V iþ1 7 6 K j · Dt i 7 FTAi 7 6 7 6 7 6 7 6 1 7 6 1 7 T 7 7 6 V 7 6 i · Dt þ H A 0 7 £ 6 iþ1 7 ¼ 6 i i i 7 ð12Þ 1 7 6 7 6 7 7 .. .. 7 6 .. 7 6 7 7 6 . 7 6 . . 7 4 7 6 5 4 5 5 j j T V ii · Dt i þ H j A i mj · Dti iþ1

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where the submatrices C, D, F, F T, G are obtained as a result of discretisation of the terms:
 Z I Z I Z 1 ›A ›A dV; dP; 7£ 7 £ A dV; s s7V dV; s s7V dP; m ›t ›t VC
752

respectively, and the rows HTj ; KTj ; are obtained from Z Z ›A dP; s s7V dP; ›t Pj Pj where the index j is a number of prescribed currents iji for the multiconducting system in ith iterative step. Each scalar m is defined as a sum of the row K T. Vector V defines scalar potential inside analysed region and the scalars V 1 ;...V j are defined as the boundary potentials for multiconducting system. The system of equation (12) is symmetric and positively defined. Using preconditioned conjugate gradient (PCG) algorithm, it quickly gives a solution of unknown potentials A and V and power losses derived from calculated potentials. Numerical example An electromagnetic part of a gear which is encountered in an industrial application has been analysed. The cross section of the analysed device is shown in Figure 4. It consists of a stator, rotor and four windings as massive conductors. Exclusively for the skin effect simulation in the massive windings the stator material is taken as ferromagnetic with relative permeability m ¼ 20; 000: Similarly, rotor has a relative permeability m ¼ 10: The windings are made from copper with conductivity s ¼ 56 £ 106 S=m: In Figure 4 only two windings groups are visible. Figure 2 shows one phase windings where the current flows in different directions (marked as ^ and (). The current distribution and power losses in four massive windings (Figure 5) are calculated.

Figure 4. The geometry of analysed system. Radial dimensions in mm. The model is 3 mm thick

Power losses analysis

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Figure 5. The current distribution at t ¼ 2 £ 102 5 s in (a) first, (b) second, (c) third, and (d) fourth massive winding from Figure 4(b)

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Figure 6. The time functions of power losses in (a) first, (b) second, (c) third and (d) fourth massive winding from Figure 4(b)

Calculated absolute value of a current density is very uneven. The skin effect enlarges the current density in the windings especially in upper windings. Because of current density unevenness, the windings resistance should be greater than for dc analyse’s. The situation explains power losses analysis where transient power losses for each winding are compared with dc power losses. The transient power losses in the windings depend on time function of applied current (Figure 6). Very small rising time and huge current value make a strong skin effect which increases power losses. Obtained power losses can be compared with dc case shown in Figure 7, where R ¼ 1:995 £ 1025 V: After current density and power losses simulation the resistances of individual windings can be obtained. The resistances are found by total power losses calculation of each winding P¼

Z

9£1025

0

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755

jJj2 dt s

and compared with dc losses P dc ¼

Z

9£1025

Ri 2 ðtÞ dt:

0

The resistance calculations are given in Table I. The skin effect significantly increases windings resistance. These tremendous resistance changes can be a reason of an excessive energy losses and device heating. The power losses should be reduced by the skin effect minimisation. There are two

Figure 7. The time function of dc power losses

Winding Resistance (V)

1 3.203 £ 102 5

2 3.823 £ 102 5

3 4.876 £ 102 5

4 6.465 £ 102 5

dc 1.995 £ 102 5

Table I. The resistance comparison

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propositions to overcome the problem. As shown in Figure 8, the massive windings are divided into two and four parts creating a winding groups. A reduction of the skin effect is achieved. The calculated resistances of individual winding groups are compared and presented in Tables II and III. Conclusion In this work, a method for considering the special class of an eddy current problem is presented. The case of a prescribed current in conductive materials is discussed. The multiconductor system is modelled by 3D finite element method. The differential equations are expressed in terms of a magnetic vector potential and an electric scalar potential. An efficient numerical technique for solving the problem is widely described with reducing complication of an iterative procedure. A numerical example as a part of electromagnetic gear is presented. The calculations and the proposed methodology offer a powerful tool for the analysis and optimisation of the current and the power loss distribution within complicated full 3D structures of conductors with the current excitation.

Figure 8. The massive windings divided into two and four parts

Table II. The resistance comparison for windings splitted into two parts

Winding group Resistance (V)

1 2.964 £ 102 5

2 3.203 £ 102 5

3 3.544 £ 102 5

4 4.083 £ 102 5

dc 2.100 £ 102 5

Table III. The resistance comparison for windings splitted into four parts

Winding group Resistance (V)

1 2.769 £ 102 5

2 2.965 £ 102 5

3 3.258 £ 102 5

4 3.700 £ 102 5

dc 2.175 £ 102 5

References Badics, Z. (1992), “Transient eddy current field of current forced three-dimensional conductors”, IEEE Trans. Mag., Vol. 28, pp. 1232-4. Biro, O. and Preis, K. (1989), “On the use of the magnetic vector potential in the finite element analysis of three dimensional eddy current”, IEEE Trans. Mag., Vol. 25 No. 4, pp. 3145-59. Biro, O., Bohm, P., Preis, K. and Wachutka, G. (2000), “Edge finite element analysis of transient skin effect problem”, IEEE Trans. Mag., Vol. 36, pp. 835-9. Gaier, Ch. and Haas, H. (1996), “A finite edge element method for transient skin effect in loaded multiconductor systems”, IEEE Trans. Mag., Vol. 32, pp. 820-3. Mayergoyz, I. (1993), “A new approach to the calculation of three-dimensional skin effect problems”, IEEE Trans. Mag., Vol. 19, pp. 2198-200. Nakata, T., Takahashi, N., Fujiwara, K., Muramatsu, K. and Cheng, Z.G. (1988), “Comparison of various methods for 3D eddy current analysis”, IEEE Trans. Mag., Vol. 24, pp. 3159-62. Patecki, A., Szyman´ski, G. and Jotter, M. (1998), “3D eddy current model for the relay in steady state”, IEEE Trans. Mag., Vol. 34, pp. 3588-91.

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Finite element analysis of the magnetorheological fluid brake transients Wojciech Szela˛g Institute of Industrial Electrical Engineering, Poznan´ University of Technology, Poznan´, Poland Keywords Newton-Raphson method, Fluid dynamics, Finite element anaylsis Abstract Deals with coupled electromagnetic, hydrodynamic, thermodynamic and mechanical motion phenomena in magnetorheological fluid brake. Presents the governing equations of these phenomena. The numerical implementation of the mathematical model is based on the finite element method and a step-by-step algorithm. In order to include non-linearity, the Newton-Raphson process has been adopted. The method has been successfully adapted to the analysis of the coupled phenomena in the magnetorheological fluid brake. Present the results of the analysis and measurements.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 758-766 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540692

Introduction Magnetorheological fluids (or simply MR fluids) were invented by Jacob Rabinow in the late 1930s (Rosensweig, 1985). Their characteristic features involve a change in viscosity upon the application of a magnetic field. A change in viscosity is inseparably connected with a change of yield stresses in the fluid. The relationship between the yield stress t0 and the magnetic flux density B for Lord MRF-132LD is shown in Figure 1. The stress changes during the increase and decrease of magnetic flux density occur in microseconds (Carlson et al., 1996; www.rheonetic.com). The fluids retain their properties in the temperature range from 2 40 to 1508C. Relative magnetic permeability of the fluid is small, mr , 10: Owing to their properties, MR fluids are useful for the efficient control of the transmission of torques and forces. They are used, among others, in rotary brakes, clutches, and rotary and linear dampers. The working principle of magnetorheological electromagnetic transducers is based on the fact that viscosity changes when the fluid is exposed to a magnetic field. Naturally, the viscosity of the fluid and the stresses developed within it are related to the magnitude of the applied magnetic field. The fluid’s viscosity and the stresses within it increase with the growth of the field and so does the yield strength counteracting the motion of moveable parts in the transducers. MR devices are studied in many renowned scientific centres throughout the world. The research focuses on the analyses of the operating states of existing devices and on the methods of improving their functional parameters, but altogether new designs are also under constant development (Carlson et al., 1996; Szela˛g, 2002; Verardi and Cardoso, 1998; www.rheonetic.com). This paper proposes a mathematical model of coupled electromagnetic, hydrodynamic, thermodynamic and mechanic motion phenomena that can be applied to analyse how MRF electromechanical brakes operate.

Coupled phenomena model The phenomena observed in magnetorheological brakes need to be analysed in terms of fields. In the brake the velocity field of the fluid depends on the angular velocity of the rotor and distribution of the yield stress in the fluid. These stresses are function of the magnetic field distribution. The yield strength counteract the motion of the rotor in the brake. Therefore, the velocity field of the fluid and the mechanical stress field are coupled with the electromagnetic field. Moreover, the magnetic permeability m, the conductivity g of the region with eddy currents, the resistances of the windings and the yield stress of the fluid are functions of the temperature. The field coupling makes the analysis of the phenomena in the transducer highly complicated. What renders it even more intricate is the changing character of those fields and the non-linear character of the equations describing them. So far, there are no comprehensive approaches to the elaborate problem of solving the time dependent coupled field phenomena. Most papers on the subject usually deal with the field analysis of some selected phenomena observed in transducers (Besbes et al., 1996; Bird et al., 1960; Chung, 1978; Demenko, 1994; Hammand, 2000). This paper is an attempt to build a model of coupled phenomena in magnetorheological transducers. The focus is on electromagnetic, hydrodynamic, thermodynamic phenomena and on the dynamics of movable elements in the brake. In this paper, a magnetorheological brake with axial symmetry is considered (Figure 2). A cylindrical coordinate system r, z, q was applied. In this case, the equation describing the transient electromagnetic field in the brake can be expressed as (Nowak, 1998; Szela˛g, 2002)

 › 1 ›w › 1 ›w g ›w : ð1Þ þ ¼J2 ›r m l ›r ›z m l › z l ›t

Finite element analysis

759

Here l ¼ 2pr; w ¼ 2prAq ; where Aq is the magnetic vector potential, J ¼ i=s is the current density in the winding, i is the winding current, s is the cross-section area of the conductor, m is the magnetic permeability, g is the conductivity of the region with the eddy currents. For the MRF, g ¼ 0: In general, the transient electromagnetic field in MRF devices is voltage-excited. This means that the currents i in the windings are not known in advance, i.e. prior to the electromagnetic field calculation (Nowak, 1998; Szela˛g, n.d.). Therefore, it is necessary to consider the equations of the electric circuit of the device. The set of independent loop equations may be written as

Figure 1. The yield stress t0¼ f(B) for MRF 132LD

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Figure 2. The brake with MRF

u ¼ Ri þ

d C; dt

ð2Þ

where u is the vector of supply voltages, i is the vector of loop currents, R is the matrix of loop resistances, C is the flux linkage vector. The vector C is calculated by means of the field model. The phenomenological approach was used to describe fluid dynamics. In this approach, the fluid is treated as a non-conducting continuum of properties determined by density r, dynamic viscosity n and magnetic permeability m (Bird et al., 1960; Rosensweig, 1985; Verardi and Cardoso, 1998). In the hydrodynamic model, the laminar flow of a non-compressible fluid with no mass sources is investigated (Bird et al., 1960; Chung, 1978; Rosensweig, 1985). It is assumed that the gravitational forces acting on the fluid are negligible compared to the forces causing its motion in the transducer. The motion of the liquid in the q-direction is caused by the motion of the rotor. For such conditions, the differential equation of the motion of the fluid may be written as (Szela˛g, 2002)

 › nz ›f › nz ›f r ›f : ð3Þ þ ¼ ›r l ›r ›z l ›z l ›t Here f ¼ 2pr nq ; where nq is the component of velocity n in the q-direction, r is the fluid density and nz is the equivalent dynamic viscosity of the fluid. The description of problem (3) should be completed by non-slip boundary conditions nq ¼ r v and nq ¼ 0 on the surface of the rotor and the frame, respectively, where v is the angular velocity of the rotor. In order to determine the equivalent dynamic viscosity of the fluid, physical properties of the fluid were considered. MRFs belong to the non-Newtonian group of fluids. The properties of such fluids can be described by the Bingham model (Bird et al., 1960; Nouar and Frigaard, 2001). A typical characteristic family t ¼ t ðD; BÞ for a one-dimensional fluid model is shown in Figure 3, where D is the velocity of deformation. The fluid behaves like an elastic body for t # t0 ðBÞ; and like a body

Finite element analysis

761 Figure 3. The shear stress in MRF

of plastic viscosity hp for t . t0 ðBÞ; hp ¼ tgðbÞ (Figure 3). In the elaborated two-dimensional model of a MR fluid, the equivalent dynamic viscosity of the fluid (Szela˛g, 2002)

nz ¼ hp þ t0 ðBÞ=kDk for ktk . t0 ðBÞ;

ð4aÞ

nz ¼ 1 for ktk # t0 ðBÞ:

ð4bÞ

The yield stress t0(B) in equation (4) is determined on the basis of the distribution of the magnetic flux density obtained from equations (1) and (2). The norms kDk; ktk of the deformation tensor D and of the stress tensor t (Hammand, 2000; Nouar and Frigaard, 2001; Szela˛g, 2002) can be expressed as !1=2 !1=2 2 X 2 2 X 2 1X 1X 2 2 kDk ¼ D ; ktk ¼ t ; ð5aÞ 2 i¼1 j¼1 i;j 2 i¼1 j¼1 i;j where Di;j ¼ 0

for ktk # t0 ðBÞ;

D ¼ 0:5½7n þ ð7nÞT
for ktk . t0 ðBÞ;

ti;j ¼ ðhp þ t0 ðBÞ=kDkÞDi;j

for ktk . t0 ðBÞ:

ð5bÞ ð5cÞ ð5dÞ

In general, the magnetic permeability m, the conductivity g of the region with eddy currents, the resistances of the windings, and the dynamic viscosity of the fluid are functions of the temperature. Therefore, the equation describing the distribution of the temperature u must be considered. This equation in the cylindrical coordinates may by written as (Bird et al., 1960)

 1 › ›u › ›u ›u kr k 2 p: ð6Þ þ ¼ rC r ›r ›z ›r ›z ›t where k ¼ f ðuÞ is the thermal conductivity, and C is the heat capacity. The quantity p is the heat source resulting from dissipation of the electrical energy in the conductors and the mechanical energy in the viscous fluid. These heat sources can be expressed as (Hammand, 2000; Hedia, 1997)

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8 > J 2 =g for the winding and the region with eddy currents > > < ( 
) ›  nq  2 ›nq 2 p¼ nz r þ for the fluid > > > ›r r ›z :

ð7Þ

The description of problem (6) has been extended by a natural boundary condition on the surface of the frame of the brake (Hedia, 1997). It has been assumed that the heat flux qn at the boundary is proportional to the temperature difference ðu 2 u0 Þ between the surface of the brake and the surrounding region: qn ¼ k ›u=›n ¼ 2hðu 2 u0 Þ: In this equation, h is the convection heat transfer coefficient and ›=›n is the derivative in the direction of the outgoing normal to the boundary surface. When analysing the performance of a MRF electromechanical brake, equations (1)-(3) and (6) describing the electromagnetic, hydrodynamic and thermodynamic phenomena must be solved with the equation of dynamics of its movable elements. For the brake, it assumes the following form Jb

dv þ TB þ T0 ¼ Tz dt

ð8Þ

where Jb is the moment of inertia; TB is the braking torque associated with the occurrence of magnetic field in the brake, T0 is the braking torque produced in the brake when magnetic field is absent and Tz is the driving torque. The braking torque TB can be determined using the equation T B ¼ trðtq þ teq Þ ds:

ð9Þ

s

The vectors tq, teq in this equation describe the stress in the fluid and the electromagnetic stress acting in the direction q at a tangent to the external surface of the brake rotor. Finite element formulation Equations (1)-(3), (6) are coupled through the viscosity function nz ¼ nz ðB; kDk; uÞ; the total braking torque T B ¼ T B ðB; kDk; uÞ; the conductivity g(u), the permeability m(B,u), heat sources p and through the boundary condition vq ¼ r v: Therefore, these equations should be solved simultaneously. In order to solve coupled equations the finite element method and a “step-by-step” procedure was used (Demenko, 1994; Szela˛g, n.d.). The backward difference scheme was also applied. The finite element and time discretisation lead to the following system of non-linear algebraic matrix equations 3 " #" # 2 wn ðDtÞ21 G wn21 2w S n þ ðDtÞ21 G 5; ð10Þ ¼4 in 2Dtu n 2 w T wn21 2DtR 2w T ½S 0n þ ðDtÞ21 G 0
fn ¼ ðDtÞ21 G 0 fn21 ;

ð11Þ

½S 00n þ ðDtÞ21 G 00
Qn ¼ ðDtÞ21 G 00 Qn21 þ P ;

ð12Þ

where n denotes the number of time-steps, Dt is the time-step, S, S 0 , S 00 are the magnetic, hydrodynamic and thermodynamic stiffness matrices, w , f, Q are the vectors of the nodal potentials w, f and u, respectively, w T is the matrix that transforms the potentials w into the flux linkages with the windings, G is the matrix of conductances of elementary rings formed by the mesh, G 0 is the matrix whose elements depend on the dimensions of the elementary rings and fluid density r, G 00 is the matrix whose elements depend on the dimensions of the elementary rings and heat capacity C, P is the vector of the nodal heat sources. Motion equation (8) is approximated by the explicit difference formula (Szela˛g, n.d., 2002) J b ðanþ1 2 2an þ an21 Þ=ðDtÞ2 ¼ T z;n 2 T B;n 2 T 0 ;

ð13Þ

where a is the position of the rotor, T z;n ¼ T z ðt n Þ; and T B;n ¼ T B ðtn Þ: The angular velocity v of the rotor may be calculated according to the formula vðt n þ 0:5DtÞ ¼ ðanþ1 2 an Þ=Dt: The braking torque TB,n is described by formula (9). In the considered brake the component Bq of the magnetic flux density B is equal to zero. Therefore, in equation (9) the component teq of Maxwell stress tensor is equal to zero. Equations (10)-(12) are non-linear. In order to solve these equations the Newton iterative method was used (Besbes et al., 1996; Szela˛g, n.d.). Results The presented algorithm for solving the equations within the field model of phenomena was applied in a computer program that simulates coupled phenomena in a MR brake. The transients in the prototype of the electromagnetic brake built at Poznan´ University of Technology were considered. The brake is shown in Figure 2. This is a cylindrical-rotor brake system. Magnetic field is excited by a ring coil in a stator. The 132LD MRF produced by Lord Corporation was used in the brake. The diameter and the length of the rotor are 26.8 and 27 mm, respectively. The maximum braking torque is c. 1.7 N m. The elaborated program was used to determine the electromagnetic field, the velocity field of the fluid and the distributions of the temperature when constant voltage is applied to the winding of the brake. It was assumed that the rotor’s angular velocity v equals 50 rad/s and that the delay values at which the fluid reacts to the changes in the magnetic field are to be neglected. It was assumed that the magnetic and temperature fields were calculated using the same mesh. In the region with the MR fluid only a part of this mesh was used. The influence of the density of this mesh on the results of the calculations was analysed. The density was increased until the difference between the two consecutively calculated values of current or torque were observed. Finally, the region was divided into about 23,000 triangular elements. The calculations have been performed on a computer with Pentium IV 1.6 GHz processor. The time of the coupled fields calculations for one time step was about 50 s. The length of the time step was chosen as equal to 0.00005 s. With such time step length the calculations lasted very long. Therefore, in the calculations presented in this paper, in order to reach the steady state,

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the elements concerning time derivatives were neglected in equations (10)-(13). Selected examples of the distributions of magnetic field lines, the respective distributions of lines connecting the points with identical velocity values and the distributions of the isothermal lines are shown in Figure 4. The calculated torque-time TB(t) and current-time i(t) characteristics are shown in Figure 5. In order to verify the calculations, the braking torque TB(t) and the current i(t) were measured on the prototype of the brake. The results are shown in Figure 6. Good concordance between the calculations and measurements was achieved. The results of the calculations indicate the relevance of this simulation method in the designing process of magnetorheological brakes. Conclusions This paper presents a field model of coupled phenomena in an electromechanical brake with MR fluids. The algorithm for solving the equations of the model was suggested. A computer program was written which enables simulating coupled

Figure 4. Distributions of: (a) the magnetic field lines, (b) the lines n ¼ const, (c) the isothermal lines for: t ¼ 0.0001 s; t ¼ 0.01 s and for steady-state, respectively

Finite element analysis

765 Figure 5. Calculated torque-time TB(t) and current-time i(t) characteristics

Figure 6. Measured torque-time TB(t) and current-time i(t) characteristics

phenomena in magnetorheological brakes. The program proved useful in simulating the transient magnetic field, the velocity field of the fluid and the distribution of the temperature in magnetorheological brake. In the analysis the non-linear properties of materials, the eddy currents induced in solid elements and the rotor movement were considered. The model of coupled phenomena shown above together with the calculation software enable a more detailed analysis of the phenomena in magnetorheological transducers than is the case with analytic models. The approach presented in the paper is very useful in designing devices in which the MR fluids are used as a working medium. References Besbes, M., Ren, Z. and Razek, A. (1996), “Finite element analysis of magneto-mechanical coupled phenomena in magnetostrictive materials”, IEEE Trans. Magn., Vol. 32 No. 3, pp. 1058-61. Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (1960), Transport Phenomena, Wiley, New York, NY. Carlson, J.D., Catanizarite, D.M. and Clair, K.A. (1996), “Commercial magneto-rheological fluid device”, Proc. 5th Int. Conf. ER Fluids, MR Suspensions and Associated Technology, Singapore, pp. 20-8. Chung, T.J. (1978), Finite Element Analysis in Fluid Dynamics, McGraw-Hill, New York, NY. Demenko, A. (1994), “Time stepping FE analysis of electric motor drives with semiconductor converter”, IEEE Trans. Magn., Vol. 30 No. 5, pp. 3264-7.

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Hammand, K.J. (2000), “The effect of hydrodynamic conditions on heat transfer in a complex viscoplastic flow field”, International Journal of Heat and Mass Transfer, Vol. 43, pp. 945-62. Hedia, H., 1997 Mode´lisation non line´aire des effets thermiques dans les syste`mes magne´todynamiques These de doctorat Universite de Liege. Nouar, C. and Frigaard, I.A. (2001), “Nonlinear stability of Poiseuilla flow of Bingham fluid: theoretical results and comparison with phenomenological criteria”, Journal of Non-Newtonian Fluid Mechanic, No. 100, pp. 127-49. Nowak, L. (1998), “Simulation of the dynamics of electromagnetic driving device for comet ground penetrator”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3146-9. Rosensweig, R.E. (1985), Ferrohydrodynamics, Cambridge University Press, Cambridge, MA. Szela˛g, W. (n.d.), “Demagnetization effects due to armature transient currents in the permanent magnet self starting synchronous motor”, EMF 2000, Gent, pp. 93-4. Szela˛g, W. (2002), “Finite element analysis of coupled phenomena magnetorheological fluid devices”, Proceedings of XVI Symposium on Electromagnetic Phenomena in Nonlinear Circuits, Leuven, pp. 5-10. Verardi, S.L. and Cardoso, J.R. (1998), “A solution of two-dimensional magnetohydrodynamic flow using the finite element method”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3134-7.

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Magnetic stimulation of knee – mathematical model

Magnetic stimulation of knee

Bartosz Sawicki, Jacek Starzyn´ski and Stanisław Wincenciak Institute of Theory of Electrical Engineering, Measurement and Information Systems, Warsaw University of Technology, Poland

767

Andrzej Krawczyk Institute of Electrical Engineering, Warsaw, Poland

Mladen Trlep University of Maribor, Slovenia Keywords Finite element analysis, Bones, Vectors, Mathematical modelling Abstract Arthritis, the illness of the bones, is one of the diseases which especially attack the knee joint. Magnetic stimulation is a very promising treatment, although not very clear as to its physical background. Deals with the mathematical simulation of the therapeutical technique, i.e. the magnetic stimulation method. Considers the low-frequency magnetic field. To consider eddy ~ and magnetic scalar potential V . currents one uses the pair of potentials: electric vector potential T Since the problem is of low frequency and the electric conductivity of biological tissues is very small, consideration of electric vector potential only is quite satisfactory.

1. Introduction Computer simulation of external very low frequency electromagnetic field applied to human body can help to understand nature of some medical treatment. Successful application of such field for treatment of heavy wounds, broken bones and some neural diseases has been widely reported (Glinka et al., 2002; Pipitone and Scott, 2001; Zucchini et al., 2002), but it is still not completely clear how the electromagnetic stimulation works and how it should be applied. We consider only extremely low frequency (ELF) external magnetic field, which is used in therapy. So the human knee as area of interest can be thought homogeneous regarding magnetic structure (equal magnetic permeability m0) and heterogeneous with respect to electric values. Thus, one has to distinguish the areas of different conductivities: muscle tissue and bones. The three-dimensional mesh has been created on the basis of real cross-sections of human body which are available in the visible human project. The realistic model confines the element of leg (ca 20 cm) and it consists of: muscle tissue, femoral bone, calf bone, tibial bone and knee cap. The whole area of interest has been divided into 930,000 elements, which gives about 165,000 nodes (Figure 1). Exact investigation of electromagnetic field in human body needs fine models of the stimulated domain which is usually very complicated. In the authors’ opinion use of FEM allows us to construct the best model of human body for low frequency field analysis. The boundary problem of external magnetic stimulation is, however, usually difficult for FEM which basically needs to limit analysed domain. The paper is due to the State Research Committee (KBN) Grant No. 4T10A02322.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 767-773 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540700

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768 Figure 1. Realistic model of healthy knee

However, most of the models of electromagnetic stimulation includes field in open space. Problems are caused by difficulties with establishing boundary conditions for “open air” coil and human body placed in its vicinity. In previous works (Starzyn´ski et al., 1999, 2002) authors have proposed methodology which allows us to restrict FEM model only to the stimulated body. ~ 2 V formulation which can for such problems be reduced to electric vector It uses T ~ For this potential it is possible to set certain boundary condition on the potential T: external surface of the body (conducting region) (Starzyn´ski et al., 2002). Application of this model is, however, restricted to the domains of low conductivity. It is no problem if the human head or healthy knee (Figure 1) has to be modeled, but simulation of electromagnetic stimulation of broken bones may demand more sophisticated method. Metallic implants widely used for fixing broken bones may substantially change field distribution. In this paper, we like to present the extension of the previously proposed method, to simulate small highly-conducting inclusions within the low-conducting region. Our method is optimised to handle effectively (minimising computational costs) large low-conducting domains placed in extremely low frequency electromagnetic field. 2. Mathematical model The analysed domain is shown in Figure 2. The conducting (g1 < 1 S/m) region 1 bounded by G1 and contains sub-region 2 with high (g2 < 107 S/m) conductivity bounded by G2. Region 1 is surrounded by (possibly in-finite) dielectric domain ðg3 ¼ 0Þ in which external sources of low frequency magnetic field Hs can be placed. The magnetic field can be expressed as: ~ e; ~¼H ~s þ H H

ð1Þ

~ e can be expressed by electric vector ~ s is excited by the external sources and H where H ~ and magnetic scalar potential V as potential T ~ 2 7V: ~e ¼ T H Using Maxwell equations for harmonic fields

ð2Þ

Magnetic stimulation of knee 769 Figure 2. Notation for the model

~ 7 £ E~ ¼ 2jvmH;

ð3Þ

~¼0 7·B

ð4Þ

~ we can obtain the following set of differential equations and definition ~J ¼ 7 £ T; ~ V: for T, 1 ~ ¼ 2jvmH ~ s þ jvm7V; ~ þ jvmT 7£ 7£T g

ð5Þ

~ 7 · ðm7VÞ ¼ 7 · mT

ð6Þ

For diamagnetic domain with constant magnetic permeability equations (5) and (6) can be simplified: 1 ~ ¼ 2jvm0 H ~ s þ jvm0 7V; ~ þ jvm0 T 7£ 7£T g

ð7Þ

~ 7 · ð7VÞ ¼ 7 · T:

ð8Þ

~ we use Coulomb gauge 7 · T ~ ¼ 0: Linked with To obtain an unique solution for T ~ proper boundary conditions, it assures 7 · T ¼ 0 in conducting domain (bounded by G2 ), what reduces equation (8) to Laplace equation: 72 V ¼ 0

ð9Þ

We enforce Coulomb gauge by adding a penalty term to the left side of equation (7):
 1 1 ~ ¼ 2jvm0 H ~ s þ jvm0 7V: ~ ~ 7 £ 7 £ T 2 7 7 · T þ jvm0 T g g

ð10Þ

~ which have to be enforced on G1 were presented in the The boundary conditions for T work of Starzyn´ski et al. (2002). It has been shown there, that if vg1 is small, then

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vm07V can be neglected at all. In our problem, however, the value of g2 requires calculation of eddy currents induced in region 2. On G2 the following condition must be satisfied:

 ›V 2 ›V 1 m0 H sn2 þ T n 2 ¼ m0 H sn1 2 ; ›n ›n

770

ð11Þ

~ s ; respectively, in low where Hsn1, Hsn2 are normal components of magnetic field H conducting area (g1) and high conducting inset (g2) and V1, and V2 are the respective scalar magnetic potentials. Normal direction is positive if points are outside of the boundary. Having in our case H sn1 ¼ H sn2 ¼ H sn we can evaluate (11) to

›V1 ›V2 2 ¼ T n: ›n ›n

ð12Þ

By analogy with the electrostatic field we can use equation (12) to propose the following solution of equation (9): V1 ¼

I

~ n · dG ~ I T n dG T ¼ ; G2 4pr G2 4pr

ð13Þ

where G2 is the boundary of the conducting region. Thus, we can calculate the magnetic field induced in the conducting region as ~e ¼ T ~ 2 7V2 ; H

~e ¼ T ~þ H or ~e ¼ T ~þ H

I

~ n · dG ~ T 1~r ; 2 G2 4pr

I

T n dG ~ 1: 2 r G2 4pr

ð14Þ

ð15Þ

ð16Þ

~e where 1~r is the unit vector pointing from the surface element dG to the point where H is calculated, and r is the distance between this point and dG. Magnetic field vector should be expressed differently in every sub-domain (according to equations (1) and (2)): I

T 2n dS ~ 1 2 r G2 4pr I T 2n dS ~ ~ ~ ~ H2 ¼ Hs þ T2 þ 1 2 r G2 4pr ~1 ¼ H ~s þ H

ð17Þ

ð18Þ

~3 ¼ H ~s H

ð19Þ

~ 1 is the magnetic field in region bounded by G1, H ~ 2 is the magnetic field where H ~ in region bounded by G2, and H3 is the magnetic field in region bounded by G3. ~ in the conducting domain, To calculate distribution of the electric vector potential T the following equation has to be solved
 I 1 1 T n dS ~ ~ ~ s: ~ ~ 1 ¼ 2jvm0 H ð20Þ 7 £ 7 £ T 2 7 7 · T þ jvm0 T þ 2 r g g G2 4pr The system of algebraic equation obtained after application of FEM to the above equation will be non-symmetric. For this reason, it can be more convenient to solve instead, iteratively equation (20) with integral term I T n dS ~ 1 2 r G2 4pr moved to the right-hand side and updated after every iteration. 2.1 Iterative algorithm Only in the domain bounded by G2, full version of equation (20) has to be solved. In bigger domain bounded by G1 it is sufficient to solve a simpler equation:

 I 1 1 T 2n dS ~ ~ ~ ~ 1 7 £ 7 £ T1 2 7 7 · T1 ¼ 2jvm0 Hs þ ð21Þ 2 r g g G2 4pr Algorithm (1) In the first step the field in high-conducting region is calculated. Basically it needs to solve equation (20), but we prefer to have symmetric system of algebraic equations and thus, we use the following equation (obtained from equation (20)) instead:

 I 1 T 2n dS ~ ~ 2 ¼ 2jvm0 H ~s þ ~2 2 7 1 7 · T ~ 2 þ jvm0 T 1 ð22Þ 7£ 7£T 2 r g2 g2 G2 4pr ~ s ; solve equation (22), We start with right-hand side term equal to 2jvm0 H and then update iteratively right-hand side of equation (22) with the boundary ~ 2 : We can use simple iteration or algorithm with right integral of calculated T side relaxation ratio a where

ðnÞ PTðnÞ 2 ¼f

ð23Þ

  f ðnÞ ¼ af ðnÞ T2ðn21Þ þ ð1 2 aÞf ðn21Þ ;

ð24Þ

~ as The convergence of iteration is fast, especially if we use previous values of T the initial solution for iterative solver (Bi-Conjugate gradient algorithm) of algebraic system.

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Solving equation (22) we assume that that eddy currents in low-conducting region g1! g2 can be neglected and thus, it is sufficient to restrict calculations to region 2. (2) Having computed T2n on boundary G2 we can solve equation (21) in whole area bounded by G1. We remember that the field in region 2 (the highconducting one) was already calculated in step 1 and thus, right side of equation (21) is now equal to zero in area with conductivity g2, and non-zero only in low conducting region. (3) In the third step, the eddy current density vector should be calculated as ~1 ~J1 ¼ 7 £ T

in region 2;

ð25Þ

and ~J2 ¼ 7 £ ðT ~1 þ T ~ 2Þ

in region 1:

ð26Þ

3. Conclusion Presented methodology allows us to calculate the eddy current distribution inside the healthy knee model (Figure 3). External homogeneous magnetic field was used as excitation, which is thought to be the most effective in therapy. According to the results we can find out that eddy current values are different in different parts of bones. The stronger current flows near skin layer than in internal parts of bone. The knee cap is stimulated the most. The simple yet effective methodology, which allows us to estimate the electromagnetic field of low frequency within the human body in the presence of small metallic objects. Presented iterative algorithm will be used in the future for models with insets. The demand for such method was caused by the need of simulation of electromagnetic stimulation used for healing bones.

Figure 3. Eddy currents density in the knee model

References Glinka, M., Sieron´, A., Birkner, E. and Grzybek, H. (2002), “The influence of magnetic fields on the primary healing of incisional wounds in rats”, Electromagnetic Biology and Medicine, Vol. 21 No. 2, pp. 169-84. Pipitone, N. and Scott, D.L. (2001), “Magnetic pulse treatment for knee osteoarthritis: a randomized, double-blind, placebo-controlled study”, Current Medical Research and Opinion, Vol. 17, pp. 190-6. Starzyn´ski, J., Krawczyk, A., Sikora, R. and Zyss, T. (1999), “Optimal design of the transcranial magnetic stimulation system”, International Symposium on Applied Electromagnetics and Mechanics’99, Pavia, Italy. Starzyn´ski, J., Sawicki, B., Wincenciak, S., Krawczyk, A. and Zyss, T. (2002), “Simulation of magnetic stimulation of the brain”, IEEE Transactions on Magnetics, Vol. 38 No. 2. Zucchini, P. et al. (2002), “In vivo effects of low frequency low energy pulsing electromagnetic fields on gene expression during the inflammation phase of bone repair”, Electromagnetic Biology and Medicine, Vol. 21 No. 3, pp. 197-208. Further reading Buechler, D.N., Christen, D.A., Durney, C.H. and Simon, B. (2001), “Calculation of electric fields induced in the human knee by a coil applicator”, Bioelectromagnetics, Vol. 22, pp. 224-31.

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2D harmonic analysis of the cogging torque in synchronous permanent magnet machines M. Łukaniszyn, M. Jagiela, R. Wro´bel and K. Latawiec Department of Electrical Engineering and Automatic Control, Technical University of Opole, Poland Keywords Magnetic devices, Torque, Flux density, Fourier transforms Abstract Presents an approach to determine sources of cogging torque harmonics in permanent magnet electrical machines on the basis of variations of air-gap magnetic flux density with time and space. The magnetic flux density is determined from the two-dimensional (2D) finite element model and decomposed into the double Fourier series through the 2D fast Fourier transform (FFT). The real trigonometric form of the Fourier series is used for the purpose to identify those space and time harmonics of magnetic flux density whose involvement in the cogging torque is the greatest relative contribution. Carries out calculations for a symmetric permanent magnet brushless machine for several rotor eccentricities and imbalances.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 774-782 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540719

1. Introduction Recent progress in the construction of low-cost high-performance machines has in particular involved PM machines with surface-mounted magnets. They are built in various configurations but radial flux machines are most popular (Bianci and Bologhnani, 2002; Salon et al., 2000; Zhu and Howe, 1992). In addition to mechanical, aerodynamic and electronic noises, low-speed PM machines are sources of detrimental magnetic effects, the most prominent of which is the cogging torque. The cogging torque arises from interaction of MMF harmonics with variations in the air-gap permeance. It thus occurs in almost all types of the machines in which air-gap is not constant. Knowledge of cogging torque sources is of great importance and it supports the design process in terms of minimization of this disadvantageous effect. Various approaches to analyzing the cogging torque, using either analytical or semi-numerical methods, are presented in the literature (Salon et al., 2000; Zhu and Howe, 1992). In these methods, the air-gap field is determined from the product of the magnetic field produced by the magnets and the relative air-gap permeance. In order to simplify the problem, the air-gap permeance is usually determined from an assumed field pattern. Additionally, infinite permeability of iron and simplified slot geometry are assumed in these methods. These drawbacks are eliminated in the finite element method, which is currently most popular due to its high capability of modeling of the natural geometry and physical properties of various objects ( Bianci and Bologhnani, 2002). In this paper, first experiences with a new approach to identify sources of cogging torque harmonics using the two-dimensional (2D) Fourier series and the time-stepping finite element method are presented. The presented algorithm as well as the elaborated The authors are grateful to the Foundation for Polish Science Warsaw, Poland for support of this research.

numerical tool enable us to determine which harmonics are responsible for the cogging torque in cylindrical PM machines. This can be of high importance when designing low-cogging torque machines. 2. PM machine and its model The calculations are carried out for a prototype low-power brushless PM motor built on the basis of a commercial asynchronous motor. An outline of its configuration and distribution of magnetic flux is shown in Figures 1 and 2, respectively. Its technical specifications are as follows: 450 W, six poles, 36 slots. Owing to low construction costs of the motor, the stator skewing is not applied, so the 2D domain can be considered for sufficiently accurate modeling of the magnetic field.

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Figure 1. Symmetric permanent magnet machine with six slots per one pole pitch

Figure 2. Distribution of magnetic flux over the motor cross-section

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To account for variations of magnetic flux density in the motor air-gap with space and time, the finite element steady-state model is applied. The magnetic field description in the considered 2D domain has the following form: 8

 > < › 1 ›A þ › 1 ›A ¼ 2J in V ›x m ›x ›y m ›y ; ð1Þ > : A ¼ 0 on G where A is the z-component of the magnetic vector potential, m is the magnetic permeability and J is the z-component of the current density vector. Calculations of the magnetic field are carried out at no-load operation with the rotor movement from 0 to 60 mechanical degrees and time-step equal to 0.5 mechanical degrees and distance between space-sampling points equal to 0.5 mechanical degrees. So, in order to obtain the same numbers of space and time, i.e. quadratic field data matrices, the time-related data were accordingly reflected. Figures 3 and 4 show variations of normal (Br) and tangential (Bc) components of air-gap magnetic flux density with space and time. 3. 2D harmonic analysis of cogging torque The 2D torque formula using Maxwell stress tensor is the basis for the developed new algorithm. Considering the uniform distribution of magnetic field along the z-direction and the end boundary conditions, the torque T can be written in the familiar form: Z r 2 2p B c B r d c; ð2Þ T¼L m0 0 where L is the stack length, r the radius of integration path and m0 the magnetic permeability of air. Br and Bc components can be written as functions of space and time using the double Fourier series in the following form:

Figure 3. Time and space variations of the normal component of magnetic flux density for balanced case

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Figure 4. Time and space variations of the tangential component of magnetic flux density for balanced case

Br ¼

1 X 1 X

bnm r cos ðnvt þ mpc þ lnm Þ;

ð3Þ

n¼1 m¼1

Bc ¼

1 X 1 X

bklc cos ðkvt þ lpc þ lkl Þ:

ð4Þ

k¼1 l¼1

where v is the angular frequency of the operation, p the number of pole-pairs, l the phase shift. Figures 5 and 6 show the 2D Fourier spectra calculated for the respective

Figure 5. 2D Fourier spectrum for the normal component of magnetic flux density

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Figure 6. 2D Fourier spectrum for the tangential component of magnetic flux density

magnetic flux density components shown in Figures 3 and 4. These are quite similar for both ones except of the magnitudes. Harmonics that travel with rotor are located across the diagonal of the spectra. These are the odd harmonics due to the magnets. The even harmonics that can be attributed to the slotting are located on the left (space) side of the diagonal. When expressions (3) and (4) are sulistituted in equation (2), the cross products between the harmonics that belong to expressions (3) and (4) are created. The harmonics terms can add or subtract from each other thus giving a number of different frequencies. For all the products, the integral (2) is non-zero for limited number of cases only. Generally, the cogging torque pulsates with a frequency that is the least common multiplier ( LCM ) of a number of basic harmonic due to the slotting and that due to the permanent magnets. Also lower harmonics have their share in the cogging torque magnitude. In order to distinguish the cases for which the integral (2) of the harmonics is non-zero, the following procedure is applied. (1) The 2D fast Fourier transform (FFT) for the normal and tangential components of magnetic flux is computed. (2) Only dominant magnitudes are selected from the full 2D spectra, i.e. not less than 0.02 T. To reduce the computation time a sparse harmonic pattern is thus created with corresponding numbers and phases. (3) For each selected magnitude from the spectrum for the normal component of magnetic flux density, the product is created with each one for the tangential component. (4) For each rotor instant, each product of the two cosine functions is integrated according to equation (2). Such an integral can vary with time according to a sum or difference between the respective harmonic terms. Pure DC components can also be produced that must cancel each other so, these can be rejected at once.

(5) From time variations of each non-zero integral, the maximum Tmax is selected for the calculation of relative contribution to the cogging torque magnitude. (6) Calculation results of the computation are listed, i.e. harmonic numbers, phases and relative magnitudes of the cogging torque produced. To validate the described procedure, calculations were carried out for the considered PM machine with two different static rotor eccentricities as well as for the balanced case. Calculation results are presented as program reports in a tabular form. The meanings of the symbols listed in tables are as follows. The first columns contain the harmonic numbers, e.g. 39S3T £ 39S39T means that the 39th space and the third time harmonics from the normal component of magnetic flux density is multiplied by the 39th space and 39th time harmonics from the tangential component. Next two columns contain the magnitudes of these harmonics, e.g. BN ¼ 0:0443 denotes the amplitude of the normal component equal to 0.0443 T, whereas BT ¼ 0:0233 is the magnitude of the tangential component equal to 0.0233 T. Respective phases are given in the next two columns, e.g. PN ¼ 280:97 for the normal component and PT ¼ 20:04 for the tangential one. The last column contains relative magnitudes of the cogging torque produced by the corresponding products, e.g. T=T max ¼ 26:07 percent. A number of products of harmonics that result in non-zero cogging torque is quite high. However, a number of the products which result in significant magnitudes of the cogging torque, in terms of the ratio T/Tmax exceeding 2 percent, is quite low. Finally, for any unbalanced as well as for the balanced case, the cogging torque is composed of only selected dominant harmonics which are listed in the tables.

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3.1 Balanced case It is clear from the analysis carried out that two most significant harmonic numbers can be distinguished (Table I). These are 39S39T, which can be attributed to the field from the magnets and 39S3T, which can be attributed to the slotting. Their time terms subtract from each other to give the basic cogging torque harmonic being the LCM of the slot number (36) and the number of poles (six). In addition to these dominant harmonics, there are a few higher ones which may be of some importance. In practice, the 75S75T can be attributed to the magnetic field and the 75S3T to the slotting. These harmonics are responsible for the 72nd harmonic in the cogging torque variation. Since the magnetic circuit is symmetric, there are no lower harmonic numbers than the basic one in the cogging torque variation. Figure 6 shows the comparison between the cogging torque determined only from the indicated magnetic flux harmonics with

39S3T 75S3T 3S39T 38S39T 39S39T 39S39T 39S39T 40S39T 75S75T

£ £ £ £ £ £ £ £ £

39S39T 75S75T 3S3T 39S3T 38S3T 39S3T 40S3T 39S3T 75S3T

BN

BT

PN

PT

T/Tmax (percent)

0.0443 0.0304 0.0025 0.0024 0.0456 0.0456 0.0456 0.0025 0.0094

0.0235 0.0075 0.0727 0.0457 0.0025 0.0457 0.0025 0.0457 0.0318

2 80.97 2 72.02 2 99.01 87.76 89.97 89.97 89.97 2 92.28 90.40

20.04 0.41 20.03 9.02 8.41 9.02 2171.77 9.02 18.01

26.07 5.77 4.53 2.31 2.40 52.05 2.38 2.40 7.52

Table I. Program report for balanced case

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that obtained from accurate magnetic field calculations. It clearly proves the correct selection of significant magnetic flux density harmonics. 3.2 Ten percent of static imbalance When the rotor static eccentricity is 0.1 mm, the main difference as compared to the balanced case can be observed for lower harmonics numbers (than the basic one) Table II). Figure 7 compares the cogging torque obtained from the identified magnetic flux density harmonics with that obtained from accurate magnetic field calculations. 3.3 Thirty percent of static imbalance Results of identification for the 30 percent static imbalance are listed in Table III. There is a number of additional harmonics that introduce various pulsations into the cogging torque variation. Both higher and lower numbers of the cogging torque harmonics are generated due to a large rotor eccentricity. Figures 8 and 9 compare the cogging torque obtained from the identified magnetic flux harmonics with that obtained from accurate magnetic field calculations.

Table II. Program report for 10 percent unbalanced case

Figure 7. Cogging torque for the balanced case

3S3T 3S3T 3S3T 3S3T 38S3T 39S3T 75S3T 3S39T 38S39T 39S39T 39S39T 39S39T 40S39T 75S75T

£ £ £ £ £ £ £ £ £ £ £ £ £ £

14S15T 16S15T 204S21T 22S21T 39S39T 39S39T 75S75T 3S3T 39S3T 38S3T 39S3T 40S3T 39S3T 75S3T

BN

BT

PN

PT

T/Tmax (percent)

1.0580 1.0580 1.0580 1.0580 0.0062 0.0450 0.0312 0.0025 0.0026 0.0457 0.0457 0.0457 0.0027 0.0094

0.0025 0.0023 0.0025 0.0025 0.0234 0.0234 0.0075 0.0728 0.0464 0.0064 0.0464 0.0030 0.0464 0.0326

90.00 90.00 90.00 90.00 2 16.92 2 81.41 2 73.53 2 99.27 110.82 89.82 89.82 89.82 2 67.33 90.11

104.89 2 73.04 112.67 2 65.58 0.23 0.23 0.72 0.02 8.60 75.24 8.60 2135.31 8.60 16.53

7.23 6.18 5.74 4.98 3.08 26.44 5.89 4.54 2.50 6.16 53.04 2.94 2.68 7.74

3S3T 3S3T 3S3T 3S3T 3S3T 3S3T 3S3T 3S3T 3S3T 3S3T 38S3T 39S3T 39S3T 39S3T 40S3T 74S3T 75S3T 15S15T 15S15T 15S15T 15S15T 15S15T 21S21T 21S21T 21S21T 33S33T 3S39T 38S39T 39S39T 39S39T 39S39T 40S39T 75S75T 75S75T

£ 14S15T £ 16S15T £ 20S21T £ 22S21T £ 26S27T £ 28S27T £ 32S33T £ 34S33T £ 38S39T £ 40S39T £ 39S39T £ 38S39T £ 39S39T £ 40S39T £ 39S39T £ 75S75T £ 75S75T £ 2S3T £ 4S3T £ 38S3T £ 20S21T £ 22S21T £ 38S3T £ 14S15T £ 16S15T £ 38S3T £ 3S3T £ 39S3T £ 38S3T £ 39S3T £ 40S3T £ 39S3T £ 74S3T £ 75S3T

BN

BT

PN

PT

T/Tmax (percent)

1.0582 1.0582 1.0582 1.0582 1.0582 1.0582 1.0582 1.0582 1.0582 1.0582 0.0159 0.0456 0.0456 0.0456 0.0045 0.0186 0.0314 0.1561 0.1561 0.1561 0.1561 0.1561 0.1203 0.1203 0.1203 0.0336 0.0025 0.0040 0.0467 0.0467 0.0467 0.0044 0.0097 0.0097

0.0071 0.0066 0.0072 0.0070 0.0021 0.0021 0.0026 0.0024 0.0037 0.0037 0.0224 0.0037 0.0224 0.0037 0.0224 0.0071 0.0071 0.0079 0.0059 0.0167 0.0072 0.0070 0.0167 0.0071 0.0066 0.0167 0.0726 0.0472 0.0167 0.0472 0.0048 0.0472 0.0200 0.0332

89.99 89.99 89.99 89.99 89.99 89.99 89.99 89.99 89.99 89.99 2 1.27 2 82.17 2 82.17 2 82.17 159.23 6.48 2 75.84 2 90.11 2 90.11 2 90.11 2 90.11 2 90.11 2 90.21 2 90.21 2 90.21 89.62 2 99.18 142.39 89.51 89.51 89.51 2 36.08 89.52 89.52

94.83 2 84.63 97.35 2 82.31 100.65 2 78.06 2 76.30 104.32 2 72.44 108.96 0.85 2 72.44 0.85 108.96 0.85 1.42 1.42 2 87.18 91.46 89.74 97.35 2 82.31 89.74 94.83 2 84.63 89.74 0.14 7.88 89.74 7.88 2 112.59 7.88 97.10 14.17

21.38 17.53 16.18 14.25 3.46 3.35 3.54 3.11 4.36 4.07 7.49 3.59 25.65 3.58 2.13 2.68 5.65 3.38 2.70 3.40 5.29 3.60 3.16 2.84 3.71 2.51 4.49 3.96 16.40 55.13 4.72 4.35 3.94 8.15

2D harmonic analysis

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Table III. Program report for 30 percent unbalanced case

Figure 8. Cogging torque for 10 percent static imbalance

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4. Conclusions In this paper, a new method for the diagnosis of sources of the cogging torque harmonics in radial-field permanent magnet machines has been presented. The algorithm has been successfully applied in the cogging torque analysis due to the rotor eccentricity for several static imbalances. In addition to the balanced case for the considered cases, the space and time harmonics were properly identified and their contributions to the cogging torque magnitude were assessed. This algorithm can also be adopted for the analysis of any other quantity such as the electromagnetic torque or radial force. These can be analyzed with respect to the geometry, e.g. permanent magnet width, pole shifting, inhomogeneous magnetization or pole distribution, etc., thus giving rise to the formulation of design indications. References Bianci, N. and Bologhnani, S. (2002), “Reducing torque ripple in PM synchronous motors by pole shifting”, ICEM 2002, Espoo, Finland, pp. 1222-6. Salon, S., Sivasubramaniam, K. and Tukenmez-Ergene (2000), “An approach for determining the source of torque harmonics in brushless DC motors”, ICEM 2000, Espoo, Finland, pp. 1707-11. Zhu, Z. and Howe, D. (1992), “Analytical prediction of the cogging torque in radial-field permanent magnet brush-less motors”, IEEE Trans. Magn., Vol. 28 No. 2, pp. 1371-4.

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dynamic radial Determination of a dynamic radial Aactive magnetic bearing model active magnetic bearing model using the finite element method Bosˇtjan Polajzˇer, Gorazd Sˇtumberger and Drago Dolinar

783

Faculty of Electrical Engineering and Computer Science, University of Maribor, Maribor, Slovenia

Kay Hameyer Institut fu¨r Elektrische Maschinen, RWTH Aachen, Aachen, Germany Keywords Magnetic fields, Modelling, Nonlinear control systems Abstract The dynamic model of radial active magnetic bearings, which is based on the current and position dependent partial derivatives of flux linkages and radial force characteristics, is determined using the finite element method. In this way, magnetic nonlinearities and cross-coupling effects are considered more completely than in similar dynamic models. The presented results show that magnetic nonlinearities and cross-coupling effects can change the electromotive forces considerably. These disturbing effects have been determined and can be incorporated into the real-time realization of nonlinear control in order to achieve cross-coupling compensations.

1. Introduction Active magnetic bearings are a system of controlled electromagnets, which enable contact-less suspension of a rotor (Schweitzer et al., 1994). Two radial bearings and one axial bearing are used to control the five degrees of freedom of the rotor, while an independent driving motor is used to control the sixth degree of freedom. No friction, no lubrication, precise position control and vibration damping make active magnetic bearings (AMBs) particularly appropriate and desirable in high-speed rotating machines. Technical applications include compressors, centrifuges and precise machine tools. The electromagnets of the discussed radial AMBs are placed on the common iron core (Sˇtumberger et al., 2000). Their behavior is, therefore, magnetically nonlinear. Moreover, the individual electromagnets are magnetically coupled. An extended dynamic AMB model is determined in this paper using the finite element method (FEM). The parameterization coupling model of the discussed radial AMBs is derived in this way. The presented dynamic AMB model is based on partial derivatives of flux linkages and radial force characteristics and, therefore, describes magnetic nonlinearities and cross-coupling effects more completely than similar dynamic AMB models (Antila et al., 1998; Sˇtumberger et al., 2000). Moreover, it is appropriate for nonlinear control design and is compact and fast enough for the real-time realization. FEM-computed force is compared with the measured force, while the flux linkages were not measured due to mechanical problems with rotor fixation. The current and position dependent partial derivatives of flux linkages are calculated by analytical derivations of the continuous approximation functions of the FEM-computed flux linkages. The impact of magnetic nonlinearities and cross-coupling effects on

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the properties of the discussed radial AMBs is then evaluated based on the performed calculations. 2. Dynamic AMB model The dynamic AMB model is according to the circuit model presented in Figure 1 given by equations (1) and (2), where u1, u2, u3 and u4 are the supply voltages, I0 is the constant bias current, ixD and iyD are the control currents in the x- and y-axis. c1, c2, c3 and c4 are the flux linkages of the corresponding electromagnets. R stands for the coil resistances. Fx and Fy are the radial force components in the x- and y-axis, m is the mass of the rotor. 3 2 ›c ›c 3 2 1 1 › c1 › c1 6 ›ixD ›iyD 7 6 ›x ›y 7 7 7 6 6 3 2 3 2 7 72 3 6 6 2 3 I 0 þ ixD u1 6 ›c2 ›c2 7 dixD 6 ›c2 ›c2 7 dx 7 7 7 6 6 6 7 6 6 ›ixD ›iyD 7 6 dt 7 6 ›x 6 7 6 u2 7 6 I 0 2 ixD 7 ›y 7 76 7 6 dt 7 7 6 6 7 6 7 6 7 þ 26 ›c ›c 7 6 di 7 þ 6 ›c ›c 7 6 dy 7 ð1Þ 6 7 ¼ R6 3 7 4 yD 5 374 6 3 6 3 5 6 u3 7 6 I 0 þ iyD 7 7 7 5 6 6 4 5 4 › i › i › x › y 7 7 6 6 dt dt yD I 0 2 iyD u4 7 7 6 xD 6 6 ›c ›c 7 6 › c4 › c4 7 4 45 5 4 4 ›x ›y ›ixD ›iyD 3 d2 x 2 3 6 dt 2 7 F x ðixD ; xÞ 7 6 1 7 6 4 5 6 d2 y 7 ¼ m F ði ; yÞ y yD 5 4 dt 2 2

Figure 1. The circuit AMB model

ð2Þ

The current and position dependent partial derivatives of the flux linkages required in equations (1) are calculated by analytical derivations of the continuous approximation functions of FEM-computed flux linkages. The force characteristics Fx(ixD,x) and Fx(iyD,y) required in equations (2) are determined by FEM. In this way, the obtained dynamic AMB model (1), (2) is described in terms of parameterization coupling. When considering the symmetry in geometry (Figure 2), and the differential driving mode of currents i1 ¼ I 0 þ ixD ; i2 ¼ I 0 2 ixD ; i3 ¼ I 0 þ iyD and i4 ¼ I 0 2 iyD ; the interaction between electromagnets in the x-axis (no. 1 and no. 2) and electromagnets in the y-axis (no. 3 and no. 4) can be expressed as equations (3) and (4).

› c1 › c3 ¼ ; ›ixD ›iyD

›c1 › c 3 ¼ ; ›iyD ›ixD

›c 1 › c2 › c3 › c4 ¼2 ¼ ¼2 ; ›x ›x ›y ›y

› c2 › c4 ¼ ; ›ixD ›iyD

› c2 › c 4 ¼ ›iyD ›ixD

› c1 › c2 › c3 › c4 ¼2 ¼ ¼2 ›y ›y ›x ›x

A dynamic radial active magnetic bearing model 785

ð3Þ

ð4Þ

The electromotive forces (EMFs) due to the magnetic nonlinearities are reflected in terms like ›c3 =›iyD and ›c3 =›y; which are normally given as constant inductance and speed coefficient, respectively (Schweitzer et al., 1994). In the work of Antila et al. (1998) magnetic nonlinearities are partially considered with dynamic inductance. However, the EMFs due to cross-coupling effects, which are reflected in terms like ›c1 =›iyD and ›c1 =›y; are neglected by Antila et al. (1998). The dynamic AMB model (1), (2), therefore, describes magnetic nonlinearities and cross-coupling effects more completely than similar dynamic models. Furthermore, it is appropriate for nonlinear control design and is compact and fast enough for the real-time realization.

Figure 2. The geometry and field distribution of the discussed radial AMBs

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3. FEM computation of flux linkage and radial force characteristics Magneto-static computation was performed by 2D FEM. The geometry and magnetic field distribution of the discussed radial AMBs is shown in Figure 2. The flux linkage characteristics c1(ixD, iyD, x,y), c2(ixD, iyD, x,y), c3(ixD, iyD, x,y) and c4(ixD, iyD, x,y) were calculated in the entire operating range from the average values of the magnetic vector potential in the stator coils. The radial force characteristics Fx(ixD, x) and Fy(iyD, y) were also calculated in the entire operating range by the Maxwell’s stress tensor method, where integration was performed over a contour placed along the middle layer of the three-layer air gap mesh. The obtained results were incorporated into the extended dynamic AMB model (1) and (2). The parameterization coupling model is derived in this way.

4. Results The magnetic properties of the rotor surface changed due to the manufacturing process of the rotor steel sheets. Therefore, the magnetic air gap became larger than the geometric one. In order to obtain good agreement between the calculated and measured forces in the linear region, the air gap was increased in the FEM computation from 0.4 to 0.45 mm. The increase in the air gap of 0.05 mm can be compared with the findings by Antila et al. (1998). A good agreement between the FEM-computed and the measured radial force characteristics can be seen in Figure 3(a) and (b). The current and position dependent partial derivatives of flux linkages, shown in Figure 3(c)-(f), were calculated by analytical derivations of the continuous approximation functions. In the results shown in Figure 3(c) and (d) the influence of magnetic nonlinearities can be seen, while the influence of magnetic cross-coupling effects can be seen in the results shown in Figure 3(e) and (f). Based on the obtained results the ratio ð›c1 =›iyD Þ=ð›c3 =›iyD Þ; as well as the ratio ð›c1 =›yÞ=ð›c3 =›yÞ was calculated inside the operating range. From the performed comparison, it is established that due to magnetic nonlinearities and cross-coupling effects the EMFs can vary in a range of up to 12 percent.

5. Conclusion The extended dynamic AMB model is presented in this paper. It is based on the FEM-computed current and position dependent partial derivatives of flux linkages and radial force characteristics. The parameterization coupling model of the discussed radial AMBs is derived in this way. The obtained dynamic AMB model, therefore, considers magnetic nonlinearities and cross-coupling effects more completely than similar dynamic AMB models. The results of the performed calculations show that inside the operating range of the discussed radial AMBs, the EMFs can vary due to magnetic nonlinearities and cross-coupling effects in a range of up to 12 percent. These disturbing effects deteriorate the static and dynamic performances of the overall system. In order to improve the system dynamics, the obtained results have to be incorporated into the real-time realization of nonlinear control.

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Figure 3. Results for the case: x ¼ 0 mm, ixD ¼ 0 A and I0 ¼ 5 A: calculated and measured force (a and b) and flux linkage partial derivatives (c, d, e and f)

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References Antila, M., Lantto, E. and Arkkio, A. (1998), “Determination of forces and linearized parameters of radial active magnetic bearings by finite element technique”, IEEE Trans. Magn., Vol. 34 No. 3, pp. 684-94. Schweitzer, G., Bleuler, H. and Traxler, A. (1994), Active Magnetic Bearings, Vdf Hochschulverlag AG an der ETH Zu¨rich, Zu¨rich. Sˇtumberger, G., Dolinar, D., Pahner, U. and Hameyer, K. (2000), “Optimization of radial active magnetic bearings using the finite element technique and the differential evolution algorithm”, IEEE Trans. Magn., Vol. 36 No. 4, pp. 1009-13.

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Electromagnetic forming A coupled numerical electromagnetic-mechanical-electrical approach compared to measurements

Electromagnetic forming

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A. Giannoglou, A. Kladas and J. Tegopoulos Electric Power Division, Faculty of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece

A. Koumoutsos, D. Manolakos and A. Mamalis Manufacturing Technology Division, Faculty of Mechanical Engineering, National Technical University of Athens, Athens, Greece Keywords Electromagnetic fields, Finite element analysis, Manufacturing systems, Numerical analysis Abstract Undertakes an analysis of electromagnetic forming process. Despite the fact that it is an old process, it is able to treat current problems of advanced manufacturing technology. Primary emphasis is placed on presentation of the physical phenomena, which govern the process, as well as their numerical representation by means of simplified electrical equivalent circuits and fully coupled fields approach of the electromagnetic-mechanical-electric phenomena involved. Compares the numerical results with measurements. Finally, draws conclusions and perspectives for future work.

1. Introduction The basic electromagnetic forming process is the compression of a metal tube inside a solenoid coil connected with a capacitor via a switch (simplified equivalent circuit shown in Figure 1(a)). The analysis of this case can fully reveal the physical principles of the process (Stadelmaier, 2000) and also its basic technological aspects ( Plum, 1996). The analytical treatment of the process in terms of equivalent circuits as well as field analysis method has been established since the end of 1970s. The main disadvantage of such approaches is that they cannot be easily applied to more complicated coil geometries, which are often used in industrial applications. In the present paper, electromagnetic forming is modeled by both elaborated equivalent circuits and coupled field approaches. The latter method involves implementation of the well-known commercial F.E. Code ANSYS Multiphysics 6th edition. The main advantage of this code is that it can simultaneously treat both the electromagnetic and mechanical aspects of the process in the same environment. Special attention is paid in order to be easy to apply the chosen modeling solutions to the more complicated industrial forming coils. 2. Modeling procedure Both electric equivalent circuit analysis and coupled electromagnetic-mechanical-electrical field approaches have been used for the process analysis. The latter technique is sophisticated and the problem configuration and assumptions adopted are extensively described hereafter.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 789-799 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540737

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Figure 1. The equivalent circuit of electromagnetic forming process: (a) circuit topology; and (b) computer implementation

2.1 Electric equivalent circuit All data, except material properties, are derived from Bednarski (1985). The layout of the forming unit is presented in Figure 3(a). The main load is the coil current. The equivalent circuit shown in Figure 1(a) has been implemented in detail by using the Matalab/Simulink software as shown in Figure 1(b). The simulated and measured time variation of the coil current are given in Figure 2(a) and (b), respectively. It may be noted that the somewhat simplified approach of equivalent circuit with distributed parameters provides sufficient accuracy, as far as the coil current is concerned. An approximate response of this circuit, is given by the equation:

Electromagnetic forming

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Figure 2. The coil current time variation of electromagnetic forming process: (a) simulated; and (b) measured

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I ðtÞ ¼ I 0 expð2gtÞ sinðvd tÞ

ð1Þ

This kind of input load is advantageous, because, in practice, it can be easily measured by means of a Rogowski coil. The constitutive behavior of the workpiece is shown in Figure 3(b). In the following, d is the skin effect depth expressed by the well-known equation (2) and m0 ¼ 4p £ 1027 H=m: The current of the coil is applied in the skin depth, which is considered to be developed only in its inner surface. This approximation needs further verification and it seems to be the main disadvantage of applying current as input load. However, according to Go¨bl (1978), no current flow is considered in the outer surface of a coil without an outer metal screen. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 re d¼ mr m0 vd

ð2Þ

The assumptions and restrictions, which the specific case study meets towards the variety of electromagnetic forming conditions, are the following. . The workpiece thickness (hw) is bigger than its skin depth (d). . The workpiece material is non-magnetic.

Figure 3. (a) Schematic of the coil and workpiece semi-cross-section; and (b) the constitutive behavior of the workpiece

.

.

. . .

. .

The current frequency ( f ) is lower than 30 kHz, thus no electromagnetic wave effects are considered. The workpiece is co-axially placed inside the coil and also symmetrically with respect to its length. The temperature dependence of material properties is ignored. The strain-rate hardening of workpiece is ignored. The air resistance in workpiece deformation is ignored. The analysis is valid in case vacuum is established inside the coil. Gravity is neglected. No contact effect is considered.

2.2 Analytical evaluation of the magnetic field The magnetic flux density in the air-gap between the coil and the workpiece is given by the equation (Bednarski, 1985): h .  ð0:5Dinc Þ2 2 ð0:5Dow Þ2 BðtÞ ¼ B1 ðtÞ ð0:5Dinc Þ2 h i i ð3Þ w w 2hdw 2 2hdw þ 2dw ð0:5Dow 2 dw Þ 2 ð0:5Dow 2 hw 2 d2 Þe þ ð0:5Dow 2 hw Þ e where 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 2 6ð0:5Dinc þ hc Þ þ ð0:5Dinc þ hc Þ þ ðlc =2Þ 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 ðtÞ ¼ m0 ½lN =ð2hc Þ
ln4 5 I ðtÞ 0:5Dinc þ ð0:5Dinc Þ2 þ ðlc =2Þ2

ð4Þ

and l ¼ 0:44 is the filling factor for the specific coil geometry. Equations (3) and (4) do not take into account the magnetic field variation due to the workpiece deformation. 2.3 Field model description The model (Figure 4) is 3D while the coil is considered single-turned for modeling simplification, so the input current should be corrected with a factor, in order the induced magnetic field to be approximately the same in both cases: NI ðtÞ 1I eq ðtÞ < ) I eq ðtÞ < 0:5NI ðtÞ lc 0:5lc

ð5Þ

Every entity of the model is meshed with elements of appropriate degrees of freedom depending on the calculations that should be executed in each of them. SOLID 62 is a coupled field magneto-structural element, which encounters the simultaneous treatment of both the electromagnetic and mechanical aspects of the process. The magnitude of the surrounding air volume is selected rather arbitrarily but big enough in order to ensure safe results for the magnetic field. The dimensionality of this entity as well as its mesh density needs optimization. Load and boundary conditions are both applied to nodes. The current is applied in transient analysis in increments of 1 ms and steps of 0.25 ms. Small time-stepping

Electromagnetic forming

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Figure 4. The finite element model layout: (a) the whole model; and (b) a detail including the workpiece and the coil. The depicted coordinate system in all figures of the model as well as results is the global Cartesian

encounters the solution convergence, since plastic strain increment greater than 15 percent in each step is not permitted. Initially, the mechanical part of the process is ignored (i.e. the workpiece is meshed with SOLID97 elements of Ax, Ay, Az, volt degrees of freedom). The maximum magnetic flux density in the gap between the coil and the undeformed workpiece is numerically evaluated and from the following equations, it may be estimated and verified whether the resulting magnetic pressure is appropriate to efficiently deform the workpiece of the specific yield strength (Figure 5(a) and (b)). It is notable that electromagnetic part of the problem is purely linear and the solution period is considerably short. The required magnetic pressure to begin the circumferential deformation of the workpiece is derived from the workpiece equation of motion and given by Bednarski (1985) and Plum (1996):

Electromagnetic forming

795

Figure 5. (a) Magnetic flux density distribution in the gap between the coil and the workpiece; and (b) eddy current distribution induced in the workpiece

p¼a

sYS hw Dow =2

ð6Þ

where a [ ð1; 10
taking into account the inertia effects, the workpiece length, the workpiece anisotropy as well as strain and strain-rate hardening, the thickness stresses because the workpiece is not thin-walled [i.e. hw =ð0:5Dow Þ . 0:1
: From equation (6), the appropriate value of the workpiece yield strength may roughly be approximated by:

sYS <

1 Dow B2max 4a hw m0

ð7Þ

3. Results and discussion According to equation (7), the workpiece yield strength should be lower than 342 MPa and higher than 34.2 MPa, in order to be efficiently deformed. The selection of the optimum electrical parameters to attain the desirable deformation

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Figure 6. (a) Lorentz’s forces distribution developed in the workpiece; and (b) Von-Mises stress distribution in the workpiece

of the workpiece is one of the most demanding problems of electromagnetic forming process. The maximum value of the magnetic flux density in the gap between coil and workpiece, evaluated by analytical calculations (Bednarski, 1985), is Bmax ¼ 12:81 T at 29 ms. This value agrees with the numerical results. There is a stress concentration near the location of tangential translation constraints. These boundary conditions may act like reflection surfaces for the stress waves, which do not really exist (Balanethiram et al., 1994) Figure 6(a) and (b). This problem should be studied more deeply. Typical deformation velocities developed in electromagnetic forming processes lie in the 100 m/s order of magnitude. Deformation velocity is a critical parameter for

studying the inertia effects occurred during process (Balanethiram et al., 1994) (Figure 7(a) and (b)). Despite the fact that the magnetic field in gap between the coil and the workpiece is reduced at the end of the coil, the Lorentz’ s forces developed in the workpiece and the resultant deformation is higher in this area. This phenomenon is due

Electromagnetic forming

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Figure 7. (a) The radial deformation velocity of the workpiece, in the middle of its length with time; and (b) final equivalent plastic strain distribution in the workpiece

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Figure 8. (a) The workpiece equivalent strain vs time, 0.5 mm under its outer surface; and (b) the tangential strain-rate during workpiece deformation, in the middle of its length

to the end effect and it may be affected by the ratio of the coil to the workpiece length. From Figure 8(a), it is obvious that there is no need to model the process beyond the first half pulse of the electric current, since the plastic deformation of the workpiece is fulfilled in this period and the subsequent pulses are insufficient to deform the workpiece. Deformation results could not be obtained from similar 2D modeling of the process (Chunfeng et al., 2002). These results are very useful for

planning the industrial fabrication of workpieces (Oliveira et al., 2001). The strain-rate of the workpiece (Figure 8(b)) is higher than 102 3 s2 1; thus, the strain-rate hardening is not negligible (Balanethiram et al., 1994). 4. Conclusions Both equivalent circuit and coupled field approaches have been used for electromagnetic forming process analysis. In the latter method the electromagnetic and the mechanical aspects are simultaneously treated in the same modeling environment. In this way, process becomes friendlier for industrial application, because it can be analyzed like the conventional sheet metal forming processes. The results have shown that the strain-rate of the workpiece is higher than 102 3 s2 1; thus, the strain-rate hardening is not negligible, the workpiece final shape is tubby and the tangential translation constraints may act like reflection surfaces for the stress waves. References Balanethiram, V.S., Hu, X., Altynova, M. and Daehn, G.S. (1994), “Hyperplasticity: enhanced formability at high rates”, J. Mat. Proc. Tech., Vol. 45 Nos 1-4, pp. 595-600. Bednarski, T. (1985), “Magnetic reducing of thin-walled tubes”, Proc. 3rd Seminar on Metal Forming, Gyo¨r, Hungary, pp. 19-33. Chunfeng, L., Zhiheng, Z., Jianhui, L., Yongzhi, W. and Yuying, Y. (2002), “Numerical simulation of the magnetic pressure in tube electromagnetic bulging”, J. Mat. Proc. Tech., Vol. 123, pp. 225-8. Go¨bl, N. (1978) “Unified calculating method of equivalent circuits of the electromagnetic forming tools”, PhD thesis, Technical University of Budapest. Oliveira, D.A., Worswick, M.J. and Finn, M. (2001), “Simulation of electromagnetic forming of aluminium alloy sheet”, SAE Trans. J. Mat. Manuf., Vol. 110, pp. 687-95. Plum, M.M. (1996), “Electromagnetic forming”, Metals Handbook, 9th ed., Maxwell Laboratories Inc., Metals Park, OH, Vol. 14. Stadelmaier, H.H. (2000), “Magnetic properties of materials”, Mat. Sci. Eng., Vol. A287, pp. 138-45.

Electromagnetic forming

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800 Received September 2002 Revised May 2003 Accepted May 2003

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2D harmonic balance FE modelling of electromagnetic devices coupled to nonlinear circuits J. Gyselinck, P. Dular, C. Geuzaine and W. Legros Department of Electrical Engineering, Institut Montefiore, University of Lie`ge, Lie`ge, Belgium Keywords Finite element analysis, Nonlinear control systems, Harmonics, Frequency multipliers Abstract This paper deals with the two-dimensional finite element analysis in the frequency domain of saturated electromagnetic devices coupled to electrical circuits comprising nonlinear resistive and inductive components. The resulting system of nonlinear algebraic equations is solved straightforwardly by means of the Newton-Raphson method. As an application example we consider a three-phase transformer feeding a nonlinear RL load through a six-pulse diode rectifier. The harmonic balance results are compared to those obtained with time-stepping and the computational cost is briefly discussed.

Introduction The steady-state finite element (FE) analysis of electromagnetic devices can be carried out either in the time domain or the frequency domain. For reasons of efficiency and ease of implementation, the first approach, also referred to as time stepping, is mostly preferred to the latter, the harmonic balance (HB) approach (De Gersem et al., 2001; Yamada and Bessho, 1988). Furthermore, the coupling of the FE model to a nonlinear circuit has – to the best of our knowledge – not yet been considered in a HB simulation. In this paper, the easy-to-implement HB-FE approach proposed by Gyselinck et al. (2002) is extended to the nonlinear components in the electrical circuit that is coupled to the FE model. The method is elaborated for a two-dimensional (2D) FE model and an electrical circuit that comprises nonlinear resistive and inductive components. Herein, the commonly used magnetic vector potential formulation and the current loop method are adopted (Lombard and Meunier, 1992). However, the same approach can be followed in many other cases, e.g. for 3D FE models with various formulations.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 800-812 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410510785

This is a revised and enhanced version of a paper which was originally presented as a conference contribution at the XVII Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC), held in Leuven, Belgium, on 1-3 July 2002. This is one of a small selection of papers from the Symposium to appear in the current and future issues of COMPEL. The research was carried out in the frame of the Inter-University Attraction Poles for fundamental research funded by the Belgian State. P. Dular and C. Geuzaine are Research Associate and Postdoctoral Researcher, respectively, with the Belgian Fund for Scientific Research (F.N.R.S.).

Equations in time domain 2D FE model We consider a FE domain V in the xy-plane. The magnetic field h(x, y, t) and the magnetic induction b(x, y, t) in V have a zero z-component. They are linked by the magnetic constitutive law h ¼ hðbÞ. The current density j(x, y, t) is directed along the z-axis: j ¼ jðx; y; tÞ1z : By introducing the magnetic vector potential aðx; y; tÞ ¼ aðx; y; tÞ1 z ; such that b ¼ curl a; the magnetic Gauss law div b ¼ 0 automatically holds. On the basis of a FE discretisation of V, a set of basis functions a n(x, y), e.g. associated with the #n nodes of the FE mesh, is defined. The magnetic potential a(x, y, t) is then approximated as aðx; y; tÞ ¼

#n X

an ðtÞan ðx; yÞ;

ð1Þ

n¼1

where an(t) are the nodal values of the magnetic vector potential. Ampe`re’s law curl h ¼ j is weakly imposed by weighing it in V with the #n basis functions am(x, y)1z ð1 # m # #nÞ: After partial integration, one thus obtains the following #n equations in terms of the #n degrees of freedom an(t): Z Z curl ðam 1 z Þ · hðbÞ dV ¼ am j dV; ð2Þ V V |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflffl ffl{zfflfflfflfflfflfflffl} Am ðtÞ

J m ðtÞ

where, for the sake of conciseness, the line integral on the boundary ›V and the corresponding boundary conditions are disregarded. When using (first-order) nodal basis functions, the left- and right-handside of equation (2), denoted as Am(t) and Jm(t), respectively, can be interpreted as the circulation of the magnetic field along an elementary contour around node m and the mth nodal current, respectively. Electrical circuit We consider an electrical circuit that comprises a number of conductors in the FE domain and a number of lumped components, viz #E voltage sources, #R resistive components and #L inductive components. Current sources and capacitive components are for the sake of brevity not considered. By introducing loop currents associated with a set of #l independent oriented current loops in the electrical circuit, Kirchhoff’s current law automatically holds. Kirchhoff’s voltage law produces #l voltage equations, which can be written as follows: ukE ¼ ukR þ ukL þ ukFE ;

ð3Þ

where ukE, ukR and ukL are the voltages in the kth current loop (1 # k # #l ) due to the voltage sources, and resistive and the inductive components, respectively, and where ukFE is due to the flux linkage of the conductors in the 2D FE model. The resistance and the end-winding inductance of the FE conductors are considered by means of lumped resistive and inductive components in the electrical circuit. The current in the lumped components can be expressed in terms of the loop currents il (1 # l # #l ) and the topology matrices D E, D R and D L. For instance, the current iRc in the cth resistive component (1 # c # #R) is given by

2D harmonic balance FE modelling 801

COMPEL 23,3

802

iRc ðtÞ ¼

#l X

DRlc il ðtÞ;

ð4Þ

l¼1

where DRlc is nonzero, and equal to ^ 1, only if the cth resistive component is present in the lth current loop, with the same or the opposite reference orientation. The voltage in the kth current loop due to the #E voltage sources is a given function of time ukE ðtÞ ¼

#E X

DEkc uEc ðtÞ;

ð5Þ

c¼1

where uEc(t) is the voltage across the cth voltage source. The voltage in the kth current loop due to the #R resistive components is ukR ðtÞ ¼

#R X

DRkc uRc ðiRc ðtÞÞ;

ð6Þ

c¼1

where uRc ¼ uRc ðiRc Þ is the linear or the nonlinear voltage-current characteristic of the cth resistive component. The voltage in the kth current loop due to the #L inductive components is ukL ðtÞ ¼

#L X

DLkc

c¼1

d fLc ðiLc ðtÞÞ; dt

ð7Þ

where fLc ¼ fLc ðiLc Þ is the linear or the nonlinear flux-current characteristic of the cth inductive component. Coupling of the FE model and the electrical circuit We consider only stranded conductors in the FE model, though massive conductors (displaying skin effect) can be readily included in the analysis (Lombard and Meunier, 1992). The current density jðx; y; t) in V may thus be written in terms of the loop currents il: jðx; y; tÞ ¼

#l X

jl ðx; yÞil ðtÞ;

ð8Þ

l¼1

where the functions jl ðx; y) are given. The right-handside Jm(t) in equation (2) can be written as J m ðtÞ ¼

#l X l¼1

J ml il ðtÞ

with J ml ¼

Z

am ðx; yÞjl ðx; yÞ dV: V

The induced voltage ukFE in the kth current loop is given by:

ð9Þ

ukFE

 Z  X #n dfkFE d dan lz ¼ ; ¼ jk ðx; yÞaðx; y; tÞ dV ¼ J nk dt dt dt V n¼1

ð10Þ

2D harmonic balance FE modelling

where lz is the active length of the 2D FE model along the z-axis. Time domain approach For a given voltage excitation and given initial conditions (at t ¼ 0), the system of #n + #l algebraic and differential equations (2) and (3) can be solved in the time domain. The time discretisation is commonly performed with the so-called u-method, which produces a system of nonlinear #n + #l algebraic equations for each time step from instant ti to instant tiþ1 ¼ ti þ Dt (Dular and Kuo-Peng, 2002). These nonlinear systems can be efficiently solved by means of the Newton-Raphson (NR) method. The u-method amounts to the Crank-Nicholson method if u ¼ 1=2, and to the backward Euler method if u ¼ 1: An intermediate value for u can be adopted so as to compromise between accuracy (u ¼ 1=2) and the elimination of spurious oscillations that may occur when strongly nonlinear components, e.g. power electronic components, are present in the electrical circuit (u ¼ 1) (Dular and Kuo-Peng, 2002). In the example given in this paper, the diodes of a rectifier are modelled as nonlinear resistances having a piecewise constant (differential) resistance, viz 102 1 V and 105 V in the conducting and nonconducting state, respectively. By simply setting u ¼ 1 for the voltage equations of the loops that contain diodes and u ¼ 0:55 for all the other equations, oscillations are suppressed or significantly damped. When the voltage excitation is time periodic and only the steady-state behaviour of the system is of interest, the time domain approach nevertheless requires to step through the transient phenomenon before reaching the quasi steady-state. In some cases, the transient phenomenon may decay very slowly. Consider, for example, the magnetising flux and currents in transformers and rotating machines, which are governed by a large time constant. It was observed that the transient can be effectively shortened by applying the excitation voltages gradually. Hereto, they are multiplied by a relaxation function f relax ðtÞ ¼ ð1 2 cosðpt=T relax ÞÞ=2 during a properly chosen initial time interval [0,Trelax].

Harmonic balance approach Harmonic balance equations The fundamental frequency and period of the time periodic analysis are denoted by f and T ¼ 1=f , respectively. The multi-harmonic time discretisation consists in approximating the #n magnetic potential nodal values an(t) and the #l loop currents il ðt) by a truncated Fourier series comprising a dc-term and nf frequencies f q ð1 # q # nf Þ; the latter being nonzero multiples of f. The corresponding 2nf þ 1 basis functions are pffiffiffi pffiffiffi ð11Þ H 0 ðtÞ ¼ 1; H 2q21 ðtÞ ¼ 2 cosð2pf q tÞ; H 2q ðtÞ ¼ 2 2 sinð2pf q tÞ; This is a set of orthonormal basis functions:

803

COMPEL 23,3

1 T

Z

(

T

H k ðtÞH l ðtÞ dt ¼ dk;l ¼

0

1;

if k ¼ l;

0;

if k – l:

ð12Þ

The degrees of freedom an(t) and il ðt) are thus written as

804 an ðtÞ ¼

2nf X

aðnlÞ H l ðtÞ

and

l¼0

il ðtÞ ¼

2nf X

iðl lÞ H l ðtÞ:

ð13Þ

l¼0

By weighing the #n + #l equations (2) and (3) with the 2nf þ 1 basis functions Hk(t) in the time interval [0, T ], a system of (2nf þ 1) (#n þ #l) nonlinear algebraic equations in terms of the (2nf þ 1) (#n + #l) unknowns aðnlÞ and ilðlÞ is obtained: Z Z 1 T 1 T Am ðtÞH k ðtÞ dt ¼ J m ðtÞH k ðtÞ dt; T 0 T 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} AðmkÞ

ð14Þ

J ðmkÞ

Z Z Z 1 T 1 T 1 T ukE ðtÞH k ðtÞ dt ¼ ukR ðtÞH k ðtÞ dt þ ukL ðtÞH k ðtÞ dt T 0 T 0 T 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} kÞ uðkE

kÞ uðkR

þ

uðkLkÞ

Z 1 T ukFE ðtÞH k ðtÞ dt; T 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð15Þ

kÞ uðkFE

with 1 # m # #n and 1 # k # #l. kÞ The harmonic components of the applied voltages, uðkE ; are the given excitation of the system, whereas the other terms in equations (14) and (15) depend, in a linear or nonlinear way, on the harmonic components of either the nodal values of the vector potential, aðnlÞ ; or the current loops, iðl lÞ : NR method The iterative resolution of this system by means of the NR method is elaborated in the following. For the pth NR iteration ðp ¼ 1; 2; . . .Þ; the equations are linearised around lÞ lÞ lÞ and iðl;p21 ; and the increments Daðn;p and Diðl;plÞ follow from the the earlier solution aðn;p21 resolution of the linearised system: " # 2nf #n #l X X ›AðmkÞ ðlÞ X ›J ðmkÞ ðlÞ kÞ kÞ Dan;p 2 Dil;p ¼ 2Aðm;p21 þ J ðm;p21 ðl Þ ðl Þ › a › i n l¼0 n¼1 l¼1 l

ð16Þ

" 2nf kÞ #l X X ›uðkR l¼0

l¼1

›iðl lÞ

þ

›uðkLkÞ ›iðl lÞ

! Diðl;plÞ

þ

kÞ #n X ›uðkFE

#

2D harmonic balance FE modelling

lÞ Daðn;p ðl Þ › a n n¼1

kÞ kÞ kÞ kÞ ¼ uðkE 2 uðkR;p21 2 uðkL;p21 2 uðkFE;p21 ;

ð17Þ

with 0 # k # 2nf ; 1 # m # #n and 1 # k # #l. The right-handside terms (except the kÞ lÞ ) and the derivatives on the left-handside need to be evaluated for aðn;p21 constant uðkE ðl Þ and Dil;p21 : The pth solution is then given by lÞ lÞ lÞ aðn;p ¼ aðn;p21 þ Daðn;p

and

lÞ iðl;plÞ ¼ iðl;p21 þ Diðl;plÞ :

ð18Þ

lÞ and iðl;0lÞ ; is usually taken to be identically zero. A properly The initial solution, aðn;0 chosen nonzero initial condition may allow us to reduce the number of NR iterations.

Evaluation of the different derivatives in equations (16) and (17) The derivative of AðmkÞ with respect to aðnlÞ depends on the magnetic constitutive law h ¼ hðbÞ that applies in V, and, more precisely, on the differential reluctivity tensor ›h=›b; i.e. derivative of the magnetic field vector h with respect to the induction vector b. Let us, for sake of simplicity, consider a nonhysteretic nonlinear isotropic material. The scalar reluctivity n(b) is a single-valued function of the magnitude of b; and the differential reluctivity tensor can be written as

›h dn ¼ n1 þ 2 2 b b; ›b db

ð19Þ

where b b is the dyadic square of b (Gyselinck et al., 2002). In the xy coordinate system, the matrix representation of the reluctivity tensor is 2 ›h ›h 3 " # " # x x   ›b ›by 1 0 ›h dn bx bx bx by 6 x 7 þ2 2 : ð20Þ ¼ 4 ›hy ›hy 5 ¼ n ›b db by bx by by 0 1 ›bx

›by

In linear isotropic media, the reluctivity n is a constant scalar and the second term in the right-handside of equation (19) vanishes. Considering that

›hðtÞ ›h ›b ›an ›h ¼ ¼ curl ðan 1 z ÞH l ðtÞ; ðl Þ ð l Þ › b › an › an ›b ›a n it follows that the derivative of AðmkÞ with respect to aðnlÞ can be written as Z   ›AðmkÞ ¼ curl ðam 1 z Þ· nðdk;lÞ curl ðan 1 z Þ dV; ðl Þ › an V

ð21Þ

ð22Þ

where the harmonic differential reluctivity tensor ndðk;lÞ depends on the variation of

805

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the differential reluctivity tensor ›h/›b in [0,T ], which in turn depends on the harmonic content of the induction b(t):

nðdk;lÞ ðb 0 ; . . .; b2nf Þ ¼

806

1 T

Z 0

T

›h H k ðtÞH l ðtÞ dt: ›b

ð23Þ

kÞ One easily finds that the derivative of uðkR with respect to ilðlÞ can be written as kÞ ›uðkR

›iðl lÞ

¼

#R X c¼1

DRkc DRlc

›uðRckÞ ›iðRclÞ

ð24Þ

;

where the #R harmonic differential resitances depend on the variation of the differential resistances in [0, T ]:

›uðRckÞ ›iðRclÞ

1 ¼ T

Z 0

T

duRc H k ðtÞH l ðtÞ dt: diRc

ð25Þ

From equation (12) it follows that for linear resistive components, the left-handside in equation (25) can be simplified to dk,lRc, where Rc is the constant resistance. The kth harmonic component of the voltage in the kth current loop due to the lumped inductances can be written as uðkLkÞ

¼

#L X c¼1

DLkc

1 T

Z 0

T

#L X dfLc 21 dt ¼ H k ðtÞ DLkc T dt c¼1

Z

T

0

dH k fLc ðtÞ dt; dt

ð26Þ

where the time derivative of the flux fLc can be eliminated due to partial integration and the periodicity, H k ð0ÞfLc ð0Þ ¼ H k ðTÞfLc ðTÞ: Further, given

›fLc dfLc ›iLc ›il dfLc L ¼ ¼ D H l ðtÞ; diLc ›il ›ilðlÞ diLc lc ›ilðlÞ

ð27Þ

the derivative of uðkLkÞ with respect to iðl lÞ can be expressed as

›uðkLkÞ ›iðl lÞ

¼

#L X

DLkc DLlc X ðLck;lÞ ;

ð28Þ

c¼1

ðk;lÞ where the #L harmonic differential reactances X Lc follow from the variation of the respective differential inductances in [0,T ]: Z 1 T dfLc dH k ðk;lÞ H l ðtÞ dt: ð29Þ X Lc ¼ 2 T 0 diLc dt

If the cth inductive component is linear, equation (29) can be simplified to X ðLck;lÞ ¼ vk;l Lc ; where Lc is the constant inductance and where vk,l is defined as follows:

vk;l

Z Z 1 T dH k 1 T dH l H l ðtÞ dt ¼ dt ¼2 H k ðtÞ T 0 dt T 0 dt 8 22pf q ; if k ¼ 2q 2 1 and l ¼ 2q; > > < if k ¼ 2q and l ¼ 2q 2 1; ¼ 2pf q ; > > : 0; in all other cases:

2D harmonic balance FE modelling ð30Þ

807

In the general nonlinear case, considering arbitrary functions h(b), uRc(iLc) and fLc(iLc), the time integration over [0,T ] in equations (23), (25) and (29) cannot be performed analytically. A simple numerically integration scheme consists in considering a sufficiently large number of equidistant time instants ti in [0,T ], each having an equal weight. kÞ with respect to iðl kÞ and aðnlÞ ; Remains the constant derivative of J ðmkÞ and uðkFE respectively. The former can, on account of equations (9) and (10), be written as J ðmkÞ

¼

#l X l¼1

kÞ uðkFE

#n X

1 J ml T

1 ¼ J nk T n¼1

Z 0

T

Z

T

H k ðtÞil ðtÞ dt ¼

0

#l X

J ml iðl kÞ ;

ð31Þ

l¼1

2nf #n X X dan dt ¼ H k ðtÞ J nk vk;l aðnlÞ : dt l¼0 n¼1

ð32Þ

The constant derivatives are thus given by

›J ðmkÞ ¼ dk;l J ml ›iðl lÞ

and

kÞ ›uðkFE

›aðnlÞ

¼ vk;l J nk :

ð33Þ

Application example In order to validate the above proposed method, we consider a fictitious 50 Hz, 380 V/380 V, 10 kV A three-phase five-limb transformer feeding a nonlinear RL load either directly (unbalanced load) or via a six-pulse rectifier (Figure 1). The nonlinear RL load consists of a resistance of 20 V in series with a saturable inductance fðiÞ ¼ f0 a tanði=i0 Þ þ L0 i; with f0 ¼ 0:7 Wb, i0 ¼ 1:5 A and L0 ¼ 2:6 mH. The primary windings are connected to three balanced 220 V/rms 50 Hz voltage sources.

Figure 1. Electrical circuit for the two operations considered

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808

The resistances of the six diodes are Ron¼ 102 1 V and Roff ¼ 105 V. All six transformer windings have 170 turns. Their resistance is either 0.73 V (primary phases) or 0.54 V (secondary phases). The end-winding inductance is neglected. Exploiting the symmetry, only half of the transformer cross-section is discretised (Figure 2). The FE mesh has 892 first-order triangular elements (#n ¼ 551). Both the axial length lz and the width of the three central limbs are 60 mm. For the saturable core we take nðbÞ ¼ 100 þ 10 expð1:8b 2 Þ: The Dirichlet condition a ¼ 0 is imposed on the outer core boundary. All calculations are carried out on a Pentium III 750 MHz. Both the time-stepping and the HB systems of algebraic equations are solved by means of GMRES with ILU preconditioning, after renumbering with the reverse Cuthill McKee algorithm (Sparskit, 2003). For each calculation, the fill-in and dropping parameters of the preconditioning are set in order to minimize the computational cost. In the nonlinear HB calculations, the fill-in of the Jacobian matrix (average number of nonzero entries per row) increases with the number of considered frequencies. For large nf, the fill-in due to the nonlinear region in the FE model may become cumbersome. Indeed, consider a first-order triangular element in a saturable region. It produces a full ð6nf þ 3Þ £ ð6nf þ 3Þ matrix in a HB simulation, compared to only 3 £ 3 contribution in a time-stepping simulation. Operation without rectification The interval [0, 8T ] is time-stepped with T relax ¼ 5T and Dt ¼ T=200; requiring a computation time of 573 s (on an average 3.4 NR iterations per time step and 0.1 s per NR iteration). The waveform at quasi-steady-state of the current in the RL load and of the induction in the leftmost limb are shown in Figures 3 and 4, respectively. One verifies that besides the fundamental frequency f, only odd harmonics ð2k þ 1Þf are present. The distortion of the load current (Figure 3) is almost entirely due to the saturation of the lumped inductance, whereas the distortion of the induction (Figure 4) is due to the saturation of the transformer core. The magnetising (or noload) current is much smaller than the load current: the peak value of the former is about 0.4 A (compared to 20 A). Also note that an asymmetrically distorted current is not abnormal in a three phase system (Gyselinck et al., 2002). HB calculations are carried out considering first only the fundamental frequency (denoted by HB 1), and then gradually expanding the spectrum with odd harmonics, up to the ninth harmonic 9f. The fifth and last calculation is denoted by HB 1,3,5,7,9. Some current and induction waveforms are shown in Figures 3 and 4. Figure 5 shows some harmonic components of the flux pattern. A satisfactory convergence towards the time-stepping results is observed.

Figure 2. 2D FE model of three phase transformer

2D harmonic balance FE modelling 809

Figure 3. Current in the RL load obtained

Figure 4. Vertical component of the induction in leftmost limb

Figure 5. Components of the flux pattern

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810

Figure 6. Current in a secondary winding

Figure 7. Current in a diode

The computation time for these five HB calculations is 7.6, 35, 98, 220 and 529 s, respectively (12 or 13 NR iterations). If the NR process is initialised with the previous HB solution (i.e. having one frequency less), the calculation time for HB 1; 3 to HB 1; 3; 5; 7; 9 is 17, 40, 71 and 126 s (with only 6, 5, 4 and 4 iterations). Operation with rectification First, the transformer core is assumed to have a constant reluctivity n ¼ nðb ¼ 1:5 TÞ: The interval ½0; 8T
is time-stepped with T relax ¼ 5T and Dt ¼ T=400; requiring a total computation time of 939 s (on an average 2.9 NR iterations per time step). The waveform of the current in a secondary winding, in a diode and in the RL load are shown in Figures 6-8, respectively. The dc-term and all harmonics are present in the currents. Even in the presence of diodes in the electrical circuit, the NR process is observed to converge well. Five HB calculations are carried out: HB 0-1 (dc-term and f ), HB 0-3 (dc-term and f, 2f and 3f ), HB 0-6, HB 0-12 and HB 0-18. Some HB results are presented in Figures 6-8. One observes that from HB 0-6 on, the agreement with the time-stepping results

2D harmonic balance FE modelling 811

Figure 8. Current in the RL load (sixth of period)

becomes more and more acceptable. These five HB simulations require 16, 16, 19, 19 and 26 NR iterations, respectively, and 4.8, 11.3, 30, 87 and 280 s. The time-stepping simulation and some HB simulations (HB 0-1, HB 0-3 and HB 0-6) are now carried out with a saturable core. The time-stepping simulation requires 1,130 s (in average 3.1 NR iterations per time step). Because of the large fill-in, the HB simulations are much more expensive than earlier: 12 s for HB 0-1, 104 s for HB 0-3, and 683 s for HB 0-6. Note that the thus obtained currents differ only slightly from those obtained earlier (with a linear FE model), and that furthermore the moderate distortion of the magnetising flux (see Figure 4) could be modelled with a low number of frequencies. Conclusion The HB analysis of a 2D FE model of an electromagnetic device coupled to an electrical circuit comprising of nonlinear resistive and inductive components has been studied. The resulting system of nonlinear algebraic equations is solved straightforwardly by means of the NR method. The proposed method has been applied successfully to a three-phase transformer feeding a nonlinear RL load through a six-pulse rectifier. If a moderate number of frequencies suffices to obtain a good accuracy, the HB-FE method can be a very interesting alternative to time-stepping. Otherwise, the fill-in due to the saturable region in the FE domain may significantly reduce the attractiveness of the HB approach. More efficient algebraic solvers are then to be found or developed. References De Gersem, H., Vande Sande, H. and Hameyer, K. (2001), “Strong coupled multi-harmonic finite element simulation package”, COMPEL, Vol. 20 No. 2, pp. 535-46. Dular, P. and Kuo-Peng, P. (2002), “An efficient time dicretization procedure for finite element-electronic circuit equation coupling”, COMPEL, Vol. 21 No. 2, pp. 274-85.

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Gyselinck, J., Dular, P., Geuzaine, C. and Legros, W. (2002), “Harmonic balance finite element modelling of electromagnetic devices: a novel approach”, IEEE Trans. Magn., Vol. 38 No. 6, pp. 521-4. Lombard, P. and Meunier, G. (1992), “A general method for electric and magnetic coupled problem in 2D and magnetodynamic domain”, IEEE Trans. Magn., Vol. 28 No. 2, pp. 1291-4. SPARSKIT (2003), a basic tool-kit for sparse matrix computations, http://www.cs.umn.edu/ Research/arpa/SPARSKIT/sparskit.html Yamada, S. and Bessho, K. (1988), “Harmonic field calculation by the combination of finite element analysis and harmonic balance method”, IEEE Trans. Magn., Vol. 24 No. 6, pp. 2588-90.

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Finite element analysis of coupled phenomena in magnetorheological fluid devices Wojciech Szela˛g Institute of Industrial Electrical Engineering, Poznan´ University of Technology, Poznan´, Poland Keywords Couplers, Electromagnetism, Fluids, Finite element analysis

Finite element analysis

813 Received September 2002 Revised February 2003 Accepted February 2003

Abstract This paper deals with coupled electromagnetic, hydrodynamic and mechanical motion phenomena in magnetorheological fluid devices. The governing equations of these phenomena are presented. The numerical implementation of the mathematical model is based on the finite element method and a step-by-step algorithm. In order to include non-linearity, the Newton-Raphson process has been adopted. A prototype of an electromagnetic brake has been built at the Poznan´ University of Technology. The method has been successfully adapted to the analysis of this brake. The results of the analysis are presented.

1. Introduction The demand for electromechanical transducers with upgraded functional parameters both in their steady and transient state has been growing in recent years. The research on how to improve these parameters is conducted in various directions. One of them involves the use of magnetorheological fluids (MRFs), with their physical properties changing under the influence of magnetic fields, for electromechanical converter applications. MRFs are microstructured fluids that consist of a suspension of magnetically polarizable particles in a non-conducting liquid (Carlson et al., 1996; Rosensweig, 1985)[1]. The most often used liquid is synthetic oil. The particles are made of a ferromagnetic material and have a size of few microns. Each of them is a separate domain with a permanent magnetic field. MRFs contain from 20 to 80 per cent of this type of particles, by weight. A characteristic feature of MRFs is that under the influence of an external magnetic field, the initially unordered particles become oriented and stick together to form particle chains in the fluid. On the macroscopic scale this process results in a significant increase in the dynamic viscosity of MRF yielding a considerable increase in shear stress (Carlson et al., 1996). The relationship between the yield stress t0 and the magnetic flux density B for the MRF of 132LD type is shown in Figure 1. The change of stress while increasing or decreasing the magnetic flux density occurs in microseconds (Carlson et al., 1996; Rosensweig, 1985). If an external field is not present, the fluid acts like a normal engine oil. Upon applying an external magnetic field, This is a revised and enhanced verson of a paper which was originally presented as a conference contribution at the XVII Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC), held in Leuven, Belgium, on 1-3 July 2002. This is one of a small selection of papers from the Symposium to appear in the current and future issues of COMPEL. The work presented in this paper was supported by the Polish Government through a special grant No. 8 T10A 006 20.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 813-824 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410510776

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814 Figure 1. The yield stress t0¼ f(B) for MRF 132LD

the magnetic moments of the particles are arranged along the magnetic field’s force lines and become less susceptible to thermal motion. For a magnetic field strength of 200-300 kA/m, the fluid assumes the same consistency as frozen butter. The fluid maintains its properties in the temperature range 2 40 to 1508C. The magnetisation characteristics of a MRF are non-linear (Figure 2). The magnetised particles in the fluid attract each other in the absence of an external magnetic field. The particles are coated with a surface active agent to prevent their agglomeration under the influence of these forces. In an immovable MR fluid the ferromagnetic particles cannot be dispersed in the liquid in a stable state for a long time due to their large size. Having been put under the influence of the gravitational force for a couple of days, the particles produce a sediment at low volume fraction. These unfavourable effects disappear as soon as the fluid is put in motion. Owing to their properties, MRFs are useful to efficient control of the transmission of torques and forces. They are used among others in rotary brakes, clutches, couplings, rotary dampers, linear dampers and tension control devices (Carlson et al., 1996; Szela˛g, 2001) [1]. This paper proposes a mathematical model of coupled electromagnetic, hydrodynamic and mechanic motion phenomena that can be applied to analyse the operation of MRF electromechanical devices. 2. Electromagnetic brakes and dampers with MRFs The scope of this paper has been limited to a short description of the construction and properties of electromechanical elements with MRFs such as brakes and vibration dampers. Their working principle is based on the phenomenon of stress changes in liquids under the influence of a magnetic field. As the field increases so does shear stress, and, as a result, the viscosity of the liquid. This increases the force acting against the movement of the mobile parts of the device. A field change is triggered off by changing the current in the coils exciting the magnetic field. The structure of a MRF brake is shown in Figure 3 (Szela˛g, 2001). This is a cylindrical-rotor brake system. The magnetic field is excited by a ring coil in a stator. The 132LD MRF produced by Lord Corporation has been used in the brake. The diameter and length of the frame are 140 and 80 mm, respectively. Up to 25 N m braking torque can be obtained in it. An advantage of this brake system is the small amount of electrical power consumed by the inducing coil, not exceeding a few watts.

Finite element analysis

815 Figure 2. B -H curve for MRF 132LD at T ¼ 208C

Figure 3. The brake with MRF

The structure of a model linear vibration damper is shown in Figure 4 (Szela˛g, 2001). It has the following properties: damping force F ¼ 15 kN; piston diameter d ¼ 50 mm; piston stroke l ¼ 50 mm; rated supply voltage U ¼ 4 V; gap length d ¼ 0:1 mm; rated current I ¼ 1:27 A: Its chambers are filled with a MRF fluid. An external force trying to change the position of the piston causes the formation of a pressure difference in the chambers of the damper. This force affects the intensity of fluid flow in the gap between the frame and piston. Since the cross-surface area of the gap is much smaller than the surface area of the piston, it creates a large hydraulic resistance to the fluid. As a result, a large force that opposes the motion of the piston and whose size depends on the viscosity of the fluid is created. This force can be controlled by changing the magnetic flux density in the gap. The magnetic field in the gap depends on the current in the exciting coil. The generated force increases significantly if a large number of exciting coils is placed in the damper. 3. Coupled phenomena model Coupled electromagnetic, hydrodynamic and mechanical phenomena have been considered. Other phenomena, such as heat and torsional stress, have not been considered.

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The field model of slow-changing electromagnetic phenomena in electromechanical devices with a MRF fluid consists of the equations describing: the magnetic field 7£H ¼ J;

ð1Þ

7·B ¼ 0;

ð2Þ

7 £ E ¼ 2›B=›t;

ð3Þ

7·J ¼ 0;

ð4Þ

and the electric field

where H is the magnetic field strength vector, B ¼ mH is the magnetic flux density vector, m is the magnetic permeability, E is the electric field intensity vector, J ¼ gE is the current density vector, and g is the conductivity of the medium. After introducing the magnetic vector potential A specified by the following relationship B ¼ 7 £ A; the above-mentioned equations describing the transient electromagnetic field in a non-linear conductive and moving medium can be expressed as (Demenko, 1994; Szela˛g, 2000) 7 £ ð1=m7 £ AÞ ¼ J :

ð5aÞ

In the region with eddy current (equation (5a)) should be completed by additional equations J ¼ 2g ›A=›t þ g7V e ;

ð5bÞ

7 · ð2g ›A=›t þ g7V e Þ ¼ 0:

ð5cÞ

Here, Ve is the electric potential. The eddy current in the windings composed of thin stranded conductors are neglected. Therefore, the current density vector in the winding is calculated from equation J ¼ ni=S c

ð5dÞ

where Sc is the cross-sectional area of the conductor, i is the winding current, and n is the unit normal vector to the cross-sectional plane.

Figure 4. The linear vibration damper with MRF

In general, the transient electromagnetic field in MRF devices is voltage-excited. This means that the currents i in the windings are not known in advance, i.e. prior to the electromagnetic field calculation (Nowak, 1998; Szela˛g, 2000). Therefore, it is necessary to consider the equations of the electric circuit of the device. The set of independent loop equations may be written as u ¼ Ri þ

d C; dt

ð6Þ

where u is the vector of supply voltages, i is the vector of loop currents, R is the matrix of loop resistances, and C is the flux linkage vector. The vector C is calculated by means of the field model. The phenomenological approach has been used to describe fluid dynamics. In this approach, the fluid is treated as a non-conducting continuum of properties determined by density r, dynamic viscosity n, magnetic permeability m, and thermal conductivity l (Bird et al., 1960; Rosensweig, 1985; Verardi and Cardoso, 1998). In the hydrodynamic model, the laminar flow of a non-compressible fluid with no mass sources is investigated (Bird et al., 1960; Chung, 1978; Rosensweig, 1985). It is assumed that the gravitational forces acting on the fluid are negligible compared to the forces causing its motion in the transducer. The motion of the liquid is caused by the motion of the rotor. It is also assumed that the internal energy and temperature of the fluid are constant. For such conditions, the flow continuity equation can be written as 7·v ¼ 0

ð7Þ

and the resulting differential equation of motion is

r

›v þ rðv·7Þv ¼ 7·tw ; ›t

ð8Þ

where v is the velocity vector, r is the fluid density, tw is the stress tensor with elements dependent on pressure, velocity and magnetic field. Equations (7) and (8) do not contain any information about the physical properties of the fluid. In order to consider these properties, the relationship between the stress tensor and the pressure, velocity v and magnetic fields B must be formed (Nouar and Frigaard, 2001; Rosensweig, 1985). One of the important properties, demanded from all models, is the symmetry of the stress tensor. This stems from the isotropy of the continuum. MRFs belong to the non-Newtonian group of fluids. The properties of such fluids are described by the following tensor equation (Bird et al., 1960; Nouar and Frigaard, 2001)

tw ¼ 2pI þ t

ð9Þ

where t is the stress tensor resulting from the heterogeneity of the velocity and yield stress t0(B) induced by a magnetic field, I is the unit matrix, and p is the pressure. The heterogeneity of the velocity field is determined by the rate of deformation tensor D D ¼ 0:5½7v þ ð7vÞT
:

ð10Þ

Finite element analysis

817

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A typical characteristic family t ¼ t ðD; BÞ for a one-dimensional fluid model is shown in Figure 5. The fluid behaves like an elastic body for t # t0 ðBÞ; and like a body of plastic viscosity hp for t . t0 ðBÞ; hp ¼ tgðbÞ (Figure 5). The properties of such a fluid can be described by the Bingham model (Bird et al., 1960; Hammand, 2000). In the elaborated three-dimensional model of a MRF the elements of the stress tensor t are determined from the equation

ti; j ¼ ðhp þ t0 ðBÞ=kDkÞDi; j

for ktk . t0 ðBÞ

ð11Þ

where i; j ¼ 1; 2; 3; kDk; ktk are the norms of the tensor D and t (Hammand, 2000; Nouar and Frigaard, 2001) kDk ¼

3 X 3 1X D2 2 i¼1 j¼1 i;j

!1=2

Di; j ¼ 0

;

ktk ¼

3 X 3 1X t2 2 i¼1 j¼1 i;j

for ktk # t0 ðBÞ:

!1=2 ;

ð12aÞ

ð12bÞ

The differential motion equation of the fluid (8), including the relationships (9)-(12), assumes the following form

r

›v þ rðv·7Þv ¼ 27p þ 7·ðnz DÞ ›t

ð13Þ

where nz is the equivalent dynamic viscosity of the fluid. From equation (11) we obtain

nz ¼ hp þ t0 ðBÞ=kDk for ktk . t0 ðBÞ;

ð14aÞ

nz ¼ 1 for ktk # t0 ðBÞ:

ð14bÞ

The yield stress t0(B), appearing in equation (14) is determined on the basis of the distribution of the magnetic flux density obtained from equations (5) and (6). When analysing the performance of MRF electromechanical transducers, equations (5), (6), (7) and (13) describing the electromagnetic and hydrodynamic phenomena must

Figure 5. The shear stress in MRF

be solved with the equation of dynamics of its movable elements. For the brake it assumes the following form

Finite element analysis

dv þ T0 ¼ Tz ð15Þ dt where Jb is the moment of inertia; v is the angular velocity of the rotor, T0 is the braking torque, i.e. torque produced by brake, and Tz is the driving torque. The braking torque is a result of the yield stress in the fluid as well as the electromagnetic forces acting on the movable elements. These forces are calculated on the basis of the Maxwell surface stress tensor te (Rosensweig, 1985; Szela˛g, 2000). The total braking torque can be determined using the equation

819

Jb

T 0 ¼ ts rðtq þ teq Þ ds:

ð16Þ

The vectors tq, teq in this equation describe the stress in the fluid and the electromagnetic stress acting in the direction q at a tangent to the external surface of the brake rotor. 4. Finite element formulation In this paper, a magnetorheological brake with axial symmetry is considered (Figure 6). The cylindrical coordinate system r, z, q has been applied. In this case, equation (5) describing the transient electromagnetic field in the region Vb can be expressed as (Nowak, 1998)

Figure 6. Cross section of the brake

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 › 1 ›w › 1 ›w g ›w : þ ¼J2 ›r m l ›r ›z m l › z l ›t

ð17Þ

Here l ¼ 2pr; w ¼ 2prAq ; where Aq is the magnetic vector potential, J is the current density in the winding, and g is the conductivity of the region with eddy current. For the MRF g ¼ 0: On the boundary of the considered region w ¼ 0: It has been assumed that the flow of an incompressible MRF is laminar and is caused by the rotor motion (Figure 6). Thus, the fluid moves in the q-direction and there is no pressure gradient in this direction (Bird et al., 1960). For these assumptions equations (7) and (13) may by written as follows

 › nz ›f › nz ›f r ›f : ð18Þ þ ¼ ›r l ›r ›z l ›z l ›t Here f ¼ 2prvq ; where vq is the component of velocity v in the q-direction.The description of the problem should be completed by equation (14) and nonslip boundary conditions vq ¼ r v and vq ¼ 0 on the surface of the rotor and frame, respectively. In order to solve equations (6), (14) (17) and (18) the finite element method and a “step-by-step” procedure have been used (Demenko, 1994; Szela˛g, 2000). The backward difference scheme has been applied. The finite element and time discretisation lead to the following system of non-linear algebraic matrix equations 3 # 2 " #" wn ðDtÞ21 G wn21 2w S n þ ðDtÞ21 G 5; ð19Þ ¼4 in 2Dtu n 2 w T w n21 2DtR 2w T ½S 0n þ ðDtÞ21 G 0
fn ¼ ðDtÞ21 G 0 fn21 ;

ð20Þ

where n denotes the number of time-step, Dt is the time-step, S and S 0 are the magnetic and hydrodynamic stiffness matrices, w and f are the vectors of the nodal potentials w and f, respectively, w T is the matrix that transforms the potentials w into the flux linkages with the windings, G is the matrix of conductances of elementary rings formed by the mesh, G 0 is the matrix whose elements depend on the dimensions of the elementary rings and fluid density r. Motion equation (15) is approximated by the explicit difference formula (Szela˛g, 2000) J b ðanþ1 2 2an þ an21 Þ=ðDtÞ2 ¼ T z;n 2 T 0;n ;

ð21Þ

where a is the position of the rotor, T z;n ¼ T z ðt n Þ; T 0;n ¼ T 0 ðt n Þ: The angular velocity v of the rotor may be calculated according to the formula vðt n þ 0:5DtÞ ¼ ðanþ1 2 an Þ=Dt: The braking torque T0,n is described by formula (16). In the considered brake the component Bq of the magnetic flux density B is equal to zero. Therefore, in formula (16) the component of Maxwell stress tensor in the q direction is equal to zero, teq ¼ 0: Equation (20) is coupled through the viscosity function nz (B) with equation (19) and through the boundary condition vq ¼ r v with equation (21). Therefore these equations should be solved simultaneously. The Newton iterative method has been used to solve the above non-linear equations (Besbes et al., 1996; Szela˛g, 2000). The Jacobian matrix

in the Newton procedure has been factorised as the product LL T of triangular matrices. 5. Results and conclusions The presented method has been applied to the analysis of the steady-state and transients of a prototype of an electromagnetic brake. The cross-section of the brake is shown in Figure 6. The dimensions of the rotor are: diameter d ¼ 26:8 mm; length L ¼ 27 mm: The 132LD MRF produced by Lord Corporation has been used in the brake. For the steady-state, the calculations have been performed for different values of the voltage u applied to the winding and different values of the angular velocity v. The influence of the current i in the winding on the velocity profile along the line AB (Figure 6) is shown in Figure 7. The magnetic field distribution in the brake for u ¼ 2:6 V is shown in Figure 8. Figure 9 shows the distributions of constant velocity contours for two values of the current i. In order to verify the calculations, the braking torque T0 has been measured on the prototype of the brake. The results of the calculations and measurements of characteristics T0(i) for v ¼ const (Figure 10) indicate the relevance of this simulation method in the designing process of the magnetorheological brakes.

Finite element analysis

821

Figure 7. Velocity profile along the line AB for (a) v ¼ 2 rad/s, and (b) v ¼ 150 rad/s

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Figure 8. The magnetic field distribution in the brake

Figure 9. The distribution of constant velocity contours for v ¼ 150 rad/s; (a) i ¼ 0 A, and (b) i ¼ 0.2 A

Transients during a step change in the voltage u from 0 to 2.6 V have been considered. The delay between the changes of the magnetic flux density and viscosity of the fluid has been neglected. The angular velocity of the rotor is assumed to be constant, v ¼ 150 rad=s: The torque-time T0(t) and current-time i(t) characteristics have been analysed. These characteristics are shown in Figure 11. Time constant of T0(t) characteristics is more than one-order of magnitude greater than the time constant of the i(t) characteristics. The elaborated algorithm and software can be effective tools for the simulation of the steady-state and transients of an MRF electromagnetic brake. They provide for analyses of coupled electromagnetic, hydrodynamic and mechanical phenomena in the brake. In the paper saturation, eddy currents, rotor movements, non-linear properties and the flow of the fluid have been considered.

Finite element analysis

823 Figure 10. Torque-current characteristic of the brake

Figure 11. Time characteristic of the brake

Note 1. WWW.rheonetic.com References Besbes, M., Ren, Z. and Razek, A. (1996), “Finite element analysis of magneto-mechanical coupled phenomena in magnetostrictive materials”, IEEE Trans. Magn., Vol. 32 No. 3, pp. 1058-61. Bird, R.B., Stewart, W.E. and Lightfoot, E.N. (1960), Transport Phenomena, Wiley, New York, NY. Carlson, J.D., Catanizarite, D.M. and Clair, K.A. (1996), “Commercial magneto-rheological fluid device”, Proc. of 5th Int. Conf. on ER Fluids, MR Suspensions and Associated Technology, Singapore, pp. 20-8. Chung, T.J. (1978), Finite Element Analysis in Fluid Dynamics, McGraw-Hill, New York, NY. Demenko, A. (1994), “Time stepping FE analysis of electric motor drives with semiconductor converter”, IEEE Trans. Magn., Vol. 30 No. 5, pp. 3264-7. Hammand, K.J. (2000), “The effect of hydrodynamic conditions on heat transfer in a complex viscoplastic flow field”, International Journal of Heat and Mass Transfer, Vol. 43, pp. 945-62. Nouar, C. and Frigaard, I.A. (2001), “Nonlinear stability of Poiseuilla flow of Bingham fluid: theoretical results and comparison with phenomenological criteria”, Journal of Non-Newtonian Fluid Mechanics, Vol. 100, pp. 127-49.

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Nowak, L. (1998), “Simulation of the dynamics of electromagnetic driving device for comet ground penetrator”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3146-9. Rosensweig, R.E. (1985), Ferrohydrodynamics, Cambridge University Press, Cambridge. Szela˛g, W. (2000), “Demagnetization effects due to armature transient currents in the permanent magnet self starting synchronous motor”, EMF’2000, 17-19 May, Gent, pp. 93-4. Szela˛g, W. (2001), “The electromagnetic devices with the magnetic fluid”, The Scientific Papers of Electrical and Control Engineering Faculty, Technical University of Gdan´sk, (in Polish) No. 6, pp. 151-5. Verardi, S.L. and Cardoso, J.R. (1998), “A solution of two-dimensional magnetohydrodynamic flow using the finite element method”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3134-7.

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The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Comparison of the Preisach and Jiles-Atherton models to take hysteresis phenomenon into account in finite element analysis Abdelkader Benabou, Ste´phane Cle´net and Francis Piriou Laboratoire d ‘Electrotechnique et d’Electronique de Puissance de Lille, Universite´ des Sciences et Technologies de Lille, Villeneuve d’Ascq, France

Preisach and Jiles-Atherton models 825 Received September 2002 Revised January 2003 Accepted January 2003

Keywords Finite element analysis, Electromagnetism, Energy Abstract In this communication, the Preisach and Jiles-Atherton models are studied to take hysteresis phenomenon into account in finite element analysis. First, the models and their identification procedure are briefly developed. Then, their implementation in the finite element code is presented. Finally, their performances are compared with an electromagnetic system made of soft magnetic composite. Current and iron losses are calculated and compared with the experimental results.

1. Introduction Hysteresis phenomenon modelling is very useful in electrical engineering, especially for ferromagnetic material modelling. So, many hysteresis models have been developed and two of them are widely used in the static case: the Preisach model (Preisach, 1935) and the Jiles-Atherton ( J-A) model ( Jiles and Atherton, 1986). The first one is based on a mathematical description of the material behaviour and the second one on a physical description using energy balance. The implementation of the magnetic constitutive relationship in field computation allows us to have a more accurate description of an electromagnetic system. According to the used formulation, scalar (f) or vector (A) potential, the hysteresis model must be chosen, respectively, with the magnetic field H or the magnetic flux density B as entry. In this communication, we compare the accuracy of both the models cited earlier with B as entry in the case of A-formulation. First, we present both models and their identification procedures. Implementation of such models in finite element analysis is briefly described. Then, these models are applied to study a coil with a soft magnetic composite (SMC) core (Cros and Viarouge, 2002a). 2. The models The constitutive relationship of a magnetic material can be described by: B ¼ m0 ðH þ M Þ

ð1Þ

This is a revised and enhanced version of a paper which was originally presented as a conference contribution at the XVII Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC), held in Leuven, Belgium, on 1-3 July 2002. This is one of a small selection of papers from the Symposium to appear in the current and future issues of COMPEL.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 825-834 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410510794

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where B is the magnetic flux density, H the magnetic field, M the magnetisation and m0 the air permeability. In this section, the Preisach and the J-A models are presented. In their original form, they give the magnetisation M versus the magnetic field H. The constitutive relationship B (H) of the material is then calculated using equation (1). The use of the A-formulation requires a model with B as entry, which can be deduced from the original form of the Preisach (Park et al., 1993) and J-A (Sadowski et al., 2001) models. In this work, the use of both the models is restricted to the study of isotropic magnetic materials and in the case of a quasi-static behaviour. Moreover, we assume that the magnetic field does not rotate. In this context, B and H are collinear and a scalar model linking their modulus is sufficient to represent the material behaviour. 2.1 The Preisach model A ferromagnetic material can be described as a set of commutators ga, b, which have a rectangular shape and two switching fields a and b, respectively, the upper and the lower switching fields (Figure 1) (Park et al., 1993). In this model, a ferromagnetic material is characterised by a density distribution p(a, b) of these commutators. Applying some physical constraints, parameters a and b, we can define the Preisach triangle D shown in Figure 1. All couples (a, b) belong to the triangle D. A ferromagnetic material state is characterised by a given distribution of couples (a, b). The total magnetisation is given by: ZZ pða; bÞga;b da db ð2Þ M ¼ M sat D

Introducing, the Everett function (Everett, 1955): ZZ pða; bÞ da db Eðx; yÞ ¼ M sat

ð3Þ

Tðx; yÞ

with T(x, y) the right-angled triangle in the Preisach triangle with (x, y) the right angle coordinates, we can write: M ðBÞ ¼ M ðBm Þ ^ 2EðB; Bm Þ

ð4Þ

where Bm is the last return point (i.e. extremum) of the magnetic flux density. Then, from the Everett function, the magnetisation can be calculated without any numerical derivation or integration, provided that the magnetic flux density extrema are known.

Figure 1. (a) Elementary magnetic commutator; and (b) Preisach triangle, D

2.2 The J-A model The original J-A model presented by Jiles and Atherton (1986) gives the magnetisation M versus the external magnetic field H. The model is based on the magnetic material response without hysteresis losses. This is the anhysteretic behaviour which Man(H ) curve can be described with a modified Langevin equation:

 He a M an ðH Þ ¼ M sat coth ð5Þ 2 He a where He¼ H+aM is the effective field experienced by the domains, a is the mean field parameter representing inter-domain coupling. The constant a is linked to the temperature. By considering the losses induced by domain wall motions, the energy dissipated through pinning during a domain wall displacement is calculated ( Jiles and Atherton, 1986). The magnetisation energy is assumed to be the difference between the energy that would be obtained in the anhysteretic case minus the energy due to the losses induced by domain walls motions. Consequently, the differential susceptibility of the irreversible magnetisation Mirr can be written as: dM irr M an 2 M irr ¼ dBe m 0 kd

ð6Þ

where the constant k is linked to the average pinning sites energy. The parameter d takes the value +1 when dH =dt $ 0 and 2 1 when dH =dt , 0: However, during the magnetisation process, domain walls do not only jump from one pinning site to another: they are flexible and bend when being held on pinning sites. Domain wall bending is associated with reversible changes in the magnetisation process (Jiles and Atherton, 1986). Assuming that the total magnetisation is the sum of the reversible and irreversible components, we have the following expression: M ¼ M irr þ cðM an 2 M irr Þ

ð7Þ

where the reversibility parameter c[ [0,1]. As we are interested here in the M (B) model, the differentiation of equation (7) with respect to B and using Be¼ m0He lead to the differential equation (8) (Sadowski et al., 2001): dM an irr ð1 2 cÞ dM dM dBe þ c dBe ¼ dM an irr dB 1 þ m0 ð1 2 cÞð1 2 aÞ dM dBe þ m0 cð1 2 aÞ dBe

ð8Þ

Five parameters a, a, k, c and Msat have to be determined from the experimental results. 3. Identification procedures 3.1 The Everett function The Preisach model is fully determined by the Everett function. Determination of this function is achieved using the experimental measures. Having a set of experimental centred minor loops, we can calculate the corresponding Everett functions using an interpolation method. This one must respect the Everett function continuity on the whole studied domain and then for hysteresis curves. The proposed method detailed by Cle´net and Piriou (2000) satisfies these conditions and is accurate.

Preisach and Jiles-Atherton models 827

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However, the experimental measure procedure has to be made carefully, especially for high excitation fields where the hysteresis loops have to be rigorously centred otherwise the interpolation method fails. 3.2 The J-A model parameters Determination of parameters c, a, k and a requires an iterative procedure presented by Jiles et al. (1992) using a major hysteresis loop, the first magnetisation and the anhysteretic curves. This method is numerically sensitive and does not systematically converge. We use a slightly different procedure (Benabou, 2001). This first calculation step gives a good estimation of the parameter values but these can still be improved. Then, in a second step, an optimisation procedure to improve the five parameters is used. In fact, this enables us to have a set of parameters close to the best solution (in the sense of the chosen objective function) which makes easier the convergence of the optimisation procedure. This procedure is carried out using a set of centred hysteresis loops as presented by Cle´net et al. (2001). 3.3 Application to the SMC material Both identification procedures have been applied to the SMC material on a torus sample. For the Everett function identification, 20 centred minor loops were measured. The 3D Everett function is given in Figure 2. We can see that this function is quite smooth. This is linked to the fact that, at the coercive field of the SMC material hysteresis loop, there is no abrupt variation of the slope. For the J-A parameters, five centred hysteresis loops were used for the optimisation procedure. The obtained parameters are given in Table I. In Figure 3, a comparison between a measured loop and simulated ones with both the models is shown. They give good results for a wide range of the magnetic field magnitude (0 to 8,000 A/m). 4. Finite element analysis In magnetostatics, the equations to be solved on a domain D bounded by a surface S, are:

Figure 2. Three-dimensional Everett function for the SMC material

div B ¼ 0

curl H ¼ J

and n £ H ¼ 0

on

n·B ¼ 0

on

Sb

ð9Þ

Sh

with Sb and Sh two complementary parts of S, J the current density and n the outward normal vector of S. To consider the material behaviour, a constitutive relationship, denoted by H ¼ f ðBÞ; is added. To model ferromagnetic material, this relationship can be one of the two models presented earlier (Section 2). All these equations are generally solved using a potential formulation. In the 2D case, the vector potential formulation is generally preferred to the scalar potential one, then we have: curl½f ðcurl AÞ
¼ J

n£A¼0

on S b

and n £ H ¼ 0

on S h

Preisach and Jiles-Atherton models 829

ð10Þ

where A represents the magnetic vector potential (i.e. B ¼ curl A). To be solved, equation (10) can be discretised using the finite element method. In the case of hysteretic constitutive relationship, it leads to a non-linear system. The numerical solution of equation (10) including hysteresis cannot be done with the same method as the one used with univoc functions (Newton-Raphson scheme for example). We choose the fixed-point method already presented by Bottauscio et al. (1995). The hysteretic constitutive relationship is then rewritten under the form: Parameter

Value

a a (A/m) k (A/m) c Msat (A/m)

1.83 £ 102 3 1,642 1,865 0.79 1,122,626

Table I. J-A parameters for the SMC material

Figure 3. Hysteresis loops calculated from both the models and the corresponding measured loop

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H ¼ f ðBÞ ¼ nFP B 2 M FP ðBÞ

ð11Þ

The reluctivity n FP is a constant and must respect some conditions to have convergence (Ionita et al., 1996). Nevertheless, the studied hysteretic models are scalar models whereas we need a vectorial model in equation (11). Then, to evolve this latter, as it was written earlier, we assume that B and H are collinear (Section 2). The magnitude of H is calculated from the one of B, by means of the hysteresis model, and the direction of H is the same as the one of B (obtained by curl A). Consequently, the magnetisation MFP has the same direction as n FPB. Its magnitude is obtained by calculating M PF ¼ f ðBÞ 2 n FP B: Finally, the equation to be solved can be written as: curl n FP curl A ¼ J þ curl M FP

ð12Þ

The discretisation with nodal shape functions of equation (12) using the Galerkin method leads to the matrix system: ½S FP
½A
¼ ½ J
þ ½M FP


ð13Þ

where the vector [A ] represents the nodal values of the vector potential, [SFP] a square matrix, [MFP] and [J ] vectors which considers the magnetisation MPF and the current density J. One can note that the matrix [SFP] is constant because the permeability nFP is constant as well. The non-linearities introduced by the ferromagnetic materials are reported in the source term [MFP] which depend on B (i.e. A). To take the coupling with the external circuit into account, vector [J ] can be expressed as a function of the coil current i (we suppose to have only one coil): ½J
¼ ½D
i

ð14Þ

The coil is supposed to be formed by thin conductors, skin effects can be neglected. Under this condition, the distribution of the current density is given in the windings. To represent this, a vector of turns density N is defined (Piriou and Razek, 1992). Its modulus is given by the ratio of turns number to the coil section and null everywhere else. Under these conditions, the terminal voltage u is linked to the vector potential using the Faraday’s law: ZZ  d A·N dD ð15Þ u ¼ Ri þ dt D where R is the resistance of the coil winding. After the discretisation, introducing a vector [G ] the previous equation becomes: u ¼ Ri þ ½G


d ½A
dt

ð16Þ

Gathering equations (13) and (16), the final system is: "

S PF

2D

0

R

#"

A i

#

# " # " # " 0 M PF d A þ ¼ þ 0 G 0 dt i u "

0

0

#

ð17Þ

5. Experimental and simulation results 5.1 The studied system To compare the performances of both the models in finite element analysis, we study a coil with an SMC core (Cros et al., 2002b). The geometry of the system is shown in Figure 4. The coil is supplied by a 60 Hz-90 V RMS sinusoidal voltage. We aim at comparing in the steady-state current, iron losses and local evolution of B(H) curve. For this purpose, three different meshes of the electromagnetic system have been considered. The first mesh M1 has 1,284 elements, the second M2 2,987 elements and the last M3 5,448 elements. This will show the influence of the quality of the mesh.

Preisach and Jiles-Atherton models 831

5.2 Comparison of the results First of all, we can compare the B(H) curves obtained with both the models. For example, hysteresis loops for the point P1 (Figure 4) are shown in Figure 5.

Figure 4. Geometry of the studied system (mm)

Figure 5. Hysteresis loops calculated from both the models for the same location

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It appears clearly that both the models give similar results. Tests have been made elsewhere in the system and both the models gave still close results. The measured current is compared with the current obtained by field calculation. The current RMS values are reported in Table II for the three meshes. The mesh M3 gives results very close to those of mesh M2, then only current waveshapes for M1 and M2 are given in Figures 6 and 7, respectively. First, we notice that mesh M2 gives a slightly less important magnitude for the current than mesh M1. Influence of the hysteresis model is more pronounced in the case of mesh M1. In fact, for this mesh, the J-A model gives results with an error of 9 per cent and the Preisach model an error of 6.5 per cent in comparison with the experience. For the mesh M2, both the models give an error of 13 per cent. The gap between the experimental and simulation results increases with the mesh refinement. There are two error sources during the process to obtain a numerical model: . the modelling error which represents the difference between the actual device and the mathematical model equation (10), . the numerical errors which are the difference between the numerical model equation (17) and the mathematical one. In Figure 8, a representation of the evolutions of both errors versus the mesh quality (i.e. the number of elements) is shown. The modelling error is constant as it depends only on the mathematical equations, whereas the numerical error decreases with the mesh quality.

Measurements

Table II. Comparison between experience and calculation

Figure 6. Experimental and calculated currents for mesh M1

IRMS¼ 0.76 A IRMS M1 (A) IRMS M2 (A) IRMS M3 (A) ILexp¼4.3 W IL1 (W) IL2 (W)

J-A model

Preisach model

0.69 0.66 0.66

0.71 0.67 0.66

4.3 4.3

4.17 4.38

Preisach and Jiles-Atherton models 833 Figure 7. Experimental and calculated currents for mesh M2

Figure 8. Evolution of the error sources

Then, in the case of mesh M1, the numerical and the modelling errors make up for each other. Instead, with meshes M2 and M3, the numerical error is weak and almost only the modelling errors remain (see Figure 8). Iron losses are also given in Table II in the case of mesh M3. Iron losses denoted as IL1 are those obtained from magnetic quantities, i.e. from the sum of iron losses in each element of the mesh. Iron losses denoted as IL2 are those obtained from electrical quantities, i.e. IL2 ¼ kuðtÞiðtÞ 2 Ri 2 ðtÞl: Both methods IL1 and IL2 give close results. So, both can be used to estimate the iron losses. It must also be noticed that, for a given model, iron losses calculated from electrical quantities have the same values for the three meshes. Then, to estimate iron losses in our case, a fine mesh is not required. Computation times for both the models are presented in Table III. It shows clearly that the Preisach model is more time-consuming than the J-A model. This is more Mesh M1 M2 M3

No. of elements

J-A model

Preisach model

1,284 2,987 5,448

1 1 1

2.8 3.1 3.9

Table III. Computation time ratio (J-A model is the reference)

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important for the finest mesh M3. In this case, the Preisach model is almost four times more time-consuming than the J-A model. 6. Conclusion In this work, we have compared the J-A and Preisach models implemented in a finite element analysis with a SMC core coil. These models are well adapted for electrical devices with no rotating field. The studied system, a coil with revolution axis symmetry, satisfies this assumption. Both the models give similar results for the local behaviour (B(H) curve) and global quantities (current, iron losses). Moreover, the J-A model, that is easier to implement and faster, has shown results similar to those of the Preisach model for the studied system. References Benabou, A. (2001), “Identification et optimisation des parame`tres du mode`le de Jiles-Atherton pour la mode´lisation de l’hyste´re´sis magne´tique”, JCGE’01, 13-14 November, Nancy, France, pp. 229-34. Bottauscio, O., Chiarabaglio, D., Chiampi, M. and Repetto, M. (1995), “A hysteretic periodic magnetic field solution using Preisach and fixed point technique”, IEEE Trans. Magnetics, Vol. 31 No. 6, pp. 3548-50. Cle´net, S. and Piriou, F. (2000), “Identification de la fonction d’Everett pour le mode`le de Preisach”, MGE 2000, 13-14 December, Lille, France, pp. 71-4. Cle´net, S., Cros, J., Piriou, F., Viarouge, P. and Lefebvre, L.P. (2001), “Determination of losses local distribution for transformer optimal designing”, Compel, Vol. 20 No. 1, pp. 187-204. Cros, J. and Viarouge, P. (2002a), “Design of inductors and transformers with soft magnetic composites”, ICEM’02, CDROM, paper No 514, Bruges, Belgium. Cros, J., Perin, A.J. and Viarouge, P. (2002b), “Soft magnetic composites for electromagnetic components in lighting applications”, IAS 2002, 13-18 October, Pittsburgh, Pennsylvania, USA. Everett, D. (1955), “A general approach to hysteresis”, Trans. Faraday Soc., Vol. 51, pp. 1551-7. Ionita, V., Cranganu-Cretu, B. and Iona, B. (1996), “Quasi-stationary magnetic field computation in hysteretic media”, IEEE Trans. Magnetics, Vol. 32 No. 3, pp. 1128-31. Jiles, D.C. and Atherton, D.L. (1986), “Theory of ferromagnetic hysteresis”, Journal of Magnetism and Magnetic Materials, Vol. 61, pp. 48-60. Jiles, D.C., Thoelke, J.B. and Devine, M.K. (1992), “Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis”, IEEE Trans. on Magnetics, Vol. 28, pp. 27-35. Park, G.S., Hahn, S.Y., Lee, K.S. and Jung, H.K. (1993), “Formulation of the Everett function using least square method”, IEEE Trans. Magnetics, Vol. 29 No. 2, pp. 1542-5. Piriou, F. and Razek, A. (1992), “Finite element analysis in electromagnetic systems accounting for electric circuit equations”, IEEE Trans. Magnetics, Vol. 28, pp. 1295-8. Preisach, F. (1935), “U¨ber die magnetische nachwirfung”, Zeitschrift fu¨r Physik, Vol. 94, pp. 277-302. Sadowski, N., Batistela, N.J., Bastos, J.P.A. and Lajoie-Mazenc, M. (2001), “An inverse Jiles-Atherton model to take into account hysteresis in time stepping finite element calculations”, Compumag 2001, 2-5 July, France, Evian, Vol. 4, pp. 246-7.

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Error bounds for the FEM numerical solution of non-linear field problems Ioan R. Ciric Department of Electrical and Computer Engineering, The University of Manitoba, Winnipeg, Canada

Theodor Maghiar

Error bounds for the FEM solution

835 Received September 2002 Revised January 2003 Accepted January 2003

Department of Electrical Engineering, University of Oradea, Oradea, Romania

Florea Hantila Department of Electrical Engineering, “Politehnica” University of Bucharest, Bucharest, Romania

Costin Ifrim Ecoair Corp., Hamden, Connecticut, USA Keywords Error analysis, Magnetic fields, Field testing Abstract A bound for a norm of the difference between the computed and exact solution vectors for static, stationary or quasistationary non-linear magnetic fields is derived by employing the polarization fixed point iterative method. At each iteration step, the linearized field is computed by using the finite element method. The error introduced in the iterative procedure is controlled by the number of iterations, while the error due to the chosen discretization mesh is evaluated on the basis of the hypercircle principle.

Introduction Real world non-linear electromagnetic field problems can only be solved by numerical methods implemented using certain discretizations of the region considered. A simple procedure for evaluating approximately the computation errors involves the discontinuities in the normal or tangential components of some vector field quantities, which theoretically should be continuous. A more consistent procedure for determining the solution accuracy employs certain positive definite quantities that tend to zero as the numerical solution approaches the exact one. These quantities are called error estimators. In the case of static or stationary linear fields, the hypercircle principle (Bossavit, 1999; Synge, 1957) allows to obtain the “distance” between the numerical and the exact solution vectors, i.e. a certain norm of the difference between them. Namely, if H *, B * are the exact solution vectors for a general linear field problem described by This is a revised and enhanced version of a paper which was originally presented as a conference contribution at the XVII Symposium on Electromagnetic Phenomena in Nonlinear Circuits (EPNC), held in Leuven, Belgium, on 1-3 July 2002. This is one of a small selection of papers from the Symposium to appear in the current and future issues of COMPEL.

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 835-844 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410510802

COMPEL 23,3

7£H* ¼ J

ð1Þ

7·B * ¼ r

ð2Þ

¼

B * ¼ mH *

836

ð3Þ ¼

under specified boundary conditions, where J, r and m are given functions of ¼ position, and H and B, with B – mH ; are vector functions satisfying separately the equations 7£H ¼J

ð4Þ

7·B ¼ r

ð5Þ


¼


¼
B* 2 B þ mH
¼ 1 kB 2 m H kn

2 2 n

ð6Þ

and

then

and ¼

kB* 2 Bkn # kB 2 mH kn

ð7Þ

where kX kn is the norm defined by 2 kX kn

¼

Z

¼

X · ðnX Þ dV

ð8Þ

V

¼

with n being, in the case of magnetic field problems, for instance, the symmetric and ¼ positive definite tensor equal to the inverse of the permeability tensor m and with the integral performed over the region V considered. A number of techniques were developed (Bossavit, 1999; Marmin et al., 2000; Marques et al., 2000) based on the hypercircle concept, yielding criteria for evaluating local errors, which are useful in the FEM treatment of linear field problems. On the other hand, for fields in non-linear magnetic media an appropriate error estimator is (Li et al., 1995; Marmin et al., 1998; Rikabi et al., 1988) Z H · B dV ð9Þ LðH ; BÞ ¼ W ðBÞ þ W * ðH Þ 2 V

with W ðBÞ ¼

W * ðH Þ ¼

Z Z

B 0

F ðB Þ · dB

V

Br

Z Z

H

F V

0

where F specifies the constitutive relation

21

0

0



ðH Þ · dH

dV

0

ð10Þ

 dV

ð11Þ

H ¼ F ðBÞ

ð12Þ

›F i ›F j ¼ ›Bj ›Bi

ð13Þ

with

where the subscripts denote the Cartesian components of B and F, and B r ; F 21 ð0Þ represents the remnant magnetic flux density vector. If the constitutive relation is strictly monotonic, i.e. ðB 0 2 B 00 Þ · ðF ðB 0 Þ 2 F ðB 00 ÞÞ . 0;

for any

B 0 – B 00

ð14Þ

then expression (9) is convex (Rikabi et al., 1988), which is a very useful property for numerical computation. A similar error estimator constructed from relation (12) is ð15Þ

1 ¼ kH 2 F ðBÞkm

which, in the linear case, gives the error in the hypercircle principle sense. In this paper, we propose a procedure for deriving error bounds for the solution of non-linear field problems, based on the polarization fixed point iterative method (PFPM). At each iteration step, the linearized field problem is solved by the finite element method (FEM). The error due to the application of the FEM is evaluated using the hypercircle principle. PFPM Consider the magnetic field equations in V 7£H ¼J

ð16Þ

7·B ¼ 0

ð17Þ

with the constitutive relation (12). We assume that at each point P in V the function F satisfies the Lipschitz condition, i.e. jF ðB 0 Þ 2 F ðB 00 Þj , LðPÞjB 0 2 B 00 j;

for any B 0 ; B 00

ð18Þ

and is also uniformly monotonic, ðB 0 2 B 00 Þ · ðF ðB 0 Þ 2 F ðB 00 ÞÞ . lðPÞðB 0 2 B 00 Þ2 ;

for any B 0 ; B 00

ð19Þ

where L(P) has an upper bound and l(P ) has a positive lower bound in V. For instance, in an isotropic medium, where H ¼ FðBÞ

B B

for

B–0

and H ¼0 we have

for

B ¼ 0;

Error bounds for the FEM solution

837

COMPEL 23,3

FðB0 Þ 2 FðB00 Þ ; nmax ðPÞ B0 2 B00 B0 ; B00

ð20Þ

FðB0 Þ 2 FðB00 Þ lðPÞ ¼ inf ; nmin ðPÞ: 0 00 B ;B B0 2 B00 0 00

ð21Þ

LðPÞ ¼ sup B0 –B00

838

B –B

In the PFPM, relation (12) is replaced by ¼

B ¼ mH þ I

ð22Þ

¼ with m constant and the non-linearity hidden in the polarization I (Hantila, 1975; Hantila et al., 2000), ¼

I ¼ B 2 mF ðBÞ ; GðBÞ:

ð23Þ

¼

The value of m can be chosen (Hantila, 1974) such that the function G defined by equation (23) is a contraction, i.e. kGðB 0 Þ 2 GðB 00 Þkn # ukB 0 2 B 00 kn ;

for any B 0 ; B 00

ð24Þ

where u , 1: In the case of an isotropic medium, one can choose at any point mðPÞ , 2mmin ðPÞ and then, the contraction factor u is:    mðPÞ mðPÞ ; 21 : ð25Þ u ¼ sup max 1 2 mmax ðPÞ mmin ðPÞ P[V ¼ Since mmin is greater than the permeability of free space m0, m in equations (22) and (23) can be chosen to be m0. In this case, m0 u¼12 mM

where

mM ¼ supmmax ðPÞ: P[V

On the other hand, if m(P) is chosen such that 1 nmin ðPÞ þ nmax ðPÞ ; nðPÞ ¼ 2 mðPÞ

ð26Þ

with nmin ðPÞ ¼ 1=mmax ðPÞ and nmax ðPÞ ¼ 1=mmin ðPÞ, then u takes its smallest value,

mmax ðPÞ 2 mmin ðPÞ : P[V mmax ðPÞ þ mmin ðPÞ

ð27Þ

u ¼ sup

¼

In the case of anisotropic materials, the choice of the tensor m is more elaborated (Hantila, 1974). If, for instance, the relationship B 2H can be decomposed into independent relationships along orthogonal directions, then the above expressions for

the contraction factor can be used for each of the components (see the example considered below). The PFPM consists of the following iterative process: (1) a value for I (0) is chosen; (2) at each successive step n, n $ 1, B (n) and H (n) are computed from the linear equations 7 £ H ðnÞ ¼ J ;

7 · B ðnÞ ¼ 0;

¼ B ðnÞ ¼ m H ðnÞ þ I ðn21Þ ;

ð28Þ

(3) the new polarization I (n) is corrected according to the non-linear equation (23), I ðnÞ ¼ GðB ðnÞ Þ: Steps (2) and (3) are repeated until kDI ðnÞ kn ¼ kI ðnÞ 2 I ðn21Þ kn is sufficiently small. It should be remarked that for any I there is one and only one field B ¼ Z ðI Þ that verifies equation (28) and that the function Z is non-expansive, i.e. kZ ðI 0 Þ 2 Z ðI 00 Þkn # kI 0 2 I 00 kn :

ð29Þ

The above scheme is a Picard-Banach procedure for computing the fixed point of the composition G (Z (I )) of the function G with Z, which has the same contraction factor as G(B ). A dual formulation can be used for the treatment of the above non-linear field problem. Equations (22) and (23) are replaced by ¼

H ¼ nB 2 M

ð30Þ

with the non-linearity contained in the magnetization M, ¼

M ¼ nF 21 ðH Þ 2 H ; G 0 ðH Þ:

ð31Þ

¼

Again, the tensor n is chosen such that G0 is a contraction. For instance, in the case of isotropic media nðPÞ , 2nmin ðPÞ; i.e. mðPÞ . mmax ðPÞ=2: The function H ¼ Z 0 ðM Þ is also non-expansive. For an isotropic medium, the smallest contraction factor is

u ¼ sup

nmax ðPÞ 2 nmin ðPÞ þ nmin ðPÞ

P[V nmax ðPÞ

ð32Þ

which is the same as that given in equation (27) and is obtained with

mðPÞ ¼

mmin ðPÞ þ mmax ðPÞ : 2

ð33Þ

It can be shown (Saaty, 1981) that, after n iterations in PFPM, the errors with respect to the exact solution vectors H *, B * of equations (16) and (17) are evaluated as 1 kDI ðnÞ kn 12u

ð34Þ

kH ðnÞ 2 F ðB ðnÞ Þkm ¼ kDI ðnÞ kn

ð35Þ

kB* 2 B ðnÞ kn #

Error bounds for the FEM solution

839

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kH * 2 H ðnÞ km #

1 kDI ðnÞ kn : 12u

ð36Þ

Equation (35) gives the relation between the error estimator (15) and the error of the polarization computed numerically by the PFPM. Applying a selected numerical method to solve the linear problem (28), we obtain an approximate value of the magnetic flux density, ðn21Þ Þ: B ðnÞ ap ¼ Z ap ðI

ð37Þ

ðnÞ I ðnÞ ap ¼ GðB ap Þ

ð38Þ

From equation (23)

and


ðnÞ ðn21Þ
þ I 2 I kDI ðnÞ kn ¼
I ðnÞ 2 I ðnÞ
ap ap

n




 


ðnÞ
ðn21Þ
#
GðB ðnÞ Þ 2 G B ap 2 I

þ
I ðnÞ ap n








ðnÞ ðn21Þ
# u
B ðnÞ 2 B ðnÞ þ I 2 I


: ap ap n

ð39Þ n

n

Thus, kDI ðnÞ kn in equations (34)-(36) can be expressed as where

kDI ðnÞ kn # u11ðnÞ þ 1ðnÞ 2

ð40Þ





11ðnÞ ¼
B ðnÞ 2 B ðnÞ ap


ð41Þ n

is due to the approximation introduced by the numerical method used for solving the linear field problem and


ðnÞ ðn21Þ
ð42Þ 1ðnÞ
2 ¼
I ap 2 I n

is due to the iterative process used for the treatment of non-linearity. Error due to FEM application In this section, we derive the error 11ðnÞ in equation (41), introduced by employing the FEM as the selected numerical method for the solution of linear field problem in equation (28), at the end of the iterative process. Let L0 be the space of the magnetic field intensity H that satisfies 7 £ H ¼ 0 with a zero tangential component, H t ¼ 0; on the section S0 of the region boundary. Let L00 be the space of the flux density B that satisfies 7 · B ¼ 0 with zero normal component, B n ¼ 0; on the remaining section S 00 of the boundary. In what follows, we analyze the magnetic field H, B with H [ L 0 and B [ L 00 ; verifying the constitutive relation ¼ B 2m H ¼ I:

It is obvious that, for such a field, we have

ð43Þ

¼ kB; H l ¼ kB; m H ln ¼ 0

ð44Þ

where the inner product is defined by kC; Dln ¼

Z

¼

C · ðnDÞ dV:

Error bounds for the FEM solution

ð45Þ

V ¼

Equation (43) shows that the fields B and 2mH are the projections of I on the ¼ subspaces L00 and mL 0 ; respectively. These projections can be found by minimizing with respect to X the “distances” kI 2 X kn between I and X [ L 00 and between I and ¼ ¼ X [ mL 0 ; respectively, which gives X ¼ B in the first case and X ¼ 2mH in the second. Since 2

2

841

2

kI 2 X kn ¼ kI kn 2 2kI ; X ln þ kX kn ; we only need minimizing the functional 2

FðX Þ ¼ 22kI ; X ln þ kX kn : ¼

ð46Þ ¼

When X [ mL 0 ; we use the scalar potential formulation, with X ¼ 2m7F and the boundary condition F ¼ 0 on the surface S0 . When X [ L 00 ; we use the vector potential formulation, with X ¼ 7 £ A and with the boundary condition At ¼ 0 on S00 . The minimization of the functional in equation (46) is performed by using the FEM. This is done in the finite-dimensional subspace La00 of L00 or in the finite-dimensional ¼ 0 ¼ 0 ¼ 0 Lb of m L (L00 is shown in Figure 1 by a horizontal plane and m L by a subspace m normal straight line). The projection Ba of the polarization I on the subspace La00 is equal to the projection of B on the same subspace, B a ¼ P a ðBÞ: Obviously, Pa is non-expansive. Therefore, the application of the FEM to obtain the approximate solution Ba reduces to the computation B a ¼ Z ap ðI Þ ¼ P a ðZ ðI ÞÞ: Thus, the computation scheme in the PFPM is ... ! I

ðn21Þ Z

Pa

G


!B ðnÞ
!B ðnÞ !I ðnÞ ! . . . : a


Figure 1. Projections of I on the ¼ subspaces L00 and m L 0 used in the FEM computation

COMPEL 23,3

Since Pa and Z are non-expansive, the above scheme generates a Picard-Banach sequence of the contractive composition GðP a ðZ ðI ÞÞÞ of G, Pa, and Z, which is convergent. The polarization I is corrected as I ¼ GðB a Þ: The dual numerical scheme of the PFPM is Pb

Z0

842

G0

. . . ! M ðn21Þ
!H ðnÞ
!H ðnÞ !M ðnÞ ! . . . b
where the projection Hb of the magnetization M on the finite-dimensional subspace Lb0 is equal to the projection of H on the same subspace, H b ¼ P b ðH Þ: In order to compute the error associated with the FEM, based on the hypercircle ^ with B^ ¼ B 2 I : Now, equations (41) and (7) yield principle, we use the field H ; B; ðnÞ ðnÞ ðnÞ ðnÞ 2 B ðnÞ 2 B ðnÞ kn 1ðnÞ _ a kn # kB a 2 B ap kn ¼ kB 1 ¼ kB

a

ð47Þ

where ¼

ðnÞ ðnÞ ðn21Þ B _ a ¼ mH _a þI ¼

ð48Þ ¼

with mH_ aðnÞ computed by minimizing the functional (46) in the subspace mL0b and with ¼ the value of m equal to that used in equation (43). The iterative process in the PFPM is performed by using Ba in the first scheme, for instance, and is ended when 1ðnÞ 2 in equation (42) is smaller than a chosen value. After the last iteration, we compute H_ aðnÞ ; ðnÞ ðnÞ B _ a ; and thus evaluate 11 from equation (47). Numerical example In order to illustrate our method, we consider in this paper a simple system that contains a permanent magnet, whose square cross section is shown in Figure 2. The permanent magnet occupies a square section with a side length which is three quarters of that of the entire region and the rest of the region is of non-magnetic material. The B 2 H characteristic corresponding to the magnetization direction of the permanent magnet is plotted in Figure 3, with the relative permeabilities mrmax ¼ 16 and mrmin ¼ 1:2; the relative permeability in the transverse direction is constant and is equal to 1.2. The optimum contraction factor calculated from equation (27) is u¼ 0.86, corresponding to a value of relative permeability in the direction of magnetization mr ¼ 1=nr ¼ 2:23 in equation (26). The discretization mesh used and the flux density lines for one of the cases considered are shown in Figure 2. The corresponding global error in equation (34), normalized with respect to the remnant flux density, for various numbers of nodes of the discretization mesh, is given in Table I. Conclusions A method for calculating a bound for the error in the numerical solution of static, stationary and quasistationary non-linear magnetic field problems, has been derived. Although in this paper we employ the PFPM for treating the non-linearity, one can use any method to compute an approximate value of B. From the B 2H characteristic, we find the inverse of permeability in equation (26) and the optimum contraction factor in equation (27). Then, we obtain the polarization I ; I ðn21Þ ¼ GðBÞ in equation (23), which is needed in equation (28). Only one step of the PFPM is necessary to compute ðnÞ ðnÞ B ðnÞ a in equation (47), that is B ap in equation (37), and then I ap from equation (38) ðnÞ from equation (48). With these quantities, we determine the errors 11ðnÞ in and B _ a

Error bounds for the FEM solution

843

Figure 2. Discretization mesh with 441 nodes/800 triangles and flux lines

Figure 3. B 2 H characteristic of the permanent magnet

Number of nodes 81 441 1,089 2,025 3,249 4,761 6,561 9,409

Relative error (percent) 65.64 29.86 19.73 14.85 11.96 10.04 8.67 7.35

Table I. Relative error in flux density for the smallest contraction factor

COMPEL 23,3

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equation (41), 1ðnÞ 2 in equation (42), the global error from equation (40) and finally, the “distance” between the approximate solution and the exact solution (equations (34)-(36)). The dual PFPM scheme, where H and M are used, can be employed in a similar manner. It should be noted that, when the difference between the values of L(P ) and l(P ) (equations (20) and (21)) increases, then the contraction factor u (equation (27)) and the error bounds increase. A more refined mesh and a greater number of iterations are needed to obtain a smaller error. In the case of linear media, the procedure presented in this paper reduces to the classical hypercircle technique. References Bossavit, A. (1999), “A posteriori error bounds by ‘local corrections’ using the dual mesh”, IEEE Trans. Magn., Vol. 35 No. 3, pp. 1350-3. Hantila, F. (1974), “Mathematical models of the relation between B and H for nonlinear media”, Rev. Roum. Sci. Techn.-Electrotechn. et Energ., Vol. 19 No. 3, pp. 429-48. Hantila, F. (1975), “A method of solving magnetic field in nonlinear media”, Rev. Roum. Sci. Techn.-Electrotechn. et Energ., Vol. 20 No. 3, pp. 397-407. Hantila, F., Preda, G. and Vasiliu, M. (2000), “Polarization method for static fields”, IEEE Trans. Magn., Vol. 36 No. 4, pp. 672-5. Li, C., Ren, Z. and Razek, A. (1995), “Application of complementary formulations and adaptive mesh refinements to non-linear magnetostatic problems”, IEEE Trans. Magn., Vol. 31 No. 3, pp. 1376-9. Marmin, F., Clenet, S., Bouillault, Fr. and Piriou, F. (2000), “Calculation of complementary solution in 2D finite element method application to error estimator”, IEEE Trans. Magn., Vol. 36 No. 4, pp. 1583-6. Marmin, F., Clenet, S., Piriou, F. and Bussy, P. (1998), “Error estimation of finite element solution in non-linear magnetostatic 2D problems”, IEEE Trans. Magn., Vol. 34 No. 5, pp. 3268-71. Marques, G., Clenet, S. and Piriou, F. (2000), “Error estimators in 3D linear magnetostatics”, IEEE Trans. Magn., Vol. 36 No. 4, pp. 1588-91. Rikabi, J., Bryant, C.F. and Freeman, E.M. (1988), “An error-based approach to complementary formulations of static field solutions”, International Journal for Numerical Methods in Engineering, Vol. 26, pp. 1963-87. Saaty, T.I. (1981), Modern Nonlinear Equations, Dover, New York, NY. Synge, J.L. (1957), The Hypercircle in Mathematical Physics, Cambridge University Press, Cambridge.