Thermal Modelling of Permanent Magnet Motor for Traction

by

Wallerand Faivre d’Arcier Laurent Sérillon

Master Thesis

Supervisor:

Juliette Soulard, KTH

Royal Institute of Technology

Department of Electrical Engineering Electrical Machines and Power Electronics

Stockholm 2007

XR-EE-EME 2007:010

2

ÉCOLE NAVALE et

GROUPE des ÉCOLES du POULMIC _______________

PROMOTION 2005 _______________

PROJET DE FIN D’ÉTUDE

GÉNIE ENERGETIQUE

_____________

THERMAL MODELLING OF PERMANENT MAGNET MOTOR FOR TRACTION

EV2 FAIVRE d’ARCIER & SERILLON

3

ACKNOWLEDGEMENTS

Our final scientific project was written at the School of Electrical Engineering, Division of Electrical Machines and Power Electronics, Royal Institute of Technology (KTH), Stockholm, in collaboration with Bombardier Transportation. It lasted from 1st October to 14th December 2007. The authors would like to express their gratitude to Doctor Juliette Soulard of KTH for her guidance and advice in the course of this project. They also wish to thank Florence Meier and Alexander Stening of KTH for excellent assistance during the whole work. We are grateful to Jean-Frédéric Charpentier of École Navale (France) for his contribution to the preliminary research. The authors also would like to thank Pascale Sammartano of Ecole Navale (France) for her help with the report. Laurent and Wallerand Stockholm, 14/12/2007 Photography in second page taken from http://www.evworld.com/images/bombardier_regina.jpg

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5

THERMAL MODELLING OF PERMANENT MAGNET MOTOR FOR TRACTION

Elèves : EV2 FAIVRE d’ARCIER, EV2 SERILLON, EN 05 V.A. : Génie Energétique (GE) Organisme d’accueil : KTH, département machines électriques & électronique de puissance, Suède. Chef de projet : Docteur J. SOULARD, KTH. Pilote de Projet : J-F CHARPENTIER, Maître de Conférences, École navale. Soutenance le 29 janvier 2008. Résumé – Présentation du sujet (but – cahier des charges – solutions) : Ce projet a pour objectif d’étudier le comportement thermique d’un moteur à aimants permanents pour la traction dans le cadre d’une collaboration avec Bombardier Transportation. Le cahier des charges prévoyait des modélisations diverses (inductances, pertes, températures) avec des simulations électromagnétiques et thermiques. Après un temps d’adaptation nécessaire à l’apprentissage du fonctionnement du logiciel Flux2D, une part importante de notre travail s’est orientée vers l’étude des performances de ce moteur à travers des simulations électromagnétiques par éléments finis avec l’exploitation et l’analyse des résultats obtenus. Dans un deuxième temps, un modèle thermique du moteur a été développé et utilisé pour étudier son comportement en régime permanent et lors d’un court-circuit. Abstract: The aim of the project was to study the thermal behaviour of a permanent magnet motor for traction in collaboration with Bombardier Transportation. Electromagnetic and thermal modelling (inductances, loss, temperature) with analytical and finite-element tools were planned. After learning to use the software Flux2D, our work was based first on the performance of the motor with the help of FEM simulation. In a second time, a thermal network has been developed and used in order to study the behaviour of the motor at steady-state and during a short-circuit. Key words: Interior PM-motor – Iron Loss –Electric Traction – Thermal Modelling – Finite Element Method (FEM).

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7

TABLE OF CONTENTS Pages

Notations 8 INTRODUCTION 11 1. FEM SIMULATION OF THE TRACTION MOTOR 13

1.1. Geometry of the motor 13

1.2. Description of the material 14

1.3. Results of FEM simulation 15

2. THERMAL MODELLING OF THE MOTOR AT STEADY-STATE 27

2.1. Heat transfer theory 27

2.2. Modelling of the rotor at steady-state 28 2.3. Modelling of motor slot at steady-state 31

2.4. Thermal steady-state analysis 34 3. TRANSIENT THERMAL INVESTIGATION OF THE MOTOR 38 3.1. Global transient analysis 38 3.2. Simulation from cold state to steady-state at maximum torque 39 3.3. Short circuit analysis 39 CONCLUSIONS AND FUTURE WORK 41 Bibliography 43 Appendix A: Definition of the thermal conductivities and resistances 44 Appendix B: Definition of the thermal resistances for slot model and calculation of the thermal resistances 46 Appendix C: Gantt diagram of the project 47

8

Notations

List of Abbreviations AC: Alternative Current DC: Direct Current FEM: Finite Element Method MMF: Magneto-Motive Force PM: Permanent Magnet List of Symbols Agap Equivalent gap area between cooling frame and stator yoke [m2] Bm Magnet flux density [T] Br Remanent flux density of magnet [T] Bstator Maximum value of fundamental flux density in the stator [T] bss Stator slot width [mm] Bteeth Maximum value of fundamental flux density in the teeth [T] dlam Lamination thickness [mm] ERMS RMS voltage [V] f Electric frequency [Hz] I Amplitude of phase current [A] Id d-axis component of the phase current I [A] Iq q-axis component of the phase current I [A] kexc Excess loss coefficient [-] kFe Lamination stacking factor [-] kh Hysteresis loss coefficient [-] Ld d-axis inductance [H] Lq q-axis inductance [H] ns Number of conductors per slot per phase [-] p Number of poles [-] P Nominal power [W] Pcond Copper losses [W] Piron Iron losses [W] Protor Rotor losses [W] Pstator Stator losses [W] Pstator teeth Stator teeth losses [W] Pstator yoke Stator yoke losses [W] Rcond Resistance of one conductor [Ω] S Area of the heat transfer [m2] T Torque [Nm]

9

β Electrical angle between the current and the magnet flux vector [rad] λiron Thermal conductivity of iron material [W/mK] λmag Thermal conductivity of magnet material [W/mK] μ0 Permeability of free space [H/m] μr Relative permeability of material [H/m] ψm Magnet flux linkage [T] σiron Lamination conductivity [(Ωm)-1]

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INTRODUCTION

Replacing induction motors by permanent magnet (PM) motors in traction drives of trains raises some questions like its behaviour when it is submitted to failure such as short-circuit.

Due to the application (transport), this PM motor must be electromagnetically and thermally characterized in order to offer reliable service to the passengers. The magnetic properties of permanent magnet material being dependant of temperature, the performance of the motor is also depending on the thermal behaviour.

During faulty conditions like phase short-circuit or turn-to-turn fault, it is essential to predict what is happening.

The first part of the project presents the results of simulations obtained by FEM program used to estimate the electromagnetic behaviour of the PM motor at maximum torque and nominal speed. These simulations are made with the help of the software Flux, from Cedrat.

The second and more original part of this report shows the steps followed to develop models to investigate the thermal behaviour of the motor at steady-state. Thermal models of the rotor and the motor slot are especially developed in order to study locally the temperature pattern.

The third part investigates the thermal behaviour of the motor at transient conditions, first from cold state to steady-state at maximum torque and then during a short circuit.

Photography taken from http://www.gronataget.se/upload/Bildarkiv/Mittenbilder/Regina01_150MellanSmal.jpg

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CHAPTER 1 FEM SIMULATION OF THE TRACTION MOTOR

This section describes the basic setup of the Finite Element (FE) model run with the dedicate software Flux, Cedrat. The model of the 4-pole permanent magnet motor given by Bombardier Transportation is simulated in order to verify some of the performance and obtain loss for thermal modelling. 1.1. GEOMETRY OF THE MOTOR Figure 1 shows a cross-section of a motor similar to the investigated one. The studied motor has four poles. Its geometry is periodic (180º) and anti-symmetric (north and south poles under one pole pair). Therefore only ¼ of the machine needs to be described in FEM model. 12 stator slots grouped in MA, PA, PB, and MC are taken into account. PA, PB represent the positive parts of coils of A and B while MA and MC symbolize the negative parts of coils A and C.

Figure 1: Cross-section of the type of PM motor that was investigated [8]

Stator Iron Yoke

Air gap

Shaft Iron Rotor Iron Core

Air in rotor cooling channels

Inner North Magnet Pole

Outer North Magnet Pole

Stator teeths

Stator slots

Stator slot wedges

14

The air gap of the studied motor has a variable thickness. It is divided into three equal areas shown in figure 2. Air gap 1 represents the air which is attached to the stator, and stays fixed. Air gap 2 is the air which rotates at the same speed as rotor. Air gap 3 is called the compressible air which is located between rotating and constant air. It is remeshed at each time step.

Figure 2: Illustration showing the air gap between rotor and stator

1.2. DESCRIPTION OF THE MATERIAL

The non-magnetic materials are modelled as vacuum. Moreover, Flux has a linear model for permanent magnet material with a constant remanent flux density Br and a constant value of the relative permeability. The characteristic curve of this type of material is shown in figure 3.

B Br = 0,9 T slope µr = 1,03 µ0H

Figure 3: Characteristic of linear magnet material

The stator and the rotor iron laminations are made of the electrical steel M600-65A. The non-linear iron lamination material is defined by a saturation curve which is shown in figure 4. This curve B(H) is characterized by a list of B(H) values available in the material database of the software.

Air gap 1 Air gap 3 Air gap 2

Stator Rotor

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Figure 4: B(H) curve of the iron lamination on M600-65A 1.3. FEM SIMULATION OF THE TRACTION MOTOR In this section, the performance of the PM motor is investigated. The aim is to examine the performance at maximum torque and nominal speed taking into consideration iron saturation and leakage flux. The simulations are implemented, first, by checking the no-load voltage at 1200 rpm and the cogging torque. Then, by fixing the stator current, the corresponding rotor position and therefore the current angle β which maximizes the output torque are found. Finally, the maximum torque at nominal speed is simulated a correct stator current.

1.3.1. No-load simulation This section is presenting the no-load performance of the motor. A forced rotation speed of 1 200 rpm is set for the rotor. As there is no current into the stator, the flux density created by the magnets in the air gap and iron parts and the induced voltage can be calculated. To obtain the no-load voltage of the motor, an electrical circuit is linked to the windings of the Flux model as shown in figure 5. This figure represents 2 coils for phase A, B_PA and B_MA, which represent positive and negative conductors of the coil, and 1 coil for phase B and phase C (PB and MC respectively). The four resistances are set to 105Ω to prevent any current and RAB is acting as voltmeter in order to measure the voltage between phases.

Slope µr = 400

B=1.3T

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Figure 5: Electrical circuit for the no-load voltage simulation

i) No-load equi-flux lines Figure 6 shows the distribution of the equi-flux lines in the motor at no-load condition. It is possible to check the flux density in the main parts of the motor according to the density of flux lines (table 1). The magnets and magnet slots have voluntarily been erased from the figure.

Figure 6: Equi-flux lines at no-load condition

RC

RA

RB

RBC

B_PA B_MA

B_PB

B_MC

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ii) Air gap flux density at no-load Figure 7 shows the resulting air gap flux density created by the magnets for the complete motor. The corresponding spectrum is shown in figure 8.

Figure 7: Air gap flux density at no-load

The air gap flux density presents some specific points. At points A, B and C, the flux density has different values because of the variation of the air gap thickness. The thicker the air gap is, the more the reluctance, the less the flux density is. Consequently, the air gap thickness is the cause of the ABC triangle. Besides, the oscillations between points A and C are due to the slotting in the stator. The distance between two oscillations (points D and E) is corresponding to the tooth pitch (distance between two teeth).

Figure 8: Spectrum of air gap flux density

A CB

D E

Curve for spectrum

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The FEM values of the fundamental component of the air gap flux density produced by the magnet, the maximum flux density in the stator teeth and yoke and the maximum flux density in the rotor yoke are summarised in table 1. FEM Max. fundamental component of the air gap flux [T] 0.5 T Max. flux density in stator teeth [T] 0.688 T Max. flux density in stator yoke [T] 1 T Max. flux density in rotor yoke [T] 1.9 T

Table 1: FEM flux densities at no-load iii) No-load voltage Figures 9a and 9b show respectively the no-load phase voltage and the corresponding spectrum. The fundamental component of the simulated phase voltage obtained with FEM at 1200 rpm is 247V.

-300

-199,999

-99,999

0

100

199,999

300

10 20 30 40 50

(E-3) s.

Volt

50

100

150

200

250

300

0 10 20 30

Volt

Figure 9a: No-load phase voltage Figure 9b: Harmonic contents of no-load voltage

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φ

Figures 10a and 10b show the phase-to-phase voltage at no-load conditions with its spectrum analysis. The maximum value of the fundamental component of this voltage is 427V.

-500

0

500

10 20 30 40 50

(E-3) s.

Volt

100

200

300

400

0 10 20 30

Volt

Figure 10a: No-load phase-to-phase voltage Figure 10b: Harmonic contents of no-load

phase-to-phase voltage iv) Performances According to Flux, the RMS phase to phase voltage at 1200 rpm is 302V. As the phase to phase voltage is proportional to the rotor speed, the voltage at 6000 rpm is deduced and equal to 1510V which is according to the required specifications. However, the same simulation should also be run with cold magnets (high remanent flux density).

1.3.2. Current angle of maximum torque The aim of this section is to find the current angle β which maximizes the mechanical torque for a given value of the stator current. This current angle refers to the electric angle between magnet flux linkage (ψm) and stator current vector in the rotor reference frame.

Figure 11: Phasor diagram for a PM motor (neglecting the stator resistance) [3]

q jωLqIq

jωLdId

Iq

Id

Is

Ψm

E = jω Ψm

Us = jω Ψs

β

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Ld and Lq are the d- and q-axis inductances while Id and Iq are the d- and q-axis currents respectively.

To find the current angle which produces maximum torque, the angle of the stator current vector β must be controlled in such a way that the summation of the flux alignment torque and the reluctance torque becomes maximal. The stator current vector is fixed and kept constant while the rotor is rotated at nominal speed for a full electrical cycle.

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600

Speed (rad/s)

Torq

ue (N

.m)

Figure 12: Torque characteristic in function of speed

The nominal speed is 1200 rpm (125 rad/s) and the nominal torque is 2000 N.m, which corresponds to a nominal power of 250 kW. To obtain an initial value of the current, it is assumed that the torque is only produced by the magnet reacting with the stator currents in q-axis. Then, it can be written: RMSRMS IEP **3= (1) The value of ERMS is set to 175V as found previously in the no-load simulation. IRMS is then equal to 480A.

Nominal point

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The stator currents are imposed to the stator by direct current sources. Because the current is assumed symmetrical, only two sources are defined in the electrical circuit. The values are set as:

0=ai

RMSb Ii23

=

RMSc Ii23

−=

The coupled electrical circuit is presented in figure 13.

Figure 13: Electrical circuit for finding the optimum current angle

A FEM simulation is run for one electrical period by choosing the time-step in such a way that the rotor turns one mechanical degree after each iteration (which is equal to two electrical degrees, since the number of poles is p = 4). The torque varies with the rotor position θ as shown in figure 14. Then, by keeping the fundamental and the second harmonics, the angle corresponding to the maximum torque is obtained. Figure 14 shows the torque as a function of mechanical rotor angle with IRMS = 262A. The angle for which the torque is maximum is deduced from the curve and it is equal to 107°. The fundamental, the second, the third and the fourth harmonics and their summation are shown in the same figure 14. The usual synchronous machine torque expression with salient pole is:

( )( )qdqdqm IILLIpT −+Ψ=4

3 (2)

Where p is the number of poles. The first term of the total torque is representative of the magnet torque (interaction current-magnet). The second term is representative of the reluctance torque. The fourth harmonic is probably due to the variable thickness of the air gap.

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The summation gives the angle when the torque reaches its maximum value: this angle is equal to θ = 107° mechanical. This corresponds to a value of β = 214°.

‐2500

‐2000

‐1500

‐1000

‐500

0

500

1000

1500

2000

2500

1 16 30 45 59 73 88 102 117 131 145 160 174 189 203

Mechanica l Ang le (deg )

Torque (Nm)

S imulated Torque

1s t harmonic

2nd harmonic

3rd harmonic

4th harmonic

S ummation of harmonics

Figure 14: 1st, 2nd, 3rd, 4th harmonics and their summation of torque respect to rotor angle

With a current equal to 480A, the maximum torque is 3600Nm. This value is much higher than the specified torque. Consequently, the current IRMS is reduced to 262A to obtain the required torque.

1.3.3. Steady-state at maximum torque and nominal speed

This section shows a study of the motor at steady-state, with the nominal speed and with the maximum angle so that the maximum torque is obtained.

Figure 15: Simulated maximum torque at optimal current angle and nominal speed

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Figure 15 shows the torque as a function of time at the optimal current angle. The mean value of the torque is equal to 510 N.m for only ¼ of the motor. Consequently, the mean value of the torque for the entire motor is equal to 2040 N.m. Moreover, this simulation allows to obtain the values of the flux density in the stator and in the teeth. These values are equal to:

• Bstator = 1.65T • Bteeth = 1.75T

Figure 16 shows the resulting air gap flux density created by the magnets and the currents for the complete motor at load.

Figure 16: Air gap flux density at load

1.3.4. Investigation of the iron losses The magnetic losses or iron losses are calculated in the post-processing module of Flux. The calculations use Bertotti’s formula (see equation 3). The values of the coefficients in the equation are depending on the material of the concerned regions in the stator. The iron loss coefficients for M600-65A iron lamination are listed in table 2. M600-65A is a common material with a lamination thickness of 0.65 mm. Hysteresis loss coefficient kh (Ws.T-2.m-3) 242 Conductivity σiron (Ωm)-1 2.78*106

Excess loss coefficient kexc (W.(s/T)3/2/m3) -0.138 Thickness of the lamination dlam (m) 6.5*10-4

Stacking factor kFe 1 Electrical frequency f (Hz) 40

Table 2: Iron loss coefficients for M600-65A

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The iron losses are created in a magnetic material by the variation of the flux density. They include hysteresis loss, excess loss and the eddy current loss. khyst and kexc represent the hysteresis loss coefficient and the excess loss coefficient respectively. They are calculated by referring to the information given by the lamination manufacturers for a fixed flux density (1.5T) and for a fixed frequency (50 Hz) [9]. The minus sign of kexc in table 3 is due to the model in Flux. The hysteresis loss is proportional to Bx with 1.5<x<2. Flux uses x = 2. If the "real" value of x should be lower than 2 then the excess loss coefficient is negative. Here, a negative value is used for kexc but it has no physical meaning. Bertotti’s equation for analytical calculations assuming a sinusoidal variation of B(t) gives the iron loss density:

67.8*)(*)(*)6

()(* 23

222

2 BfkBfdfBkP exclamiron

hystiron ++=σπ

(3)

i) Analytical calculations The rotor volume is calculated by subtracting to the rotor iron volume the shaft volume and the volume of the 16 magnets. The iron stator volume is calculated by subtracting to the stator volume the volume of the 48 slots. Rotor Surface (m2) 4.16*10-2

Rotor Volume (m3) 1.14*10-2

Stator Surface (m2) 8.22*10-2

Stator Volume (m3) 2.26*10-2

Shaft Volume (m3) 1.75*10-3

Iron lamination density (kg/m3) 7 750 Table 3: Geometry data.

The magnetic field is assumed to be sinusoidal in order to apply Bertotti’s equation. The stator is divided in two parts: the teeth and the yoke. According to 1.3.3, in table 4, the value of the magnetic field density is taken equal to 1.65T for the stator yoke and 1.75T for the stator teeth. Table 4 presents the analytical results. Piron (1.75T) (W/m3) 39 031 Piron (1.65T) (W/m3) 34 696 Pstator yoke (1.65T) (W) 559 Pstator teeth (1.75T) (W) 252 Pstator (W) 811

Table 4: Iron loss of stator iron at maximal load and 1200 rpm by analytical approach

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ii) Comparison with FEM Figures 17a and 17b present the iron loss for a quarter of the motor as a function of time. Figure 17a: Rotor iron loss Figure 17b: Stator iron loss The mean values of the rotor and stator iron loss which are shown in table 5. Bstator (T) 1.65 Bteeth (T) 1.75 Protor (W) 248 Pstator (W) 850

Table 5: Mean iron loss at maximal torque and 1200 rpm by FEM.

The magnetic field is rotating with the rotor at the same speed. Therefore, there should be no variation of the magnetic field in the rotor with time, so there should not be any iron loss in the rotor. Iron losses calculated by FEM could come from local variations next to the air gap. A comparison between the analytical and FEM results is presented in table 6. The obtained iron losses by FEM model are 5 % higher. Analytical FEM Pstator iron losses (W) 811 850

Table 6: Iron losses comparison

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iii) Copper loss This type of loss is related to the conductor and the copper loss in the end windings is not taken into account. In order to calculate the copper loss, equation (4) is used:

2* RMScoppercopper IRP = (4) The motor model in Flux made the assumption that the coils are in series combination. Yet in the real motor, they are in parallel combination. Therefore, only half the current imposed previously in the software goes through the conductors. Consequently, the value of I used in this part of the study is 131A. The value of copper resistance, Rcopper, is obtained by multiplying copper resistivity with the ratio length by area section. The resistivity is a function of the temperature and is calculated as:

[ ])20(10*92.31*)20()( 3 −+= − TCT coppercopper ρρ (5)

Where ρcopper (20°C) =17nΩ.m The conductors are designed to withstand 240°C; consequently the resistivity at 240°C is considered. It is equal to 29nΩ.m. Copper losses are deduced and equal to 9.54 kW for the whole motor.

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CHAPTER 2 THERMAL MODELLING OF THE MOTOR AT STEADY-STATE

Overheating of the permanent magnets can entail demagnetization and consequently, damage to the motor. So it is important to predict and control the temperature of the permanent magnets accurately. 2.1. HEAT TRANSFER THEORY When a system shows temperature gradients, heat energy transfers occur. This heat is transferred under three different phenomena: conduction, convection and radiation. These types of transfer are ruled by the first principle of thermodynamics which emphasizes that nothing is created, nothing is lost, the energy is transformed from one form into another.

2.1.1. Conduction In a solid where there is a temperature gradient, the second principle of thermodynamics emphasizes that heat flows in the direction from higher to lower temperature. Heat is transferred by conduction, with the heat flow qk characterized by Fourier’s law:

dxdTSkq kk **−= (6)

Where S is the area for the heat transfer, kk is the thermal conductivity of the solid material or the heat transfer coefficient, x represents the distance the heat is conducted and T(x) is the local temperature. Moreover, the flow of heat faces a heat resistance defined by:

kk kS

xR*

= (7)

The heat transfers in electrical motors are usually modeled with an equivalent circuit. The thermal resistances in the model are calculated from the equivalent conduction resistances of different parts of the motor. Some resistances representing heat transfer by convection are also present.

2.1.2 Convection Heat is transferred by convection when there is a temperature gradient on the surface between a solid and its surrounding medium applied on the fluid (movement). Convection can be classified as forced or natural depending on the external force. Convection is characterized by the convective heat transfer coefficient hc which depends on the fluid characteristics. The heat flow qc corresponding to convection is described by Newton’s law:

Tshq cc Δ= ** (8) Where S is the area for the heat transfer, hc is the convective heat transfer coefficient and ΔT is the difference between surface and fluid temperatures. Moreover, the thermal equivalent resistance associated to convective heat transfer is calculated according to:

cc hSR *

1= (9)

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2.2. MODELLING OF THE ROTOR AT STEADY-STATE This section describes the thermal model of the rotor. The method used to calculate the steady-state temperatures in the rotor is also presented.

2.2.1. Approximation for the PM machine rotor For thermal modelling, networks are usually used. Due to the complexity of an electrical machine, a large thermal network is necessary. Consequently, some assumptions and simplifications are considered which allow reducing the thermal network. The general ideas of the rotor thermal model were found in [8]. The geometrical symmetries of the machine can be taken into account. Indeed, the motor geometry is periodic (180º) and anti-symmetric, therefore only ¼ of the machine needs to be modelled. Figure 18 shows how the quarter of the motor is divided into different angles by taking into account the magnet geometry. The modelled part of the rotor can be finally reduced to 1/8 of the complete rotor.

Figure 18: Cross section of a quarter of a similar machine with identification of key angles [8] Moreover, the heat flow is assumed to be in the radial direction in the rotor. With this assumption, the calculations of the rotor thermal resistances are simplified.

2.2.1. Thermal equivalent network Considering the geometrical symmetries and the radial direction assumption of the heat flow, the thermal equivalent network for the rotor is deduced by observing figure 19 (corresponding to investigated motor). The simplified geometry of the PM rotor with the resistances which are taken into account is shown. The proportions are being kept between the iron and the magnet surfaces. The four positions of the magnets have been defined in accordance with the real design of the rotor.

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Figure 19: Simplified geometry of the investigated PM rotor with the thermal resistances Figure 20 shows the thermal equivalent network used for the thermal modelling of the rotor. This network has 2 nodes. One node symbolizes the air gap, the other the shaft.

Figure 20: Thermal equivalent network for the rotor thermal modelling The thermal resistances between the nodes describe the conductive character of the heat flow. The thermal resistance value depends on geometry and material property through the thermal conductivity.

M2 M3

M4

M1 I1

I2 I3

I4 I5

I6

I7

I8

Air gap

30

2.2.3. Basic theory about thermal resistance calculation In a cylindrical section, the expression of the radial thermal resistance is:

θλL

rr

R

ext

th

⎟⎟⎠

⎞⎜⎜⎝

⎛

= int

ln (10)

Where rext is the outer radius, rint is the inner radius, λ is the thermal conductivity of the material, θ is the angular span and L is the active length of the machine. When n resistances are connected in series, then the total equivalent resistance is equal to:

∑=

=n

iitotal RR

1 (11)

When n resistances are connected in parallel, then the total equivalent resistance is equal to: 1

1

1−

=⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

n

i itotal R

R (12)

2.2.4. Calculation of the equivalent resistances With the previous assumptions, the model is limited to 1/8 of the rotor. So, the equivalent resistance corresponding to figure 17 will be marked Rtotal_8 while the total equivalent resistance corresponding to the entire rotor will be defined as Rtotal_rotor. The relation between these two resistances is:

8__ *81

totalrotortotal RR = (13)

The thermal resistance Rtot1 is defined as the equivalent resistance for the angle θ1 of the rotor and is equal to:

2111 IMItot RRRR ++= (14) In the same way, Rtot2, Rtot3 and Rtot4, corresponding to the angles θ2, θ3 and θ4 respectively, are deduced:

534232 IMIMItot RRRRRR ++++= (15a)

7463 IMItot RRRR ++= (15b)

84 Itot RR = (15c) The expressions defining each of these resistances are summarized in Appendix 1. As a parallel combination of four resistances, the expression of Rtotal_8, is deduced:

14

18_

1−

=⎟⎟⎠

⎞⎜⎜⎝

⎛= ∑

i totitotal R

R (16)

For the investigated motor, Rtotal_8 = 0.2K/W.

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2.3. MODELLING OF PM MACHINE SLOT AT STEADY-STATE The aim of this section is to determine the spreading of temperature in the stator slot, insulation and copper. A stator slot consists of ns conductors, knowing that each conductor has copper losses Pcond, equal to:

2* RMScondcond IRP = (17)

2.3.1. Simulation with one conductor First, the ns conductors can be considered as one solid block of copper with the total copper losses nsRcondI2 as shown in figures 21a and 21b. Around it, there is insulation with a thermal conductivity equal to 0.2 W/K. The copper temperature is assumed to be between 230 and 240°C. The other assumptions are:

• End windings are no considered (heat exchange with internal air through convection) • The heat flow is assumed to flow in three different directions from the copper

To both teeth on each side of the slot (horizontally in figure 21b) To the stator yoke (vertically in figure 21b)

Figure 21a: Scheme of stator slot Figure 21b: Model of the slot with copper

conductor and insulation

The corresponding thermal network is shown in figure 22.

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Figure 22: Thermal network of the stator slot The thermal resistance Rthbottom is defined as the equivalent resistance from the conductors to the stator yoke and consists of a series combination of three resistances. Its expression is therefore:

212019 ththththbottom RRRR ++= (18) In the same way, Rside1 and Rside2 are deduced:

ironththththside RRRRR +++= 1514131 (19) ironththththside RRRRR +++= 1817162 (20)

Moreover, Rthside1 and Rthside2 are equal and consequently, they can be modeled by only a resistance, noted Rthside whose value is:

22

21 thsidethsidethside

RRR == (21)

The equivalent thermal network is reduced with two resistances, Rthside and Rthbottom, in parallel combination with a source term P equal to nsRI2. Rthslot is defined by the equation:

thbottomthside

thbottomthsidethslot RR

RRR+

=*

(22)

At steady-state (maximum torque), the difference of the temperature between the stator yoke and copper is given by:

2* RInRT sthslot=Δ (23) All the data and expressions of the different resistance are available in Appendix B. If the temperature of the slot is fixed at 240°C, and the copper loss is equal to 100W, the temperature difference is 63.5°C. Consequently, the mean temperature in the yoke is equal to 176.5°C.

33

2.3.2 Simulation with ns conductors in the slot The simulation with one conductor gives a general idea of the temperature gap between the coils and the iron teeth. However, this modelisation assumes all the conductors are at the same temperature and that there is no gradient in the teeth. So, a new more detailed network has to been developed. In the new design, each conductor is studied separately. In the new network, shown in figure 23, resistances between each conductor have been included.

Figure 23: Thermal network with ns=6 conductors in the slot With this new network, some conclusions can be drawn:

• The difference of temperature between conductors in the traction motor is equal to around 20°C. The nearest conductor of the stator yoke has a better exchange area and consequently, its temperature is less high than the one of the other conductors.

• In the stator teeth, a constant increase of temperature is observed from the stator yoke to the rotor edge. The difference of temperature between top and bottom of tooth is 15°C.

• The copper loss is cooled away mostly through the teeth and slightly through the stator yoke.

34

2.4. THERMAL STEADY-STATE ANALYSIS OF THE MOTOR 2.4.1. Theory about matrix resolution Due to the high number of nodes and current sources, the analytical resolution is more complicated. Consequently, in this part, a matrix method is used in order to solve the issue. This type of resolution has been found in [8] and is described below. Moreover, an adequate thermal equivalent model for motors has been presented in [7]. It is used here with some simplifications in combination with our specially developed slot and rotor local models. In a steady-state thermal analysis, n equations are necessary to define a thermal network with n + 1 nodes. The extremity of the shaft and the coolant for the stator are supposed to be at constant temperature and are taken as reference equal to 40°C. The adapted method allows to calculate the vector temperature Θ as:

PG 1−=Θ (24) Where P is the matrix representing the loss at each node of the thermal network. The values of Θ represent the temperature rises at each node.

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nP

PP

P 2

1

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Θ

nθ

θθ

2

1

The form of the matrix G is taken from [8] and is defined as a n*n thermal conductance matrix:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

−−

=

∑

∑

∑

=

=

=

n

i innn

n

n

i i

n

n

i i

RRR

RRR

RRR

G

1 ,2,1,

,21 ,21,2

,12,11 ,1

111

111

111

Where the diagonal elements represent “the sum of the network conductances connected to node n, and G (i,j) is the negative of the thermal conductance between nodes i and j” ([8]).

2.4.2. Steady-state analysis of the motor with one solid conductor per slot Figure 24 represents the thermal equivalent network of the motor with only one conductor. It presents only five nodes (θ0, θ1, θ2, θ3, θ4). The frame and the coolant are not at the same temperature therefore a resistance, Rth0, has been introduced to complete the model. The different resistances are defined in table 7.

35

Figure 24: Thermal network of the motor with one conductor for one entire slot As five nodes are present, a 4*4 thermal matrix is required in order to define the conductance matrix.

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

++−−−

−++−

−−++−

−−+++

=

643643

67655

455422

323201

111111

111110

111111

101111

thththththth

ththththth

thththththth

thththththth

RRRRRR

RRRRR

RRRRRR

RRRRRR

G

The values of the different resistances are available in table 7.

Location Thermal resistances Value (K /W) Yoke-frame Rth1 0.157 Yoke-tooth Rth2 0.118 Slot bottom Rth3 4.87

Conductor tooth Rth4 0.375 Tooth-air gap Rth5 0.118

Wedge Rth6 6.5 Air gap-rotor-shaft Rth7 41.3 (15.4+1.2+24.7)

Coolant Rth0 0.4 (assumed) Table 7: Value of the different resistances for 1/48 of the motor

36

The high value of the resistance Rth7 is due to the air gap and length of the shaft. For the air gap resistance, only the conduction is considered as there is no rotation, it corresponds to the worst case for copper and iron cooling. A better model with forced convection should be done. This global study of the motor with one conductor per slot is an approach which allows obtaining the temperatures in all parts of the motor. A summary of these temperatures is shown in table 8, the reference temperature is kept at 40°C and the temperature of the frame is fixed at 115°C.

Nodes i Temperature (°C) 1 (yoke) 149 2 (teeth) 170

3 (air gap) 171 4 (conductor) 236

Table 8: Absolute temperatures at different nodes for copper loss fixed at 200W per slot

By studying these results, an observation can be drawn: the heat flow passing from the stator to the rotor is low in comparison to the heat evacuated by the stator yoke.

2.4.3. Steady-state analysis with ns conductors Figure 25 represents the thermal equivalent circuit with ns conductors and 2ns + 6 nodes, the reference is θ0. The iron loss is not taken into account to begin with.

Figure 25: Thermal network of the motor with ns conductors

37

Rth1 and Rth10 represent the resistance of the yoke and the shaft respectively. Rth9 is the resistance of the rotor calculated in section 2.2.1. adapted to fit with the “one-slot” stator model. The temperature is supposed to be equal between the coolant and the extremity of the shaft. Moreover Rth0 is fixed in order to keep the difference of temperature between the coolant and the frame at 80°C. The air gap is divided into three parts in order to have a more precise model. The results show that the value of the temperature in the conductors is between 223°C to 240°C. The bottom slot conductor is the coldest due to a larger exchange area with the yoke. A temperature rise is observed from the iron yoke (155°C) to the rotor edge (172°C).

2.4.4. Magnet temperatures at maximum torque Because permanent magnets have temperature sensitive properties, they can loose partially or completely their magnetic characteristics when they are submitted to high temperatures. It can entail a risk of demagnetization which damages the motor. In the studied case, 200°C is the assumed temperature working point of the magnets. With this steady-state study, and by taking a RMS current equal to 131A at the maximum torque, the temperatures of the different magnets are shown in table 9.

Different magnets Temperature (°C) Magnet M1 136.8 Magnet M2 136 Magnet M3 138.1 Magnet M4 136.9

Table 9: Temperatures of the magnets with extremity of the shaft at 40°C Besides, it was assumed that the outside temperature is supposed to be fixed at 40°C. A value of the current of 170A leads to magnets reaching 200°C at steady-state, but copper temperature would be higher than the chosen insulation class. At last, if the iron losses are taken into consideration, an increase of temperature from 1°C to 4°C is noticed.

38

CHAPTER 3 TRANSIENT THERMAL INVESTIGATION OF THE MOTOR

The steady-state modelling allows to study the final temperatures for a certain loss pattern. Nevertheless, issues like short-circuit between different conductors or between one conductor and the yoke are time dependent. That is the reason why a thermal transient model of the motor is necessary. 3.1. GLOBAL TRANSIENT ANALYSIS In this modelling, the energy stored by the materials has to be taken into account. This is represented by capacitances in the thermal network. The thermal capacitances of the motor parts are calculated with their dimensions and their physical properties. The global function for several parts is:

i

n

iith cmC *

1∑=

= (25)

Where mi is the weight and ci is the specific heat of each part i. The new matrix equation is deduced from the heat equation, which is:

)(* TapKdtdT

Δ−= (26)

In a matrix form, the above equation becomes the following one:

)(*)( 1 Θ−=Θ − GPCdtd (27)

Where Θ(t=0) takes the initial value (cold or steady-state) and where C is the n*n matrix used for modelling the capacitance of each node and defined as :

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

thn

th

C

CC

…0

01

Figure 26 represents the global thermal equivalent circuit with ns conductors and capacitances for transient investigations.

39

Figure 26: Transient thermal network with ns conductors and capacitances 3.2. SIMULATION FROM COLD STATE TO STEADY-STATE AT MAXIMUM TORQUE The different parts of the motor do no reach the steady-state at the same time. Figure 27 shows the evolution of the temperature from the cold case (all parts at 40°C) to the steady-state.

Figure 27: Evolution of the temperature from cold-state to steady-state

40

Being the site of loss source, the conductors are the first to reach the steady-state. The temperature of the conductors approaches 240°C, which means that the assumptions are globally coherent because the motor is designed to withstand this temperature. Only one conductor is colder than the others. This is due to the proximity of the yoke which cools the bottom slot coil as well. Lastly, the steady-state in the rotor needs a long time to be established because of the air gap having a high thermal equivalent resistance. 3.3. SIMULATION OF THE SHORT CIRCUIT AT t= 0.1s The aim of this section is to investigate the evolution of the temperature at different points in the motor when a short circuit occurs in two adjacent conductors in a slot, also called turn-to-turn failure. The electromagnetic transient was not simulated so no evolution of the current during short-circuit was available. However, it was assumed that during a short circuit, the current in the first conductor (only case studied) might be multiplied by 10, and consequently the losses could be multiplied by 100. The first consequence will be an increase of the temperature of the coil and there is a high probability that the insulation melts and looses its properties. In this case, the short circuit can spread to the other conductors or to the yoke. The aim is to determine the time between the beginning of the short circuit and the fusion of the insulation.

Figure 28: Temperature evolution after a short circuit in the bottom-slot conductor In this simulation, a short circuit of 1 300A is provoked in the conductor at the bottom of the slot. According to the data, the temperature reaches 300°C in 0.7 second. After 1.5 seconds, the insulation, directly in contact with the coil, is melted if a melted temperature of 370°C is taken for the insulation. From then, the physical properties are modified and the model is not valid anymore. There is around one second to disconnect current sources after the short circuit. If the current is not turned down, in a few seconds the fault could spread to the adjacent conductors in the slots. Results show that the temperatures in the teeth and in the yoke evolve really slowly compared to the time constant of the temperature increase in the bottom-slot conductor. Consequently, the other parts of the motor are not threatened.

41

CONCLUSIONS

From the results of FEM simulations of a permanent magnet motor for traction with the help of the software Flux, the value of the current required to reach the maximum torque was found. Moreover, with this value of current and the magnetic field, the copper and iron loss have been deduced. The iron loss represents about 8% of the copper loss so the iron loss could be neglected in the thermal model. Thermal models were then developed for both stator and rotor. Thermal behaviours considering steady-state and transient conditions have been investigated. At thermal steady-state under maximum torque, the temperature of the rotor is not high enough to entail problem for the magnets with the chosen assumptions. In case of a turn-to-turn circuit, it is estimated that after one second the insulation might reach a temperature leading to a spreading of the failure. The other parts of the motor will not be damaged in reason of thermal inertia, at least during the time of validation of the model. This study raises three main questions for future work:

• First, in order to obtain a more realistic model, it would have been interesting to develop a convective resistance for the air gap.

• Then, the cooling system (frame with cooling ducts to begin with) needs to be included in order to complete the model.

• Lastly, our work was based on many assumptions. The models and assumptions should be validated by comparing results with measurements done on the motor in controlled conditions.

42

43

BIBLIOGRAPHY Reference reports: [1] A. Stening, Design and Optimization of Surface-mounted Permanent Magnet Synchronous Motor for a High Cycle Industrial Cutter, Master Thesis, Royal Institute of Technology, Sweden, 2006. [2] S. Meier, Theorical Design of Surface-mounted Permanent Magnet Motors with field-weakening capability, Master Thesis, Royal Institute of Technology, Sweden, 2002. [3] S. Mohammadzadeh Babr, Design of Permanent Magnet Motors with concentrated windings for concrete cutters, Master of Science Thesis, Royal Institute of Technology, Sweden, 2007. [4] D. Staton, A. Boglietti, A. Cavagnino, Solving the more difficult aspects of electric motor thermal analysis. IEMDC’03. IEEE International. Electric Machines and Drives Conference, 2003.Volume 2, June 2003. Pages: 747-755 vol.2. [5] A. Stening, Performance Study of an Inset Permanent Magnet Motor, Internal report, KTH, Sweden, 2006. [6] F. Magnussen, C. Sadarangani, Winding factors and Joule losses of permanent magnet machines with concentrated windings. IEMDC’03.IEE International. Electric Machines and Drives Conference, 2003. Volume 1, June 2003, pages: 333-339. [7] J. Lindstrom, Thermal Model of a Permanent-Magnet Motor for a Hybrid Electric Vehicle, Chalmers University of Technology, Internal report, Göteborg, Sweden, 1999. [8] A.M. EL-Refaie, N.C. Harris, T.M. Jahns, K.M. Rahman, Thermal Analysis of Multibarrier Interior PM Synchronous Machine Using Lumped Parameter Model. IEEE Transactions on, Energy Conversion. Volume 19, Issue 2, June 2004, pages: 303-309. Websites: [9] Surahammars bruk, company website (Cogent) http://www.sura.se/ Non-oriented fully processed electrical steel, datasheet

44

APPENDIX A

A.1: Definition of the thermal conductivities λ mag: thermal conductivity of magnet material λiron: thermal conductivity of iron material

A.2: Definition of the thermal resistances

RI1: thermal resistance between shaft and M1 in θ1

RM3: thermal resistance between I4 and I5 in θ2

RM1: thermal resistance between I1 and I2 in θ1

RI5: thermal resistance between M3 and the air gap in θ2

RI2: thermal resistance between M1 and the air gap in θ1

RI6: thermal resistance between shaft and M4 in θ3

RI3: thermal resistance between shaft and M2 in θ2

RM4: thermal resistance between I6 and I7 in θ3

RM2: thermal resistance between I3 and I4 in θ2

RI7: thermal resistance between M4 and the Air gap in θ3

RI4: thermal resistance between M2 and M3 in θ2

RI8: thermal resistance between shaft and the air gap in θ4

A.3: Definition of the radius

Rshaft: distance between point O and point A R2’: distance between point O and point G R1: distance between point O and point B R2’’: distance between point O and point H R1’: distance between point O and point C R2’’’: distance between point O and point I Rext: distance between point O and point D R3: distance between point O and point K R2: distance between point O and point F R3’: distance between point O and point L

O A B C D

F G H

I

K L

45

A.4: Expressions of the thermal resistances.

iron

shaftI L

RR

Rλθ ***8

ln

1

1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

= mag

M LRR

Rλθ ***8

ln

2

''2

'''2

3

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

magM L

RR

Rλθ ***8

ln

1

1

'1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

= iron

ext

I LRR

Rλθ ***8

ln

2

'''25

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

iron

ext

I LRR

Rλθ ***8

ln

1

'12

⎟⎟⎠

⎞⎜⎜⎝

⎛

= iron

shaftI L

RR

Rλθ ***8

ln

3

3

6

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

iron

shaftI L

RR

Rλθ ***8

ln

2

2

3

⎟⎟⎠

⎞⎜⎜⎝

⎛

= mag

M LRR

Rλθ ***8

ln

3

3

'3

4

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

magM L

RR

Rλθ ***8

ln

2

2

'2

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

= iron

ext

I LRR

Rλθ ***8

ln

3

'37

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

ironI L

RR

Rλθ ***8

ln

2

'2

''2

4

⎟⎟⎠

⎞⎜⎜⎝

⎛

= iron

shaft

ext

I LRR

Rλθ ***8

ln

48

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

46

APPENDIX B

B.1: Definition of the thermal resistances for slot model

Rth13 = Rth16 = Rth19: thermal resistances through e1 thickness

Rth15 = Rth18 = Rth21: thermal resistances through e3 thickness

Rth14 = Rth17 = Rth20: thermal resistances through e2 thickness

Riron: thermal resistance of the stator iron half tooth tangentially

B.2: Calculation of the thermal resistances.

13

313 * S

eR

λ=

13

318 * S

eR

λ= condstack hLS *13=

14

214 * S

eR

λ=

19

119 * S

eRλ

= )2(* 114 ehLS condstack +=

15

115 * S

eR

λ=

19

220 * S

eRλ

= )22(* 2115 eehLS condstack ++=

15

116 * S

eR

λ=

19

321 * S

eR

λ= condstack bLS *19 =

14

217 * S

eR

λ=

47

APPENDIX C

Gantt diagram of the project