Magnetic Leakage Fields and End Region Eddy Current Power Losses in Synchronous Generators

Transcription

1 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1575 Magnetic Leakage Fields and End Region Eddy Current Power Losses in Synchronous Generators BIRGER MARCUSSON ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2017 ISSN ISBN urn:nbn:se:uu:diva

2 Dissertation presented at Uppsala University to be publicly examined in Room 2001, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Thursday, 30 November 2017 at 09:00 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Emeritus Göran Engdahl (KTH, Electrotechnical Design, School of Electrical Engineering). Abstract Marcusson, B Magnetic Leakage Fields and End Region Eddy Current Power Losses in Synchronous Generators. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala: Acta Universitatis Upsaliensis. ISBN The conversion of mechanical energy to electrical energy is done mainly with synchronous generators. They are used in hydropower generators and nuclear plants that presently account for about 80% of the electric energy production in Sweden. Because of the dominating role of the synchronous generators, it is important to minimize the power losses for efficient use of natural resources and for the economies of the electric power companies and their customers. For a synchronous machine, power loss means undesired heat production. In electric machines, there are power losses due to windage, friction in bearings, resistance in windings, remagnetization of ferromagnetic materials, and induced voltages in windings, shields and parts that are conductive but ideally should be non-conductive. The subject of this thesis is prediction of end region magnetic leakage fields in synchronous generators and the eddy current power losses they cause. The leakage fields also increase the hysteresis losses in the end regions. Magnetic flux that takes paths such that eddy current power 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 cause hot spots, discoloration and reduce the service life of the insulation on the core laminations. If unattended, damaged insulation could lead to electric contact and eddy currents induced by the main flux between the outermost laminations. That gives further heating and deterioration of the 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 have to be replaced. However, the end region leakage flux primarily causes heating close to the main stator conductors which makes the damage possible to discover by visual inspection before it has become irrepairable. Keywords: magnetic leakage fields, leakage flux, eddy currents, losses, synchronous generator Birger Marcusson, Department of Engineering Sciences, Electricity, Box 534, Uppsala University, SE Uppsala, Sweden. Birger Marcusson 2017 ISSN ISBN urn:nbn:se:uu:diva (

3 List of papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I II III IV V B. Marcusson and U. Lundin, "Axial Magnetic Fields at the Ends of a Synchronous Generator at Different Points of Operation", IEEE Transactions on Magnetics, vol. 51, no. 2, pp. 1-8, Feb B. Marcusson and U. Lundin, "Axial Magnetic Fields, Axial Force, and Losses in the Stator Core and Clamping Structure of a Synchronous Generator with Axially Displaced Stator", Electric Power Components and Systems, vol. 45, no. 4, pp , Jan B. Marcusson and U. Lundin, "Harmonically Time Varying, Traveling Electromagnetic Fields along a Plate and a Laminate with a Rectangular Cross Section, Isotropic Materials and Infinite Length", Progress In Electromagnetics Research B, Vol. 77, , 2017 B. Marcusson and U. Lundin, "Harmonically Time Varying, Traveling Electromagnetic Fields along a Laminate Approximated by a Homogeneous, Anisotropic Block with Infinite Length", Submitted to Progress In Electromagnetics Research B B. Marcusson and U. Lundin, "A Loss Model and Finite Element Analyses of the Influence of Load Angle Oscillation on Stator Eddy Current Losses in a Synchronous Generator", Submitted to Transactions on Magnetics Reprints were made with permission from the publishers.

4

5 Contents 1 Introduction Basic Synchronous Machine Terminology Background on Research on Magnetic Leakage Flux Overview of Losses in Electric Machines Review of Research on Magnetic Leakage Flux Theory Phasors The Steady State Voltage Equation for a Synchronous Generator Derivation of Rotor Reference Frame Voltage Equations of a Synchronous Generator Assumptions, Definitions and Voltage Equations in the Stator Reference Frame Inductances in the Stator Reference Frame of Salient Pole Synchronous Machines Transformation of Electrical Parameters to the Rotor Reference Frame Linearization of the Equation of Motion of Load Angle Oscillation Methods Effective Permeabilities in a Laminate Calculation of Axial Force Calculation of Losses in the Steel of the Stator Core and Clamping Structure in a Synchronous Generator The Finite Element Method Domain and Boundary Conditions in Magnetic Analysis of an Electric Machine with ANSYS Maxwell The Method of Separation of Variables Estimation of Frictional Power Loss in a Slider Bearing Estimation of Windage Power Loss Experimental Equipment and Measurements Magnetic Field Sensors and Measurements of Axial Magnetic Flux Density Results that Were Not Included in the Papers Supplements to Paper II... 49

6 4.2 Supplements to Paper V Summary of Papers Paper I Paper II Paper III Paper IV Paper V Svensk Sammanfattning Conclusions Acknowledgement Suggested Future Work Appendices A Appendices A.1 Galerkin s Method References... 73

7 1. Introduction 1.1 Basic Synchronous Machine Terminology A synchronous machine is a type of electric machine designed for conversion between electrical and mechanical energy in both directions. The machine is operating as a generator when it converts mechanical power to electrical power, 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 that is called rotor or translator depending on if the machine is (roughly) cylindrical 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 conductors) with alternating currents (AC), a laminated core of magnetic steel for magnetic field amplification, clamping parts and a frame for keeping the core and windings fixed. A conventional synchronous machine has a rotor with electromagnetic poles consisting mainly of a rotor field winding for direct current (DC) around a core of magnetic steel. The strength of the magnetic field produced by the poles increases with the rotor current, called the field current. The name synchronous machine refers to the fact that the moving part is synchronized with the magnetic field produced by the currents in the phase windings, i.e., the moving part and the magnetic fields move with the same 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 stator conductors. If the resistance is low, the current and power for a certain voltage 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 current, during each cycle, must reach its maximum after the voltage reaches its maximum. A generator operating alone (island operation) satisfies this for any field current since it determines the voltage itself within certain limits. For a generator connected to a strong power grid (network), i.e. a power grid that keeps the voltage constant, the field current must be higher than for a purely resistive load. In this situation the generator is said to be overexcited. If the load is capacitive, the current must lead the voltage. For a generator connected to a strong power grid, this can be accomplished by the generator if the field current is lower than for a purely resistive load. In this situation the generator is said to be underexcited. 7

8 Figure 1.1. Model of the salient pole synchronous generator used for experiments conducted by the author and other members of the hydropower group at the Division for Electricity. The letters N and S indicate magnetic north and south poles respectively. 1.2 Background on Research on Magnetic Leakage Flux Systems for conversion, transformation and transportation of energy are subject to unwanted generation of heat. This heat is considered as power loss if it is not desirable and economical to use it. In general, power losses mean inefficient use of energy sources, and for the electric power companies, the power losses mean losses of income from the selling of electrical energy. Therefore, it is important to minimize the power losses, wherever they occur. There 8

9 are several types of power losses in electric machines designed for conversion between electrical and mechanical energy. Examples of losses are windage power losses, friction power losses in bearings, resistive losses in windings and cables, eddy current power losses in all conductive parts, hysteresis power losses in ferromagnetic materials. The focus in this thesis is on unwanted axially directed magnetic fields, so called magnetic leakage fields, and the losses associated with them at the end regions of synchronous generators. Magnetic fields that do not take the desired paths are considered to be leakage fields. The terms magnetic leakage field and magnetic leakage flux are used in the same context. Hydro power plants and nuclear plants dominate the Swedish power production and use rotating synchronous generators. Most research published on end region power losses has been conducted on fast spinning synchronous generators, called turbogenerators or non-salient pole generators with cylindrical rotors driven by gas turbines. It has been found that the end regions of large turbogenerators become warmer when the generator is underexcited than when it is overexcited. The type of generators used in hydropower plants are salient pole synchronous generators. The initial question for the work in this thesis was if also the end region losses of salient pole synchronous generators could be sensitive to the excitation. Motivation for initiating the study of axial magnetic fields in a salient pole synchronous generator comes from two cases of overheating of the end regions of large hydropower generators in Scandinavia. In case one, which was the least severe, there was discoloration of stator core laminations of the second plate package from both ends. The suspected cause is underexcited operation combined with insufficient cooling of the second plate package. In case two, consecutive short circuits overheated the stator end regions. This resulted in degradation of the lamination insulation followed by permanent heat damages. Typical signs of core end heating due to eddy currents are discoloration of the varnish of the outermost stator laminations behind the slots for the stator conductors at both generator ends [1]. Core plate varnishes from purely to partly organic can withstand C continuously [2]. According to IEEE Std , modern inorganic insulation such as aluminum orthophosphate that can stand 500 C. The term magnetic field normally refers to the magnetic flux density. Flux density integrated over a surface is magnetic flux. At the ends of a conventional rotating, electric machine, the magnetic field spreads out into a continuum of directions and induces undesired eddy currents in any conducting parts nearby. Time varying leakage flux that passes through the end laminate in the direction perpendicular to the lamination planes induces relatively large eddy currents in the laminations for two reasons. First, according to Faraday s law, the induced electric fields are perpendicular to the magnetic field that causes 9

10 them. Second, the laminations have large, continuous areas with high conductivity. In addition, the combination of lamination and leakage flux can lead to extra high flux and hysteresis power losses in the end laminations. 1.3 Overview of Losses in Electric Machines Losses can be divided into different types depending on when, where and how they appear. The losses can also be put into different groups if they can be determined individually from measurements. The IEEE standard [3] puts losses into five such groups. They are a) Friction and windage losses at no field current, no magnetic flux, and no stator current. Windage loss can be considered as a special case of the friction loss. 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 per phase winding (times three). e) The squared field current multiplied by the direct current resistance in the field winding. Core loss is normally considered to be the same as iron loss since the stator and rotor cores are usually made of ferromagnetic materials [4], [5]. However, the core loss measured according to the standard [3] is not a pure core loss since it includes eddy current losses in stator conductors, support structure and frame of the stator, and rotor surfaces. The core loss density in the laminate material can be determined by Epstein tests according to IEC standard Iron losses in electric machines can be determined by FEA, as described in section 3.3. Stray load losses include the extra losses that are generated during load and that are not accounted for by the other mentioned losses. A large part of the stray load losses appears in stator bars because of leakage flux through the stator slots. The leakage flux makes the induced electric field different in different parts of the conductors. This gives rise to current displacement in the conductors. This, in turn, increases the resistance and the resistive losses. However, these losses are reduced by the combination of twisting and subdivision of stator conductors into thin parts/strands. Other parts of the stray load losses 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 included in the magnetic circuits. 10

11 1.4 Review of Research on Magnetic Leakage Flux Designs used for reducing losses caused by magnetic leakage fields in the stator ends of electrical machines are slitted stator teeth, laminated or nonlaminated magnetic flux screens/shunts, non-magnetic flux shields of high conductivity (usually copper), extra coating of varnish on the laminations [1], rounded or stepped stator and rotor ends at the air gap, rotor core shorter than stator core, non-magnetic material in clamping structure, and non-magnetic rotor coil retaining rings on turbogenerators. In 1927 stator teeth slitting in radial-axial planes was suggested as a way to reduce eddy current losses in the teeth [6]. The teeth are only slitted near the machine ends. The eddy current loss in the teeth can be roughly halved by one slit through the middle of the teeth if the skin effect is negligible, but in practice the tendency of the eddy currents to concentrate near the surface makes the slits somewhat less efficient. Slits make the teeth mechanically weaker and more difficult to hold in place. For mechanical reasons slits are cut obliquely in alternating directions in different plate layers. Due to the eddy currents the voltage 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 have been suggested [8]. There seems to be very little published about this. In Kuyser got a few patents on nonmagnetic, highly conductive shields to mount on the stator ends in order to deflect the magnetic field from the stator core of turbogenerators. One shield, called squirrel-cage damper, patent GB220362, was in the air gap surrounding the retaining ring. Another shield was a heavy, plane, copper alloy plate, patent GB Apart from holes for stator conductors the plate covered the whole end region from the air gap to 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 nonmagnetic cast iron clamping plate in the other end. The temperature increase, measured with thermocouples, was 39.9 C in the core end with copper plate and 71.8 C in the other core end. Approximately the latter temperature increase was also measured in the core end of a generator of the same design but with magnetic cast iron clamping rings [9]. Measurements with a flat copper screen between a piece of clamping plate and a pancake coil showed that although the screen reduced the loss in the clamping plate, the total loss in the screen and the clamping plate increased [8]. Analytical 2-D calculations showed that a highly conductive screen on the end of the stator diverts the leakage magnetic field at the cost of high loss in the screen and flux concentration in the laminate parts not covered by the screen [10]. Therefore, a screen on the stator laminate end should cover the stator teeth completely in order to avoid high axial flux concentration. Finite element simulations of an overexcited turbogenerator with power factor 0.95 showed that a plane copper shield on the clamping 11

12 plate could at best reduce the axial flux to the core by about 20 %, no matter how thick the shield was. The flux was diverted to radial skin layers in the shield before entering the core. As long as the shield thickness is so thin that the induced eddy currents in the shield give a negligible contribution to the total 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 begin do decrease approximately exponentially towards an asymptotic value if the imposed field is only axially directed [11]. A shield that is wrapped to cover also the radially inner surface of the clamping plate shields the plate and core back more but diverts more flux into the teeth. In 1974 Howe et al. [10] by means of 2-D analytical calculations concluded that the losses in the stator end were reduced by rounding or stepping the stator core end and by making the rotor shorter than the stator. Figures showing turbogenerators in [9] and [12] indicate that stator stepping was used because of the protruding retaining ring long before Howe et al. mentioned the benefits of core stepping for reduction of eddy current losses, but a reason for stepping is to get a sufficient inflow of cooling air. Finite element analyses have confirmed that core stepping give reduction of eddy current losses, but the loss reduction 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 clamping materials [12]. A negligible difference between heating of magnetic and nonmagnetic cast iron clamping plates was measured on two turbogenerators of the same design [9]. However, linear material finite element analyses of a 3.1 MW permanent magnet generator show that the losses, about 2.9 kw in clamping plates of magnetic construction steel, can be about a factor ten higher than in clamping plates of nonmagnetic stainless steel [15]. Hysteresis losses are not mentioned in [15]. An advantage with magnetic clamping rings is that they shield the stator core from leakage flux [14]. At least until 1948 Westinghouse Electric Corporation continued to build their machines with magnetic alloy steel clamping rings and magnetic rotor coil retaining rings. In 1920 people working with turbogenerators recognized that the heating of the stator end regions was increased at underexcited operation. In 1929 Kuyser mentioned that, in experiments with a turbogenerator with the stator current kept constant, the end plate temperature increased roughly 20, 45 and 70 C in the three extreme load cases zero power factor overexcited, short circuit and zero power factor underexcited [9]. The reasons for the differences in stator end region heating between leading with zero power factor, lagging with zero power factor and short circuit operation was the degree and direction of saturation of the retaining ring. At underexcitation with zero power factor the field current is zero. Flux picked up by the retaining ring flows in the 12

13 peripheral direction to the adjacent pole. Then the magnetic cross section of the retaining ring determines the magnitude of the leakage flux. It takes only a small field current to reduce the permeability of the retaining ring. Kuyser stated that at short circuit and overexcitation, the field current magnetises the retaining ring in a direction that opposes the stator ampere turns. This is not true for the axial field but true for the radial direction in the air gap and in the whole retaining ring provided that the center of the stator end windings is farther away from the stator core end than the whole end ring is. Kuyser mentioned that it had been known for some years that non-magnetic retaining rings can reduce the end leakage field and losses. Historically, the high strength of the magnetic retaining rings was a reason for using them, especially by British designers because of their relatively high safety factors [9]. He suggested that the retaining rings should be either as thin as possible or non-magnetic [9]. In 1953 Estcourt et. al. largely repeated Kuyser s explanation but with commonly used load cases and mentioning of the direction of the armature reaction in these cases. The reluctance in the peripheral direction of the retaining ring is important for the leakage flux and depends on the operation conditions [12]. Both the magnitude of the rotor (field) flux and its position relative to the main direction of the stator (armature) reaction affects the reluctance and leakage flux. At unit power factor operation, the stator flux is directed somewhere between the poles where the reluctance of the retaining ring is relatively low. At constant apparent power and terminal voltage the end iron temperature increased a factor 2-3 from overexcited to underexcited state with power factor 0.8. At constant power factor and terminal voltage, the temperature almost increased linearly with the apparent power. At constant power factor and apparent power, the temperature decreased linearly with increasing terminal voltage [12]. Nowadays strong nonmagnetic 18Mn-18Cr steel retaining rings are used [16]. According to IEEE Standard , the axial leakage fields can be high in large machines at underexcitation, especially if the end windings are long as in turbogenerators. For practical and mechanical reasons the stator core lamination of a large machine is made of overlapping plate segments. This segmentation is also a way 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 generators can be so strong that the induced eddy currents can cause voltage breakdown through 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 for testing purposes. 13

14 2. Theory 2.1 Phasors A phasor can be represented by a complex number that in turn can be represented by a vector from the origin in the complex number plane. Phasors are used to describe physical quantities that vary harmonically with time. The length of the vector is the amplitude. The angle from some reference line, usually the real axis, to the vector is the phase angle of the phasor. Phasors have been used extensively in the work leading to this thesis since all of it concerns synchronous three-phase machines at steady state or near steady state where currents, voltages and electromagnetic fields vary approximately harmonically with time. A horizontal bar above a symbol denotes a complex quantity, such as a phasor. Although the field current is constant at steady state, it can be modeled as a phasor, ī f, since it exposes each pole pitch wide sector of the stator by flux that is roughly sinusoidal both in space and time when the rotor rotates with constant angular velocity. Electromagnetic fields penetrating a material undergo phase shifts that depend on the material, the frequency and the geometry, but any phase difference between a main current, such as a phase current or the field current, and the phasor of the radial flux it creates in a region from the stator conductors to the rotor is normally assumed to be negligible, at least in linear materials and far from saturated soft magnetic materials. For soft magnetic materials the hysteresis is small. Here, it will be assumed that a current phasor is in phase, not in anti-phase, with the phasor of the flux linkage contribution it gives rise to. 2.2 The Steady State Voltage Equation for a Synchronous Generator The steady state stator voltage equation, (2.1), in the rotor reference system is of great importance for Paper I, II and V. In phasor form, the equation relates the RMS (root mean square) terminal voltage, V and the RMS stator current, Ī, to the RMS voltage, Ē, induced in the stator windings by the rotor. Parameters derived from the voltage equation are active power, P, reactive power, Q, power factor angle, ϕ, load angle, δ, current phase angle, α, stator current component Ī d in phase with the rotor d (direct) axis and Ī q in phase with the q (quadrature) axis. In Paper I, Ī d and Ī q have important roles in a phasor model for magnetic leakage fields, and in Paper V, Ī d and Ī q have important 14

15 roles in a phasor model for eddy current density. Voltage equations are easier to solve in the rotor reference frame than in the stator reference frame because the inductances are constants in the rotor reference frame. At steady state, also the other electrical parameters mentioned are constants. The voltage equation with generator sign convention used is Ē = V + jx d Ī d + jx q Ī q + RĪ (2.1) where X d is the stator d axis reactance, X q is the stator q axis reactance, R is the stator resistance per phase, and j = 1. With the d axis as the real axis, and the q axis as the imaginary axis, Ī d = I d, Ī q = ji q, and (2.1) gives I d = X q(e V q ) RV d R 2 + X d X q and I q = R(E V q)+x d V d R 2 + X d X q The active power is = X q(e V cosδ) RV sinδ) R 2 + X d X q (2.2) = R(E V cosδ)+x dv sinδ R 2 + X d X q. (2.3) P = 3Re{ V Ī } = 3(V d I d +V q I q ) 3V = R 2 + X d X q ( E(Rcosδ + X q sinδ) VR+ V 2 (X d X q )sin2δ ). (2.4) where the exponent denotes a complex conjugate. The reactive power is Q = 3Im{ V Ī } = 3(V q I d V d I q ) 3V ( = E(Xq R 2 cosδ Rsinδ) V (X d sin 2 δ + X q cos 2 δ) ). (2.5) + X d X q In case of load angle oscillations, it may be better to assume that the pole flux linkage rather than field current is approximately constant. In that case (2.1) can be replaced by the classical model, Ē = V + jx dīd + jx q Ī q + RĪ (2.6) where Ē is called q axis transient emf (electromotive force) which is proportional to the pole flux linkage, and X d =X d ω s Ldf 2 /L f is the transient d axis reactance where L df = 3/2M f is the mutual inductance between the field winding and the fictitious stator d axis winding. The expression for L df is derived in section L f is the self inductance of the field winding. Normally there is a transformer and a transmission line between the generator and a strong power grid that in simpler models is assumed to have a constant RMS voltage and is then called an infinite bus. Optionally, the terminal voltage in (2.1) or (2.6) can be replaced by the voltage of the infinite bus if the 15

16 impedance between the generator and the infinite bus is accounted for. In that case, an alternative to (2.6) is Ē = V IB + jx DĪd + jx Q Ī q + R t Ī, (2.7) where X D = X d + X tr, X Q = X q + X tr and R t = R + R tr, where X tr is the reactance and R tr is the resistance of the transformer plus transmission line. Fig. 2.1 shows a phasor diagram with rotor poles. Since ī f and the d axis are in anti-phase, it can be useful to call the rotor north pole axis in phase with ī f the field axis or simply the f axis. The choice of minus sign in Faraday s law on integral form dictates that the pole flux linkage contribution Ψ ff from ī f should lead the emf Ē it induces in the stator winding by 90. Since Ψ ff is in phase with a rotor north pole, Ē is in phase with the q axis. A load angle, δ IB, from Ē to V IB is useful in analyses of load angle oscillations since V IB has a known, constant angular speed relative to the stator reference frame. In stability analyses, there is a distinction between electrical and mechanical power. In that case, the electric power symbol is P e, and the symbol for mechanical power is P m. The active power delivered by the generator can be expressed as P e = 3R t I 2 + 3Re{ V IB Ī } 3R t = (Rt 2 + X D X Q) 2 [(X Q(E V IB cosδ IB ) R t V IB sinδ IB ) 2 +(R t (E V IB cosδ IB )+X DV IB sinδ IB ) 2 ] ) 3V IB (E (R t cosδ IB + X Q sinδ IB ) V IB R t + V IB 2 (X D X Q)sin2δ IB + Rt 2 + X D X. Q (2.8) Under the assumption that R t X D and R t X Q, (2.8) can be simplified to P e 3V ( IB X D X E X Q sinδ IB + V ) IB Q 2 (X D X Q )sin2δ IB. (2.9) Differentiation of P e with respect to δ IB gives the so called synchronizing power coefficient [19] which can be used in the equation of motion for a generator with load angle oscillations. The coefficient is dp e 3V IB ( E dδ IB X D X X Q cosδ IB +V IB (X ) D X Q )cos2δ IB. (2.10) Q 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 coefficient and just express P e in terms of the components of current and voltage for each point in time where δ IB has been determined. 16

17 Figure 2.1. Phasor diagram. 2.3 Derivation of Rotor Reference Frame Voltage Equations of a Synchronous Generator In this section the used transient and steady state voltage equations in the rotor reference frame will be derived. The steady state stator voltage equation is then obtained from the transient equations as a special case Assumptions, Definitions and Voltage Equations in the Stator 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 mutual inductances between rotor and stator and the terms in the voltage equations in the rotor reference frame depend on two choices. The first is if the d axis is leading or lagging the q axis. The second is if generator or motor reference signs/directions are used. A third choice is the ratio, N fic /N, where N is the number of turns per phase winding, and N fic is the chosen number of turns in each of the fictitious stator windings. The third choice affects the magnitudes of the mutual inductances between rotor and stator and the corresponding terms in the voltage equations in the rotor reference frame. Fig. 2.2 shows the cross section of a simplified synchronous machine. The angle θ is 17

18 used to specify the rotor position relative to the stator. The choice here is to let the q axis lead the d axis. Each current has an ideal magnetic axis that defines the positive main flux direction for positive current according to the right hand rule. Slots and rotor pole saliency combined with magnetic materials and/or eddy currents make the main direction of the flux from a current deviate from its ideal magnetic axis. On the other hand, the flux from a current could be considered 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 coil is 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 reference directions are shown. A dot in a circle denotes current direction towards the reader. 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 machine. Generator sign convention is here used for the stator windings. That means that a stator current is positive when it is directed away from the gener- 18

19 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 with impedance Z l via a transformer and transmission line with impedance Z t. The Z l components represent circuits with resistive, inductive and/or capacitive elements. The load is typically a motor. The Z t components typically represent just a resistance in series with an inductance. 19

20 The transient voltage equation in the stator reference system is v a r i a Ψ a v b 0 r i b v c v f = 0 0 r i c r f 0 0 i f d Ψ b Ψ c dt Ψ f r D 0 i D Ψ D r Q i Q Ψ Q (2.11) which can be written more compactly as [ ] vabc = v fdq [ RS 0 0 R R ][ iabc i fdq ] d dt [ ] Ψabc. (2.12) Ψ fdq where v abc = v a v b, v fdq = v f 0, i abc = i a i b, Ψ abc = Ψ a Ψ b etc.. (2.13) 0 v c i c Ψ c The flux linkages are Ψ a L a L ab L ac L af L ad L aq Ψ b L ba L b L bc L bf L bd L bq Ψ c Ψ f = L ca L cb L c L cf L cd L cq L fa L fb L fc L f L fd L fq Ψ D L Da L Db L Dc L Df L D L DQ Ψ Q L Qa L Qb L Qc L Qf L QD L Q i a i b i c i f i D i Q (2.14) which can be written more compactly as [ ] [ Ψabc LSS = Ψ fdq ][ ] L SR iabc. L RR i fdq (2.15) L RS The inductance matrix in (2.14) is symmetrical since mutual inductances are reciprocal according to the Neumann formula [22], [23]. In a salient pole synchronous machine the air gap length is not constant around the rotor. That makes all inductances that involve a stator winding dependent on the rotor angle θ that changes with time when the rotor rotates. The stator slots are normally so small compared to the rotor poles that θ does not matter for the rotor inductances. Therefore, L f, L fd, L D and L Q are independent of θ but they change with the degree of saturation of the magnetic materials. Saturation will not be considered here. The generator is assumed to be magnetically symmetrical such that there is no net flux linkage between the d and q axis rotor windings. That means that 20 L fq = L DQ = 0. (2.16)

21 2.3.2 Inductances in the Stator Reference Frame of Salient Pole Synchronous Machines It is here assumed that the poles have been shaped such that the radial air gap permeances and, therefore also the inductances, in the stator reference frame vary approximately harmonically around average values around the rotor [24]. A consequence of that is that the amplitudes of the stator self and mutual inductances are equal [25]. The inductances in the stator reference frame are L a = L s L m cos2θ (2.17) L b = L s + L m cos2(θ π 6 ) (2.18) L c = L s + L m cos2(θ + 7π 6 ) (2.19) L ab = M s L m cos2(θ π 3 ) (2.20) L ac = M s L m cos2(θ + π 3 ) (2.21) L bc = M s L m cos2θ (2.22) L af = M f sinθ (2.23) L bf = M f cos(θ π 6 ) (2.24) L cf = M f cos(θ + 7π 6 ) (2.25) L ad = M D sinθ (2.26) L bd = M D cos(θ π 6 ) (2.27) L cd = M D cos(θ + 7π 6 ) (2.28) L aq = M Q cosθ (2.29) L bq = M Q cos(θ 2π 3 ) (2.30) L cq = M Q cos(θ + 2π 3 ) (2.31) Transformation of Electrical Parameters to the Rotor Reference Frame The rotating magnetic field created by the three stator phase currents can be obtained by only two currents, i d and i q, whose windings are fixed relative to 21

22 the rotor. Such windings are fictitious since the rotor with fixed magnetization would not be able to induce any voltage in these windings. Nevertheless, the concept of d and q components of stator currents, voltages, inductances and flux linkages are useful as mentioned in section 2.2. Fictitious windings do not have to have an integer number of turns. Here it is assumed that the d and q windings each have a factor k more turns than each of the stator windings have. Together with a chosen k, the requirement that i d and i q should give the same mmf as the stator phase currents determines a transformation matrix from i a, i b, i c to i d and i q under balanced conditions, i.e. such that i a + i b + i c = 0. To get a transformation that works also under unbalanced conditions, a fictitious component i 0 = o(i a +i b +i c ) is defined. For θ from the a axis to the q axis, and for arbitrary k and o, the transformation matrix becomes P = 1 sinθ sin(θ 2π 3 ) sin(θ + 2π 3 ) cosθ cos(θ 2π k 3 ) cos(θ + 2π 3 ). (2.32) o o o The inverse of P is P 1 = 2k 3 1 sinθ cosθ 2o sin(θ 2π 3 ) cos(θ 2π 3 ). (2.33) sin(θ + 2π 3 ) cos(θ + 2π 3 ) In the special case o =1/ 2 and k = 3/2, the transformation matrix becomes orthogonal and is given by P = 2 sinθ sin(θ 2π 3 ) sin(θ + 2π 3 ) 3 cosθ cos(θ 2π 3 ) cos(θ + 2π 3 ). (2.34) The same matrix can be used for transformation of currents, voltages and flux linkages between the stator reference frame and the rotor dq reference frame according to i dq0 = Pi abc, i abc = P 1 i dq0 (2.35) v dq0 = Pv abc, v abc = P 1 v dq0 (2.36) Ψ dq0 = PΨ abc, Ψ abc = P 1 Ψ dq0 (2.37) o 1 2o where i dq0 = i d. (2.38) i q i 0 22

23 The other vectors in (2.35), (2.36) and (2.37) are defined analogously. A consequence of the orthogonality is that the transformation is power invariant, i.e. v T dq0 i dq0 =(Pv abc ) T Pi abc = v T abc PT Pi abc = v T abc i abc. (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 the left gives Ψ dq0 = PL SS P 1 i dq0 + PL SR i fdq (2.40) Ψ fdq = L RS P 1 i dq0 + L SR i fdq (2.41) which can be written as [ ] [ ][ ] Ψdq0 LS M idq0 = Ψ fdq M T L R i fdq (2.42) where M T = M T if k = ± 3/2. Equation (2.42) can be written more explicitly as Ψ d L d 0 0 L df L dd 0 i d Ψ q 0 L q L qq i q Ψ 0 Ψ f = 0 0 L i 0 L fd 0 0 L f L fd 0 i f. (2.43) Ψ D L Dq 0 0 L Df L D 0 i D Ψ Q 0 L Qq L Q i Q The values of the mutual inductances between the d and q windings and the rotor windings depend on k but not o. For arbitrary k and o, (2.43) can be written as L s + M s + 3L m M f 2k 3M D 2k 0 3M Q 2k 0 L s + M s 3L m L s 2M s km f 0 0 L f L fd 0 km D 0 0 L Df L D 0 0 km Q L Q. (2.44) which is an unsymmetrical and therefore unphysical inductance matrix unless k = ± 3/2. In spite of that, the traditional choice is k = 3/2. Some authorities and machine designers think it is advantageous for a number of reasons, partly because it gives the fictitious stator current the same magnitude as any of the real phase currents under balanced conditions [25]. However, this choice is associated with one set of equations that are valid in SI units and another that is valid in per unit [26], [20]. This can be confusing. From now on, only k = 3/2 will be used in this thesis. The minus sign in L df and L dd is a consequence of the opposite directions of the f and d axes. This, in turn, is a 23

24 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 P from the left gives Equation (2.42) in (2.45) gives v dq0 = PR S P 1 i dq0 P d dt where v dq0 = PR S P 1 i dq0 P d dt ( P 1 Ψ dq0 ) (2.45) v fdq = R R i fdq d dt Ψ fdq. (2.46) ( P 1 (L S i dq0 + Mi fdq ) ) = R S i dq0 P d dt P 1 (L S i dq0 + Mi fdq ) L S di dq0 dt P d dt P 1 = P dp 1 dθ dθ dt M di fdq dt (2.47) = ω (2.48) It can be noted that the signs in (2.48) are opposite those obtained when the q axis is lagging the d axis. Equation (2.42) in (2.46) gives v fdq = R R i fdq d dt (MT i dq0 + L R i fdq ) = R R i fdq M T di dq0 dt L R di fdq dt (2.49) According to (2.47), (2.48), (2.49), (2.42) and (2.43) the voltage equations can be written explicitly as di d v d = ri d + ω(l q i q + L qq i Q ) L d dt L di f df dt L di D dd dt v q = ri q ω(l d i d + L df i f + L dd i D ) L q di q dt L qq v 0 = ri 0 L 0 di 0 dt v f = r f i f + L f di f dt + L df di D v D = r D i D + L D dt di Q v Q = r Q i Q + L Q dt di d dt + L di D fd dt + L dd di d dt + L fd + L qq di q dt di f dt = 0 di Q dt = 0 (2.50) At steady state operation, the rotor angular velocity is constant, and the time derivatives are zero so that the damper winding currents are also zero. In that 24

25 case the first two equations in (2.50) are reduced to v d = ri d + ωl q i q v q = ri q ω(l d i d + L df i f ) (2.51) Since the phase difference is 90 between the d axis and the q axis, the fictitious current and voltage can be expressed with the phasors and ī fic = i d + ji q (2.52) v fic = v d + jv q. (2.53) The magnitude of ī fic in relation to the amplitude of i a at steady state under balanced conditions can be obtained under the assumption that and that i a = îcosωt = îcos(θ θ 0 ) θ = ωt + θ 0 (2.54) i b = îcos(ωt 2π 3 )=îcos(θ 2π 3 θ 0) i c = îcos(ωt + 2π 3 )=îcos(θ + 2π 3 θ 0). (2.55) Equation (2.55) inserted in (2.35) gives which combined with (2.52) gives that i d = 3 2k îsinθ 0 i q = 3 2k îcosθ 0 (2.56) Analogously, i fic = 3 2k î = 3 2î. (2.57) v fic = 3 3 2k ˆv = ˆv. (2.58) 2 The equations (2.57) and (2.58) combined with the RMS values I = î/ 2 and V =ˆv/ 2give i fic = 3I v fic = 3V. (2.59) 25

26 Therefore, division of (2.51) by 3 and insertion of L df = 3M f /(2k) from (2.44) with k = 3/2 converts (2.51) to equations for RMS values according to V d = ri d + ωl q I q V q = ri q ωl d I d + ω M f 2 i f (2.60) In (2.60) ωl q = X q, ωl d = X d, and if ω M f 2 i f is denoted with E, (2.60) can be written as V d = ri d + X q I q V q = ri q X d I d + E. (2.61) Finally, with r = R, Ī = I d + ji q and V = V d + jv q, the equations in (2.61) can be combined to the complex steady state stator voltage equation (2.1). 2.4 Linearization of the Equation of Motion of Load Angle Oscillation The swing equation of motion of the rotor of a synchronous generator has been used in Paper V and is given by ([25]) J dω m dt = T m T e (ω m ω m,s ) D ω m,s (2.62) where J is the mass moment of inertia, ω m is the mechanical angular speed, ω m,s is the steady state operation/synchronous value of ω m, T m is the mechanical torque on the rotor, T e is the electrical torque developed by the generator, and D is some positive damping factor that can account for the effect of damper circuits. A more general equation of motion would include damping terms due to bearing friction and windage that increase with ω m but are independent of the difference ω m ω m,s. Normally, for load angle oscillations, ω m ω m,s.in that case bearing friction and windage are approximately constant and can be included in T m. Multiplication of (2.62) by ω m gives ω m J dω m dt = 4 dω ωj p2 dt = P m P e (ω ω s ) ω ω s D P m P e (ω ω s )D (2.63) where p is the number of rotor poles, ω = p/(2ω m ) is the electrical angular frequency, and ω s is the steady state operation value of ω, P m is the net mechanical power on the rotor, P e is the electrical power developed by the 26

27 generator. For numerical solution, (2.63) can be written as a system of two equations, dθ dt =ω dω dt = P (2.64) p2 m P e (ω ω s )D. 4ωJ In what follows, δ IB is the load angle to the infinite bus, δ IB,0 is the initial steady state value of δ IB, and β = δ IB - δ IB,0. A phasor diagram with angles is given in Paper V. Equation (2.64) can, with use of Ω=ω ω s =dβ/dt, be rewritten in terms of β as dβ dt =Ω dω dt = d2 β dt 2 = P p2 m P e D dβ dt 4(ω s + dβ dt )J. (2.65) In the case of small β, (2.65) can be linearized around β = 0. Typically, dβ/dt << ω s if the rotor oscillates at a natural frequency. Consequently, ω s + dβ dt ω s = constant which can be used in front of J in (2.65). With the abbreviation k for p 2 /(4ω s J), the linearized electrical power can be expressed as P e =P e,s + dp e(β=0) dβ β where P e,s is the steady state value. Similarly, the linearized mechanical power can be written as P m =P m,s + ΔP m. Considering that the steady state parts of P e and P m cancel each other, the linearized equation of motion of a generator connected to a strong grid becomes the same as for a mass point connected to a damper and a spring, i.e. d 2 β(t) dt 2 + 2C dβ(t) dt + Gβ(t)=F(t)=kΔP m (t) (2.66) where 2C = kd and G = k dp e(β=0) dβ = k dp e(δ IB =δ IB,0 ) dδ IB is a torsional stiffness divided by J at β =0.Gcan be calculated with help of the synchronizing power coefficient (2.10). The solution of (2.66) is β = β p + β h (2.67) where β p is a particular solution and β h is a solution of (2.66) with F replaced by 0 on the right hand side. That does not mean that β h is independent of F, as will be clear from (2.73) and (2.74) below. With a damping less than the critical, there are oscillations, and β h is β h =(Acosω n t + Bsinω n t)e Ct (2.68) where ω n = G C 2 (2.69) 27

28 is the load dependent, natural angular frequency of β without the disturbance F(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)=F 0 sinω F t, the particular solution is β p = ± F 0 r sin(ω Ft + φ) (2.70) where r = (ωf 2 G)2 + 4C 2 ωf 2 (2.71) and φ = arctan 2Cω F ω 2 F G (2.72) with plus sign in (2.70) if G>ωF 2 and minus sign if G<ω2 F. The coefficients of β h in (2.68) are and A = β(0)+f 0 2Cω F r 2 (2.73) ( dβ(0) B = + β(0)c + F ) 0ω F 1 dt r 2 (2C 2 + ωf 2 G) (2.74) ω n where β(0) and dβ(0) dt are initial values. A value of ω F that gives a local maximum of the amplitude of β is a resonance angular frequency, ω r. When the damping approaches zero, both ω n and ω r approach G, and the amplitude of β approaches infinity. When there is damping, the transient, represented by β h, becomes negligible sooner or later. Then β p remains with resonance angular frequency ω r = G 2C 2. This inserted in (2.70) gives that the amplitude of β p at ω r is F 0 ˆβ p,r = 2C G C = F 0. (2.75) 2 2Cω n The period of the load angle oscillation is approximately t p =2π/ω n. 28

29 3. Methods Some of the methods used to produce the results in the papers are here presented in more detail than in the papers. 3.1 Effective Permeabilities in a Laminate In electric machines, the laminated cores consist of sheets of electric steel insulated from each other with relatively thin layers of varnish or oxide. Fig. 3.1 shows a laminate piece consisting of one steel layer with thickness t s and one varnish layer with thickness t v. Figure 3.1. A piece of a laminate with fluxes parallel to each other and to the lamination planes For the purpose of deriving an effective laminate permeability in directions parallel to the laminations, the magnetic fluxes, Φ s in the steel and Φ v in the varnish, are assumed to be homogeneously distributed in each material layer and directed in the same direction parallel to the laminations. It is also assumed that there is no significant current density, J, in the laminate. In that case, Ampere s law with line integration of the magnetic field strength, H, 29

30 along the boundary Γ in Fig. 3.1 gives H dl = J ds = H s Δx H v Δx = R s Φ s R v Φ v = 0 (3.1) Γ S where S in the surface integral is a surface bounded by Γ, and R s = Δx A s μ s, R v = Δx A v μ v (3.2) are the reluctances of the steel and varnish layers. The flux multiplied by the reluctance the flux meets is the magnetic potential difference between the two ends separated by the distance Δx. According to (3.1), the potential difference is the same in both material layers. From this, an effective reluctance parallel to the lamination planes can be defined such that it gives a magnetic potential difference in a homogeneous approximation of the laminate that is the same as the magnetic potential difference in the real laminate, i.e. (Φ s + Φ v )R i,ef f = Φ s R s, i = x or y (3.3) with Δl R i,ef f =, i = x or y (3.4) (A s + A v )μ i,ef f 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 μ i,ef f = μ st s + μ v t v t s +t v = kμ s +(1 k)μ v kμ s, i = x or y (3.5) in the lamination planes where k is the stacking factor defined by k = t s t s +t v. (3.6) The inverse of the reluctance is called permeance. The expression for μ i,ef f is more directly obtained by defining an effective permeance to be equal to the sum of the plate permeance and the varnish permeance, i.e. i,ef f = 1 R i,ef f = 1 R s + 1 R v, i = x or y. (3.7) For simplicity the approximation at the end of (3.5) can be used when the stator is not very saturated. Equation (3.5) motivates scaling down the B values of the B H curves in the laminate plane by k. This has been used in simulations at low field current. This implies e.g. that if the flux density in a steel plate is B s, the flux density in the FE model is k B s. If there are eddy currents in the laminate planes (3.5) is only approximately correct. In the following derivation of the effective permeability in the stacking direction, conservation 30

31 of flux is assumed in that direction. The series reluctance of one plate and one varnish layer is then set equal to an effective reluctance R z,ef f = R s + R v = t s +t v = t s + t v (3.8) Aμ z,ef f Aμ s Aμ v which gives the effective laminate permeability μ z,ef f = t s +t v t s μ s + t v μ v. (3.9) In simulations with saturation described in Paper II, the nonlinear, anisotropic lamination model as described in [27] has been used with a stacking factor. It should 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 Force In Paper II the finite element program ANSYS Maxwell 3D has been used for calculation of the axial force on the stator by means of the principle of virtual work [28], [29], [30]. According to the manual of ANSYS Maxwell only the change of coenergy W co at constant current in the virtually distorted elements on the boundary of the virtually moved object is taken into account in the force F z = W co z = constant current z V H ( B dh)dv (3.10) 0 where H is the magnetic field intensity. For comparison the axial force has also been calculated by integration of the Maxwell stress tensor T over some surface 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 outward directed and location dependent unit normal vector ˆn in cylindrical coordinates is F = S 1 μ 0 S 1 μ 0 T ˆndS = B 2 r 1 2 B2 B r B ϕ B r B z B ϕ B r B 2 ϕ 1 2 B2 B ϕ B z B z B r B z B ϕ B 2 z 1 2 B2 S ((B ˆn)B ˆn 2 B2 )ds ˆn r ˆn ϕ ds = ˆn z (3.11) where B is the magnetic flux density. The surface S was chosen to be the boundary of a hollow cylinder sector enclosing the modeled stator sector. All 31

32 parts of the boundary except the cylindrical surface close to the stator wall in the 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 large but with opposite signs because of the periodicity and the opposing normal directions. The axial force on the outer radial surface is zero because of the tangential field boundary condition. The axial forces on the boundary end planes at constant z are pointing towards the machine. The total axial force is the sum of the contributions from the inner radial surface and the two end planes at constant z. 3.3 Calculation of Losses in the Steel of the Stator Core and Clamping Structure in a Synchronous Generator An overview of iron loss models can be found in [33]. Single valued magnetization data curves and tabulated core loss data from a steel manufacturer have been used in the simulations for calculation of core loss after the magnetic field has been calculated because, until recently, it has not been possible to make use of hysteresis data in the field calculation ANSYS Maxwell. One model of harmonically time varying iron power loss density in a silicon steel laminate is the loss separation modell [33], p = c h f ˆB 2 + c e f 2 ˆB 2 + c ex f 1.5 ˆB 1.5 (3.12) where f is the frequency and ˆB is the peak magnetic flux density. From left to right the terms are hysteresis loss density, p h, eddy current loss density, p e, and excess loss density, p ex. Excess loss density is a term that was added in order to explain the difference between measured and computationally predicted losses in silicon steel [33]. At a given frequency the model can be expressed as p = a ˆB 2 + b ˆB 1.5. (3.13) The coefficients a and b can be determined by fitting the model to measured loss density. A comparison between (3.12) and (3.13) shows that the first term in (3.13) is the sum of hysteresis loss density and eddy current loss density whereas the second term in (3.13) must be excess loss density. The eddy current loss density caused by flux parallel to the lamination planes can be approximated by p e = σ 6 (π fhˆb) 2 (3.14) where σ is the conductivity and h is the plate thickness [34]. The hysteresis loss density can then be estimated by a ˆB 2 p e. Fitting p = a ˆB 2 + b ˆB 1.5 to data for steel M350-50A from Surahammars Bruk AB [35] or Cogent via Ernest Matagne [36] gives a negative b with a relatively small magnitude that gives a hardly noticeable difference to the fit. In addition, 32

33 a positive b is expected [33]. Hence, the available loss data does not seem to support the distinction of excess loss from the other losses. Furthermore, the uncertainty is large in given loss data. At 1.5 T and 60 Hz the loss density can be 30 % higher than the typical value for steel M350-50A [36]. Therefore the simplified model, p( ˆB)=a ˆB 2, have been used in Paper II. Fig. 3.2 shows fits to 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 used in Paper II. First the core loss density given in units W/kg was converted to W/m 3 by ρ steel t steel + ρ varnish t varnish ρ steel t steel p = p m ρ laminate = p m p m = p m kρ steel t steel +t varnish t steel +t varnish (3.15) where p m is the given core loss density, and k = 0.500/0.543 is the assumed stacking factor of the stator laminate in Paper II. The coefficient, a, was determined as the solution of a least squares problem according to n i=1 (a ˆB 2 i p i) 2 = 0 a = n i=1 ˆB 2 i p i a n i=1 ˆB 4 (3.16) i where n is the number of data points, and p i is the core loss density at peak magnetic flux density ˆB i. The data from [35] give a W/(T 2 m 3 ), and the data from [36] give a 9685 W/(T 2 m 3 ). 40 Total loss density (kw/m 3 ) P Cogent P C.fit P Sura P S.fit Flux density (T) Figure 3.2. The core loss density of a laminate of steel M350-50A manufactured by Surahammars Bruk AB [35], a subsidiary of Cogent Power. The core loss in a volume V is approximately P C = a ˆB 2 dv. (3.17) This in turn can be approximated by P C = max V V ab(t) 2 dv (3.18) 33

34 where B(t) is the instantaneous value of B at time t. In Paper II the core loss was estimated according to (3.18). This underestimates the loss since ˆB is not reached simultaneously everywhere within a volume that is not infinitely small. However, the slot pitch wide slices of the teeth and yoke are relatively small and such that the volume integral of the square of the instantaneous flux density in each slice oscillates between a minimum and a much (, five to nine times,) 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 leakage fields. These eddy current losses have been calculated both in the laminated stator core and in selected slices of the clamping structure by p E = J J/σdV (3.19) where J is the eddy current density. V 3.4 The Finite Element Method The finite element method (FEM) is a numerical method for solving (partial) differential equations with boundary conditions. The strength of the method is that the domain, i.e. the region in which the solution is calculated, and its contents can have complicated shapes. Some solvers allow nonlinear and anisotropic materials. In common for FE methods is that the domain is divided into 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 an edge, they share the value of the degree of freedom in that location. Values between nodes and/or edges are approximated by interpolation. On a fundamental level, the idea of the FEM is that even if the field changes much across the domain, the field change across a subdomain goes to zero if the size of the subdomain goes to zero. The solution u m in element number m is approximated by a linear combination of simple functions called base functions and can be expressed as u N m(r,t)= u m, j s m, j (r,t) (3.20) j=1 where, s m, j is a base function, one per degree of freedom. Each base function is 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 cubic polynomial. The coefficients u m, j are to be determined by the FEM so that u m approximates u in the space filled by element m. The coefficients of the polynomials are uniquely determined by the solution of an equation system considering the boundary conditions on the outermost elements, continuity 34

35 conditions at the borders between the elements, and the partial differential equation 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 of FEM. Galerkin s and Rayleigh-Ritz s methods are classical ways to obtain an approximate solution to a differential equation with boundary conditions. In classical 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 parameter the solver calculates is the magnetic vector potential, A, in all magnetic problems. After A has been determined, the magnetic flux density can be calculated from B = A. In 3-D magnetic problems a current vector potential is used in current conducting regions, and a scalar potential of the magnetic field strength, H, is used in the whole domain [37] Domain and Boundary Conditions in Magnetic Analysis of an Electric Machine with ANSYS Maxwell The magnetic field is important both inside the machine materials and the surrounding medium, for simplicity assumed to be air. Therefore a FE calculation domain contains at least a part of the machine and air. For simulation of cylindrical 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 mesh is fixed relative to the rotor. Outside the faceted cylinder the mesh is fixed relative 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 of the machine, it is sufficient to include just one spatial period of the machine and air in the calculation domain. Periodic (matching) boundary conditions can then be applied to the two surfaces, called master and slave, that are one spatial period from each other. Fig. 4.1 in section 4.1 shows an example where the spatial period spans four pole pitches. The domain can be reduced further to contain only half a spatial period if antiperiodic boundary conditions can be used. This is usually the case in 2-D FE simulations as long as the number of stator slots per pole and phase is an integer. Antiperiodic conditions force the magnetic field vector in an arbitrary point on the slave surface to have the same magnitude and opposite direction as the corresponding (mirror) point on the master surface located half a spatial period from the slave surface. This requires that the geometry of the slave surface must match the geometry of the master surface exactly. In 3-D the coil ends also must be considered. In some machine types they are such that at least one spatial period of the machine must be modeled. In 3-D the domain can be halved if the machine has a geometrical and magnetic symmetry plane along which the motion takes place. Boundary conditions must be used on the boundary of the calculation domain. 35

36 Some boundary conditions, like vector potential in 2-D, can also be used on objects within the domain. Boundary conditions available in transient magnetic analysis in ANSYS Maxwell 2D are vector potential, balloon, odd symmetry, even symmetry, master and slave. The balloon condition is a way to simulate that the flux lines can extend to infinity. Even symmetry gives normal flux, odd symmetry gives tangential flux. In a 2-D problem type with currents only in the z direction, the vector potential,a, only has a z component, A z. This can be set to any constant on the boundary in order to force tangential flux there. Setting A z = constant on the boundary forces A z to be perpendicular to the boundary. This together with B = A and A z ẑza z = 0 (3.21) implies that B is parallel to the boundary. If the problem is symmetric about the z axis and the currents flow only in the ϕ direction, the vector potential has only the component, A ϕ. In this case A ϕ ˆϕA ϕ = A ϕ ρ A ϕ ρ (3.22) which is not zero in general. The only constant value A ϕ can have on the whole boundary that forces tangential flux is zero. In many 3-D cases the only boundary conditions the ANSYS Maxwell user need to apply in transient magnetic and harmonic eddy current FE analysis (FEA) of rotating electric machines are the periodic or antiperiodic conditions. Tangential H conditions in 2-D are expressed as Dirichlet conditions on the magnetic vector potential. Tangential H conditions in 3-D are automatically fulfilled unless some other boundary condition is applied. The tangential H condition can be expressed as a homogeneous Neumann condition on the scalar potential Ω, i.e. H ˆn = Ω ˆn = Ω/ n = 0 (3.23) where ˆn is a unit normal vector to the boundary. The Maxwell 3D user can apply current insulating boundary conditions in order to avoid modeling and meshing thin gaps between conductors. Balloon boundary conditions are not available in 3-D. 3.5 The Method of Separation of Variables The method of separation of variables is also called Fourier s method. It is a method for solving boundary value problems that are linear partial differential equations with boundary conditions. The solution is a field, i.e. a function of space position r = (x,y,z) and, in some cases, time, t. The method is supported 36

37 by the fact that a square integrable function can be expressed as a Fourier sum that, when the number of terms goes to infinity, converges pointwise towards the function almost everywhere. [38], [39], [40]. Most of the following text about Fourier s method is based on information in [41]. In order for a boundary value problem to have a unique solution, the problem must be specified in a way that depends on the type of equation. Partial differential equations are classified as hyperbolic as the wave equation, parabolic as the diffusion equation or elliptic as Helmholtz equation. For uniqueness of the solution in a closed domain, these types of equations need different initial conditions (in the whole domain) but the geometrical boundary conditions are of the same types. They are Dirichlet, Neumann or Robin (Churchill) conditions. A Dirichlet condition is a specification of the field value, a Neumann condition is a specification of the normal derivative of the field, and a Robin condition is a specification of a linear combination of the field value and the normal derivative. Different types of boundary conditions can be applied to different parts of the boundary. For Laplace s equation, solutions can differ by a constant if only Neumann conditions are used. Fourier s method is used in geometries for which there are coordinate systems such that, on every part of the the boundary, one and only one coordinate is constant. Each term in a Fourier series can be expressed as the product of functions, each of which depend on only one variable. Of the functions in the product, one can be considered as a Fourier coefficient, and the others are eigenfunctions. A Fourier series is thus a linear combination of composite or single variable eigenfunctions. This is analogous to expressing a vector as a linear combination of base vectors. The eigenfunctions depend on the differential equation, the type of boundary conditions and the geometry. The eigenfunctions used in each Fourier series are a subset of a complete set of eigenfunctions. That means that it is possible to approximate any square integrable function arbitrarily well in norm with a finite linear combination of eigenfunctions from the complete set in the domain where the function is defined. Eigenfunctions used in Fourier series are pairwise orthogonal. Because of the orthogonality, each Fourier coefficient depends only on the eigenfunctions in the same term in the Fourier series. Therefore the relative magnitudes of the Fourier coefficients can be interpreted as measures of the relative importance of modes of which the Fourier series is composed. Examples of functions that can be parts of complete, orthogonal eigenfunction systems are the sine, cosine, Bessel, spherical Bessel and spherical harmonic functions. An eigenfunction, X, is in general, if not always, chosen to satisfy boundary conditions such that X =0or X X n =0or n + αx = 0. Such conditions are called homogeneous. They are special because they are automatically satisfied by functions expressed as linear combinations of the eigenfunctions. Fourier s 37

38 method gives the solution as one Fourier series for time independent problems with periodic and/or homogeneous boundary conditions on all parts of the boundary except at most two where one and the same coordinate is constant. A boundary condition function at a constant coordinate is expressed as Fourier series of eigenfunctions that depend on another coordinate and the types of boundary conditions on boundaries where another coordinate is constant. If the problem has nonperiodic inhomogeneous boundary conditions on two or more parts of the boundary, the field can be written as a sum of Fourier series such that each series satisfies the boundary condition requirements mentioned above for one series. This is possible because of the linearity of the equations. If all geometrical boundary conditions are periodic and/or homogeneous, the solution of a time dependent problem can be expressed as one Fourier series. If a time dependent problem has at least one nonperiodic inhomogeneous boundary condition, the field can, in the first step, be expressed as the sum of two contributions. The first contribution satisfies all geometrical boundary conditions and e.g. Laplace s equation. The second contribution satisfies the time dependent problem with periodic and/or homogeneous geometrical boundary conditions. The field that satisfies Laplace s equation can in turn be written as a sum of contributions such that each contribution satisfies the boundary requirements for one Fourier series. A truncated Fourier series of a function, f, has the same values as the eigenfunctions 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 sum does not converge towards f on the boundary no matter how many terms are included in the series, even if f is continuous [41]. It may be considered as an example of Gibbs phenomenon [41] although it is perhaps mostly associated with a discontinuity of f [42]. In the author s numerical examples the maximum overshoot of a Fourier sum appears to converge towards max( 2Si(x) π ) % where Si(x) = x du. 0 sinu u Of special interest for the work in Paper 3 in this thesis is the time harmonic complex form of the wave equation, also called reduced wave equation since the time dependence has been removed. The equation is a special case of the homogeneous Helmholtz equation, 2 f + κ f = 0. For a non-negative and non-zero κ one can distinguish between two cases of this equation. In the first case κ is an eigenvalue which is a geometrical parameter that can have an infinite number of values corresponding to an infinite number of eigenfunctions. In the other case κ is a complex number that depends only on the frequency and the material properties. In this case the solution is unique for a fixed material and frequency [43]. The uniqueness is not surprising considering the uniqueness of the solution of the time dependent wave equation and the equivalence of the time dependent and the reduced wave equations for harmonically time varying fields. The removal of the time dependence from the 38

39 time dependent wave equation is possible under two conditions. First, the material properties must be constant so that the time dependent wave equation is linear, and second, the time dependence of the fields must be harmonic which is possible when the equation is linear. When the equation is linear, a linear combination of solutions, corresponding to e.g. different initial conditions, is also a solution to the equation alone. In particular, if one solution is u(r,t) = û(r)cos(ωt φ(r)), another solution is v(r,t) =ˆv(r)sin(ωt φ(r)). A complex linear combination of these solutions is w(r,t) = ŵ(r)e j(ωt φ(r)). After this solution has been inserted into the time dependent Maxwell s equations or the wave equation, and the time differentiations have been performed, one can 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 is the time independent equations in complex form. Suitable for the complex wave equation are boundary conditions in complex form [44]. It can be noted that in the special case of harmonic time variation with given frequency, initial conditions on both the field and its first time derivative would overdetermine the field. 3.6 Estimation of Frictional Power Loss in a Slider Bearing Tilting pad thrust bearings are common in hydropower generators because of their ability to carry high load at low friction [45]. A tilting pad bearing is a kind of slider bearing. In Paper II a the frictional power loss in a tilting pad thrust bearing was estimated with one dimensional theory of combined Poiseuille and Couette flow [46]. The flow is the result of a horizontal plane surface moving with speed U in the x direction and a plane plate inclined and at rest relative to the horizontal plane. Fig. 3.3 shows the wedge into which the 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, (h i h o )/L, is of the order of 10 4 in a tilting pad bearing [46]. The normal force is ( ) hi f n h o F n = ηul2 W h 2 (3.24) o where η is the viscosity, W is the width of the solid bodies in the y direction perpendicular to the flow. The frictional force is F f = ηul2 W h o f f ( hi h o ). (3.25) The functions f n and f f are given in detail in [46]but with other names. For Paper II it was sufficient to note that for a given ratio h i /h o, h o from (3.24 inserted into (3.25) gives that F f F n. Since also the frictional power loss, 39

40 Figure 3.3. Lubricant dragged to the right through the wedge creates a lifting force and frictional force on the inclined surface. P f, is proportional to F f, the conclusion is that P f = k F n where k depends on h i /h o, η, U, L, W and some, possibly weighted, average radius, r av of the bearing such that P f = r av F f. 3.7 Estimation of Windage Power Loss In the Discussion of Paper II, (3.26) was the basis for the statement that the windage power loss can be reduced by an increased air gap. The air gap gets very large for the part of the rotor that protrudes from the stator bore after axial displacement 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 smooth cylindrical bore can be estimated by [47] P w = 2πρr4 ω 3 L, Re = ρωrδ (3.26) Re η 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 gap between the cylinders, and η is the viscosity. For turbulent flow (3.26) is modified by replacement of 2/Re by the skin friction coefficient, C d, that can be determined iteratively from 1 = ln(Re C d ) (3.27) Cd C d decreases when Re increases. The formulas for the smooth cylinders heavily underestimate the windage losses in real machines. Therefore windage loss in rotating electric machines is usually determined from experiments combined with computational fluid dynamics (CFD) simulations [48]. 40

41 3.8 Experimental Equipment and Measurements Information about the experimental generator is given in Paper I, [49]. An asynchronous motor rated 75 kw and controlled by a frequency converter supplied the generator with mechanical power. Stator phase voltages, stator phase currents, field current, and axial flux densities were measured with sensors with voltage outputs. The output signals were collected with LabVIEW and data acquisition equipment from National Instruments [50]. LabVIEW was also used for calculation of electrical parameters such as phase angles, instantaneous active and reactive power. That information was used during the tests for setting the points of operation and for recording the data when the generator was reasonably stable. A problem was load angle oscillations at high field current when the generator was connected to the power grid. All sensor outputs were voltages. The signals were recorded with a sampling frequency of 5 khz during 2-6 s for each point of operation. Shielded cables were used for the output voltages from the sensors. The shielded sensor cables were connected to interface boxes made in the laboratory. Each such box was connected to a NI 9205 module. The field current was set by a laboratory DC power supply of model EA-PS W. The torque was controlled via a NI 9265 module connected to the frequency converter and powered by a A 12 V,7.2 Ah Lead-Acid battery Magnetic Field Sensors and Measurements of Axial Magnetic Flux Density The axial magnetic flux density on the stator clamping structure was measured by three each of the linear output Hall effect integrated circuit chip transducers of types Honeywell SS94A2 and SS94A2D connected in parallel for a common supply voltage, V s. Each chip has three pins, +, - and output. Laser trimmed thin film and thick film resistors compensate for temperature variations and minimize sensitivity variations from one device to the next. Measured sensor input resistance including cables, contacts and solders was Ω. A68Ω resistor and a 11.1 V Lithium polymer battery was connected in series with the sensors to supply the sensors with about 8 V. The output voltage, V o, of the sensors is a linear function of the magnetic flux density in a V o span proportional to V s. The linearity error of a sensor curve of V o as a function of flux density is defined as the maximum V o deviation between the sensor 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 output voltage, V o,0 V s /2. The axial magnetic flux density, B z, for each sensor was calculated by B z (t)=(v o V o,0 )/s, s = ΔV o ΔB = 0.625V s ΔB (3.28) 41

42 where s is the sensitivity, ΔV o is the span and ΔB is the range according to the manufacturer s data sheet. See Table 3.1. Under the assumption that the axial flux density varied sinusoidally with time, the null was approximated by the time average V o,av of V o for the work in Paper I. To approximate V o,0 with V o,av can be a significant source of error considering that the pole shoes have different axial lengths, almost an air gap from the shortest with 142 plates to the longest with 146 plates of thickness 2 mm. More important than the differences between pole lengths is the differences between the axial coordinate (z) levels of the lower pole ends where the sensors were mounted. The difference between the axial coordinate levels was found to be up to six mm. On the other hand, asymmetry of the rotor was not included in FEA. Therefore, it is not obvious that the approximation of V o,0 would increase the disagreement between measured and simulated B z. The pole shoe lengths were measured after 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 SS94A2D Supply voltage, V s (V) 6.6 to to 12.6 Supply current (ma) 13 to to 30 Max output current (ma) 1 1 Response time (μs) 3 3 Span of V o V s V s Flux density range (mt) -50 to to 250 Sensitivity (V/T) Linearity error (% of span) -1.5 to 0, typ to 0, typ V o (V) at 0 T, V s = 8 V DC, 25 C 4.00± ±0.04 Temperature error (% of span/ C) of null ±0.02 ±0.007 Temperature error (% of span/ C) of gain ±0.02 ±0.02 Temperature range ( C) -40 to to 125 The Honeywell sensors were calibrated after the main measurements, but the calibration data was used for the work in Paper II. The calibration would not have been conducted if the similarity between measurements and FEA had been good in Fig. 3 in Paper II [51]. When individual sensitivities and nulls were used for each sensor, there was improvement in the similarities between measurements and FEA with respect to the slopes of the curves in Fig. 3 in Paper II but hardly any improvement in the absolute values of the differences between measured and simulated axial flux densities. Hence, it seems like the main cause of the differences is not the asymmetry of the rotor poles. Calibration showed that the sensors were almost identical, as expected from the specifications from the manufacturer. Fig. 3.4 for one SS94A2 sensor and Fig. 3.5 for one SS94A2D sensor show V o versus flux density measured by a 42

43 teslameter. Fig. 3.6 shows the null versus V s for the calibrated sensors. Fig. 3.7 and Fig. 3.8 show the sensitivities of the SS94A2 sensors and the SS94A2D sensors respectively. 5 Output voltage (V) V r1 7.0V V r1 7.5V V r1 8.0V V r1 8.5V Magnetic Flux Density (T) Figure 3.4. Sensor output voltage as a function of magnetic flux density measured by a teslameter at four different values of supply voltage for a sensor of type SS94A2. Equipment for the calibration was 1) Lake Shore Model 410 Teslameter with a transverse probe, accuracy ±2 % from 0 to 2 T, 2) LabVIEW for reading sensor signals and supply voltage, 3) One 12 V, 7.2 Ah Lead-acid battery to supply the sensors, 4) One Vishay Spectrol Model turns, 5 kω potentiometer for adjustment of supply voltage, 5) Two brick shaped permanent magnets, 6) Two L shaped pieces of magnetic steel, 7) Blocks of magnetic and non magnetic materials for adjustment of air gaps in the magnetic circuit. The experimental setup is shown in Fig For each air gap size in the test, the magnetic flux density was measured and noted on a piece of paper along a line in the middle of the large, upper air gap. Then the sensor blocks were tested one 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 supply voltage cables were intertwined but not shielded. Intertwining of the voltage supply cables reduced unwanted induced voltages with a torque independent period of about 0.73 s in the supply voltage. One hypothesis is that the noise was induced by leakage flux from the asynchronous machine that could work with 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

44 5 Output voltage (V) V t1 7.0V V t1 7.5V V t1 8.0V V t1 8.5V Magnetic Flux Density (T) Figure 3.5. Sensor output voltage as a function of magnetic flux density measured by a teslameter at four different values of supply voltage for a sensor of type SS94A2D. 4.4 Quiescent output voltage (V/T) Supply voltage (V) V q r1 V q r2 V q r3 V q t1 V q t2 V q t3 Figure 3.6. Quiescent output voltage as a function of supply voltage for all calibrated sensors. motor in a compressor [52]. However, the 0.73 s period noise amplitude in the supply voltage was only about 1 per mille and negligible for the sensor 44

45 56 54 Sensitivity (V/T) s r1 s r2 s r Supply voltage (V) Figure 3.7. Sensitivity as a function of supply voltage for the calibrated SS94A2 sensors Sensitivity (V/T) s t1 s t2 s t Supply voltage (V) Figure 3.8. Sensitivity as a function of supply voltage for the calibrated SS94A2D sensors. output. Fig shows the circuit board and the batteries that supplied first set of cheap (not Honeywell) Hall sensors that failed already during prelimi- 45

46 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 aluminium block. For the lowest fields, the aluminium pipe in the background was used on top of the aluminium block. nary measurements. For flexibility and protection, the Honeywell sensor chips were encapsulated in epoxy blocks. Wood blocks with a socket just big enough for a sensor block were glued to the lower clamping ring for stable and centered location of the sensor blocks on the clamping ring. Sensor blocks on the clamping fingers were fixed by wedges. The fixtures of four of the sensors on the lower clamping structure of the generator are shown in Fig 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 component when a rotor north pole was right in front of them and the stator current was zero, i.e. the generator was at no load. The z axis points upwards along the shaft. During all measurements the rotor was rotating anticlockwise as seen from above. This corresponds to the phase sequence a (blue), b (yellow) and c (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

47 Figure Circuit board for connection of Hall effect integrated circuit chips to a supply voltage. tical planes spanned by the shaft and the magnetic axes of phase c. Fig shows the locations of the sensors for measurement of axial flux density. In the sensor names the letter r refers to the clamping ring, and t refers to clamping finger tip or tooth. 47

48 Figure Slotted wood blocks and wedges fixed the sensor blocks to the clamping structure of the generator. Figure Positions of the Hall sensors for measurement of axial magnetic flux density. Sensor positions r1, r2, t1 and t2 were also used in simulations. 48

49 4. Results that Were Not Included in the Papers 4.1 Supplements to Paper II For Paper II [51], some figures intended for or suitable for the topics had to be skipped because of limitations on the number of figures the journal could allow. Here, some of those figures have been included. The FEA model used for calculation of B z 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 for calculation of B z on the clamping structure. Although the chosen constant permeability in the clamping structure made simulated and measured RMS B z agree well at no load and I f 12 A in figure 2 in Paper II, the agreement is not as good at load. Hysteresis could have something to do with that. Alternatively, the insulation between the stator lamination plates is not very good. Since hysteresis and macroscopic eddy currents cause damping and phase change of the electromagnetic field components, a small simulation study of the effect of macroscopic eddy currents on amplitude 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 in the fingers and only 3-5 in the ring it was possible to get good agreement between simulations and measurements as shown in Fig. 4.2, but the used low 49

50 permeability in the ring can be put in question since a permanent magnet is attracted to it roughly about as strongly as to the outer wall of the laminated stator core. Furthermore, the agreement was only checked at no load and I f 12 A. Without eddy currents in the simulations, B z would have its extreme values approximately on the d axis. The inclusion of eddy currents makes B z lag the d axis. A higher axial conductivity gives more damping, a steeper slope of the top of the B r curve and a larger simulated delay (lag) of B z on the ring than in measurements. In light of this study some loss phenomenon seems to be a plausible explanation for the fact that B z on the ring was lagging the stator current Ī at load with low load angles in measurements described in Paper I, [49]. By simple vector addition one could otherwise expect the phase of B z on the clamping structure to be between the phases of Ī f and Ī in every load case Flux Density (T) B r s B z t1 s 10 B z r1 s B r m B z t1 m 10 B z r1 m Time (ms) Figure 4.2. Calculated and measured radial flux density B r in the air gap in front of slice c, axial flux density B zr1 on Ring and B zt1 on the tip of F1i at no load with I f 12 A and the stator raised 7 mm. Last subscript meaning: s = simulation, m = measurement. Concerning the calculation of optimal axial rotor displacement, Fig. 4.3 shows a simplified picture of the axial forces in an electric machine. The magnetic force versus rotor displacement in Fig. 4.4 is just a mirrored version of Fig. 4 in Paper II. For accurate determination of the optimal rotor displacement, cubic polynomial fits were made to the magnetic force, F m, and the stator iron loss within a small z interval where the optimal displacement could be expected. Fig. 4.5 shows the frictional loss, the stator iron loss and their sum when the frictional loss is assumed to be 0.3% of the rated power. The optimal displacement, mm in the example, is the point where the slope 50

51 of the total loss is zero. The polynomial fit of the magnetic force was used in the definition of the frictional loss fit. Figure 4.3. Simplified picture for the definition of the axial forces and the axial coordinate. 4.2 Supplements to Paper V Fig. 4.6, 4.7 and 4.8 show the instantaneous eddy current loss density expressed in W/kg steel at steady state with power factor 1, overexcitation with β = β max =16, and overexcitation with β = β max =16 respectively. Considering the low assumed effective relative permeability of the stator core in the z direction, the stacking factor is about 0.9 = V steel /V lam. The loss in the laminate is assumed to be only in the steel, not the varnish. The density of 51

52 2 1.5 F z F z Fit Axial Force (kn) Axial Displacement of Rotor (mm) Figure 4.4. Axial magnetic force versus axial rotor displacement. electrical steel is assumed to be 7650 kg/m 3. The loss is P = p W/m 3,lam V lam = p W/kg,steel ρ steel V steel (4.1) This implies that the loss density in the laminate expressed in W/kg steel is p W/kg,steel = p W/m 3,lam V lam ρ steel V steel = p W/m 3,lam 6885kg/m 3 (4.2) For the solid steel clamping structure, the density has been assumed to be 7860 kg/m 3 which gives the loss density p W/kg,clamp = p W/m 3,clamp /7860 kg/m3. 52

53 Loss (W) P f P f fit P I P I fit P I+f Axial Displacement of Rotor (mm) Figure 4.5. Thrust bearing frictional loss, P f, stator iron loss, P I, and the sum of the losses versus axial rotor displacement. The frictional loss is assumed to be 0.3% of the rated power. 53

54 Figure 4.6. Eddy current loss density in the stator steel at a time point at steady state with power factor 1. Figure 4.7. Eddy current loss density in the stator steel at a time point when β = β max =16 at overexcitation. 54

55 Figure 4.8. Eddy current loss density in the stator steel at a time point when β = β max =16 at underexcitation. 55