ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2017

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1575

Magnetic Leakage Fields and EndRegion Eddy Current PowerLosses in Synchronous Generators

BIRGER MARCUSSON

ISSN 1651-6214ISBN 978-91-513-0103-7urn:nbn:se:uu:diva-331182

Dissertation presented at Uppsala University to be publicly examined in Room 2001,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Thursday, 30 November 2017 at 09:00for the degree of Doctor of Philosophy. The examination will be conducted in English.Faculty examiner: Professor Emeritus Göran Engdahl (KTH, Electrotechnical Design, Schoolof Electrical Engineering).

AbstractMarcusson, B. 2017. Magnetic Leakage Fields and End Region Eddy Current Power Lossesin Synchronous Generators. Digital Comprehensive Summaries of Uppsala Dissertations fromthe Faculty of Science and Technology 1575. 76 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-513-0103-7.

The conversion of mechanical energy to electrical energy is done mainly with synchronousgenerators. They are used in hydropower generators and nuclear plants that presently account forabout 80% of the electric energy production in Sweden. Because of the dominating role of thesynchronous generators, it is important to minimize the power losses for efficient use of naturalresources and for the economies of the electric power companies and their customers. For asynchronous machine, power loss means undesired heat production. In electric machines, thereare power losses due to windage, friction in bearings, resistance in windings, remagnetization offerromagnetic materials, and induced voltages in windings, shields and parts that are conductivebut ideally should be non-conductive.

The subject of this thesis is prediction of end region magnetic leakage fields in synchronousgenerators and the eddy current power losses they cause. The leakage fields also increase thehysteresis losses in the end regions. Magnetic flux that takes paths such that eddy currentpower losses increase in end regions of synchronous generators is considered to be leakage flux.Although only a small fraction of the total magnetic flux is end region leakage flux, it can causehot spots, discoloration and reduce the service life of the insulation on the core laminations. Ifunattended, damaged insulation could lead to electric contact and eddy currents induced by themain flux between the outermost laminations. That gives further heating and deterioration ofthe insulation of laminations deeper into the core. In a severe case, the core can melt locally,cause a cavity, buckling and a short circuit of the main conductors. The whole stator may haveto be replaced. However, the end region leakage flux primarily causes heating close to the mainstator conductors which makes the damage possible to discover by visual inspection before ithas become irrepairable.

Keywords: magnetic leakage fields, leakage flux, eddy currents, losses, synchronous generator

Birger Marcusson, Department of Engineering Sciences, Electricity, Box 534, UppsalaUniversity, SE-75121 Uppsala, Sweden.

© Birger Marcusson 2017

ISSN 1651-6214ISBN 978-91-513-0103-7urn:nbn:se:uu:diva-331182 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-331182)

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I B. Marcusson and U. Lundin, "Axial Magnetic Fields at the Ends of aSynchronous Generator at Different Points of Operation", IEEETransactions on Magnetics, vol. 51, no. 2, pp. 1-8, Feb. 2015

II B. Marcusson and U. Lundin, "Axial Magnetic Fields, Axial Force,and Losses in the Stator Core and Clamping Structure of aSynchronous Generator with Axially Displaced Stator", Electric PowerComponents and Systems, vol. 45, no. 4, pp. 410-419, Jan. 2017

III B. Marcusson and U. Lundin, "Harmonically Time Varying, TravelingElectromagnetic Fields along a Plate and a Laminate with aRectangular Cross Section, Isotropic Materials and Infinite Length",Progress In Electromagnetics Research B, Vol. 77, 117-136, 2017

IV B. Marcusson and U. Lundin, "Harmonically Time Varying, TravelingElectromagnetic Fields along a Laminate Approximated by aHomogeneous, Anisotropic Block with Infinite Length", Submitted toProgress In Electromagnetics Research B

V B. Marcusson and U. Lundin, "A Loss Model and Finite ElementAnalyses of the Influence of Load Angle Oscillation on Stator EddyCurrent Losses in a Synchronous Generator", Submitted toTransactions on Magnetics

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Basic Synchronous Machine Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Background on Research on Magnetic Leakage Flux . . . . . . . . . . . . . . . . . . . 81.3 Overview of Losses in Electric Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Review of Research on Magnetic Leakage Flux . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 The Steady State Voltage Equation for a Synchronous

Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Derivation of Rotor Reference Frame Voltage Equations of a

Synchronous Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.1 Assumptions, Definitions and Voltage Equations in the

Stator Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2 Inductances in the Stator Reference Frame of Salient

Pole Synchronous Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Transformation of Electrical Parameters to the Rotor

Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Linearization of the Equation of Motion of Load Angle

Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Effective Permeabilities in a Laminate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Calculation of Axial Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.3 Calculation of Losses in the Steel of the Stator Core and

Clamping Structure in a Synchronous Generator . . . . . . . . . . . . . . . . . . . . . . . . 323.4 The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Domain and Boundary Conditions in MagneticAnalysis of an Electric Machine with ANSYSMaxwell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.5 The Method of Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.6 Estimation of Frictional Power Loss in a Slider Bearing . . . . . . . . . . . . 393.7 Estimation of Windage Power Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8 Experimental Equipment and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.8.1 Magnetic Field Sensors and Measurements of AxialMagnetic Flux Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Results that Were Not Included in the Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1 Supplements to Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Supplements to Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5 Summary of Papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Svensk Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

8 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

9 Suggested Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71A.1 Galerkin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

1. Introduction

1.1 Basic Synchronous Machine TerminologyA synchronous machine is a type of electric machine designed for conversionbetween electrical and mechanical energy in both directions. The machineis operating as a generator when it converts mechanical power to electricalpower, and as a motor when it converts electrical power to mechanical power.The machine consists of a stationary part, called stator, and a moving part thatis called rotor or translator depending on if the machine is (roughly) cylindri-cal or linear. Fig. 1.1 shows a so called salient pole synchronous generator.The main parts of the stator are the three phase windings (main stator con-ductors) with alternating currents (AC), a laminated core of magnetic steel formagnetic field amplification, clamping parts and a frame for keeping the coreand windings fixed. A conventional synchronous machine has a rotor withelectromagnetic poles consisting mainly of a rotor field winding for directcurrent (DC) around a core of magnetic steel. The strength of the magneticfield produced by the poles increases with the rotor current, called the fieldcurrent. The name synchronous machine refers to the fact that the movingpart is synchronized with the magnetic field produced by the currents in thephase windings, i.e., the moving part and the magnetic fields move with thesame velocity or rotate with the same angular velocity under normal operation.The point of operation of a synchronous generator depends on the resistance,inductance and capacitance of the load connected to the generator via the sta-tor conductors. If the resistance is low, the current and power for a certainvoltage must be high. If the load is inductive, the current must lag the voltage.For harmonically time varying currents and voltages this means that the cur-rent, during each cycle, must reach its maximum after the voltage reaches itsmaximum. A generator operating alone (island operation) satisfies this for anyfield current since it determines the voltage itself within certain limits. For agenerator connected to a strong power grid (network), i.e. a power grid thatkeeps the voltage constant, the field current must be higher than for a purelyresistive load. In this situation the generator is said to be overexcited. If theload is capacitive, the current must lead the voltage. For a generator connectedto a strong power grid, this can be accomplished by the generator if the fieldcurrent is lower than for a purely resistive load. In this situation the generatoris said to be underexcited.

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Figure 1.1. Model of the salient pole synchronous generator used for experiments con-ducted by the author and other members of the hydropower group at the Division forElectricity. The letters N and S indicate magnetic north and south poles respectively.

1.2 Background on Research on Magnetic Leakage FluxSystems for conversion, transformation and transportation of energy are sub-ject to unwanted generation of heat. This heat is considered as power loss if itis not desirable and economical to use it. In general, power losses mean inef-ficient use of energy sources, and for the electric power companies, the powerlosses mean losses of income from the selling of electrical energy. There-fore, it is important to minimize the power losses, wherever they occur. There

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are several types of power losses in electric machines designed for conversionbetween electrical and mechanical energy. Examples of losses are windagepower losses, friction power losses in bearings, resistive losses in windingsand cables, eddy current power losses in all conductive parts, hysteresis powerlosses in ferromagnetic materials. The focus in this thesis is on unwanted axi-ally directed magnetic fields, so called magnetic leakage fields, and the lossesassociated with them at the end regions of synchronous generators. Magneticfields that do not take the desired paths are considered to be leakage fields. Theterms magnetic leakage field and magnetic leakage flux are used in the samecontext. Hydro power plants and nuclear plants dominate the Swedish powerproduction and use rotating synchronous generators. Most research publishedon end region power losses has been conducted on fast spinning synchronousgenerators, called turbogenerators or non-salient pole generators with cylin-drical rotors driven by gas turbines. It has been found that the end regions oflarge turbogenerators become warmer when the generator is underexcited thanwhen it is overexcited. The type of generators used in hydropower plants aresalient pole synchronous generators. The initial question for the work in thisthesis was if also the end region losses of salient pole synchronous generatorscould be sensitive to the excitation.

Motivation for initiating the study of axial magnetic fields in a salient polesynchronous generator comes from two cases of overheating of the end regionsof large hydropower generators in Scandinavia. In case one, which was theleast severe, there was discoloration of stator core laminations of the secondplate package from both ends. The suspected cause is underexcited operationcombined with insufficient cooling of the second plate package. In case two,consecutive short circuits overheated the stator end regions. This resulted indegradation of the lamination insulation followed by permanent heat damages.

Typical signs of core end heating due to eddy currents are discoloration ofthe varnish of the outermost stator laminations behind the slots for the statorconductors at both generator ends [1]. Core plate varnishes from purely topartly organic can withstand 180-300◦C continuously [2]. According to IEEEStd 56-2016, modern inorganic insulation such as aluminum orthophosphatethat can stand 500◦C.

The term magnetic field normally refers to the magnetic flux density. Fluxdensity integrated over a surface is magnetic flux. At the ends of a conven-tional rotating, electric machine, the magnetic field spreads out into a contin-uum of directions and induces undesired eddy currents in any conducting partsnearby. Time varying leakage flux that passes through the end laminate in thedirection perpendicular to the lamination planes induces relatively large eddycurrents in the laminations for two reasons. First, according to Faraday’s law,the induced electric fields are perpendicular to the magnetic field that causes

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them. Second, the laminations have large, continuous areas with high conduc-tivity. In addition, the combination of lamination and leakage flux can lead toextra high flux and hysteresis power losses in the end laminations.

1.3 Overview of Losses in Electric MachinesLosses can be divided into different types depending on when, where and howthey appear. The losses can also be put into different groups if they can be de-termined individually from measurements. The IEEE standard 115-2009 [3]puts losses into five such groups. They are

a) Friction and windage losses at no field current, no magnetic flux, and nostator current. Windage loss can be considered as a special case of the frictionloss.b) Core loss at non-zero field current, and open stator circuit.c) Stray load loss at non-zero field current, stator current at short circuit.d) The squared stator current multiplied by the direct current resistance perphase winding (times three).e) The squared field current multiplied by the direct current resistance in thefield winding.

Core loss is normally considered to be the same as iron loss since the statorand rotor cores are usually made of ferromagnetic materials [4], [5]. How-ever, the core loss measured according to the standard [3] is not a pure coreloss since it includes eddy current losses in stator conductors, support struc-ture and frame of the stator, and rotor surfaces. The core loss density in thelaminate material can be determined by Epstein tests according to IEC stan-dard 60404-2. Iron losses in electric machines can be determined by FEA, asdescribed in section 3.3.

Stray load losses include the extra losses that are generated during loadand that are not accounted for by the other mentioned losses. A large part ofthe stray load losses appears in stator bars because of leakage flux throughthe stator slots. The leakage flux makes the induced electric field different indifferent parts of the conductors. This gives rise to current displacement inthe conductors. This, in turn, increases the resistance and the resistive losses.However, these losses are reduced by the combination of twisting and subdi-vision of stator conductors into thin parts/strands. Other parts of the stray loadlosses are the result of eddy currents in pole faces and parts such as frames,housings, cover plates and clamping plates that are not intended to be includedin the magnetic circuits.

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1.4 Review of Research on Magnetic Leakage FluxDesigns used for reducing losses caused by magnetic leakage fields in thestator ends of electrical machines are slitted stator teeth, laminated or non-laminated magnetic flux screens/shunts, non-magnetic flux shields of highconductivity (usually copper), extra coating of varnish on the laminations [1],rounded or stepped stator and rotor ends at the air gap, rotor core shorter thanstator core, non-magnetic material in clamping structure, and non-magneticrotor coil retaining rings on turbogenerators.

In 1927 stator teeth slitting in radial-axial planes was suggested as a way toreduce eddy current losses in the teeth [6]. The teeth are only slitted near themachine ends. The eddy current loss in the teeth can be roughly halved by oneslit through the middle of the teeth if the skin effect is negligible, but in prac-tice the tendency of the eddy currents to concentrate near the surface makesthe slits somewhat less efficient. Slits make the teeth mechanically weaker andmore difficult to hold in place. For mechanical reasons slits are cut obliquelyin alternating directions in different plate layers. Due to the eddy currents thevoltage across a slit can be at least 1-2 V which stresses the insulation [7].

Laminated magnetic screens for protection of the core clamping plates havebeen suggested [8]. There seems to be very little published about this. In 1924-1927 Kuyser got a few patents on nonmagnetic, highly conductive shields tomount on the stator ends in order to deflect the magnetic field from the sta-tor core of turbogenerators. One shield, called squirrel-cage damper, patentGB220362, was in the air gap surrounding the retaining ring. Another shieldwas a heavy, plane, copper alloy plate, patent GB220362. Apart from holesfor stator conductors the plate covered the whole end region from the air gapto the back of the core. The plate served as both clamping structure and shield.One generator was equipped with such a shield in one end and a nonmagneticcast iron clamping plate in the other end. The temperature increase, mea-sured with thermocouples, was 39.9◦C in the core end with copper plate and71.8◦C in the other core end. Approximately the latter temperature increasewas also measured in the core end of a generator of the same design but withmagnetic cast iron clamping rings [9]. Measurements with a flat copper screenbetween a piece of clamping plate and a pancake coil showed that although thescreen reduced the loss in the clamping plate, the total loss in the screen andthe clamping plate increased [8]. Analytical 2-D calculations showed that ahighly conductive screen on the end of the stator diverts the leakage magneticfield at the cost of high loss in the screen and flux concentration in the laminateparts not covered by the screen [10]. Therefore, a screen on the stator lami-nate end should cover the stator teeth completely in order to avoid high axialflux concentration. Finite element simulations of an overexcited turbogenera-tor with power factor 0.95 showed that a plane copper shield on the clamping

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plate could at best reduce the axial flux to the core by about 20 %, no matterhow thick the shield was. The flux was diverted to radial skin layers in theshield before entering the core. As long as the shield thickness is so thin thatthe induced eddy currents in the shield give a negligible contribution to the to-tal field, the losses in the shield increase in proportion to the shield thickness.When the thickness increases above a fraction of a skin depth the losses begindo decrease approximately exponentially towards an asymptotic value if theimposed field is only axially directed [11]. A shield that is wrapped to coveralso the radially inner surface of the clamping plate shields the plate and coreback more but diverts more flux into the teeth.

In 1974 Howe et al. [10] by means of 2-D analytical calculations concludedthat the losses in the stator end were reduced by rounding or stepping the sta-tor core end and by making the rotor shorter than the stator. Figures showingturbogenerators in [9] and [12] indicate that stator stepping was used becauseof the protruding retaining ring long before Howe et al. mentioned the benefitsof core stepping for reduction of eddy current losses, but a reason for steppingis to get a sufficient inflow of cooling air. Finite element analyses have con-firmed that core stepping give reduction of eddy current losses, but the lossreduction at load is not as large as at no load [13].

Before 1920 no generators were built with nonmagnetic end structures [14].At least from about 1925 some manufacturers used nonmagnetic clampingmaterials [12]. A negligible difference between heating of magnetic and non-magnetic cast iron clamping plates was measured on two turbogenerators ofthe same design [9]. However, linear material finite element analyses of a3.1 MW permanent magnet generator show that the losses, about 2.9 kW inclamping plates of magnetic construction steel, can be about a factor ten higherthan in clamping plates of nonmagnetic stainless steel [15]. Hysteresis lossesare not mentioned in [15]. An advantage with magnetic clamping rings is thatthey shield the stator core from leakage flux [14]. At least until 1948 Westing-house Electric Corporation continued to build their machines with magneticalloy steel clamping rings and magnetic rotor coil retaining rings.

In 1920 people working with turbogenerators recognized that the heatingof the stator end regions was increased at underexcited operation. In 1929Kuyser mentioned that, in experiments with a turbogenerator with the statorcurrent kept constant, the end plate temperature increased roughly 20, 45 and70◦C in the three extreme load cases zero power factor overexcited, short cir-cuit and zero power factor underexcited [9]. The reasons for the differencesin stator end region heating between leading with zero power factor, laggingwith zero power factor and short circuit operation was the degree and directionof saturation of the retaining ring. At underexcitation with zero power factorthe field current is zero. Flux picked up by the retaining ring flows in the

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peripheral direction to the adjacent pole. Then the magnetic cross section ofthe retaining ring determines the magnitude of the leakage flux. It takes onlya small field current to reduce the permeability of the retaining ring. Kuyserstated that at short circuit and overexcitation, the field current magnetises theretaining ring in a direction that opposes the stator ampere turns. This is nottrue for the axial field but true for the radial direction in the air gap and inthe whole retaining ring provided that the center of the stator end windingsis farther away from the stator core end than the whole end ring is. Kuysermentioned that it had been known for some years that non-magnetic retain-ing rings can reduce the end leakage field and losses. Historically, the highstrength of the magnetic retaining rings was a reason for using them, espe-cially by British designers because of their relatively high safety factors [9].He suggested that the retaining rings should be either as thin as possible ornon-magnetic [9]. In 1953 Estcourt et. al. largely repeated Kuyser’s expla-nation but with commonly used load cases and mentioning of the direction ofthe armature reaction in these cases. The reluctance in the peripheral direc-tion of the retaining ring is important for the leakage flux and depends on theoperation conditions [12]. Both the magnitude of the rotor (field) flux and itsposition relative to the main direction of the stator (armature) reaction affectsthe reluctance and leakage flux. At unit power factor operation, the stator fluxis directed somewhere between the poles where the reluctance of the retainingring is relatively low. At constant apparent power and terminal voltage the endiron temperature increased a factor 2-3 from overexcited to underexcited statewith power factor 0.8. At constant power factor and terminal voltage, the tem-perature almost increased linearly with the apparent power. At constant powerfactor and apparent power, the temperature decreased linearly with increas-ing terminal voltage [12]. Nowadays strong nonmagnetic 18Mn-18Cr steelretaining rings are used [16]. According to IEEE Standard 56-2016, the axialleakage fields can be high in large machines at underexcitation, especially ifthe end windings are long as in turbogenerators.

For practical and mechanical reasons the stator core lamination of a largemachine is made of overlapping plate segments. This segmentation is also away to reduce eddy currents induced by axial leakage flux in the stator core[17]. However, the axial leakage fields at the ends of short-circuit generatorscan be so strong that the induced eddy currents can cause voltage breakdownthrough the insulation along the gaps between the segments in the same layer[18]. A short-circuit generator is used for generating a high power pulse fortesting purposes.

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2. Theory

2.1 PhasorsA phasor can be represented by a complex number that in turn can be rep-resented by a vector from the origin in the complex number plane. Phasorsare used to describe physical quantities that vary harmonically with time. Thelength of the vector is the amplitude. The angle from some reference line, usu-ally the real axis, to the vector is the phase angle of the phasor. Phasors havebeen used extensively in the work leading to this thesis since all of it concernssynchronous three-phase machines at steady state or near steady state wherecurrents, voltages and electromagnetic fields vary approximately harmonicallywith time. A horizontal bar above a symbol denotes a complex quantity, suchas a phasor. Although the field current is constant at steady state, it can bemodeled as a phasor, i f , since it exposes each pole pitch wide sector of thestator by flux that is roughly sinusoidal both in space and time when the ro-tor rotates with constant angular velocity. Electromagnetic fields penetratinga material undergo phase shifts that depend on the material, the frequencyand the geometry, but any phase difference between a main current, such as aphase current or the field current, and the phasor of the radial flux it createsin a region from the stator conductors to the rotor is normally assumed to benegligible, at least in linear materials and far from saturated soft magnetic ma-terials. For soft magnetic materials the hysteresis is small. Here, it will beassumed that a current phasor is in phase, not in anti-phase, with the phasor ofthe flux linkage contribution it gives rise to.

2.2 The Steady State Voltage Equation for aSynchronous Generator

The steady state stator voltage equation, (2.1), in the rotor reference system isof great importance for Paper I, II and V. In phasor form, the equation relatesthe RMS (root mean square) terminal voltage, V and the RMS stator current,I, to the RMS voltage, E, induced in the stator windings by the rotor. Param-eters derived from the voltage equation are active power, P, reactive power,Q, power factor angle, ϕ , load angle, δ , current phase angle, α , stator cur-rent component Id in phase with the rotor d (direct) axis and Iq in phase withthe q (quadrature) axis. In Paper I, Id and Iq have important roles in a phasormodel for magnetic leakage fields, and in Paper V, Id and Iq have important

14

roles in a phasor model for eddy current density. Voltage equations are easierto solve in the rotor reference frame than in the stator reference frame becausethe inductances are constants in the rotor reference frame. At steady state, alsothe other electrical parameters mentioned are constants. The voltage equationwith generator sign convention used is

E = V + jXdId + jXqIq +RI (2.1)

where Xd is the stator d axis reactance, Xq is the stator q axis reactance, R isthe stator resistance per phase, and j =

√−1. With the d axis as the real axis,and the q axis as the imaginary axis, Id = Id , Iq = jIq, and (2.1) gives

Id =Xq(E −Vq)−RVd

R2+XdXq=

Xq(E −V cosδ )−RV sinδ )R2+XdXq

(2.2)

and

Iq =R(E −Vq)+XdVd

R2+XdXq=

R(E −V cosδ )+XdV sinδR2+XdXq

. (2.3)

The active power is

P = 3Re{V I∗}= 3(VdId +VqIq)

=3V

R2+XdXq

(E(Rcosδ +Xq sinδ )−V R+

V2(Xd −Xq)sin2δ

). (2.4)

where the exponent ∗ denotes a complex conjugate. The reactive power is

Q = 3Im{V I∗}= 3(VqId −VdIq)

=3V

R2+XdXq

(E(Xq cosδ −Rsinδ )−V (Xd sin2 δ +Xq cos2 δ )

). (2.5)

In case of load angle oscillations, it may be better to assume that the poleflux linkage rather than field current is approximately constant. In that case(2.1) can be replaced by the classical model,

E ′=V + jX ′d Id + jXqIq +RI (2.6)

where E ′ is called q axis transient emf (electromotive force) which is propor-tional to the pole flux linkage, and X ′

d =Xd −ωsL2d f /L f is the transient d axisreactance where Ld f =−√

3/2Mf is the mutual inductance between the fieldwinding and the fictitious stator d axis winding. The expression for Ld f isderived in section 2.3.3. L f is the self inductance of the field winding. Nor-mally there is a transformer and a transmission line between the generatorand a strong power grid that in simpler models is assumed to have a constantRMS voltage and is then called an infinite bus. Optionally, the terminal volt-age in (2.1) or (2.6) can be replaced by the voltage of the infinite bus if the

15

impedance between the generator and the infinite bus is accounted for. In thatcase, an alternative to (2.6) is

E ′ = VIB + jX ′DId + jXQIq +Rt I, (2.7)

where X ′D = X ′

d + Xtr, XQ = Xq + Xtr and Rt = R + Rtr, where Xtr is the reactanceand Rtr is the resistance of the transformer plus transmission line.Fig. 2.1 shows a phasor diagram with rotor poles. Since i f and the d axis

are in anti-phase, it can be useful to call the rotor north pole axis in phase withi f the field axis or simply the f axis. The choice of minus sign in Faraday’slaw on integral form dictates that the pole flux linkage contribution Ψ f f fromi f should lead the emf E it induces in the stator winding by 90◦. Since Ψ f f isin phase with a rotor north pole, E is in phase with the q axis. A load angle,δIB, from E ′ to VIB is useful in analyses of load angle oscillations since VIB hasa known, constant angular speed relative to the stator reference frame. In sta-bility analyses, there is a distinction between electrical and mechanical power.In that case, the electric power symbol is Pe, and the symbol for mechanicalpower is Pm. The active power delivered by the generator can be expressed as

Pe = 3RtI2+3Re{VIBI∗}=

3Rt

(R2t +X ′DXQ)2

[(XQ(E ′ −VIB cosδIB)−RtVIB sinδIB)2

+(Rt(E ′ −VIB cosδIB)+X ′DVIB sinδIB)

2]

+3VIB

(E ′(Rt cosδIB +XQ sinδIB)−VIBRt +

VIB2 (X ′

D −XQ)sin2δIB

)R2t +X ′

DXQ.

(2.8)

Under the assumption that Rt � X ′D and Rt � XQ, (2.8) can be simplified to

Pe ≈ 3VIB

X ′DXQ

(E ′XQ sinδIB +

VIB

2(X ′

D −XQ)sin2δIB

). (2.9)

Differentiation of Pe with respect to δIB gives the so called synchronizingpower coefficient [19] which can be used in the equation of motion for a gen-erator with load angle oscillations. The coefficient is

dPe

dδIB≈ 3VIB

X ′DXQ

(E ′XQ cosδIB +VIB(X ′

D −XQ)cos2δIB). (2.10)

For the linearized, analytical solution in section 2.4 the coefficient is needed.However, for a numerical solution, it may be more convenient to skip the co-efficient and just express Pe in terms of the components of current and voltagefor each point in time where δIB has been determined.

16

Figure 2.1. Phasor diagram.

2.3 Derivation of Rotor Reference Frame VoltageEquations of a Synchronous Generator

In this section the used transient and steady state voltage equations in the rotorreference frame will be derived. The steady state stator voltage equation isthen obtained from the transient equations as a special case.

2.3.1 Assumptions, Definitions and Voltage Equations in theStator Reference Frame

The derivation of the voltage equations follows essentially the steps of [20]and [21] but with partly other reference directions. The signs of the mutualinductances between rotor and stator and the terms in the voltage equationsin the rotor reference frame depend on two choices. The first is if the d axisis leading or lagging the q axis. The second is if generator or motor refer-ence signs/directions are used. A third choice is the ratio, Nf ic/N, where Nis the number of turns per phase winding, and Nf ic is the chosen number ofturns in each of the fictitious stator windings. The third choice affects themagnitudes of the mutual inductances between rotor and stator and the corre-sponding terms in the voltage equations in the rotor reference frame. Fig. 2.2shows the cross section of a simplified synchronous machine. The angle θ is

17

used to specify the rotor position relative to the stator. The choice here is to letthe q axis lead the d axis. Each current has an ideal magnetic axis that definesthe positive main flux direction for positive current according to the right handrule. Slots and rotor pole saliency combined with magnetic materials and/oreddy currents make the main direction of the flux from a current deviate fromits ideal magnetic axis. On the other hand, the flux from a current could beconsidered to be the flux in absence of magnetic materials and eddy currents.From that point of view, the main flux direction created by a current in a coilis exactly the direction of the ideal magnetic axis. Soft magnetic materials are,however, not independent sources of magnetic flux.

Figure 2.2. Cross section of a simplified synchronous machine. Positive current ref-erence directions are shown. A dot in a circle denotes current direction towards thereader. A cross in a circle denotes current direction away from the reader.

Fig. 2.3 shows equivalent circuits in the stator reference frame for the ma-chine. Generator sign convention is here used for the stator windings. Thatmeans that a stator current is positive when it is directed away from the gener-

18

ator. Then the induced voltage is minus the time derivative of the flux linkage.A consequence of the minus sign is that each main flux (linkage) phasor is 90◦ahead of (leading) the emf phasor (electromotive force) it induces in the stator.

Figure 2.3. Equivalent circuits of a synchronous machine connected to load withimpedance Zl via a transformer and transmission line with impedance Zt . The Zl com-ponents represent circuits with resistive, inductive and/or capacitive elements. Theload is typically a motor. The Zt components typically represent just a resistance inseries with an inductance.

19

The transient voltage equation in the stator reference system is⎡⎢⎢⎢⎢⎢⎢⎣

vavbvc−v f00

⎤⎥⎥⎥⎥⎥⎥⎦=−

⎡⎢⎢⎢⎢⎢⎢⎣

r 0 0 0 0 00 r 0 0 0 00 0 r 0 0 00 0 0 r f 0 00 0 0 0 rD 00 0 0 0 0 rQ

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

iaibici fiDiQ

⎤⎥⎥⎥⎥⎥⎥⎦− ddt

⎡⎢⎢⎢⎢⎢⎢⎣

ΨaΨbΨcΨ fΨDΨQ

⎤⎥⎥⎥⎥⎥⎥⎦

(2.11)

which can be written more compactly as[

vabcv f DQ

]=−

[RS 00 RR

][iabci f DQ

]− ddt

[ΨΨΨabcΨΨΨ f DQ

]. (2.12)

where

vabc =

⎡⎣va

vbvc

⎤⎦ , v f DQ =

⎡⎣−v f00

⎤⎦ , iabc =

⎡⎣ia

ibic

⎤⎦ , ΨΨΨabc =

⎡⎣Ψa

ΨbΨc

⎤⎦etc.. (2.13)

The flux linkages are⎡⎢⎢⎢⎢⎢⎢⎣

ΨaΨbΨcΨ fΨDΨQ

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

La Lab Lac La f LaD LaQLba Lb Lbc Lb f LbD LbQLca Lcb Lc Lc f LcD LcQL f a L f b L f c L f L f D L f QLDa LDb LDc LD f LD LDQLQa LQb LQc LQ f LQD LQ

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

iaibici fiDiQ

⎤⎥⎥⎥⎥⎥⎥⎦

(2.14)

which can be written more compactly as[

ΨΨΨabcΨΨΨ f DQ

]=

[LSS LSRLRS LRR

][iabci f DQ

]. (2.15)

The inductance matrix in (2.14) is symmetrical since mutual inductances arereciprocal according to the Neumann formula [22], [23]. In a salient polesynchronous machine the air gap length is not constant around the rotor. Thatmakes all inductances that involve a stator winding dependent on the rotorangle θ that changes with time when the rotor rotates. The stator slots arenormally so small compared to the rotor poles that θ does not matter for therotor inductances. Therefore, L f , L f D, LD and LQ are independent of θ butthey change with the degree of saturation of the magnetic materials. Saturationwill not be considered here. The generator is assumed to be magneticallysymmetrical such that there is no net flux linkage between the d and q axisrotor windings. That means that

L f Q = LDQ = 0. (2.16)

20

2.3.2 Inductances in the Stator Reference Frame of Salient PoleSynchronous Machines

It is here assumed that the poles have been shaped such that the radial air gappermeances and, therefore also the inductances, in the stator reference framevary approximately harmonically around average values around the rotor [24].A consequence of that is that the amplitudes of the stator self and mutualinductances are equal [25]. The inductances in the stator reference frame are

La = Ls −Lm cos2θ (2.17)

Lb = Ls +Lm cos2(θ − π6) (2.18)

Lc = Ls +Lm cos2(θ +7π6) (2.19)

Lab =−Ms −Lm cos2(θ − π3) (2.20)

Lac =−Ms −Lm cos2(θ +π3) (2.21)

Lbc =−Ms −Lm cos2θ (2.22)

La f =−Mf sinθ (2.23)

Lb f = Mf cos(θ − π6) (2.24)

Lc f = Mf cos(θ +7π6) (2.25)

LaD =−MD sinθ (2.26)

LbD = MD cos(θ − π6) (2.27)

LcD = MD cos(θ +7π6) (2.28)

LaQ = MQ cosθ (2.29)

LbQ = MQ cos(θ − 2π3) (2.30)

LcQ = MQ cos(θ +2π3) (2.31)

2.3.3 Transformation of Electrical Parameters to the RotorReference Frame

The rotating magnetic field created by the three stator phase currents can beobtained by only two currents, id and iq, whose windings are fixed relative to

21

the rotor. Such windings are fictitious since the rotor with fixed magnetizationwould not be able to induce any voltage in these windings. Nevertheless, theconcept of d and q components of stator currents, voltages, inductances andflux linkages are useful as mentioned in section 2.2. Fictitious windings donot have to have an integer number of turns. Here it is assumed that the d andq windings each have a factor k more turns than each of the stator windingshave. Together with a chosen k, the requirement that id and iq should givethe same mmf as the stator phase currents determines a transformation matrixfrom ia, ib, ic to id and iq under balanced conditions, i.e. such that ia + ib + ic= 0. To get a transformation that works also under unbalanced conditions, afictitious component i0 = o(ia+ ib+ ic) is defined. For θ from the a axis to theq axis, and for arbitrary k and o, the transformation matrix becomes

P =1k

⎡⎢⎣sinθ sin(θ − 2π

3 ) sin(θ + 2π3 )

cosθ cos(θ − 2π3 ) cos(θ + 2π

3 )

o o o

⎤⎥⎦ . (2.32)

The inverse of P is

P−1 =2k3

⎡⎢⎣

sinθ cosθ 12o

sin(θ − 2π3 ) cos(θ − 2π

3 )12o

sin(θ + 2π3 ) cos(θ + 2π

3 )12o

⎤⎥⎦ . (2.33)

In the special case o = 1/√2 and k =

√3/2, the transformation matrix becomes

orthogonal and is given by

P =23

⎡⎢⎢⎣sinθ sin(θ − 2π

3 ) sin(θ + 2π3 )

cosθ cos(θ − 2π3 ) cos(θ + 2π

3 )1√2

1√2

1√2

⎤⎥⎥⎦ . (2.34)

The same matrix can be used for transformation of currents, voltages and fluxlinkages between the stator reference frame and the rotor dq reference frameaccording to

idq0 = Piabc,⇒ iabc = P−1idq0 (2.35)

vdq0 = Pvabc,⇒ vabc = P−1vdq0 (2.36)

ΨΨΨdq0 = PΨΨΨabc,⇒ ΨΨΨabc = P−1ΨΨΨdq0 (2.37)

where

idq0 =

⎡⎣id

iqi0

⎤⎦ . (2.38)

22

The other vectors in (2.35), (2.36) and (2.37) are defined analogously. A con-sequence of the orthogonality is that the transformation is power invariant,i.e.

vTdq0idq0 = (Pvabc)

T Piabc = vTabcPT Piabc = vT

abciabc. (2.39)

where the exponent T denotes a transpose operation. Insertion of (2.35) and(2.37) into (2.15) followed by multiplication of the first row with P from theleft gives

ΨΨΨdq0 = PLSSP−1idq0+PLSRi f DQ (2.40)

ΨΨΨ f DQ = LRSP−1idq0+LSRi f DQ (2.41)

which can be written as [ΨΨΨdq0ΨΨΨ f DQ

]=

[LS MMT LR

][idq0i f DQ

](2.42)

whereMT =MT if k =±√3/2. Equation (2.42) can be written more explicitly

as ⎡⎢⎢⎢⎢⎢⎢⎣

ΨdΨqΨ0Ψ fΨDΨQ

⎤⎥⎥⎥⎥⎥⎥⎦=

⎡⎢⎢⎢⎢⎢⎢⎣

Ld 0 0 Ld f LdD 00 Lq 0 0 0 LqQ0 0 L0 0 0 0

L f d 0 0 L f L f D 0LDq 0 0 LD f LD 00 LQq 0 0 0 LQ

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

idiqi0i fiDiQ

⎤⎥⎥⎥⎥⎥⎥⎦. (2.43)

The values of the mutual inductances between the d and q windings and therotor windings depend on k but not o. For arbitrary k and o, (2.43) can bewritten as

⎡⎢⎢⎢⎢⎢⎢⎣

Ls +Ms +3Lm2 0 0 −3Mf

2k −3MD2k 0

0 Ls +Ms − 3Lm2 0 0 0 3MQ

2k0 0 Ls −2Ms 0 0 0

−kMf 0 0 L f L f D 0−kMD 0 0 LD f LD 00 kMQ 0 0 0 LQ

⎤⎥⎥⎥⎥⎥⎥⎦. (2.44)

which is an unsymmetrical and therefore unphysical inductance matrix unlessk =±√

3/2. In spite of that, the traditional choice is k = 3/2. Some authoritiesand machine designers think it is advantageous for a number of reasons, partlybecause it gives the fictitious stator current the same magnitude as any of thereal phase currents under balanced conditions [25]. However, this choice isassociated with one set of equations that are valid in SI units and another thatis valid in per unit [26], [20]. This can be confusing. From now on, onlyk =

√3/2 will be used in this thesis. The minus sign in Ld f and LdD is a

consequence of the opposite directions of the f and d axes. This, in turn, is a

23

consequence of the choice of the q axis leading the d axis. Insertion of (2.35),(2.36) and (2.37) into (2.12) followed by multiplication of the first row with Pfrom the left gives

vdq0 =−PRSP−1idq0−Pddt

(P−1ΨΨΨdq0

)(2.45)

v f DQ =−RRi f DQ − ddt

ΨΨΨ f DQ. (2.46)

Equation (2.42) in (2.45) gives

vdq0 =−PRSP−1idq0−Pddt

(P−1(LSidq0+Mi f DQ)

)

=−RSidq0−Pddt

P−1(LSidq0+Mi f DQ)−LSdidq0

dt−M

di f DQ

dt(2.47)

where

Pddt

P−1 = PdP−1

dθdθdt

= ω

⎡⎣0 −1 01 0 00 0 0

⎤⎦ . (2.48)

It can be noted that the signs in (2.48) are opposite those obtained when the qaxis is lagging the d axis. Equation (2.42) in (2.46) gives

v f DQ =−RRi f DQ − ddt(MT idq0+LRi f DQ)

=−RRi f DQ −MT didq0

dt−LR

di f DQ

dt(2.49)

According to (2.47), (2.48), (2.49), (2.42) and (2.43) the voltage equations canbe written explicitly as

vd =−rid +ω(Lqiq +LqQiQ)−Lddiddt

−Ld fdi f

dt−LdD

diDdt

vq =−riq −ω(Ldid +Ld f i f +LdDiD)−Lqdiqdt

−LqQdiQdt

v0 =−ri0−L0di0dt

v f = r f i f +L fdi f

dt+Ld f

diddt

+L f DdiDdt

vD = rDiD +LDdiDdt

+LdDdiddt

+L f Ddi f

dt= 0

vQ = rQiQ +LQdiQdt

+LqQdiqdt

= 0 (2.50)

At steady state operation, the rotor angular velocity is constant, and the timederivatives are zero so that the damper winding currents are also zero. In that

24

case the first two equations in (2.50) are reduced to

vd =−rid +ωLqiqvq =−riq −ω(Ldid +Ld f i f ) (2.51)

Since the phase difference is 90◦ between the d axis and the q axis, the ficti-tious current and voltage can be expressed with the phasors

i f ic = id + jiq (2.52)

andv f ic = vd + jvq. (2.53)

The magnitude of i f ic in relation to the amplitude of ia at steady state underbalanced conditions can be obtained under the assumption that

θ = ωt +θ0 (2.54)

and that

ia = icosωt = icos(θ −θ0)

ib = icos(ωt − 2π3) = icos(θ − 2π

3−θ0)

ic = icos(ωt +2π3) = icos(θ +

2π3

−θ0). (2.55)

Equation (2.55) inserted in (2.35) gives

id =32k

isinθ0

iq =32k

icosθ0 (2.56)

which combined with (2.52) gives that

i f ic =32k

i =

√32

i. (2.57)

Analogously,

v f ic =32k

v =

√32

v. (2.58)

The equations (2.57) and (2.58) combined with the RMS values I = i/√2 and

V = v/√2 give

i f ic =√3I

v f ic =√3V. (2.59)

25

Therefore, division of (2.51) by√3 and insertion of Ld f = −3Mf /(2k) from

(2.44) with k =√3/2 converts (2.51) to equations for RMS values according

to

Vd =−rId +ωLqIq

Vq =−rIq −ωLdId +ωMf√2

i f (2.60)

In (2.60) ωLq = Xq, ωLd = Xd , and if ω Mf√2i f is denoted with E, (2.60) can be

written as

Vd =−rId +XqIq

Vq =−rIq −XdId +E. (2.61)

Finally, with r = R, I = Id + jIq and V = Vd + jVq, the equations in (2.61) canbe combined to the complex steady state stator voltage equation (2.1).

2.4 Linearization of the Equation of Motion of LoadAngle Oscillation

The swing equation of motion of the rotor of a synchronous generator has beenused in Paper V and is given by ([25])

Jdωm

dt= Tm −Te − (ωm −ωm,s)

Dωm,s

(2.62)

where J is the mass moment of inertia, ωm is the mechanical angular speed,ωm,s is the steady state operation/synchronous value of ωm, Tm is the mechan-ical torque on the rotor, Te is the electrical torque developed by the generator,andD is some positive damping factor that can account for the effect of dampercircuits. A more general equation of motion would include damping terms dueto bearing friction and windage that increase with ωm but are independent ofthe difference ωm−ωm,s. Normally, for load angle oscillations, ωm ≈ ωm,s. Inthat case bearing friction and windage are approximately constant and can beincluded in Tm. Multiplication of (2.62) by ωm gives

ωmJdωm

dt=4p2

ωJdωdt

= Pm −Pe − (ω −ωs)ωωs

D ≈ Pm −Pe − (ω −ωs)D (2.63)

where p is the number of rotor poles, ω = p/(2ωm) is the electrical angu-lar frequency, and ωs is the steady state operation value of ω , Pm is the netmechanical power on the rotor, Pe is the electrical power developed by the

26

generator. For numerical solution, (2.63) can be written as a system of twoequations,

dθdt

=ω

dωdt

= p2Pm −Pe − (ω −ωs)D

4ωJ.

(2.64)

In what follows, δIB is the load angle to the infinite bus, δIB,0 is the initialsteady state value of δIB, and β = δIB - δIB,0. A phasor diagram with anglesis given in Paper V. Equation (2.64) can, with use of Ω=ω −ωs=dβ/dt, berewritten in terms of β as

dβdt

=Ω

dΩdt

=d2βdt2

= p2Pm −Pe −D dβ

dt

4(ωs +dβdt )J

.(2.65)

In the case of small β , (2.65) can be linearized around β = 0. Typically,dβ/dt << ωs if the rotor oscillates at a natural frequency. Consequently,ωs +

dβdt ≈ ωs = constant which can be used in front of J in (2.65). With the

abbreviation k for p2/(4ωsJ), the linearized electrical power can be expressedas Pe=Pe,s +

dPe(β=0)dβ β where Pe,s is the steady state value. Similarly, the lin-

earized mechanical power can be written as Pm=Pm,s+ΔPm. Considering thatthe steady state parts of Pe and Pm cancel each other, the linearized equationof motion of a generator connected to a strong grid becomes the same as for amass point connected to a damper and a spring, i.e.

d2β (t)dt2

+2Cdβ (t)

dt+Gβ (t) = F(t) = kΔPm(t) (2.66)

where 2C = kD and G = k dPe(β=0)dβ = k dPe(δIB=δIB,0)

dδIBis a torsional stiffness di-

vided by J at β = 0. G can be calculated with help of the synchronizing powercoefficient (2.10). The solution of (2.66) is

β = βp +βh (2.67)

where βp is a particular solution and βh is a solution of (2.66) with F replacedby 0 on the right hand side. That does not mean that βh is independent of F ,as will be clear from (2.73) and (2.74) below. With a damping less than thecritical, there are oscillations, and βh is

βh = (Acosωnt +Bsinωnt)e−Ct (2.68)

whereωn =

√G−C2 (2.69)

27

is the load dependent, natural angular frequency of β without the disturbanceF(t), A and B are coefficients determined by initial conditions on β and dβ/dt.In the special case of a sinusoidal disturbance, F(t)=F0 sinωFt, the particularsolution is

βp =±F0rsin(ωFt +φ) (2.70)

wherer =

√(ω2

F −G)2+4C2ω2F (2.71)

andφ = arctan

2CωF

ω2F −G

(2.72)

with plus sign in (2.70) if G>ω2F and minus sign if G<ω2

F . The coefficientsof βh in (2.68) are

A = β (0)+F02CωF

r2(2.73)

and

B =

(dβ (0)

dt+β (0)C+

F0ωF

r2(2C2+ω2

F −G)

)1

ωn(2.74)

where β (0) and dβ (0)dt are initial values. A value of ωF that gives a local max-

imum of the amplitude of β is a resonance angular frequency, ωr. When thedamping approaches zero, both ωn and ωr approach

√G, and the amplitude

of β approaches infinity. When there is damping, the transient, represented byβh, becomes negligible sooner or later. Then βp remains with resonance angu-lar frequency ωr =

√G−2C2. This inserted in (2.70) gives that the amplitude

of βp at ωr is

βp,r =F0

2C√

G−C2=

F02Cωn

. (2.75)

The period of the load angle oscillation is approximately tp=2π/ωn.

28

3. Methods

Some of the methods used to produce the results in the papers are here pre-sented in more detail than in the papers.

3.1 Effective Permeabilities in a LaminateIn electric machines, the laminated cores consist of sheets of electric steel in-sulated from each other with relatively thin layers of varnish or oxide. Fig. 3.1shows a laminate piece consisting of one steel layer with thickness ts and onevarnish layer with thickness tv.

Figure 3.1. A piece of a laminate with fluxes parallel to each other and to the lamina-tion planes

For the purpose of deriving an effective laminate permeability in directionsparallel to the laminations, the magnetic fluxes, Φs in the steel and Φv in thevarnish, are assumed to be homogeneously distributed in each material layerand directed in the same direction parallel to the laminations. It is also as-sumed that there is no significant current density, J, in the laminate. In thatcase, Ampere’s law with line integration of the magnetic field strength, H,

29

along the boundary Γ in Fig. 3.1 gives∮

ΓH · dl =

∫S

J · dS = HsΔx−HvΔx = ℜsΦs −ℜvΦv = 0 (3.1)

where S in the surface integral is a surface bounded by Γ, and

ℜs =Δx

Asμs, ℜv =

ΔxAvμv

(3.2)

are the reluctances of the steel and varnish layers. The flux multiplied by thereluctance the flux meets is the magnetic potential difference between the twoends separated by the distance Δx. According to (3.1), the potential differenceis the same in both material layers. From this, an effective reluctance parallelto the lamination planes can be defined such that it gives a magnetic potentialdifference in a homogeneous approximation of the laminate that is the sameas the magnetic potential difference in the real laminate, i.e.

(Φs +Φv)ℜi,e f f = Φsℜs, i= x or y (3.3)

withℜi,e f f =

Δl(As +Av)μi,e f f

, i= x or y (3.4)

under the assumption that the z direction is the stacking direction. Equations(3.1), (3.3), (3.4) and B = μH give the effective permeability

μi,e f f =μsts +μvtv

ts + tv= kμs +(1− k)μv ≈ kμs, i= x or y (3.5)

in the lamination planes where k is the stacking factor defined by

k =ts

ts + tv. (3.6)

The inverse of the reluctance is called permeance. The expression for μi,e f f ismore directly obtained by defining an effective permeance to be equal to thesum of the plate permeance and the varnish permeance, i.e.

℘i,e f f =1

ℜi,e f f=1

ℜs+1

ℜv, i= x or y. (3.7)

For simplicity the approximation at the end of (3.5) can be used when the statoris not very saturated. Equation (3.5) motivates scaling down the B values ofthe B−H curves in the laminate plane by k. This has been used in simulationsat low field current. This implies e.g. that if the flux density in a steel plateis Bs, the flux density in the FE model is k ·Bs. If there are eddy currentsin the laminate planes (3.5) is only approximately correct. In the followingderivation of the effective permeability in the stacking direction, conservation

30

of flux is assumed in that direction. The series reluctance of one plate and onevarnish layer is then set equal to an effective reluctance

ℜz,e f f = ℜs +ℜv =ts + tv

Aμz,e f f=

tsAμs

+tv

Aμv(3.8)

which gives the effective laminate permeability

μz,e f f =ts + tvtsμs+ tv

μv

. (3.9)

In simulations with saturation described in Paper II, the nonlinear, anisotropiclamination model as described in [27] has been used with a stacking factor. Itshould be pointed out that (3.5) and (3.9) are used in the model described in[27] whether or not the preconditions used in the derivations are met.

3.2 Calculation of Axial ForceIn Paper II the finite element program ANSYS Maxwell 3D has been used forcalculation of the axial force on the stator by means of the principle of virtualwork [28], [29], [30]. According to the manual of ANSYS Maxwell only thechange of coenergyWco at constant current in the virtually distorted elementson the boundary of the virtually moved object is taken into account in the force

FFFz =∂Wco

∂ z

∣∣∣constant current

=∂∂ z

∫V(∫ H

0BBB ·dHHH)dV (3.10)

where H is the magnetic field intensity. For comparison the axial force hasalso been calculated by integration of the Maxwell stress tensor TTT over somesurface that encloses the object and no other source of magnetic field [31],[32]. The magnetic force on an object bounded by surface S with an outwarddirected and location dependent unit normal vector n in cylindrical coordinatesis

FFF =

∫S

TTT · ndS =

1μ0

∫S

⎡⎣B2r − 1

2B2 BrBϕ BrBzBϕBr B2ϕ − 1

2B2 BϕBz

BzBr BzBϕ B2z − 12B2

⎤⎦⎡⎣ nr

nϕnz

⎤⎦dS =

1μ0

∫S((BBB · n)BBB− n

2B2)dS

(3.11)

where B is the magnetic flux density. The surface S was chosen to be theboundary of a hollow cylinder sector enclosing the modeled stator sector. All

31

parts of the boundary except the cylindrical surface close to the stator wall inthe air gap coincide with boundaries of the FE model. According to (3.11)the axial forces on the two boundary planes at constant ϕ are equally largebut with opposite signs because of the periodicity and the opposing normaldirections. The axial force on the outer radial surface is zero because of thetangential field boundary condition. The axial forces on the boundary endplanes at constant z are pointing towards the machine. The total axial forceis the sum of the contributions from the inner radial surface and the two endplanes at constant z.

3.3 Calculation of Losses in the Steel of the Stator Coreand Clamping Structure in a Synchronous Generator

An overview of iron loss models can be found in [33]. Single valued mag-netization data curves and tabulated core loss data from a steel manufacturerhave been used in the simulations for calculation of core loss after the mag-netic field has been calculated because, until recently, it has not been possibleto make use of hysteresis data in the field calculation ANSYS Maxwell. Onemodel of harmonically time varying iron power loss density in a silicon steellaminate is the loss separation modell [33],

p = ch f B2+ ce f 2B2+ cex f 1.5B1.5 (3.12)

where f is the frequency and B is the peak magnetic flux density. From left toright the terms are hysteresis loss density, ph, eddy current loss density, pe, andexcess loss density, pex. Excess loss density is a term that was added in order toexplain the difference between measured and computationally predicted lossesin silicon steel [33]. At a given frequency the model can be expressed as

p = aB2+bB1.5. (3.13)

The coefficients a and b can be determined by fitting the model to measuredloss density. A comparison between (3.12) and (3.13) shows that the first termin (3.13) is the sum of hysteresis loss density and eddy current loss densitywhereas the second term in (3.13) must be excess loss density. The eddycurrent loss density caused by flux parallel to the lamination planes can beapproximated by

pe =σ6(π f hB)2 (3.14)

where σ is the conductivity and h is the plate thickness [34]. The hysteresisloss density can then be estimated by aB2− pe.Fitting p = aB2+ bB1.5 to data for steel M350-50A from Surahammars BrukAB [35] or Cogent via Ernest Matagne [36] gives a negative bwith a relativelysmall magnitude that gives a hardly noticeable difference to the fit. In addition,

32

a positive b is expected [33]. Hence, the available loss data does not seem tosupport the distinction of excess loss from the other losses. Furthermore, theuncertainty is large in given loss data. At 1.5 T and 60 Hz the loss density canbe 30 % higher than the typical value for steel M350-50A [36]. Therefore thesimplified model, p(B) = aB2, have been used in Paper II. Fig. 3.2 shows fitsto two sets of data, one from [35] and one from [36]. The fit to the data from[36] fits the loss model better but the larger loss density from [35] was usedin Paper II. First the core loss density given in units W/kg was converted toW/m3 by

p= pmρlaminate = pmρsteeltsteel +ρvarnishtvarnish

tsteel + tvarnish≈ pm

ρsteeltsteel

tsteel + tvarnish= pmkρsteel

(3.15)where pm is the given core loss density, and k = 0.500/0.543 is the assumedstacking factor of the stator laminate in Paper II. The coefficient, a, was deter-mined as the solution of a least squares problem according to

∂ ∑ni=1(aB2i − pi)

2

∂a= 0⇒ a =

∑ni=1 B2i pi

∑ni=1 B4i

(3.16)

where n is the number of data points, and pi is the core loss density at peakmagnetic flux density Bi. The data from [35] give a ≈ 10825 W/(T2 m3), andthe data from [36] give a ≈ 9685 W/(T2 m3).

0 0.5 1 1.5Flux density (T)

0

10

20

30

40

Tota

l los

s de

nsity

(kW

/m3 )

PCogent

PC.fitPSura

PS.fit

Figure 3.2. The core loss density of a laminate of steel M350-50A manufactured bySurahammars Bruk AB [35], a subsidiary of Cogent Power.

The core loss in a volume V is approximately

PC =∫

VaB2dV. (3.17)

This in turn can be approximated by

PC = max∫

VaB(t)2dV (3.18)

33

where B(t) is the instantaneous value of B at time t. In Paper II the core losswas estimated according to (3.18). This underestimates the loss since B isnot reached simultaneously everywhere within a volume that is not infinitelysmall. However, the slot pitch wide slices of the teeth and yoke are relativelysmall and such that the volume integral of the square of the instantaneous fluxdensity in each slice oscillates between a minimum and a much (, five to ninetimes,) larger maximum. This somewhat justifies the approximation.Since the core loss density has been measured with an Epstein frame, the

measured loss include very little eddy current losses from magnetic leakagefields. These eddy current losses have been calculated both in the laminatedstator core and in selected slices of the clamping structure by

pE =∫

VJJJ · JJJ/σdV (3.19)

where J is the eddy current density.

3.4 The Finite Element MethodThe finite element method (FEM) is a numerical method for solving (partial)differential equations with boundary conditions. The strength of the methodis that the domain, i.e. the region in which the solution is calculated, andits contents can have complicated shapes. Some solvers allow nonlinear andanisotropic materials. In common for FE methods is that the domain is dividedinto subdomains, so called finite elements. Each element has boundary points,so called nodes, in which the element can be connected to other elements.Mathematically, a connection means that where elements share a node or anedge, they share the value of the degree of freedom in that location. Valuesbetween nodes and/or edges are approximated by interpolation. On a funda-mental level, the idea of the FEM is that even if the field changes much acrossthe domain, the field change across a subdomain goes to zero if the size ofthe subdomain goes to zero. The solution um in element number m is approx-imated by a linear combination of simple functions called base functions andcan be expressed as

u∗m(rrr, t) =N

∑j=1

um, jsm, j(rrr, t) (3.20)

where, sm, j is a base function, one per degree of freedom. Each base functionis zero everywhere except locally around a node or edge of a finite element.Where a base function is nonzero, it is usually a linear, quadratic or cubicpolynomial. The coefficients um, j are to be determined by the FEM so thatu∗m approximates u in the space filled by element m. The coefficients of thepolynomials are uniquely determined by the solution of an equation systemconsidering the boundary conditions on the outermost elements, continuity

34

conditions at the borders between the elements, and the partial differentialequation for each element.The FEM solution is approximative but can converge to the exact solution

when the element sizes are decreased. There are different formulations ofFEM. Galerkin’s and Rayleigh-Ritz’s methods are classical ways to obtain anapproximate solution to a differential equation with boundary conditions. Inclassical FEM, Galerkin’s or Rayleigh-Ritz’s method is used for each element.Steps in FEM with Galerkin’s method are given in appendix.In the commercial FEM software ANSYS Maxwell 2-D, the field param-

eter the solver calculates is the magnetic vector potential, AAA, in all magneticproblems. After AAA has been determined, the magnetic flux density can be cal-culated from BBB = ∇×AAA. In 3-D magnetic problems a current vector potentialis used in current conducting regions, and a scalar potential of the magneticfield strength, HHH, is used in the whole domain [37].

3.4.1 Domain and Boundary Conditions in Magnetic Analysis ofan Electric Machine with ANSYS Maxwell

The magnetic field is important both inside the machine materials and the sur-rounding medium, for simplicity assumed to be air. Therefore a FE calculationdomain contains at least a part of the machine and air. For simulation of cylin-drical rotational motion the same mesh is used during the whole simulation.The rotor and some air outside it is enclosed in a faceted cylinder whose meshis fixed relative to the rotor. Outside the faceted cylinder the mesh is fixedrelative to the stator. A synchronous electric machine has a periodic structure.If the excitation or load on the machine is the same on each spatial period ofthe machine, it is sufficient to include just one spatial period of the machineand air in the calculation domain. Periodic (matching) boundary conditionscan then be applied to the two surfaces, called master and slave, that are onespatial period from each other. Fig. 4.1 in section 4.1 shows an example wherethe spatial period spans four pole pitches. The domain can be reduced furtherto contain only half a spatial period if antiperiodic boundary conditions canbe used. This is usually the case in 2-D FE simulations as long as the numberof stator slots per pole and phase is an integer. Antiperiodic conditions forcethe magnetic field vector in an arbitrary point on the slave surface to have thesame magnitude and opposite direction as the corresponding (mirror) point onthe master surface located half a spatial period from the slave surface. Thisrequires that the geometry of the slave surface must match the geometry of themaster surface exactly. In 3-D the coil ends also must be considered. In somemachine types they are such that at least one spatial period of the machinemust be modeled. In 3-D the domain can be halved if the machine has a geo-metrical and magnetic symmetry plane along which the motion takes place.Boundary conditions must be used on the boundary of the calculation domain.

35

Some boundary conditions, like vector potential in 2-D, can also be used onobjects within the domain. Boundary conditions available in transient mag-netic analysis in ANSYS Maxwell 2D are vector potential, balloon, odd sym-metry, even symmetry, master and slave. The balloon condition is a way tosimulate that the flux lines can extend to infinity. Even symmetry gives nor-mal flux, odd symmetry gives tangential flux. In a 2-D problem type withcurrents only in the z direction, the vector potential ,AAA, only has a z compo-nent, Az. This can be set to any constant on the boundary in order to forcetangential flux there. Setting Az = constant on the boundary forces ∇Az to beperpendicular to the boundary. This together with BBB = ∇×AAA and

∇Az ·∇× zzzAz = 0 (3.21)

implies that BBB is parallel to the boundary. If the problem is symmetric aboutthe z axis and the currents flow only in the ϕ direction, the vector potential hasonly the component, Aϕ . In this case

∇Aϕ ·∇× ϕϕϕAϕ =Aϕ

ρ∂Aϕ

∂ρ(3.22)

which is not zero in general. The only constant value Aϕ can have on thewhole boundary that forces tangential flux is zero.

In many 3-D cases the only boundary conditions the ANSYS Maxwell userneed to apply in transient magnetic and harmonic eddy current FE analysis(FEA) of rotating electric machines are the periodic or antiperiodic condi-tions. Tangential HHH conditions in 2-D are expressed as Dirichlet conditions onthe magnetic vector potential. Tangential HHH conditions in 3-D are automati-cally fulfilled unless some other boundary condition is applied. The tangentialHHH condition can be expressed as a homogeneous Neumann condition on thescalar potential Ω, i.e.

HHH · nnn = ∇Ω · nnn = ∂Ω/∂n = 0 (3.23)

where nnn is a unit normal vector to the boundary. The Maxwell 3D user canapply current insulating boundary conditions in order to avoid modeling andmeshing thin gaps between conductors. Balloon boundary conditions are notavailable in 3-D.

3.5 The Method of Separation of VariablesThe method of separation of variables is also called Fourier’s method. It is amethod for solving boundary value problems that are linear partial differentialequations with boundary conditions. The solution is a field, i.e. a function ofspace position rrr = (x,y,z) and, in some cases, time, t. The method is supported

36

by the fact that a square integrable function can be expressed as a Fourier sumthat, when the number of terms goes to infinity, converges pointwise towardsthe function almost everywhere. [38], [39], [40]. Most of the following textabout Fourier’s method is based on information in [41].

In order for a boundary value problem to have a unique solution, the prob-lem must be specified in a way that depends on the type of equation. Par-tial differential equations are classified as hyperbolic as the wave equation,parabolic as the diffusion equation or elliptic as Helmholtz equation. Foruniqueness of the solution in a closed domain, these types of equations needdifferent initial conditions (in the whole domain) but the geometrical bound-ary conditions are of the same types. They are Dirichlet, Neumann or Robin(Churchill) conditions. A Dirichlet condition is a specification of the fieldvalue, a Neumann condition is a specification of the normal derivative of thefield, and a Robin condition is a specification of a linear combination of thefield value and the normal derivative. Different types of boundary conditionscan be applied to different parts of the boundary. For Laplace’s equation, so-lutions can differ by a constant if only Neumann conditions are used.

Fourier’s method is used in geometries for which there are coordinate sys-tems such that, on every part of the the boundary, one and only one coordinateis constant. Each term in a Fourier series can be expressed as the productof functions, each of which depend on only one variable. Of the functionsin the product, one can be considered as a Fourier coefficient, and the othersare eigenfunctions. A Fourier series is thus a linear combination of compos-ite or single variable eigenfunctions. This is analogous to expressing a vectoras a linear combination of base vectors. The eigenfunctions depend on thedifferential equation, the type of boundary conditions and the geometry. Theeigenfunctions used in each Fourier series are a subset of a complete set ofeigenfunctions. That means that it is possible to approximate any square in-tegrable function arbitrarily well in norm with a finite linear combination ofeigenfunctions from the complete set in the domain where the function is de-fined. Eigenfunctions used in Fourier series are pairwise orthogonal. Becauseof the orthogonality, each Fourier coefficient depends only on the eigenfunc-tions in the same term in the Fourier series. Therefore the relative magnitudesof the Fourier coefficients can be interpreted as measures of the relative im-portance of modes of which the Fourier series is composed. Examples offunctions that can be parts of complete, orthogonal eigenfunction systems arethe sine, cosine, Bessel, spherical Bessel and spherical harmonic functions.

An eigenfunction, X , is in general, if not always, chosen to satisfy bound-ary conditions such that X = 0 or ∂X

∂n = 0 or∂X∂n +αX = 0. Such conditions are

called homogeneous. They are special because they are automatically satisfiedby functions expressed as linear combinations of the eigenfunctions. Fourier’s

37

method gives the solution as one Fourier series for time independent problemswith periodic and/or homogeneous boundary conditions on all parts of theboundary except at most two where one and the same coordinate is constant.A boundary condition function at a constant coordinate is expressed as Fourierseries of eigenfunctions that depend on another coordinate and the types ofboundary conditions on boundaries where another coordinate is constant. Ifthe problem has nonperiodic inhomogeneous boundary conditions on two ormore parts of the boundary, the field can be written as a sum of Fourier seriessuch that each series satisfies the boundary condition requirements mentionedabove for one series. This is possible because of the linearity of the equations.If all geometrical boundary conditions are periodic and/or homogeneous, thesolution of a time dependent problem can be expressed as one Fourier series. Ifa time dependent problem has at least one nonperiodic inhomogeneous bound-ary condition, the field can, in the first step, be expressed as the sum of twocontributions. The first contribution satisfies all geometrical boundary condi-tions and e.g. Laplace’s equation. The second contribution satisfies the timedependent problem with periodic and/or homogeneous geometrical boundaryconditions. The field that satisfies Laplace’s equation can in turn be writtenas a sum of contributions such that each contribution satisfies the boundaryrequirements for one Fourier series.A truncated Fourier series of a function, f , has the same values as the eigen-

functions have on the boundaries regardless of the value of f on the boundary.If f does not agree with the eigenfunctions on a boundary, the Fourier sumdoes not converge towards f on the boundary no matter how many terms areincluded in the series, even if f is continuous [41]. It may be considered as anexample of Gibbs phenomenon [41] although it is perhaps mostly associatedwith a discontinuity of f [42]. In the author’s numerical examples the maxi-mum overshoot of a Fourier sum appears to converge towards max(2Si(x)π )−1≈ 17.9 % where Si(x) = ∫ x

0sinu

u du.

Of special interest for the work in Paper 3 in this thesis is the time har-monic complex form of the wave equation, also called reduced wave equationsince the time dependence has been removed. The equation is a special caseof the homogeneous Helmholtz equation, ∇2 f +κ f = 0. For a non-negativeand non-zero κ one can distinguish between two cases of this equation. In thefirst case κ is an eigenvalue which is a geometrical parameter that can havean infinite number of values corresponding to an infinite number of eigen-functions. In the other case κ is a complex number that depends only on thefrequency and the material properties. In this case the solution is unique fora fixed material and frequency [43]. The uniqueness is not surprising consid-ering the uniqueness of the solution of the time dependent wave equation andthe equivalence of the time dependent and the reduced wave equations for har-monically time varying fields. The removal of the time dependence from the

38

time dependent wave equation is possible under two conditions. First, the ma-terial properties must be constant so that the time dependent wave equation islinear, and second, the time dependence of the fields must be harmonic whichis possible when the equation is linear. When the equation is linear, a linearcombination of solutions, corresponding to e.g. different initial conditions, isalso a solution to the equation alone. In particular, if one solution is u(rrr, t) =u(rrr)cos(ωt −φ(rrr)), another solution is v(rrr, t) = v(rrr)sin(ωt −φ(rrr)). A com-plex linear combination of these solutions is w(rrr, t) = w(rrr)e j(ωt−φ(rrr)). Afterthis solution has been inserted into the time dependent Maxwell’s equationsor the wave equation, and the time differentiations have been performed, onecan notice that each term in the equations are proportional to the factor e jωt .Hence the factor can be removed after the time differentiation. The result isthe time independent equations in complex form. Suitable for the complexwave equation are boundary conditions in complex form [44]. It can be notedthat in the special case of harmonic time variation with given frequency, initialconditions on both the field and its first time derivative would overdeterminethe field.

3.6 Estimation of Frictional Power Loss in a SliderBearing

Tilting pad thrust bearings are common in hydropower generators because oftheir ability to carry high load at low friction [45]. A tilting pad bearing isa kind of slider bearing. In Paper II a the frictional power loss in a tiltingpad thrust bearing was estimated with one dimensional theory of combinedPoiseuille and Couette flow [46]. The flow is the result of a horizontal planesurface moving with speedU in the x direction and a plane plate inclined andat rest relative to the horizontal plane. Fig. 3.3 shows the wedge into whichthe lubricating fluid is assumed to be dragged by the viscous frictional force.A no slip condition is assumed between the fluid and the plane solid bodies.The slope, (hi − ho)/L, is of the order of 10−4 in a tilting pad bearing [46].The normal force is

Fn =ηUL2W

h2ofn

(hi

ho

)(3.24)

where η is the viscosity,W is the width of the solid bodies in the y directionperpendicular to the flow. The frictional force is

Ff =ηUL2W

hof f

(hi

ho

). (3.25)

The functions fn and f f are given in detail in [46]but with other names. ForPaper II it was sufficient to note that for a given ratio hi/ho, ho from (3.24inserted into (3.25) gives that Ff ∝

√Fn. Since also the frictional power loss,

39

Figure 3.3. Lubricant dragged to the right through the wedge creates a lifting forceand frictional force on the inclined surface.

Pf , is proportional to Ff , the conclusion is that Pf = k√

Fn where k dependson hi/ho, η , U , L,W and some, possibly weighted, average radius, rav of thebearing such that Pf = ravFf .

3.7 Estimation of Windage Power LossIn the Discussion of Paper II, (3.26) was the basis for the statement that thewindage power loss can be reduced by an increased air gap. The air gap getsvery large for the part of the rotor that protrudes from the stator bore after axialdisplacement from a centered position if the rotor is not longer than the stator.Laminar windage loss for a smooth cylinder rotating in the center of a smoothcylindrical bore can be estimated by [47]

Pw =2πρr4ω3L

Re, Re =

ρωrδη

(3.26)

where L is the rotor length, r is the rotor radius, ω is the rotor angular velocity,ρ is the density of the fluid, Re is the Reynolds number, δ is the radial gapbetween the cylinders, and η is the viscosity. For turbulent flow (3.26) ismodified by replacement of 2/Re by the skin friction coefficient, Cd , that canbe determined iteratively from

1√Cd

= 2.04+1.768ln(Re√

Cd) (3.27)

Cd decreases when Re increases. The formulas for the smooth cylinders heav-ily underestimate the windage losses in real machines. Therefore windage lossin rotating electric machines is usually determined from experiments com-bined with computational fluid dynamics (CFD) simulations [48].

40

3.8 Experimental Equipment and MeasurementsInformation about the experimental generator is given in Paper I, [49]. Anasynchronous motor rated 75 kW and controlled by a frequency converter sup-plied the generator with mechanical power. Stator phase voltages, stator phasecurrents, field current, and axial flux densities were measured with sensorswith voltage outputs. The output signals were collected with LabVIEW anddata acquisition equipment from National Instruments [50]. LabVIEW wasalso used for calculation of electrical parameters such as phase angles, instan-taneous active and reactive power. That information was used during the testsfor setting the points of operation and for recording the data when the genera-tor was reasonably stable. A problem was load angle oscillations at high fieldcurrent when the generator was connected to the power grid. All sensor out-puts were voltages. The signals were recorded with a sampling frequency of 5kHz during 2-6 s for each point of operation. Shielded cables were used for theoutput voltages from the sensors. The shielded sensor cables were connectedto interface boxes made in the laboratory. Each such box was connected to aNI 9205 module. The field current was set by a laboratory DC power supplyof model EA-PS 8160-170 10000W. The torque was controlled via a NI 9265module connected to the frequency converter and powered by a A 12 V, 7.2Ah Lead-Acid battery.

3.8.1 Magnetic Field Sensors and Measurements of AxialMagnetic Flux Density

The axial magnetic flux density on the stator clamping structure was measuredby three each of the linear output Hall effect integrated circuit chip transducersof types Honeywell SS94A2 and SS94A2D connected in parallel for a com-mon supply voltage, Vs. Each chip has three pins, +, - and output. Lasertrimmed thin film and thick film resistors compensate for temperature varia-tions and minimize sensitivity variations from one device to the next. Mea-sured sensor input resistance including cables, contacts and solders was 994-1052Ω. A 68Ω resistor and a 11.1 V Lithium polymer battery was connectedin series with the sensors to supply the sensors with about 8 V. The outputvoltage, Vo, of the sensors is a linear function of the magnetic flux density ina Vo span proportional to Vs. The linearity error of a sensor curve of Vo as afunction of flux density is defined as the maximum Vo deviation between thesensor curve and a line that passes through the endpoints of the sensor curve.The output voltage at zero magnetic field is called null or quiescent outputvoltage, Vo,0 ≈ Vs/2. The axial magnetic flux density, Bz, for each sensor wascalculated by

Bz(t) = (Vo −Vo,0)/s, s =ΔVo

ΔB=0.625Vs

ΔB(3.28)

41

where s is the sensitivity, ΔVo is the span and ΔB is the range according tothe manufacturer’s data sheet. See Table 3.1. Under the assumption that theaxial flux density varied sinusoidally with time, the null was approximated bythe time average Vo,av of Vo for the work in Paper I. To approximate Vo,0 withVo,av can be a significant source of error considering that the pole shoes havedifferent axial lengths, almost an air gap from the shortest with 142 plates tothe longest with 146 plates of thickness 2 mm. More important than the dif-ferences between pole lengths is the differences between the axial coordinate(z) levels of the lower pole ends where the sensors were mounted. The differ-ence between the axial coordinate levels was found to be up to six mm. Onthe other hand, asymmetry of the rotor was not included in FEA. Therefore, itis not obvious that the approximation of Vo,0 would increase the disagreementbetween measured and simulated Bz. The pole shoe lengths were measuredafter the first submission of Paper I and were not accounted for in Paper I..

Table 3.1. Honeywell Hall effect integrated circuit chip sensors

Parameter SS94A2 SS94A2DSupply voltage, Vs (V) 6.6 to 12.6 6.6 to 12.6Supply current (mA) 13 to 30 13 to 30Max output current (mA) 1 1Response time (μs) 3 3Span of Vo 0.625 Vs 0.625 Vs

Flux density range (mT) -50 to 50 -250 to 250Sensitivity (V/T) 50 10Linearity error (% of span) -1.5 to 0, typ. -0.8 -1.5 to 0, typ. -0.8Vo (V) at 0 T, Vs = 8 V DC, 25◦C 4.00±0.04 4.00±0.04Temperature error (% of span/◦C) of null ±0.02 ±0.007Temperature error (% of span/◦C) of gain ±0.02 ±0.02Temperature range (◦C) -40 to 125 -40 to 125

The Honeywell sensors were calibrated after the main measurements, butthe calibration data was used for the work in Paper II. The calibration wouldnot have been conducted if the similarity between measurements and FEA hadbeen good in Fig. 3 in Paper II [51]. When individual sensitivities and nullswere used for each sensor, there was improvement in the similarities betweenmeasurements and FEA with respect to the slopes of the curves in Fig. 3 inPaper II but hardly any improvement in the absolute values of the differencesbetween measured and simulated axial flux densities. Hence, it seems likethe main cause of the differences is not the asymmetry of the rotor poles.Calibration showed that the sensors were almost identical, as expected fromthe specifications from the manufacturer. Fig. 3.4 for one SS94A2 sensor andFig. 3.5 for one SS94A2D sensor show Vo versus flux density measured by a

42

teslameter. Fig. 3.6 shows the null versusVs for the calibrated sensors. Fig. 3.7and Fig. 3.8 show the sensitivities of the SS94A2 sensors and the SS94A2Dsensors respectively.

-10 -5 0 5 10Magnetic Flux Density (T)

3

3.5

4

4.5

5

Out

put v

olta

ge (V

)

Vr1 7.0VVr1 7.5VVr1 8.0VVr1 8.5V

Figure 3.4. Sensor output voltage as a function of magnetic flux density measured bya teslameter at four different values of supply voltage for a sensor of type SS94A2.

Equipment for the calibration was 1) Lake Shore Model 410 Teslameterwith a transverse probe, accuracy ±2 % from 0 to 2 T, 2) LabVIEW for read-ing sensor signals and supply voltage, 3) One 12 V, 7.2 Ah Lead-acid batteryto supply the sensors, 4) One Vishay Spectrol Model 534 10 turns, 5 kΩ po-tentiometer for adjustment of supply voltage, 5) Two brick shaped permanentmagnets, 6) Two L shaped pieces of magnetic steel, 7) Blocks of magnetic andnon magnetic materials for adjustment of air gaps in the magnetic circuit. Theexperimental setup is shown in Fig. 3.9. For each air gap size in the test, themagnetic flux density was measured and noted on a piece of paper along a linein the middle of the large, upper air gap. Then the sensor blocks were testedone by one at the spots of the measured field.Shielded cables were used but they were split up at the sensor connection

pins and a circuit board used for connection of the supply voltage. The supplyvoltage cables were intertwined but not shielded. Intertwining of the voltagesupply cables reduced unwanted induced voltages with a torque independentperiod of about 0.73 s in the supply voltage. One hypothesis is that the noisewas induced by leakage flux from the asynchronous machine that could workwith a constant slip frequency close to 1/0.73 s at constant rotor angular speed.A magnetic field disturbance of 1 Hz has been caused by the asynchronous

43

-50 0 50Magnetic Flux Density (T)

2.5

3

3.5

4

4.5

5O

utpu

t vol

tage

(V)

Vt1 7.0VVt1 7.5VVt1 8.0VVt1 8.5V

Figure 3.5. Sensor output voltage as a function of magnetic flux density measured bya teslameter at four different values of supply voltage for a sensor of type SS94A2D.

7 7.5 8 8.5Supply voltage (V)

3.4

3.6

3.8

4

4.2

4.4

Qui

esce

nt o

utpu

t vol

tage

(V/T

)

Vq r1Vq r2Vq r3Vq t1Vq t2Vq t3

Figure 3.6. Quiescent output voltage as a function of supply voltage for all calibratedsensors.

motor in a compressor [52]. However, the 0.73 s period noise amplitude inthe supply voltage was only about 1 per mille and negligible for the sensor

44

7 7.5 8 8.5Supply voltage (V)

44

46

48

50

52

54

56Se

nsiti

vity

(V/T

)

sr1sr2sr3

Figure 3.7. Sensitivity as a function of supply voltage for the calibrated SS94A2sensors.

7 7.5 8 8.5Supply voltage (V)

8.5

9

9.5

10

10.5

11

Sens

itivi

ty (V

/T)

st1st2st3

Figure 3.8. Sensitivity as a function of supply voltage for the calibrated SS94A2Dsensors.

output. Fig. 3.10 shows the circuit board and the batteries that supplied firstset of cheap (not Honeywell) Hall sensors that failed already during prelimi-

45

Figure 3.9. Experimental setup for calibration of Hall effect integrated circuit chips.The sensors were put, one by one, at spots along the black mid line on the aluminiumblock. For the lowest fields, the aluminium pipe in the background was used on top ofthe aluminium block.

nary measurements. For flexibility and protection, the Honeywell sensor chipswere encapsulated in epoxy blocks. Wood blocks with a socket just big enoughfor a sensor block were glued to the lower clamping ring for stable and cen-tered location of the sensor blocks on the clamping ring. Sensor blocks on theclamping fingers were fixed by wedges. The fixtures of four of the sensors onthe lower clamping structure of the generator are shown in Fig. 3.11.The Hall effect sensors were mounted on the lower end of the generator

such that they were exposed to magnetic flux densities with positive z compo-nent when a rotor north pole was right in front of them and the stator currentwas zero, i.e. the generator was at no load. The z axis points upwards along theshaft. During all measurements the rotor was rotating anticlockwise as seenfrom above. This corresponds to the phase sequence a (blue), b (yellow) andc (red). The terminal voltage at no load in phase winding c was lagging 90◦behind the flux density measured by the transducers that were mounted in ver-

46

Figure 3.10. Circuit board for connection of Hall effect integrated circuit chips to asupply voltage.

tical planes spanned by the shaft and the magnetic axes of phase c. Fig. 3.12shows the locations of the sensors for measurement of axial flux density. In thesensor names the letter r refers to the clamping ring, and t refers to clampingfinger tip or tooth.

47

Figure 3.11. Slotted wood blocks and wedges fixed the sensor blocks to the clampingstructure of the generator.

Figure 3.12. Positions of the Hall sensors for measurement of axial magnetic fluxdensity. Sensor positions r1, r2, t1 and t2 were also used in simulations.

48

4. Results that Were Not Included in thePapers

4.1 Supplements to Paper IIFor Paper II [51], some figures intended for or suitable for the topics had tobe skipped because of limitations on the number of figures the journal couldallow. Here, some of those figures have been included. The FEA model usedfor calculation of Bz at load and no-load is shown in Fig. 4.1

Figure 4.1. FEA model of a generator. The air is not shown. The model was used forcalculation of Bz on the clamping structure.

Although the chosen constant permeability in the clamping structure madesimulated and measured RMS Bz agree well at no load and I f 12 A in figure 2in Paper II, the agreement is not as good at load. Hysteresis could have some-thing to do with that. Alternatively, the insulation between the stator lamina-tion plates is not very good. Since hysteresis and macroscopic eddy currentscause damping and phase change of the electromagnetic field components, asmall simulation study of the effect of macroscopic eddy currents on ampli-tude and phase was conducted at no load with the model of Fig. 1 in Paper II.With a conductivity of 8 kS/m in the axial direction in the stator laminations,conductivity 1.5 MS/m in the clamping structure, relative permeability 38 inthe fingers and only 3-5 in the ring it was possible to get good agreement be-tween simulations and measurements as shown in Fig. 4.2, but the used low

49

permeability in the ring can be put in question since a permanent magnet isattracted to it roughly about as strongly as to the outer wall of the laminatedstator core. Furthermore, the agreement was only checked at no load and I f 12A. Without eddy currents in the simulations, Bz would have its extreme valuesapproximately on the d axis. The inclusion of eddy currents makes Bz lag thed axis. A higher axial conductivity gives more damping, a steeper slope of thetop of the Br curve and a larger simulated delay (lag) of Bz on the ring thanin measurements. In light of this study some loss phenomenon seems to bea plausible explanation for the fact that Bz on the ring was lagging the statorcurrent I at load with low load angles in measurements described in Paper I,[49]. By simple vector addition one could otherwise expect the phase of Bz onthe clamping structure to be between the phases of I f and I in every load case.

0 5 10 15 20−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time (ms)

Flux

Den

sity

(T) Br s

Bz t1 s10⋅Bz r1 sBr mBz t1 m10⋅Bz r1 m

Figure 4.2. Calculated and measured radial flux density Br in the air gap in front ofslice c, axial flux density Bzr1 on Ring and Bzt1 on the tip of F1i at no load with I f 12 Aand the stator raised 7 mm. Last subscript meaning: s = simulation, m = measurement.

Concerning the calculation of optimal axial rotor displacement, Fig. 4.3shows a simplified picture of the axial forces in an electric machine. The mag-netic force versus rotor displacement in Fig. 4.4 is just a mirrored version ofFig. 4 in Paper II. For accurate determination of the optimal rotor displace-ment, cubic polynomial fits were made to the magnetic force, Fm, and thestator iron loss within a small z interval where the optimal displacement couldbe expected. Fig. 4.5 shows the frictional loss, the stator iron loss and theirsum when the frictional loss is assumed to be 0.3% of the rated power. Theoptimal displacement, -1.83 mm in the example, is the point where the slope

50

of the total loss is zero. The polynomial fit of the magnetic force was used inthe definition of the frictional loss fit.

Figure 4.3. Simplified picture for the definition of the axial forces and the axial coor-dinate.

4.2 Supplements to Paper VFig. 4.6, 4.7 and 4.8 show the instantaneous eddy current loss density ex-pressed in W/kg steel at steady state with power factor 1, overexcitation withβ = βmax = 16◦, and overexcitation with β = βmax = 16◦ respectively . Con-sidering the low assumed effective relative permeability of the stator core inthe z direction, the stacking factor is about 0.9 = Vsteel/Vlam. The loss in thelaminate is assumed to be only in the steel, not the varnish. The density of

51

-20 -15 -10 -5 0 5 10 15 20Axial Displacement of Rotor (mm)

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2Ax

ial F

orce

(kN

)FzFz Fit

Figure 4.4. Axial magnetic force versus axial rotor displacement.

electrical steel is assumed to be 7650 kg/m3. The loss is

P = pW/m3,lamVlam = pW/kg,steelρsteelVsteel (4.1)

This implies that the loss density in the laminate expressed in W/kg steel is

pW/kg,steel =pW/m3,lamVlam

ρsteelVsteel=

pW/m3,lam

6885kg/m3(4.2)

For the solid steel clamping structure, the density has been assumed to be 7860kg/m3 which gives the loss density pW/kg,clamp = pW/m3,clamp/7860 kg/m

3.

52

-20 -10 0 10 20Axial Displacement of Rotor (mm)

100

200

300

400

500

600

700

800

900

Loss

(W)

PfPf fit

PI

PI fitPI+f

Figure 4.5. Thrust bearing frictional loss, Pf , stator iron loss, PI , and the sum of thelosses versus axial rotor displacement. The frictional loss is assumed to be 0.3% ofthe rated power.

53

Figure 4.6. Eddy current loss density in the stator steel at a time point at steady statewith power factor 1.

Figure 4.7. Eddy current loss density in the stator steel at a time point when β = βmax= 16◦ at overexcitation.

54

Figure 4.8. Eddy current loss density in the stator steel at a time point when β = βmax= 16◦ at underexcitation.

55

5. Summary of Papers

Five papers have been written. What they have in common is magnetic leak-age fields and/or the eddy current power losses they cause in (salient) polesynchronous generators. Regarding laminates of electric steel, the magneticfield component in the stacking direction can be considered as leakage field.Each paper can contribute to make it easier to predict the magnetic (leakage)fields or the eddy current power losses they cause, primarily in the end regionsof the machines. Previous publications on end region losses were focused onturbogenerators. Paper I, II and V show that axial leakage fields can increasesignificantly also in the end regions of a salient pole generator at underex-citation. The results of paper V could give electrical power companies andmachine operators stronger reasons to avoid underexcitation and to use andimprove existing power stabilizing systems in order to reduce load angle os-cillations.

Paper III and IV contribute analytically to the understanding of the distri-bution of electromagnetic fields, including leakage fields, in laminated blocks.This is a step towards improved understanding the distribution of electromag-netic fields in the laminated cores of electric machines. The analytical expres-sions for the laminate blocks can indirectly be of use for the design of electricmachines via comparisons between laminated core materials that will be ex-posed to traveling wave leakage fields. Previous analytical works neglectedmagnetic fields perpendicular to the stacking direction.

5.1 Paper IPreviously published papers show that the points of operation are importantfor the leakage fields in turbogenerators. Little or nothing has previously beenpublished about the importance of the points of operation for leakage flux insalient pole synchronous machines. It is important to be able to predict themagnetic leakage fields at the ends of conventional synchronous machinessince the leakage fields cause losses. A linear steady state phasor model forthe axial magnetic flux density in the end regions of non-salient synchronousgenerators has previously been verified experimentally. This paper describesan extension of the model to salient pole synchronous generators and a method

56

for calculating the coefficients in the model. Measurements and 3D finite ele-ment simulations justify a distinction between axial flux density contributionsfrom the stator current component in phase with the direct pole axis and thestator current component in phase with the quadrature (inter pole) axis. Howthe coefficients and the axial magnetic flux density on the clamping structureat the ends of a small synchronous generator change with steady state opera-tion conditions is here shown with measurements and to some extent with 3Dfinite element simulations. The axial magnetic flux density at constant loadangle decreases when the field current increases, at least as long as the fieldcurrent is less than what is required to give a nominal no load voltage.

The main research contribution of Paper I is perhaps the idea of extensionof the phasor model from non-salient to salient pole synchronous generatorssince the model applicable for salient pole generators is more general and alsoapplicable for non-salient pole generators.

5.2 Paper IIPreviously, the effect of rotor length on axial leakage flux in the end regionsof turbogenerators has been simulated but not measured. The effect of axialdisplacement of the rotor in the stator bore could be expected to be similar tothe effect of a change of rotor length, but nothing previously published hasbeen found about the effect of axial displacement on end region losses. Thispaper contains results from two studies with an axially displaced stator of asmall synchronous generator with a vertical shaft.

In the first study, axial magnetic leakage fields in the ends of a small syn-chronous generator at load and no load were measured and simulated withfinite element analyses (FEA). The axial displacements were 0 mm, 7 mm upand 8 mm down. The upward and downward displacements are slightly lessthan an air gap. At load, measurements and FEA were conducted at three lev-els each of power, excitation and axial displacement. At no load, downwarddisplacement of a magnetized rotor in a synchronous machine with a verticalshaft exposes the lower stator end to higher magnetic flux density. At load,the displacement can locally lead to either increased or decreased axial fluxdensity depending on the point of operation. Decreased axial flux density canoccur locally at overexcitation. The leakage flux increased with the power andincreased when the excitation was decreased.

In the second study, axial force and iron losses at no load were calculatedwith nonlinear materials and a three dimensional time stepped finite elementmethod. Although the leakage fields can increase the magnetic saturation andtherefore also the hysteresis power losses in the end regions, the FEA results

57

based on the simple loss models in section 3.3 strongly indicate that the eddycurrent power loss density is much more sensitive to the leakage fields thanhysteresis power loss density is. At least at no load with a saturated, rotatingrotor core, the total iron losses in a synchronous generator increase with axialdisplacement of the rotor from the centered position in a stator bore. However,for some machines with a vertical shaft, the increased iron loss can be morethan compensated by decreased friction loss in the thrust bearing. If the fric-tion in the thrust bearing and the axial magnetic stiffness are not too low, thereis an axial rotor position at which the sum of all important losses at normaloperation is minimized. That the sum of the major losses can decrease by adownward displacement for a symmetrical machine is based on two observa-tions. The first is that the magnetic force between the stator and the rotor isattractive. That implies that the magnetic force can partly unload the thrustbearing if the rotor is displaced downward. The second is that the major typesof losses except the thrust bearing losses have local extreme values at zerodisplacement for symmetry reason. There can be an optimal downward dis-placement also for not very symmetrical machines since, at a sufficiently largedisplacement, the conductors are not able to carry higher currents in order tomaintain the required power of the machine. Paper II gives a method for esti-mation of an optimal downward rotor displacement in a symmetrical machinewith a vertical shaft.

5.3 Paper IIIPreviously published analytical expressions for electromagnetic waves in lam-inated cores are simple but neglect end effects and are based on more simpli-fying assumptions. The method of separation of variables has been used here.The first incentive for derivation of analytical expressions was the questionabout how the laminations affected the penetration depths in different direc-tions. The question is of interest for finite element analyses (FEA) since theelements should be small in areas where the fields change a lot in a shortdistance. Unlike FEA, a not too complicated mathematical expression can fa-cilitate the physical understanding by mathematically showing how materialproperties, geometrical dimensions and frequency affect the electromagneticfields. Therefore analytical expressions can be valuable complements to fi-nite element analyses and measurements. The main motive for the work pre-sented in this paper is to obtain analytical expressions that can be used whenthe boundary values are known. This paper contains derivation of propagationfactors and Fourier series for harmonically time varying, traveling electromag-netic fields in a plate and a laminate with rectangular cross sections, isotropicmaterials and infinite length. Different and quite general fields are taken intoaccount on all boundaries. Choices of boundary conditions and continuity

58

conditions between material layers are discussed. Since some continuity con-ditions imply other continuity conditions, the choice of sufficient continuityconditions was not obvious. Together, the continuity conditions used in thederivation specify or imply continuity or discontinuity for the H componentsand their spatial first order partial derivatives. Certain combinations of typesof boundary conditions, Dirichlet and Neumann, make the derivation possiblefor a laminate. Comparisons are made between results of Fourier series andfinite element calculations. For non-magnetic materials, the agreement is verygood, but for magnetic materials, the FEA solution is inaccurate within a cou-ple of finite element lengths from plate edges at interfaces between materialswith different permeabilities.

5.4 Paper IVIt is relatively demanding to model each lamination in a laminated core as aseparate object. In FEA of electric machines it is common to approximatethe laminated core by a single object with a homogeneous but anisotropic ma-terial. With such an approximation, there is no need to solve equations ofcontinuity conditions between laminations, and it is sufficient to know spa-tially smoothed out boundary fields on the laminate. Analytical expressionsthat include arbitrarily directed fields on all laminate boundaries can be usedfor calculation of the fields inside the laminate when the boundary fields areknown from, e.g., measurements. A linear laminate block could be used innon-destructive testing for comparisons between different laminates. This pa-per contains derivation of Fourier series of harmonically time varying, trav-eling electromagnetic fields in a laminate approximated by a homogeneous,anisotropic block. Because of difficulties with the anisotropy, a special caseis studied where the laminate conductivity is much lower in the stacking di-rection than in the directions perpendicular to the stacking direction. Fourierseries of the magnetic field component in the stacking direction is determinedwithout any simplifying conditions about the conductivity. The component ofthe magnetic field strength in the stacking direction is used as a source termin two-dimensional Poisson equations for the magnetic field strength in direc-tions perpendicular to the stacking direction. Sine interpolation and differentchoices of types of boundary conditions are discussed. Different alternativesubdivisions of the Poisson boundary value problems are compared. The al-ternative subdivisions differ when it comes to how much the orthogonality ofthe eigenfunctions can be used. The alternatives with less use of orthogonalitygive time consuming integrals. The Fourier series of the electric field com-ponents are expressed in the Fourier series of the magnetic field components.Results from Fourier series in the three-dimensional case are compared withresults from finite element calculations.

59

Shorted derivations of simple analytical expressions are given for both trav-eling and standing waves in two dimensions. In the cases where the laminationhas one or two boundaries in the stacking direction but no other boundaries,the results are simple expressions. They show that the time average of the eddycurrent loss density is twice as high for a traveling wave as for a standing wave.Furthermore, the eddy current loss density is proportional to the conductivityin a direction perpendicular to the stacking direction. The loss density is alsoproportional to the square of the product of the frequency, the magnetic fluxdensity and the wave length along the laminate, in a direction perpendicularto the stacking direction. Real machines are finite, but the dependence on thewave length indicates that the eddy currents can be reduced if the number ofpoles is increased for a fixed machine size. The pole pitch limits the radiusof the eddy and the magnitude of the eddy current. This is analogous to thefact that the thickness limits the magnitude and geometrical extension of eddycurrents in the case of an alternating magnetic field in the plane of a thin plate.

5.5 Paper VThe load angle of a synchronous generator connected to a power grid has aneigenfrequency that depends on the operating conditions. The existence of aneigenfrequency can make the generator sensitive to electrical and mechanicaldisturbances and motivates the use of damper windings and power stabiliz-ing systems. However, the incentives for very strict limitation of load angleoscillation decreases with decreasing generator size since a small generatordoes not have much influence on the frequency quality. Weak damping andbroadband noise in the mechanical power can be sufficient for exciting andmaintaining a load angle oscillation.

The swing equation of motion and the steady state and simplified transientvoltage equations have been solved for a synchronous generator connecteddirectly to an infinite bus in a number of cases with different load angle ampli-tude and excitation. In most cases the field current was constant. In two casesthe field voltage was constant instead of the field current. For each load angleamplitude and excitation, rotor angular speed, phase currents and field currentas functions of time were used as input data for transient FEA for calculationsof eddy current power losses in the stator steel. FEA show that the eddy cur-rent losses in the stator core and clamping structure increase as a consequenceof the load angle oscillations or, more precisely, largely because of the statesthe generator slowly passes through during the load angle oscillations. Ananalytical model based on a phasor model for eddy current density in magnet-ically linear materials is suggested. The model agrees well with finite elementresults and predicts increased losses during underexcitation. The increasededdy current power loss during load angle oscillations can be explained by

60

two facts. The first is that the stator current increases with the load angle. Thesecond is the quadratic dependence on the stator current components along thedirect and quadrature axes. The loss model is applicable at steady state as wellas at load angle oscillations for prediction of the time averaged eddy currentpower loss during a normal electrical period.

61

6. Svensk Sammanfattning

Magnetiska läckfält orsakar förluster. Därför är det viktigt att förstå vad sompåverkar läckfälten. Denna förståelse har genom åren lett till designförslagoch begränsningar för driftlägen för elektriska maskiner. Att undermagnetis-erad drift i snabbroterande synkronmaskiner gav extra uppvärmning av sta-torändarna noterades redan 1920. Detta förklarades 1929 dels med att densvaga rotorströmmen gjorde att stålet var omättat i den ring som ska hindra ro-torns härvändar från att slungas ut av centrifugalkrafter, och dels med att rotor-fältet och statorfältet samverkade med avseende på magnetiseringsriktningeni ringen vid undermagnetisering. Vattenkraftgeneratorer och andra långsamtroterande synkronmaskiner har utpräglade/utstickande magnetiska rotorpoleroch ingen härvsammanhållande ring på rotorn. Dessutom finns få eller ingapublicerade artiklar om att undermagnetisering kan ge överhettade ändpartierhos sådana maskiner. Efter att en vattenkraftgenerator i Sverige fått blånadstatorplåt i ändarna av okänd anledning och en vattenkraftgenerator i Norgefått permanenta, allvarliga uppvärmningsskador i statorändarna efter en se-rie kortslutningar fattade Svenskt Vattenkraftcentrum beslut om att inleda detdoktorandprojekt om axiella magnetiska läckfält som denna avhandling hand-lar om.

Det första delprojektet resulterade i Artikel 1. Målet var att mäta det axiellaläckfältet på en generator vid olika operationspunkter, förklara mätresultatenoch med finita elementanalyser (FEA) bekräfta mätresultaten. En tidigare pub-licerad fasvisarmodell för axiellt läckfält vid synkronmaskiner med rund rotormodifierades något för att passa maskiner med utpräglade poler. Modifierin-gen innebär att man skiljer mellan magnetfältsbidrag från statorström i fasmed rotorns direkta polaxel och magnetfältsbidrag från statorström i fas meden axel mitt i gapet mellan två poler. Detta motiveras av att luftgapsreluk-tansen varierar runt rotorn då den har utpräglade poler. FEA har bekräftatatt utpräglingen har inverkan på de axiella läckfälten. Artikel 1 innehåller enmetod för beräkning av koefficienterna i fasvisarmodellen. Med undantag avvärdet på fasvinklar och värdet på en av koefficienterna vid liten lastvinkelvisar FEA och mätresultat samma trender. Skillnaderna kan bero på mate-rialmodeller, generatorgeometri, beräknings- och mätnoggrannhet samt nog-grannhet i givarnas placering. Både FEA och mätningar visade att läckfältetpå tryckfingrar och tryckring på statorns ena ända ökade då rotorströmmenminskade vid undermagnetisering förutsatt att lastvinkeln var tillräckligt litenför att inte statorströmmen skulle dominera läckfälten. Då statorströmmen

62

dominerar läckfälten blir dessa minimala då statorströmmen är minimal.

I delprojekt 2 var det primära målet att med mätningar och FEA undersökahur det axiella läckfältet påverkas av statorns axiella position i förhållande tillrotorn. Enligt tidigare publicerade artiklar är rotorns längd i förhållande tillstatorns längd viktig för styrkan på de axiella läckfälten men inga mätningarhade gjorts. I delprojekt 2 utfördes mätningar och FEA i tre axiella positionervid tre effektlägen och tre värden på effektfaktorvinkeln. Inom mätintervallenökade läckfältet på tryckfingrarna på statorns undersida då statorn hissadesupp ca ett luftgap. Vid övermagnetisering och högsta effektläget sjönk dockläckfältet något på tryckringen då statorn hissades upp. Läckfältet ökade medeffekten och minskande magnetisering. Denna studie kompletterades medFEA av axiell kraft och järnförluster som funktion av axiell statorförskjutningvid tomgång och olinjära material. Järnförlusterna i hela statorn ökar med ax-iell förskjutning p.g.a. virvelströmmarna. Eftersom den magnetiska kraftenär attraktiv mellan rotor och stator är det möjligt att sänka lagerförlusterna ien vertikalaxlad maskin genom att sänka rotorn. En metod för uppskattningav en optimal axiell rotorsänkning för minimering av förluster föreslogs ochanvändes i ett numeriskt exempel med de järnförluster och den axiella kraftsom beräknats för den studerade maskinen.

Ämnet i delprojekt 3 är härledning av analytiska uttryck för harmoniskttidsvarierande, gående elektromagnetiska vågor med godtyckliga fältriktningarlängs en plåt och ett laminatblock med oändlig längd i vågornas färdriktning.Tidigare publicerade analytiska uttryck för elektromagnetiska vågor i lami-nat är enklare men försummar ändeffekter och baseras på mer förenklandeantaganden. Separationsmetoden har använts. Den första drivkraften till attförsöka härleda analytiska uttryck var frågan om hur lamineringen påverkadefältens inträngningsdjup i olika riktningar. Frågan är intressant vid FEA efter-som elementen bör vara små i områden där fälten ändrar sig mycket på enkort sträcka som ett skinndjup kan vara. Till skillnad från FEA kan ett intealltför komplicerat matematiskt uttryck underlätta den fysikaliska förståelsengenom att visa direkt hur materialegenskaper, geometriska mått och frekvenspåverkar de elektromagnetiska fälten. Ett starkt argument för att härleda deanalytiska uttrycken är att de i kombination med uppmätta fält på laminatytankan vara ett alternativ till FEA för den enkla geometrin. Utöver att de an-alytiska uttrycken kan användas för verifiering av numeriska beräkningspro-gram så kan de få praktisk användning i laboratorier vid jämförelser mellanolika laminat. Att vid provning utsätta laminatblock för gående elektromag-netiska vågor med de fältriktningar som förekommer i generatorer och motoreristället för alternerande magnetfält som i transformatorer och Epsteintest kanvara ett komplement för jämförelser mellan laminat, särskilt om laminaten skamonteras i ändpartierna på generatorer och motorer. Uttryck för utbrednings-faktorer härleddes. De har central betydelse för inträngningsdjupet i den mån

63

det är relevant att tala om inträngningsdjup. I 3-D ger de geometriska be-gränsningarna upphov till moder vilket ihop med att varje mod har sin egenutbredningsfaktor kan ge upphov till lokala extremvärden i plåten eller lam-inatet. Den välbekanta formeln för skinndjup δ = 1/

√π f μσ är därför bara

relevant om våglängden och geometriska mått kan försummas vilket går om deär mycket större än 1/

√π f μσ . Under härledningen av Fourierserier för mag-

netiska och elektriska fältkomponenter diskuterades val av kontinuitetsvillkoroch typer av randvillkor. Eftersom vissa kontinuitetsvillkor implicerar andrakontinuitetsvillkor var valet av tillräckliga kontinuitetsvillkor inte självklart.Ortogonaliteten hos egenfunktionerna i Fourierserierna kan användas för attbestämma alla elektromagnetiska komponenter i laminatet från randvärdenaom vissa kombinationer av Dirichlet- och Neumannvillkor används. Randvär-dena för tre av de sex elektromagnetiska komponenterna räcker för fullständigbestämning av alla elektromagnetiska komponenter i laminatet. Vilka tre kom-ponenter som måste vara kända beror på valda kombinationer av randvillko-rstyper. I fallet med två materialskikt löstes kontinuitetsekvationerna och deanalytiska fältuttrycken jämfördes med FEA. I fallet med ett godtyckligt antalmaterialskikt presenterades kontinuitetsekvationerna utan lösningar.

Delprojekt 4 handlar om samma sak som delprojekt 3 men med laminat-blocket approximerat av ett block med homogent, anisotropt material. Förde-lar med denna approximation är att den inte kräver lösning av kontinuitetsek-vationer, och att mängden data att hantera blir mycket mindre med laminatetapproximerat med ett enda, homogent block. På grund av svårigheter medanisotropin i tre dimensioner studerades specialfallet att laminatet har mycketlägre konduktivitet i stackningsriktningen än i övriga riktningar. Fältkom-ponenter i stackningsriktningen kan bestämmas utan något förenklande an-tagande om konduktiviteter. Med magnetfältet i stackningsriktningen somkällterm i tvådimensionella Poissonekvationer härleddes analytiska uttryckför magnetfältskomponenterna längs och tvärs laminatet. Olika alternativauppdelningar av randvärdesproblemen jämfördes. Sinusinterpolation och valav randvillkorstyper diskuterades. De elektriska fältkomponenterna uttrycktesmed hjälp av uttrycken för de magnetiska fältkomponenterna. Fältkomponen-terna beräknade med de analytiska uttrycken jämfördes med FEA. I tvådimen-sionella fall där laminatet har en eller två begränsningsytor i stackningsriktnin-gen men inga andra begränsningar blev resultaten enkla uttryck som visar atttidsmedelvärdet av förlusttätheten är dubbelt så hög vid gående våg som vidstående våg längs laminatytan. Förlusttätheten är proportionell mot konduk-tiviteten i en tangentiell riktning, frekvensen i kvadrat, axiella magnetfältet ikvadrat och våglängden i kvadrat. Visserligen finns inga oändligt stora elek-triska maskiner men resultaten tyder ändå på att virvelströmmarna i ändarnaav statorlaminat kan reduceras om maskinens poltal ökar utan att maskinstor-leken ändras. Våglängden begränsar virvelströmmens storlek och utbredning ilaminatplanet. Detta är analogt med att plåttjockleken begränsar virvelström-

64

mens storlek och utbredning i fallet med alternerande magnetfält i planet hosen tunn plåt.

I delprojekt 5 studerades lastvinkelpendlingens inverkan på virvelströms-förlusterna i statorn i en synkrongenerator. Lastvinkeln för en synkrongener-ator ansluten till ett starkt nät har en egenfrekvens som är beroende av oper-ationstillståndet, reaktansen, rotationströghetsmomentet, frekvensen och antalpoler. Existensen av en egenfrekvens gör lastvinkeln känslig för störningar ieffekt. I stora generatorer används dämpkretsar och reglersystem för att sta-bilisera frekvens och spänning men incitament för att använda system för attstrikt motverka lastvinkelpendling minskar med avtagande generatorstorlekför generatorer anslutna till ett starkt nät. Svag dämpning och bredbandigtbrus i den mekaniska effekten kan räcka för att sätta igång och vidmakthålla enlastvinkelpendling. Numerisk lösning av spänningsekvationer och rörelseek-vationen för rotorns rotation då generatorn är ansluten till ett starkt nät hargenomförts. Rotorvinkelhastighet, statorfasströmmar och fältström användessom indata till transienta pendlingsanalyser med förlustberäkning i FEA. Attvirvelströmsförlusterna ökar med amplituden på lastvinkelpendlingen visasdels med FEA och dels analytiskt med en förlustmodell baserad på en fasvis-armodell för virvelströmtätheten. Modellen kan användas för att förutsägavirvelströmsförlustens tidsmedelvärde i en stator under en normal elektriskperiod vid såväl stationär drift som vid lastvinkelpendling.

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7. Conclusions

When it comes to operating conditions, simple vector addition gives that whatmatters the most for the leakage flux is not the power factor, cosϕ , and not theload angle, δ , but the current magnitudes and the phase difference between theflux contributions from the rotor and the stator. The phase difference is 90◦ +δ + ϕ , as Tavner pointed out [53]. The interest in the effect of the power factor,and therefore overexcitation and underexcitation, is based on the importanceof the power factor for normal operation. It is more important to produce acertain amount of power than a certain amount of leakage flux.

At a given active power or load angle, axial leakage flux and eddy currentpower losses in the stator of a salient pole synchronous generator are higherat underexcitation than at overexcitation. A part of the explanation is thatthe phase difference between the stator and rotor magnetic field contributionsincreases with the field current. The phase difference is more than 90◦ atoverexcitation and less than 90◦ at underexcitation with sufficiently low fieldcurrent. A phase difference of less than 90◦ means cooperation, and a phasedifference of more than 90◦ means counteraction between the stator and rotorfield contributions. Another part of the explanation is that the paths of theleakage flux depend on the mentioned phase difference. At overexcitation thestator mmf counteracts and deflects the field contribution from the rotor. Thisis true for the main flux as well as the leakage flux but whereas the deflectionof the main flux has a small effect on the reluctance the main flux meets, de-flection of leakage flux away from the generator leads to leakage flux paths ofsignificantly higher reluctance because of the low permeability of the air.

A previously published phasor model for prediction of axial leakage fluxin fast spinning synchronous machines with round rotors has been generalizedby making a distinction between magnetic field contributions from the d andq components of the stator current. The distinction between these contribu-tions in a salient pole synchronous generator is justified by experiments, finiteelement simulations and the fact that the reluctance between the rotor and thestator is larger at the q axis than at the d axis in a salient pole machine.

Axial displacement of the rotor in a stator bore leads to increased eddy cur-rent losses from axial leakage flux at no load. Other iron losses are relativelyindependent of axial displacement not larger than a few air gaps. For a suf-ficiently small downward rotor displacement of a symmetrical synchronous

66

machine with a vertical shaft, the reduction of the thrust bearing losses islarger than the increase of the other major losses at normal operation. If themachine is symmetrical, there is a downward optimal rotor displacement thatminimizes the sum of the thrust bearing losses and the other losses, even ifthe friction is very low. The optimal downward rotor displacement increaseswith the friction and axial magnetic stiffness and decreases with increasingmachine weight at a given magnetic force. With a high magnetic stiffness, agiven downward displacement gives a great load reduction in the thrust bear-ing which is needed to compensate for the increasing absolute value of theslope of the iron loss curve with increased downward displacement. Locallyon the stator ends, the axial flux can increase or decrease depending on theaxial displacement and the point of operation. If the machine is not symmetri-cal, there can still be an optimal downward rotor displacement if the friction ishigh. At load, when the generator is connected to the power grid, the leakageflux increases with the power and increases when the excitation, i.e. the fieldcurrent, is reduced.

For harmonically time varying electromagnetic traveling fields along an in-finitely long laminate block, it is possible to express the field componentsinside the laminate with Fourier series that depend on the boundary values ofthree of the six electromagnetic field components. Measured boundary fieldstogether with the analytical field expressions can be an alternative to finite el-ement analyses. Laminate blocks can be used in non-destructive testing forcomparisons between laminates. With an appropriate choice of combinationsof Dirichlet and Neumann boundary conditions, all initially unknown Fouriercoefficients for any field component are collected in only one Fourier seriesper field component. This made the use of separation of variables and theorthogonality of eigenfunctions possible to use for the derivation of the ana-lytical expressions of the field components. The analytical expressions alsodepend on the choice of types of boundary conditions. This choice determineswhich three field components that must be known on any particular boundaryif the normal derivatives cannot be directly measured. For each normal deriva-tive that can be measured, it is possible to choose to use that instead of one ofthe three field components.

For an infinitely long laminate block approximated by a homogeneous,anisotropic block, it is possible to use the method of separation of variablesto derive Fourier series of the magnetic field component in the stacking di-rection. Under the precondition that the laminate conductivity is much lowerin the stacking direction than in the other directions, it is possible to deriveFourier series also for the other electromagnetic field components if the mag-netic field component in the stacking direction is used as a source term intwo-dimensional Poisson equations for the magnetic field components per-pendicular to the stacking direction. With an appropriate choice of combina-

67

tions of Dirichlet and Neumann boundary conditions, the orthogonality of theeigenfunctions can be used in the Poisson equations for obtaining equationsfor the unknown Fourier coefficients. The electromagnetic field componentsinside the laminate can be determined if three of the six electromagnetic fieldcomponents are known on the boundary. On the two laminate end planes inthe stacking direction, knowledge of the magnetic field components is suffi-cient. On the other two laminate boundaries, the choice of combination oftypes of boundary conditions determines which three field components thatmust be known unless required normal derivatives of field components can bemeasured directly. If the choice of boundary conditions leads to a sine seriesfor one of the contributions of a field component, the truncated series con-verges slowly and suffers from overshoot and ringing (Gibbs phenomenon) ifthe field component is not zero at the ends of the interval within which the sinefunctions are eigenfunctions. The overshoot and ringing can be eliminated bysine interpolation.

The average eddy current loss density during a cycle in an infinite laminatewith one boundary plane parallel to the laminate sheets is proportional to theconductivity and the square of the product of the longitudinal wave length, thefrequency and the amplitude of the normal component of the magnetic fluxdensity. In this case, the spatial average of the eddy current loss density istwice as large for a traveling wave as for a standing wave.

The load angle of a synchronous generator connected to a strong powergrid has an eigenfrequency that depends on the point of operation. The ex-istence of an eigenfrequency makes the load angle sensitive to disturbancesin the power. Load angle oscillation with constant average mechanical powerleads to a higher eddy current power loss than without oscillation. This canbe explained in two steps. First, the stator current increases with the load an-gle. Second, a model for eddy current power loss shows that the loss containspositive contributions that are proportional to the squares of the d and q com-ponents of the stator current.

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8. Acknowledgement

The research presented in this thesis was carried out as a part of "SwedishHydropower Centre - SVC". SVC has been established by the Swedish En-ergy Agency, Elforsk and Svenska Kraftnät together with Luleå University ofTechnology, The Royal Institute of Technology, Chalmers University of Tech-nology and Uppsala University.

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9. Suggested Future Work

Comparative, quantitative studies of different designs for reduction of end re-gion losses in salient pole synchronous generators are suggested.

70

A. Appendices

A.1 Galerkin’s MethodA general boundary value and initial value problem for a scalar field u, thate.g. could be a component of a vector field, is of the form

L u(rrr, t) = f (rrr, t) in a domain Ω (DE)α(rrr, t)ν ·∇u(rrr, t)+β (rrr, t)u(rrr, t) = g(rrr, t) on domain boundary Γ1 (BC1)

u on Γ2 =±u on Γ3 (BC2a)ν ·∇u(rrr, t) on Γ2 =±ν ·∇u(rrr, t) on Γ3 (BC2b)

u(rrr,0) = h1(rrr) (IC1)∂u(rrr,0)/∂ t = h2(rrr) (IC2)

(A.1)

whereL is a differential operator, rrr is a position, and t is time. Depending onthe values of α and β , (BC1) can be a Dirichlet, Neumann or Robin condition.The conditions (BC2a) and (BC2b) together are applicable for a field that isperiodic in space. That can be the case in periodic structures like electricmachines at steady state with harmonic field variation in time. The plus signis applicable if Γ2 and Γ3 are separated by one period of the structure. Theminus sign is applicable if Γ2 and Γ3 are separated by one half period of thestructure.The steps in Galerkin’s finite element method are: 1. Multiply the differ-

ential equation by a base function that is zero on the part of the boundarywhere a Dirichlet condition is applied. In case of nonzero Dirichlet condition,convert to a homogeneous problem by setting u = v+w where w is a knownfunction that equals g on the boundary. 2. Integrate by parts the expressionfrom step 1 and use boundary conditions of Neumann or Robin type. 3. In-sert the approximative solution (3.20) into the integral expressions. In caseof nonzero Dirichlet condition, insert a corresponding approximation of v∗minstead. 4. Assemble the equations of all elements into one single equationsystem. Unknown normal derivatives on internal elements cancel each otherin the assembled equation system. 5. Solve the equation system.The motivation for requiring that the base functions are zero on the bound-

ary with a Dirichlet condition has to do with the Lax-Milgram theorem thatgives conditions for existence and uniqueness of the approximative solution.An introductory course on numerical methods for partial differential equationshas been given by e.g. Braunschweig Technische Universität with tutorials

71

downloadable from [54], [55]. An example of Galerkin’s method combinedwith time stepping is given by [56].

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