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## Über dieses Buch

This book is the collection of the contributions offered at the International Symposium on Electromagnetic Fields in Electrical Engineering, ISEF '87, held in Pavia, Italy, in September 1987. The Symposium was attended by specialists engaged in both theoretical and applied research in low-frequency electromagnetism. The charming atmosphere of Pavia and its ancient university provided a very effective environment to discuss the latest results in the field and, at the same time, to enjoy the company or colleagues and friends coming from over 15 countries. The contributions have been grouped into 7 chapters devoted to fundamental problems, computer programs, transformers, rotating electrical machines, mechanical and thermal effects, various applications and synthesis, respectively. Such a classification is merely to help the reader because a few papers could be put in several chapters. Over the past two decades electromagnetic field computations have received a big impulse by the large availability of digital computers with better and better performances in speed and capacity. Many various methods have been developed but not all of them appear convenient enough for practical engineering use. In fact, the technical and industrial challenges set some principal attributes and criteria for good computation methods. They should be relatively easy to use, fit into moderately sized computers, yield useful design data, maintain flexibility with m1n1mum cost in time and effort.

## Inhaltsverzeichnis

### Introductory remarks

Some writers regard electromagnetism as a well-understood and even as a closed field of study. Maxwell’s equations can be accepted and all that is needed is to find appropriate solutions. This, however, is an onlooker’s view, practitioners know that there are many fundamental problems to be investigated. Partly this is due to the vast range of electromagnetic phenomena and the even greater range of applications. Then there is the inherent complication of the interaction of electromagnetic energy with matter. Also there are geometrical features inherent in electromagnetic fields and only recently have numerical methods become available which make it possible to study the importance of local and global geometries.

P. Hammond

### Power Losses in Electrotechnical Sheet Steel under Rotational Magnetization

In magnetic circuits of electric machines, as well as in nodes of three-phase transformer cores, rotational magnetization occurs. Calculations of iron losses in this kind of magnetization are based upon the knowledge of rotational field distribution in particular elements of the considered magnetic circuit. The rotational field inside the iron core was calculated by means of reluctance network method1. Calculations were carried out for a ring sample magnetic circuit and for a three-phase induction motor magnetic circuit. In this paper an idea of calculations and results of examinations of basic losses in a ring sample laminated core have been presented.

Jan Anuszczyk

### AC High Magnetic Field Generation Due to Eddy-Current Concentration Effect

The paper describes the high magnetic field generation by the use of eddy-current effects. The main features of the new type generator as well as some of results obtained both numerically and experimentally are presented. A short account of this research has also been presented.

K. Besso, S. Yamada, M. Kooto, T. Minamitani

### Scalar Potential Models for Magnetic Fields of Volume Current Distributions

A scalar potential function for magnetic fields in the presence of given volume current distributions is defined, by using an equivalent distribution of fictitious magnetization. The general formulation of the magnetic field problems, based on the equations and boundary conditions satisfied by this scalar potential, is presented. For practical configurations, the magnetic field due to given volume current distributions can be determined from that due to surface distributions of fictitious magnetic charge. Two examples illustrate the modelling procedure and the efficiency of the proposed method with respect to methods developed so far.

I. R. Ciric

### The Electrodynamics of Ampere and Neumann

A brief outline of the history of the Ampere-Neumann electrodynamics of metals is provided. It was developed in France and in Germany during the 19th century. The old theory is based on instantaneous action-at-a-distance. The paper points out to what extent it agrees with presently taught relativistic electromagnetic field theory and then delineates an area where the old and new theories disagree. Four groups of experiments with metallic conductors are cited which, in the area of disagreement, favor the Ampere-Neumann electrodynamics. A fifth group of experiments is mentioned which reveals a new electrodynamic force in dense arc plasma which is orders of magnitude stronger than the Lorentz force. Precisely such a force has emerged from the modern extension of the old theory. This seems to broaden the empirical basis of the Ampere-Neumann electrodynamics to include dense plasma conductors.

Peter Graneau

### A Contribution to the Theory of Skin-Effect

We can easily measure transient phenomena connected with the operation of a circuit but the knowledge of how the current increases or decreases in the circuit does not mean that we know the distribution of the current density over the cross-section of the conductors. There are no convincing experiments to prove the existing theory because we cannot measure the current density distribution in the conductors.

Jan Nasiłowski

### Introductory remarks

During the last twenty years engineers and scientists in electromagnetic fields calculation have used computer programs to solve an increasingly large number of their problems. The construction of efficient and economical numerical programs depends on a great variety of factors. This subsection consists of five papers that discuss numerically efficient algorithms for electromagnetic fields calculation in various electrodynamical systems.

R. Sikora

### Impedance and Losses in Magnetic Cylindrical Conductors

Maxwell’s equations are solved for a sinusoidal current in a cylindrical conductor. The material is assumed to have a rectangular magnetization curve of the form B = BS sign H. The magnetization process is described. Initially, the conduction, because of the shielding effect, takes place only on the outside of the conductor. As time progresses, the current circulates in an increasing section of the conductor. The problem is solved mathematically. Tables are presented to simplify the computation process. The results give the voltage drop, the voltage waveform, and the losses. Experimental results confirm the validity of the hypothesis.

Bernard Béland, Daniel Gamache

### Errors of Solution of Classical and Variational Finite Difference Method Applied to Transient Electromagnetic Field Analysis

The process of entering the electromagnetic field into ferromagnetics has a very strong influence on the dynamics of electrical machines, especially in the case of field forcing and decaying of exciting current. The other, especially important when applying pulse magnetic field, is magnetic forming of ferromagnetic1,2. The process of magnetic forming is based on applying the Maxwell pressure interacting on conductors and ferromagnetics placed within the transient magnetic field. The transient magnetic field is generated in cylindrical winding, while the formed tube is placed inside the winding. The amplitude of flux density in the workpiece wall is 20 ÷ 50 T, while the pressure is 109 N/m2. This problem has been widely solved in world literature, but magnetic permeability of the workpiece has been assumed to be constant 1,2,5. The searched model system of magnetic forming (Figure 1) has a ferromagnetic tubular core defined by electric conductivity and static magnetic permeability μst = f(r,z,t). This problem has led to a two-dimensional problem, this transient electromagnetic field is described by the following field equation: 1$$\begin{array}{*{20}{c}} {v\frac{{{{\partial }^{2}}{{A}_{\theta }}}}{{\partial {{z}^{2}}}} + \frac{{\partial v}}{{\partial z}} \cdot \frac{{\partial {{A}_{\theta }}}}{{\partial z}} + \frac{v}{r} \cdot \frac{{\partial {{A}_{\theta }}}}{{\partial r}} - \frac{{v{{A}_{\theta }}}}{{{{r}^{2}}}} + \frac{{{{A}_{\theta }}}}{r} \cdot \frac{{\partial v}}{{\partial r}} + \frac{{{{\partial }^{2}}{{A}_{\theta }}}}{{\partial {{r}^{2}}}} + \frac{{\partial {{A}_{\theta }}}}{{\partial r}} \cdot \frac{{\partial v}}{{\partial r}} = } \\ { = - {{J}_{W}}(t) + \frac{{\partial {{A}_{\theta }}}}{{\partial t}}} \\ \end{array}$$ where: $$\begin{array}{*{20}{c}} {v = \frac{1}{{{{\mu }_{{st}}}}},} & {{{A}_{\theta }} = } \\ \end{array} {{A}_{\theta }}(r,z,t).$$

K. Komçza, S. Wiak

### Analysis of Electromagnetic Shielding Problems Using the BEM

In general, the analysis of shielding problems requires calculating of electromagnetic field distribution in the regions which can be divided into the subregions with different physical properties and different material parameters.

Wojciech Krajewski

### The Use of the Boundary Element Method in Transient Open Boundary Problems

The so called “open boundary problems” (OBP) have been solved by means of various methods- Mainly, the combination of two or more methods were employed and they were usually the finite element method (FEM) and some others, like the boundary integral methods or analytical methods. Also, the OBP have been analysed by means of the FEM only, but then the artificial boundary has been put far enough from the object in question. All the approaches lack the methodological clarity and, what is more important, are cumbersome in a practical use. Moreover, the FEM, being the basis for all the above mentioned solving procedures, is not suited to the eddy current problems, especially while dealing with transient phenomena1. That is why, for the last few years there have been attemps to solve the OBP by the use of the BEM only. The harmonic (sinusoidal) problems have already been solved4 and this paper is to show how to solve transient problems. These problems are very attractive for the BEM treatment as the BEM is well-suited to both transient and open boundary problems2.

Andrzej Krawczyk

### Calculation of Electric and Magnetic Field by Means of the Method of Tubes and Slices

Variational methods are often preferred in the calculation of electromagnetic field problems. They rely on the fact that the field as a system can be described by global parameters which have a stationary value when the system is varied in a prescribed manner. The parameters are generally formulated in terms of energy and the variational method explores the behaviour of the energy near its equilibrium value. Typical energy functionals are expressed in terms of circuit quantitites such as resistance, inductance and capacitance.

J. K. Sykulski, P. Hammond

### Introductory remarks

Computer programs may be used in different aspects of practical problems concerning electromagnetic fields. One of them, perhaps the largest, deals with the application of finite element method. A second aspect in the use of computer programmes in solving specific practical problems.

J. A. Tegopoulos

### Perturbation Finite Element Method of Electromagnetic Field Calculation for Eddy-Current Coupling

This paper describes how the non-linear equation of the FEM for electromagnetic field calculation of eddy-current coupling can be turned directly into a set of linear perturbation equations by perturbation method. The set of perturbation equations can be solved directly without the need for iteration. For each and every perturbation equation, the coefficient matrices are identical and remain constant, with only the right hand column vectors being different from one another. The right hand column vectors of any perturbation equation can be calculated on the basis of the solutions of the preceding set of equations. Therefore, the method is simpler and more effective than the general FEM.

Li Biao

### Macro - Element in Cartesian Coordinates

This paper presents the utilisation of macro-element in a Cartesian coordination system. Such macro-element has been introduced and applied to the straight overlap joint analysis. Their use increases the accuracy and avoids the new subdivision at different parameter values such as the overlap length. The harmonic number is analysed and a minimum number is given.

### Analysis of Transients in Electrical Circuits Containing Initially Magnetized Ferromagnetics for Impulse Excitation

Many electrical devices have magnetic cores made of ferromagnetic materials with pronounced nonlinearity. Finding the proper values of equivalent circuit parameters for transient analysis, however, may be difficult or even virtually impossible, due to the nonlinear nature of the current skin effect in ferromagnetic cores. Beland and Gamach1, Davidson2, and MacBain3 have studied experimentally and modelled numerically the transient processes in electrical machines or their parts, taking into account the nonlinearity of the magnetic media, but these investigations have been made with assumptions limiting the shape of the electric currents flowing in the circuit. Fridman4 has proved the range of using an equivalent circuit of transformer type for fixed frequency of current flowing through the exciting winding and core thickness. Wiak5,6 and Zakrzewski5 have proved that it is possible to calculate transients in electrical circuits containing ferromagnetics, in many cases, without using an equivalent circuit idea. Instead, the contribution of the eddy-current field to the total impedance of the system is calculated throughout simultaneous solution of a nonlinear field equation in a ferromagnetic core, and the shape of the circuit is being corrected after each step of numerical iteration. In this paper, the author has successfully applied the “circuit-field method”5 to the circuits containing ferromagnetics with DC initial magnetization of a core.

Sławomir Wiak

### 3-D Reactance Calculation of Air-Core Coils with Magnetic Screening

In the previous paper5 a method for computing 3-D magnetic field of aircore coils has been proposed. These coils may be simulated by means of the four spatial solids (rectangular prisms) situated in the unbounded space as shown in Fig.l. The coils operating as air-core reactors may be suitably screened (partially or completely) with sheet iron. For other reasons they may also be located nearby the steel solids which may be considered as the magnetic screens as well. A magnetic screen is the more effective the less saturated than the sheet iron is (µs» µO). In the present paper 3-D field distribution as well as the reactance of an air-core coil are calculated having regard to the fact that nonsaturated magnetic screens can be located in a different way. In this connection: this paper may be considered as a further development of the method described in the paper5.

K. Zakrzewski, M. Łukaniszyn

### Introductory remarks

Eight papers from six countries are included in this chapter. Most of them are concerned with two-dimensional numerical analysis of flux and eddy current distributions. Forces, stray losses and temperature distributions are calculated from those results. Experiments have been done to verify the results obtained by numerical analysis. These researches are useful for the optimum design of electric machines.

T. Nakata

### A Method for Losses Evaluation in Large Power Transformer Tanks

The aim of this paper is to show how it is possible to calculate, with a good approximation, the eddy current losses in metal transformer parts. The calculation method is firstly described: it consists in the calculation of vector potential, starting from the diffusion equation discretized by means of the Finite Elements Method, with the ferromagnetic material of the tank taken as linear; the eddy current losses are then calculated. Finally, the method is applied to the analysis of two machines, respectively of 350 kVA and 370 MVA; it is also shown that in order to achieve good results, the above method requires several basic hypotheses which vary from case to case.

A. Babare, A. Di Napoli, E. Santini, G. Scendrate

### The FEM Analysis of Magnetic Field in Converter Transformers During Faults

The rectifier transformers during internal faults in valve sets undergo strong submagnetization with DC, saturating its core. Simultaneousely the nonsymetrical short circuit conditions produce large amount of currents in both windings. The core in instants of high saturation loses its “magnetic mirror” properties and the main flux finds its way as well in the nonmagnetic space out of the core. The configuration of leakage flux varies in function of time and contains the DC component fast increasing with time. This conditions greatly differ with classic understanding of leakage flux in usual transformers operating in networks during faults. The above phenomena can influence the electromagnetic field in transformer space, as well as the field of dynamic forces and the mechanical behaviour of windings. Evidently they deserve, a more detailed analysis. It must consider the magnetic properties of steel and should explain the difference of field distribution in convertor transformers in comparison to usual network units.

Michał Jabłoński, Ewa Napieralska-Juszczak

### Additional Losses in Frequency Tripler Windings

Magnetic frequency triplers under construction have mega-Watt outputs1 and accurate determination of their winding losses is essential. Methods established for rotating machines and transformers2,4,5 are inadequate for frequency multipliers which have highly distorted primary current and higher frequency output current. The additional loss caused by the magnetic flux in the cross-sections of the windings is the sum of the losses from each flux harmonic. The direct analytical method is used to calculate the losses in each wire of the windings. This method consists in solving Helmholtz’s equation with boundary conditions defined by the distribution of the magnetic field acting on the wire3. This solution makes possible calculation of the additional loss factors in the tripler windings.

### Stray-Load Losses in Yoke-Beams of Transformers

The yoke-beams, serving for clamping the yoke of the transformer core and for supporting the windings, are structural parts of complicated shape. The beams are in the stray magnetic field of the windings and the stray flux can produce considerable eddy current losses in them. Due to the large surface and efficient cooling of the beam hot spots generally do not form, however, the loss can reach or even exceed the tank losses. Thus it is advisable to assess the stray losses in the yoke-beams as well when designing the transformer. However, it is impracticable to accurately simulate such a complicated construction in the calculation and to pay attention to the non-linearity as well as the reaction of induced eddy currents at the same time. Calculation methods based on simplified models and on simplifying conditions are required. In the following two simple models are presented for the approximate calculation of the losses in yoke-beams. The losses calculated are compared to those measured on a transformer.

D. Kerényi

### Influence of Structure Geometry, Screens and Eddy Currents on the Critical Distance of Tank Wall in Power Transformers

It is generally known that at small distance of a tank wall from transformer windings the state and position of tank wall strongly effects the leakage field configuration. At larger distance this influence is practically negligible and e.g. measurements of stray losses in a tank are possible with the short-circuit test “with and without tank”. Such a “critical” tank wall distance has been investigated with both the reluctance network and the finite element method for normal large transformers with various structures and screens. The existence of such a characteristic distance and its influence on leakage field phenomena has been investigated and evaluated.

A. Savini, J. Turowski

### Numerical Analysis of Inrush Currents in Transformers

The finite element method for analysing inrush currents in transformers connected to constant voltage sources taking into account three-dimensional magnetic fields and residual magnetism is developed.The principle and the finite element formulation of the method are described, and an example of application is shown.

J. Takehara, M. Kitagawa, Nakata, N. Takahashi

### Improvement of Transformer Core Loss by Use of Low-Loss Electrical Steel

The results of an experimental investigation of the influence of high permeability grain-oriented (HGO) steel on transformer core loss are presented. The power losses of cores assembled with HGO material of grade M2H and with conventional grain-oriented material of grade M5 are compared. The aim of the investigation was to determine the influence of a number of parameters (e.g. single-phase and three-phase core forms, different corner and T-joint designs, core proportions, core dimensions, a number of laminations per stagger layer, overlap length) on the amount of loss reduction in the cores of HGO material.

Zvonimir Valković

### Currents and Forces in Conductors of Rectangular Cross Section

In this paper an analytical method is given for the determination of current density and forces in two parallel rectangular conductors of equal dimensions carrying equal and oppositely directed sinusoidal currents. The solutions are given in integral form and a method has been developed for the numerical treatment of the solutions on a digital computer. An application is also given.

P. P. Yannopoulos, J. A. Tegopoulos, M. P. Papadopoulos

### Introductory remarks

The permanently growing possibilities of computers enable one to use more and more complicated and accurate methods of analyzing the electromagnetic field in electrical machines and of calculating the machine parameters. Two approaches can be followed. The first one consists in setting up field equations and boundary conditions for a machine part, in solving these equations in an analytical way and in using the computer only in the last stage of calculation. The second one starts with the discretization of the analyzed area into surface elements by means of special programs and in the numerical solution of a large set of equations. This last approach is represented by the finite difference method and by the finite element method, based on variational principles. Another numerical method called the boundary element method consists in the discretization of the area boundaries instead of the area itself and in discribing the field by boundary integral equations converted into a numerical form. It is also possible to represent the separate parts of the magnetic and electrical circuits of an electrical machine by lumped elements and to put them together in a network, which may be solved afterwards, in an usual way.

T. Sliwinski

### Magnetic Shielding of Turbo-Generator Overspeed Test Tunnels

Turbo-generator rotors induce high eddy current losses in the surrounding steel walls when turning in an overspeed test facility with the field current switched on. By means of cylindrical shielding the losses are reduced to a very small fraction. The interesting field quantities, the power losses and the currents per pole are numerically calculated depending on the rotation frequency, on the number of poles and on the shield dimensions.

E. Ch. Andresen, W. Müller

### An Equivalent Circuit Model for Inverter-Fed Synchronous Machines

Commutation time of inverter fed synchronous motors is commonly expressed in terms of the commutation reactance Xc. A precise method of calculation is desirable especially for solid salient pole motors. A magnetic circuit method is used to model a machine cross section which is coupled to the stator windings making possible a direct coupling of the magnetic circuit with the inverter.Commutation is effectively a line-to-line short circuit superimposed on a load condition and initial studies have been made on a line-to-line short circuit without rotor motion.Calculated commutation on load agrees well with test results.

M. J. Carpenter, D. C. Macdonald

### Magnetic Field and Force of Linear Induction Motor Fed From Non-sinusoidal Source

Performances of rotating induction machines fed from a non-sinusoidal source have been investigated rather thoroughly whereas this same problem for linear machines has received much less attention 1,2,3. The first published papers report an appreciable influence of the higher time harmonics on primary losses and their negligible influence on terminal performances. This paper describes computational and experimental results relating to the influence of a non-sinusoidal supply on the magnetic field and force of a small single-sided LIM controlled with variable-voltage-fixed-freguency /VVFF/ pattern. Theoretical analysis based on the Fourier series method, validated by experimental data, confirms the conclusion about a negligible influence of the higher time harmonics in the exciting current on a LIM performance.

Ewa Gierczak, Maciej Włodarczyk, Kazimierz Adamiak

### Electromagnetic Parameters of a Turbogenerator Determined by the Finite Element Calculation

Eddy currents induced in the solid rotor of large turbogenerators influence remarkably the electromagnetic properties of the machine being investigated. The magnetic linkage of the armature winding, of the field winding and the mutual field linkage between them results from the magnetic field distribution in the air gap evaluated for the case of a harmonic current impression. The eddy current reaction in the rotor was taken into account by the finite element method in the calculation of the field distribution. Constant permeability of the core was assumed and the end effects in the rotor were neglected but the complicated structures of the slotted solid rotor were considered and the aluminum alloy wedges located between the rotor teeth. The evaluated spectral transmittances (frequency plots of the fundamental transmittances) describe the electromagnetic properties of the machine and are suitable for the designation of the approximative operational transmittances containing the equivalent electromagnetic parameters of the machine model with lumped constants.

### Analysis of the Implications of Main Field Entities On Commutation in Field Regulated D.C. Motors

The paper applies a detailed field analysis by means of the finite element method to the study of the commutation in a d.c. motor speed-regulated by main field weakening; by comparing the resulting flux density diagrams in interpole airgaps to an ideal reference diagram, more reliable predictions about commutation are drawn than from usual evaluations. The method allows also some improved criteria for design purposes to be outlined.

M. Rizzo, A. Savini, C. Zimaglia

### Introductory remarks

In the last decade many efforts have been made to precalculate the characteristic data of induction machines and to predetermine the performance more exactly by means of computer calculation, simulation and modelling. The aim is a better knowledge of what happens in induction motors at steady state and transient performance, when supplied by the mains and by frequency inverters as well. The aim is furthermore to improve the design methods and to increase the efficiency of the machines for the benefit of energy savings.

E. Andresen

### Coupling of the Finite Element and Boundary Element Methods by Iterative Technique

The magnetic field distribution in the cross-section of an induction motor at no-load is computed. An iterative assembling of the Finite Element Method with the Boundary Element Method is achieved- The field quantities of interest are found as a result of successive separated calculations of stator and of rotor fields. These fields are linked by means of the BEM input and output quantities. An algorithm of the procedure and conclusive estimation of the iteration process are shown in detail.

Krzysztof Bill, Krystyn Pawluk, Witold Perzanowski

### Effect of Magnetic Saturation On the Air-Gap Flux Density Wave in Polyphase Induction Machines

Both the performance analysis and the design of electric machines is based more and more often upon the field methods instead of circut theory. However, on the basis of the electromagnetic field theory, it is not possible to obtain analytical relations between functional parameters and structural parameters of a machine. Thus, for implementation of the field methods numerical algorithms are necessary and therefore only approximate results may be obtained. This means that the solution is of the implicit form and not of the usually desired explicit form. Indeed, the solution in the implicit form is not very useful for the design of optimal machines, because in this case we are looking, among other things, for appropriate dimensions of the machine’s magnetic circuit. That is why analytical methods for computation of magnetic circuits are still very important, even though they are worked out with some simplifying assumptions.

Mirosław I. Dąbrowski

### Hybrid Simulation of Electromagnetic Field in Squirrel-Cage Winding

The complete mathematical model of the phenomena in the electromagnetic field of electrical machines should include not only the field equations but also the Kirchhoff’s equations which describe the scheme of windings connections. The methods of the formation of such a model and of the electromagnetic field simulation in the windings composed of a series of connected coils have been given in works1,2. However, in the presented paper the author describes the method of direct simulation of the electromagnetic field in a region with squirrel-cage windings. The specific hybridism of the method consists in connection of the field and Kirchhoff’s equations, and also in application of a hybrid computer system composed with a parallel matrix processor for the field simulation and a conventional digital computer. The systems with a two-dimensional magnetic field have been considered. The sectors of end rings with parts of bars which protrude from the ferromagnetic core have been treated as concentrated resistance and inductance elements.

Andrzej Demenko

### Induction Machine with Anisotropic Multilayer Rotor Modelling the Electromagnetic and the Electrodynamic States of a Symmetrical Machine with Deep Bar Cage in Solid Iron Rotor Core

Eddy current phenomena in the deep bar cage situated in the slots of the solid iron rotor complicate immensely the analysis of transients of such symmetrical polyphase induction machines. When the end effects in the rotor including the influence of end rings are neglected, one obtains two dimensional electromagnetic field distributions. In this case the copper bars and the solid iron teeth can be substituted accurately with an anisotropic two-layer continuous secondary structure having different electromagnetic constants (permeability, conductivity) in the tangential, radial and axial direction1. The successive layers in the multilayer machine model are: isotropic air gap, the first magnetically anisotropic rotor layer substituting the slot openings and tooth-top space, the second anisotropic layer substituting the deep bars with adjacent solid iron teeth and the isotropic solid iron layer under the slots. The sinusoidal1y distributed symmetrical m1-phase winding was assumed in the stator with p pole pairs, ξ1w1 effective number of turns per phase winding and an ideal sheeted magnetic nonsatureted core. The orthogonal transformation of the stator phase current in a two phase current I1(t) expressed in new frames attached to the rotor, enables the solution of the electromagnetic field in the rotor, excited by the primary equivalent current sheet fixed relative to the rotor body $${{a}_{1}}(x,t) = - 2\sqrt {{\frac{{{{m}_{1}}}}{2}}} \frac{{{{w}_{1}}{{\xi }_{1}}}}{{\bar{p}{{\tau }_{p}}}}{{I}_{1}}(t)\sin (\frac{\pi }{{{{\tau }_{p}}}}x)$$.

### Peculiarities of Electromagnetic Field in Slots

A lot of papers have been published on the electromagnetic field in slots of electrical machines and on the slot leakage. In many books and papers the slot leakage is calculated in a very primitive way neglecting physical principles. Simplified boundary conditions at the slot opening are often assumed thus causing remarkable inaccuracies. Analytical methods are applicable for calculation of slot leakage in the case of negligible skin effect and of bar impedance with the influence of the skin effect only for simple slot shapes (e.g. circular, rectangular, trapezoidal). For often used more complicated slot forms analytical solutions do not exist or are inaccurate. Recently, numerical methods based on variational prinor ciples have been introduced for the slot leakage calculation1, 2.The practical use of the afore-mentioned methods is until now rather limited especially in the case when the skin effect exists.

### Introductory remarks

The topic of coupled fields has recently gained increasing interest. It is well known that, on one hand, heat dissipation due to applied and induced currenty accompanies the operation of practically all electromagnetic devices so that heat transfer plays a considerable role in their design; on the other hand, the electromagnetic field gives origin to mechanical stresses and, when applicable, movement in electromechanical devices. This interaction is, actually speaking, reciprocal in that temperature distribution and movement caused by mechanical stress influence, in turn, the electromagnetic field. As a consequence, the investigation of coupled fields, including electric, magnetic, thermal and mechanical fields, is complex and troublesome. Despite the availability of big computers and new methodologies, the solution of the resulting non linear time varying equations in three dimensions for any device is not yet feasible. So, at the moment either simplified structures are considered for solving the whole problem or complicated structures are analyzed, making however approximations on the formulation of the problem. Five contributions are included in the chapter.

A. Viviani

### Numerical Analysis of Effect of Material and Structural Parameters on Magnetic Bearing Performance

In recent years the phenomenon of levitation has attracted a lot of attention from engineers. It is applied in many branches of technology, e.g. crucible less melting of conductors and semiconductors, high-speed ground transportation and frictionless bearings. Although there are many ways of suspension of bodies the means based on utilisation of energy of the electromagnetic or the magnetic field are most commonly applied. This paper presents the analysis of magnetic bearing with a rotating ferromagnetic shaft.

Sławomir Iskierka

### The Finite Element Method Solution for the Stationary Two-Dimensional Thermal Problem

The problems of the heat flow and the electro-magnetic field are strongly coupled in all electrical equipments.

Grżyna Krusz

### Force Calculation in Electromagnetic Devices

In this paper the authors show how the Maxwell stress tensor can be used, in practice, in order to determine local magnetic forces distribution in an electromagnetic system. An experimental set-up has been designed in view of comparing the theoretical results with the measurements.

Yvan Lefevre, Michel Lajoie-Mazenc, Bernard Davat

### Electromagnetic Fields and Forces in a Linear Induction Motor during Direct Current Braking

Linear induction motors (LIM) find great application in many power transmission systems. Besides the motor operation the performance of the LIM during braking is also very important. The characteristics of LIM have been studied in recent years 1,2,3, but considerably less attention has been paid to the direct current braking (DC braking) performance of LIM. This paper presents the results of the analysis of LIM during DC braking with regard to the influence of the longitudinal end-effect. The relationships describing electromagnetic fields in the individual zones of the motor and forces acting on the secondary have been determined. The influence of the longitudinal end-effect on DC braking characteristics has been discussed.

Krzysztof Pieńkowski

### Numerical Analysis of Electromagnetic and Temperature Fields in Induction Heated Ferromagnetic Slabs

Induction heating is widely used for quick heating of conductive materials. Because temperature distribution of the heated body is governed by the eddy current distribution, it is very important to analyse the relation between the two distributions. For heating of ferromagnetic billets, depending on the dimension of the workpiece, typical field strengths H can range from 50 to 200 kA/m and typical supply frequency f can range from 50 to 3000 Hz. At the resulting deep levels of saturation the waveforms for flux density and especially current density become highly distorted.

A. Stochniol, V. S. Nemkov

### Introductory remarks

In this chapter papers. have been grouped dealing with specific applications of various nature. Some of the papers, however, though describing particular devices, could belong to other chapters, if one considers that generally the results obtained can be extended to a wider class of devices.

M. D’Amore

### Computer - Aided Modelling and Simulation of Fast Transient Phenomena in Large Coils

This paper deals with resonance problems due to stray capacitances in large magnets, such as those used in fusion research machines or big H.V. transformers. The paper presents the method set up to investigate frequency response of the coils of the Poloidal Field System of the RFX fusion experiment.For each coil an equivalent network is first identified which takes into account, for each turn, self and mutual inductances and stray parameters. The frequency spectrum of such a network is then numerically analysed and the corresponding resonant frequencies are found in order to identify a much simplified model with similar electrical behaviour at the coil terminals, and to allow the overall winding to be analysed at once.

P. P. Campostrini, A. Stella

### Remote Electromagnetic Propulsion in the Presence of a Metallic Shield

This paper describes an electromagnetic technique to reposition the spacer springs in CANDU reactors. The need to perform this manoeuvre in non-commissioned reactors was required subsequent to the failure of a pressure tube in a reactor at a Canadian Nuclear Generating Station. A contributing factor in the failure of the tube was the fact that the annular spacers used to maintain the coaxial configuration between the metallic pressure tube and its surrounding metallic calandria tube, had been displaced. Afterwards it was realized that displacement of the spacers had also occurred in the non-commissioned reactors. It will be recognized that the spacers were not accessible for mechanical repositioning, leaving only the possibility of an electromagnetically coupled remote repositioning procedure. This paper describes the analysis and solution of the complex coupled electromechanical problem. The mathematical analysis problem was ultimately solved using a finite element process.

R. D. Findlay, J. H. Dableh

### Application of the Harmonic Analysis Technique to Determining Eddy Currents in Conducting Plates

A novel analytical approach to predict eddy currents in a conducting plate is proposed. Based upon the knowledge of the incident flux to the rectangular plate, a curve fit to a double Fourier series is obtained. This closed formula representation is used to solve the diffusion equation using the physical boundary conditions of the plate. Limitations and applicability of the method are then discussed.

R. D. Findlay, B. Szabados, I. ElNahas, M. S. ElSobki, M. Poloujadoff

### Numerical Field Calculation of Earthing Systems

This article describes results of the long-standing research on computer aided design, analysis and calculation of earthing systems. Numerical calculations of the quasistatic current field have been carried out by the boundary element method (current simulation method). Analyses of the complex earthing systems of the large industrial plants have been based on these calculations. The results obtained by measuring models and real objects have proved of satisfactory accuracy for the numerical procedure.

### Computer-Aided Design Technique for Proportional Electromagnets

A remarkable class of electromechanical converters used in proportional hydraulic devices, at the interface between the electronic control part and the hydraulic drive part, is represented by proportional electromagnets. These are, in fact, plunger-type d.c. electromagnets with a very studied air-gap geometry and intentionally saturated ferromagnetic zones. Such a particular magnetic structure allows a static characteristic adjustment, so that the developed electromagnetic force (i) remains constant with plunger positional changes (within the work domain of the axial air gap), (ii) is proportional to the excitation current, and (iii) has a considerably increased magnitude in a limited volume.

Mircea M. Rădulescu, Vasile Iancu, Ioan-Adrian Viorel, Károly Biró

### Reluctance Network Analysis of Coupled Fields in a Reversible Electromagnetic Motor

A large number of reversible electromagnetic, thyristor controlled, linear motors (Fig.1) have been designed and produced by the Novosibirsk Institute of Electrical Engineering. These machines cover a wide range of frequencies (from 0.4 to 50 Hz) and forces (from a few to several thousand newtons) and have been applied in the chemical and mining industries and used in household equipment and professional tools1,2.

Victor R. Rais, Janusz Turowski, Marek Turowski

### Introductory remarks

It is widely known that, in electrodynamics, analysis was developed in preference. Therefore, when the geometric and material structure of a device is known and when the field sources in this device are given, then we may accomplish the analysis of field in it, Synthetizing means searching for the structure producing the desired field. Synthesis is performed in order to design some device or to discover the features of the existing device unless they are not yet known.

K. Pawluk

### The State of Art in the Synthesis of Electromagnetic Fields

One observes that many inverse problems of technical electrodynamics are recently studied. The basic ordination of the synthesis problems might be regarded from an engineering, a physical and a mathematical point of view. There is a wide diversity in formulating the synthesis problems; some of them have been discussed in this paper and a basic classification of engineering synthesis problems has been proposed. A short review of mathematical techniques has been also presented.

Krystyn Pawluk, Marek Rudnicki

### Comparative Analysis of Numerical Methods for Shape Designing

Based on the Finite Element Method (FEM) two nonlinear algorithms for the synthesis of the geometry of the investigated region have been presented. One of the algorithms used is the Variable Metric Method (VMM) and the other is based on the Sensitivity Method (SM). The simple numerical example using this methods has been presented.

Jan Sikora, Maciej Stodolski, Stanisław Wincenciak

### Synthesis of a Turbo-Generator Nonlinear Parametric Model for the Analysis Problems of Power Industry System Regimes

At the level of the national economy development achieved, the problem of improving the methods and means to plan and to control the electric power systems (EPS) is of particular concern. The solution of this problem involves the intensive use of mathematical modelling methods and modern computer facilities. In this case, to obtain the results of high quality and reliability it is necessary to have the authentic mathematical models of the EPS elements.

V. E. Tonkal, Yu. G. Blavdzevitch, N. V. Raptsun

### Backmatter

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