Ghent University Library

Fabrice Mutabazi, Nicholas Larno
permanent magnet motor Analytical model of physical properties of a high speed
Academic year 2016-2017 Faculty of Engineering and Architecture Chair: Prof. dr. ir. Luc Dupré Department of Electrical Energy, Systems and Automation
Master of Science in de industriële wetenschappen: elektrotechniek Master's dissertation submitted in order to obtain the academic degree of
Counsellor: Bert Hannon Supervisor: Prof. dr. ir. Peter Sergeant
Copyright protection
The authors give permission to make this master dissertation available for consultation and to copy parts of this master dissertation for personal use. In the case of any other use, the copyright terms have to be respected, in particular with regard to the obliga- tion to state expressly the source when quoting results from this master dissertation."
i
Fabrice Mutabazi, Nicholas Larno
permanent magnet motor Analytical model of physical properties of a high speed
Academic year 2016-2017 Faculty of Engineering and Architecture Chair: Prof. dr. ir. Luc Dupré Department of Electrical Energy, Systems and Automation
Master of Science in de industriële wetenschappen: elektrotechniek Master's dissertation submitted in order to obtain the academic degree of
Counsellor: Bert Hannon Supervisor: Prof. dr. ir. Peter Sergeant
Foreword and Acknowledgments
The subject of the master dissertation is the analytical study of physical properties in high speed electrical machines.
We would like to thank our supervisor Prof.dr.ir. Peter Sergeant and counselor ing. Bert Hannon for their insightful guidance, advice and help during the entire year. This thesis would be less accurate without their efforts. We would also like to express our gratitude to everyone who has given us support to complete this master dissertation. Especially Alyssa Vansteenbrugge and Ruben Lammertyn for reading and correcting the script.
iii
Abstract
The goal of this master dissertation is to create an analytical model to study physical properties of a high speed permanent magnet motor. The demand for electrical machines is changing. On the one hand, the demand for more efficient electrical machines is increasing due to an increasing ecological awareness. On the other hand, there is an increase in demand of electrical machines with a high power density due to a trend towards more flexible applications which demands for smaller machines. This is driving the interest for high-speed permanent magnet synchronous machines (PMSMs). Design- ing high speed electrical machines is a significantly more challenging task than traditional machines. Due to the high operating speed of the machines, rigorous mechanical, thermal and electromagnetic conditions need to be withstood. This work will focus on the electromagnetic aspect of these machines such as eddy currents, eddy current losses and torque.
the first step was to introduce an analytical method to study the physical properties of an electrical machine. An analytical method was preferred because of various reasons. First analytical models have proven to require less computational time [1], [2] and are more flexible in comparison with a finite element model. Secondly, when using readily available numerical software, less machine insight is required while developing the model. Developing an analytical model requires a more profound knowledge of the machine’s operation and the underlaying physics. Thus using an analytical model is in the authors best interest in gaining knowledge on the subject. thirdly, analytical models are easier to parametrize, which is a key feature in this work. The second step was to understand the mathematical background of the physical properties and adjust them to the used analytical model. The discussed physical properties are the machine’s torque and its components, the eddy currents and the losses they produce in the shielding cylinder and the permanent magnets. The final step was to conduct a parameter study in order to gain a better understanding of the effects of the shielding cylinder on the machine’s performance. The effect of the shielding cylinder’s thickness and conductivity on the torque and the eddy current losses is studied.
v
Analytical model of physical properties of a high speed permanent magnet motor
Nicholas Larno, Fabrice Mutabazi
Supervisor(s): Peter Sergeant, Bert Hannon
Abstract—The goal of this master dissertation is to create an analytical model to study physical properties of a high speed permanent magnet motor. The demand for electrical machines is changing. On the one hand, the demand for more efficient electrical machines is increasing due to an increasing ecological awareness. On the other hand, there is an increase in demand of electrical machines with a high power density due to a trend towards more flexible applications which demands for smaller machines. This is driving the interest for high-speed permanent magnet synchronous machines (PMSMs). Designing high speed electrical machines is a significantly more challenging task than traditional machines. Due to the high operating speed of these machines, there are rigorous mechanical, thermal and electromagnetic conditions that need to be withstood. This work will focus on the electromagnetic aspect of these machines such as eddy currents, eddy current losses and torque.
Keywords—Fourier-based modeling, eddy currents, eddy current losses, torque, high speed permanent magnet motor, shielding cylinder
I. INTRODUCTION
THE first step was to introduce an analytical method to study the physical properties of an electrical machine. An ana-
lytical method was preferred because of various reasons. First, analytical models have proven to require less computational time [1], [2] and are more flexible in comparison with a finite element model. Secondly, when using readily available numerical software, less machine insight is required while developing the model. Devel- oping an analytical model requires a more profound knowledge of the machine’s operation and the underlaying physics. Thus using an analytical model is in the authors best interest in gaining knowledge on the subject. Thirdly, analytical models are easier to parametrize which is a key feature in this work. The second step was to understand the mathematical background of the physical properties and adjust them to the used analytical model. The discussed physical properties are the machine’s torque and its components, the eddy currents and the losses they produce in the shielding cylinder and the permanent magnets. The final step was to conduct a parameter study in order to gain a better understanding of the effects of the shielding cylinder on the machine’s performance. The effect of the shielding cylinder’s thickness and conductivity on the torque and the eddy current losses is studied.
II. FOURIER-BASED MODELING
The goal of this technique is to solve Maxwell’s equations in the entire geometry. To do so Maxwell’s equations are reformulated to a single partial-differential equation using a magnetic
N. Larno and F. Mutabazi are with the Department of Electrical energy, met- als, mechanical constructions and systems, Ghent University (UGent), Gent, Belgium. (E-mail: [email protected], [email protected])
potential, that equation is referred to as the governing equation. The machine is divided into subdomains in which the governing equation is simplified and solved using the separation of variables technique. Using this technique implies that the result will contain integration constants and eigenvalues. These are determined by imposing boundary conditions, these conditions also ensure a coherent solution over the entire geometry.
III. TORQUE
In this chapter an analytical method for torque calculation is introduced. The calculation is based on the Maxwell’s stress tensor, which is briefly described in this chapter. Secondly the implementation in the Fourier-based model is presented and the two considered torque components are described. The Maxwell’s stress tensor is based on the total magnetic force F on a rigid body with volume V and surface S which is placed in a magnetic field. The derivation starts with a calculation of the total force due to electromagnetic fields on the charges and currents within a volume V calculated with the Lorentz force law:
F =
ρ(E + v × B)dV (1)
The total electromagnetic torque in a machine equipped with a shielding cylinder is a combination of two torque producing phenomena.
0 0.1 0.2 0.3 0.4 −2
1
4
7
T PM
T SC
Fig. 1. The torque and its components
First of which is the interaction of the permanent magnets and the magnetic field due to the alternating stator current. This
interaction produces the dominant torque and is referred to as TPM , the torque produced in the magnets. The other torque component is produced by the interaction of eddy currents in the shielding cylinder and the permanent magnet’s magnetic field. This component is referred to as TSC , the torque produced in the shielding cylinder. It should be noted that TPM and TSC
are fictitious components, they do not occur separately but are regarded as such to provide a better understanding of the machine’s physics.
IV. EDDY CURRENTS
Knowledge of the eddy currents induced in the shielding cylinder and the permanent magnets is crucial for the calculation of the eddy current losses, which is an important design factor for electrical machines. In a moving conductive material, the current density J is related to the electrical field E and the magnetic field B surrounding the conductor:
J = σ (E + v × B) (2)
For the permanent magnets, a constant needs to be introduced to ensure that the net current in each magnet is zero. In a 2D approximation, it is possible for currents to be closed in another magnet. Current flowing from one magnet to the other is of course not possible. Because the shielding cylinder is made of one single piece and thus short-circuited, it is possible for current loops to be closed in an other region of the shielding cylinder without violating the law of conservation of energy. Thus it is not necessary to use this spatially constant current density in the shielding cylinder.
V. EDDY CURRENT LOSSES
To compute the eddy current losses in the shielding cylinder and the permanent magnets, two different methods are used. The losses in the permanent magnets can be calculated form Joule’s formula:
P (t) =
dv (3)
Joule’s formula is not effective at computing the losses in the SC because the r-dependent part of J3(r, φ, t) contains Bessel functions, those functions can not be integrated analytically. To cope with this problem, the losses in the shielding cylinder are calculated using the Poynting vector (S):
S = E × H (4)
The Poynting vector represents the rate per unit area at which energy crosses a surface. The integration of S over a closed surface determines the energy flowing through that surface. In this case, this implies that the total power P transmitted from the stator to the rotor can be calculated by integrating the Poynting vector over the surface of a cylinder that includes the rotor.
VI. PARAMETER STUDY
A. Torque
As expected, TPM shows very little dependency of the shielding cylinders conductivity. TPM is mainly caused by the inter-
action of synchronous field components and the expectation is that these components are not affected by the shielding cylinder’s conductivity. Although when plotted separately, a small variation of TPM can be observed. The reason for these small variations is that a part of TPM is produced by the interaction between asynchronous field components at the boundary between the air gap and the slotting region. The evolution of TSC is very similar to the speed-torque characteristic of an induction machine and can be explained by regarding the shielding cylinder as the squirrel cage of an induction machine. Note that TSC is negative which implies that the asynchronous components with a slower rotational speed than the synchronous components are dominant.
0 2 4 6 8 10
x 10 7
T or
qu e
(N m
T SC
Fig. 2. Average torque as a function of the conductivity of the SC
The second parameter studied is the thickness of the shielding cylinder. The evolution of TSC can be explained by regarding the shielding cylinder as an electrical conductor and applying Pouillet’s law, which means that the shielding cylinder’s resistance decreases as the thickness increases. This explains the initial increase of TSC . As the thickness further increases, there is a decrease of both components due to the increase of the effective air gap. The shielding cylinder proves to be beneficial in reducing the ripple on the total torque as an increase of the thickness shows a significant decrease of the torque ripple of TPM . This can be interesting for applications which need a constant torque.
B. Dissipated power
The shielding cylinder proves to be beneficial in reducing total dissipated power. Because of the shielding effect less eddy current are induced in the permanent magnet which have a relatively high resistance. The eddy currents instead are induced in the SC which has a lower resistivity, thus the total dissipated power is significantly reduced as depicted on Figure 4.
REFERENCES
[1] A. Hemeida, B. Hannon, H. Vansompel, and P. Sergeant, “Comparison of three analytical methods for the precise calculation of cogging torque and torque ripple in axial flux pm machines,” MATHEMATICAL PROBLEMS IN ENGINEERING, vol. 2016, pp. 1–14, 2016.
[2] K. Ramakrishnan, M. Curti, D. Zarko, G. Mastinu, J. J. H. Paulides, and E. A. Lomonova, “A comparison study of modelling techniques for permanent magnet machines,” in 2016 Eleventh International Conference on Ecological Vehicles and Renewable Energies (EVER), pp. 1–6, April 2016.
0 1 2 3 4 5
x 10 −3
T SC
Fig. 3. Average torque as a function of the thickness of the SC
0 2.6
P ow
SC
Fig. 4. Dissipated power as a function of the conductivity of the SC
[3] B. Hannon, P. Sergeant, and L. Dupre, “Voltage Sources in 2d Fourier- Based Analytical Models of Electric Machines,” Mathematical Problems in Engineering, vol. 2015, p. e195410, Oct. 2015.
[4] Z. Q. Zhu, L. J. Wu, and Z. P. Xia, “An accurate subdomain model for magnetic field computation in slotted surface-mounted permanent-magnet machines,” IEEE Transactions on Magnetics, vol. 46, pp. 1100–1115, April 2010.
[5] T. Lubin, S. Mezani, and A. Rezzoug, “2-d exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots,” IEEE Transactions on Magnetics, vol. 47, pp. 479–492, Feb 2011.
[6] Z. J. Liu and J. T. Li, “Analytical solution of air-gap field in permanent- magnet motors taking into account the effect of pole transition over slots,” IEEE Transactions on Magnetics, vol. 43, pp. 3872–3883, Oct 2007.
[7] K. Ramakrishnan, M. Curti, D. Zarko, G. Mastinu, J. J. H. Paulides, and E. A. Lomonova, “Comparative analysis of various methods for modelling surface permanent magnet machines,” IET Electric Power Applications, vol. 11, no. 4, pp. 540–547, 2017.
[8] H. V. Xuan, D. Lahaye, H. Polinder, and J. A. Ferreira, “Influence of stator slotting on the performance of permanent-magnet machines with concentrated windings,” IEEE Transactions on Magnetics, vol. 49, pp. 929–938, Feb 2013.
[9] B. Hannon, P. Sergeant, and L. Dupre, “2d analytical torque study of slotless and slotted pmsm topologies at high-speed operation,” in Proceedings of ElectrIMACS 2014, pp. 248–231, Universitat Politecnica de Valencia, 2014.
[10] P. D. Pfister, X. Yin, and Y. Fang, “Slotted permanent-magnet machines: General analytical model of magnetic fields, torque, eddy currents, and permanent-magnet power losses including the diffusion effect,” IEEE Transactions on Magnetics, vol. 52, pp. 1–13, May 2016.
[11] F. Zhou, J. Shen, W. Fei, and R. Lin, “Study of retaining sleeve and conductive shield and their influence on rotor loss in high-speed pm bldc motors,” IEEE Transactions on Magnetics, vol. 42, pp. 3398–3400, Oct 2006.
[12] D. J. Griffiths, Introduction to Electrodynamics. Pearson Education, 2007. [13] B. Hanon, P. Sergeant, and L. Dupre, “2d analytical torque study of slotless
and slotted pmsm topologies at high-speed operation.” none. [14] B. Hannon, P. Sergeant, and L. Dupre, “2-d analytical subdomain model of
a slotted pmsm with shielding cylinder,” IEEE Transactions on Magnetics, vol. 50, pp. 1–10, July 2014.
[15] B. Hanon, P. Sergeant, and L. Dupre, “Torque and torque components in high-speed permanent-magnet synchronous machines with a shielding cylinder.” none.
[16] A. Tenconi, S. Vaschetto, and A. Vigliani, “Electrical machines for high- speed applications: Design considerations and tradeoffs,” IEEE Transac- tions on Industrial Electronics, vol. 61, pp. 3022–3029, June 2014.
[17] B. Hanon, P. Sergeant, and L. Dupre, “Two-dimensional fourier-based modeling of electric machines.” none.
[18] Z. Tian, C. Zhang, and S. Zhang, “Analytical calculation of magnetic field distribution and stator iron losses for surface-mounted permanent magnet synchronous machines,” Energies, 2017.
[19] J. J. Lee, W. H. Kim, J. S. Yu, S. Y. Yun, S. M. Kim, J. J. Lee, and J. Lee, “Comparison between concentrated and distributed winding in ipmsm for traction application,” in 2010 International Conference on Electrical Ma- chines and Systems, pp. 1172–1174, Oct 2010.
[20] M. Merdzan, J. J. H. Paulides, and E. A. Lomonova, “Comparative analysis of rotor losses in high-speed permanent magnet machines with different winding configurations considering the influence of the inverter pwm,” in 2015 Tenth International Conference on Ecological Vehicles and Renew- able Energies (EVER), pp. 1–8, March 2015.
[21] S. Niu, S. L. Ho, W. N. Fu, and J. Zhu, “Eddy current reduction in high-speed machines and eddy current loss analysis with multislice time-stepping finite-element method,” IEEE Transactions on Magnetics, vol. 48, pp. 1007–1010, Feb 2012.
[22] D. A. Co Huynh, Liping Zheng, “Losses in high speed permanent magnet machines used in microturbine applications,” J. Eng. Gas Turbines Power 131(2), 022301 (Dec 23, 2008) (6 pages), p. 6, 2008.
[23] J. Lim, Y. J. Kim, and S.-Y. Jung, “Numerical investigation on permanent- magnet eddy current loss and harmonic iron loss for pm skewed ipmsm,” Journal of Magnetics, 2011.
Analytisch model van fysische eigenschappen van een permanente magneetbekrachtigde
hogesnelheidsmotor Nicholas Larno, Fabrice Mutabazi
Supervisor(s): Peter Sergeant, Bert Hannon
Abstract— Het doel van deze masterproef is het creeren van een analytisch model om de fysische eigenschappen van een hoge snelheidsmotor met permanente magneten te bestuderen. De vraag naar elektrische machines is aan het veranderen. Aan de ene kant is de vraag naar meer ef- ficiente elektrische machines aan het stijgen door een stijgende ecologische bewustwording. Aan de andere kant is er een stijgende vraag naar elektrische machines met een hoge vermogendichtheid door een trend naar meer flexibele applicaties die kleinere machines vereisen. Dit drijft de aandacht naar permanent magneet synchrone hoge snelheidsmotoren (PMSMs). Het ontwerpen van hoge snelheidsmotoren is een aanzienlijk uitdagendere taak dan traditionele machines. Door de hoge snelheid van die motoren zijn er extreme mechanische, thermische en elektromagnetische condities die ze moeten kunnen weerstaan. Dit werk zal zich verdiepen in het elektromagnetische aspect van deze machines zoals de wervelstromen, wervelstroom- verliezen en het koppel.
Keywords—Fourier-based modeling, eddy currents, eddy current losses, torque, high speed permanent magnet motor, shielding cylinder
I. INLEIDING
DE eerste stap was het invoeren van een analytisch model om de fysische eigenschappen van een elektrische machine te
bestuderen. Een analytische methode was gekozen om verschil- lende redenen. Ten eerste hebben analytische modellen bewezen minder reken- tijd te vereisen ([1], [2]) en zijn ze meer flexibel in vergelijking met een eindige elementen methode. Ten tweede, bij gebruik van reeds beschikbare numerieke software is er minder machine inzicht vereist tijdens het ontwikkelen van het model. Het ontwikkelen van een analytisch model vereist een diepgaande kennis van de werking van de machine en de onderliggende fysica. Zo is het gebruik van een analytisch model het best voor de auteurs om kennis over het onderwerp te verwerven. Ten derde zijn analytische modellen eenvoudiger te parametri- seren, wat een belangrijk onderdeel is in dit werk. De tweede stap was het begrijpen van de wiskundige achtergrond van de fysische eigenschappen en deze toepassen op het gebruikte analytisch model. De besproken fysische eigenschappen van de machine zijn het koppel en zijn componenten, de wervelstromen en de verliezen die ze produceren in de shielding cylinder en de permanente magneten. De laatste stap was het uitvoeren van een parameter studie om een beter inzicht te krijgen in het effect van de shielding cylinder op de performantie van de machine. Het effect van de dikte en de geleidbaarheid van de shielding cylinder op het koppel en de wervelstroom verliezen werden bestudeerd.
N. Larno and F. Mutabazi are with the Department of Electrical energy, me- tals, mechanical constructions and systems, Ghent University (UGent), Gent, Belgium. (E-mail: [email protected], [email protected])
II. FOURIER-BASED MODELING
Het doel van deze techniek is het oplossen van de Max- well vergelijkingen in de volledige geometrie. Om dit te doen worden de Maxwell vergelijkingen geherformuleerd naar een partiele differentiaalvergelijking gebruikmakend van een magnetisch potentiaal, deze vergelijking wordt de governing equation genoemd. De machine is in subdomeinen ingedeeld waarin de governing equation vereenvoudigd is en kan opgelost worden door de scheiding van veranderen techniek toe te passen. Het gebruik van deze techniek zorgt ervoor dat het bekomen re- sultaat integratieconstanten en eigenwaarden zal bevatten. Deze worden bepaald door grensvoorwaarden in te stellen, deze voor- waarden zorgen ook voor een coherente oplossing over de ge- hele meetkunde.
III. KOPPEL
In dit hoofdstuk wordt een analytische methode voor het berekenen van het koppel gentroduceerd. De berekening is gebaseerd op de Maxwell stress tensor, welke uitvoerig besproken wordt in dit hoofdstuk. Ten tweede wordt de implementatie in het Fourier-based model besproken als ook de twee beschouwde koppel componenten. De Maxwell stress tensor is gebaseerd op de totale magnetische kracht F op een star lichaam met volume V en oppervlakte S dat zich in een magnetisch veld bevindt. De afleiding start met de berekening van de totale kracht veroorzaakt door elektromagnetische velden op ladingen en stromen in een volume V berekent met de Lorentz force law:
F =
ρ(E + v × B)dV (1)
Het totale elektromagnetische koppel in een machine uitge- rust met een shielding cylinder is een combinatie van twee koppel producerende fenomenen.
Het eerste fenomeen is de interactie tussen de permanente magneten en het magnetisch veld opgewekt door de wisselende stator stromen. Deze interactie produceert het dominante koppel en wordt aangeduid als TPM , het koppel geproduceerd in de magneten. De andere component wordt geproduceerd door de interactie tussen de wervelstromen in de shielding cylinder en het magnetisch veld van de permanente magneten. Deze component wordt aangeduid als TSC , het koppel geproduceerd in de shielding cylinder. Merk op dat TPM en TSC fictieve componenten zijn, ze komen niet afzonderlijk voor maar worden op
0 0.1 0.2 0.3 0.4 −2
1
4
7
T PM
T SC
Fig. 1. Het totale koppel en zijn componenten
deze manier beschouwd om de fysische achtergrond van de machine beter te begrijpen.
IV. WERVELSTROMEN
Kennis over de wervelstromen die genduceerd worden in de shielding cylinder en de permanente magneten is cruciaal voor het berekenen van de verliezen die ze teweegbrengen, deze verliezen zijn een belangrijke ontwerpfactor voor elektrische machines. In een bewegende geleider is de stroomdichtheid J gerelateerd aan het elektrisch veld E en het magnetisch veld B dat de geleider omvat:
J = σ (E + v × B) (2)
Voor de permanente magneten moet een constante toege- voegd worden om ervoor te zorgen dat de netto stroom in elke magneet nul is. In een 2D benadering is het mogelijk dat stromen zich sluiten in een andere magneet. Stroom die vloeit van de ene magneet naar de andere is natuurlijk onmogelijk. Door- dat de shielding cylinder een geheel is en dus kortgesloten is, is het mogelijk dat er stromen sluiten in een andere regio van de shielding cylinder zonder te zondigen tegen de wet van behoud van energie. Hierdoor is het niet nodig deze constante in de shielding cylinder te gebruiken.
Voor de permanente magneten moet een constante toege- voegd worden om ervoor te zorgen dat de netto stroom in elke magneet nul is. In een 2D benadering is het mogelijk dat stromen zich sluiten in een andere magneet. Stroom die vloeit van de ene magneet naar de andere is natuurlijk niet mogelijk. Door- dat de shielding cylinder een geheel is en dus kortgesloten is, is het mogelijk dat er stromen sluiten in een andere regio van de shielding cylinder zonder te zondigen tegen de wet van behoud van energie. Hierdoor is het niet nodig deze constante in de shielding cylinder te gebruiken.
V. WERVELSTROOM VERLIEZEN
Om de wervelstroom verliezen in de shielding cylinder en de permanent magneten te berekenen worden er twee methodes ge- bruikt. De verliezen in de permante magneten worden berekend
met de formule van Joule:
P (t) =
dv (3)
Deze formule is echter niet effectief om de verliezen in de shielding cylinder te berekenen doordat het r-afhankelijk deel van J3(r, φ, t) Bessel functies bevat, deze functies kunnen niet analytisch gentegreerd worden. Om dit probleem op te lossen worden de verliezen in de shielding cylinder berekent door gebruik te maken van de Poynting vector S:
S = E × H (4)
De Poynting vector representeert de hoeveelheid energie die per tijdseenheid door een eenheidsoppervlakte stroomt. De integratie van S over een gesloten oppervlak bepaalt de energie die door dat oppervlak stroomt. In dit geval betekent dit dat het totaal vermogen P dat overgedragen wordt van de stator naar de rotor berekend kan worden door de integratie van de Poynting vector over een oppervlak dat de rotor omvat.
VI. PARAMETER STUDIE
A. Koppel
Zoals verwacht toont TPM slechts een beperkte afhankelijk- heid van de geleidbaarheid van de shielding cylinder. TPM is vooral veroorzaakt door de interactie van synchrone veldcomponenten en de verwachting was dat deze componenten niet benvloed worden door de geleidbaarheid van de shielding cylinder. Maar wanneer deze afzonderlijk wordt geplot kan er toch een kleine variatie waargenomen worden. De reden voor deze variatie is dat een deel van TPM geproduceerd wordt door de interactie tussen asynchrone veldcomponenten aan de grens tussen de luchtspleet en de sloten. De evolutie van TSC is gelijkaardig aan de koppel-toerental ka- rakteristiek van een inductie machine en kan ook verklaard worden door de shielding cylinder te beschouwen als de kooian- ker van een inductie machine. Merk op dat TSC negatief is, dit houdt in dat asynchrone componenten met een tragere draaisnel- heid dan de synchrone componenten dominant zijn.
0 2 4 6 8 10
x 10 7
T or
qu e
(N m
T SC
Fig. 2. Het gemiddelde koppel in functie van de geleidbaarheid van de SC
De tweede parameter die bestudeerd wordt is de dikte van de shielding cylinder. De evolutie van TSC kan verklaard worden
door de shielding cylinder te bekijken als een elektrische geleider en de wet van Pouillet toe te passen. Dit houdt in dat de weerstand van de shielding cylinder daalt als de dikte vergroot. Dit verklaart de initiele stijging van TSC . Als de dikte verder vergroot is er een daling van beide componenten door het ver- groten van de effectieve luchtspleet. De shielding cylinder blijkt gunstig te zijn bij het verminderen van de rimpel op het totale koppel aangezien een stijging van de dikte een significante daling van de koppelrimpel van TPM teweeg brengt. Dit kan inte- ressant zijn voor applicaties die een constant koppel vereisen.
0 1 2 3 4 5
x 10 −3
T SC
Fig. 3. Het gemiddelde koppel in functie van de dike van de SC
B. Gedissipeerd vermogen
De shielding cylinder blijkt ook gunstig te zijn in het verlagen van het totaal gedissipeerd vermogen. Door het shielding effect worden er minder wervelstromen genduceerd in de permanente magneten die een relatief hoge weerstand hebben. Deze wervelstromen worden in plaats daarvan genduceerd in de shielding cylinder, welke een lagere weerstand heeft. Hierdoor wordt het totaal gedissipeerd vermogen significant gereduceerd zoals te zien is op Figuur 4.
0 2.6
P ow
SC
Fig. 4. Het gedissipeerd vermogen in functie van de geleidbaarheid van de SC
REFERENCES
[1] A. Hemeida, B. Hannon, H. Vansompel, and P. Sergeant, “Comparison of three analytical methods for the precise calculation of cogging torque and torque ripple in axial flux pm machines,” MATHEMATICAL PROBLEMS IN ENGINEERING, vol. 2016, pp. 1–14, 2016.
[2] K. Ramakrishnan, M. Curti, D. Zarko, G. Mastinu, J. J. H. Paulides, and E. A. Lomonova, “A comparison study of modelling techniques for permanent magnet machines,” in 2016 Eleventh International Conference on Ecological Vehicles and Renewable Energies (EVER), pp. 1–6, April 2016.
[3] B. Hannon, P. Sergeant, and L. Dupre, “Voltage Sources in 2d Fourier- Based Analytical Models of Electric Machines,” Mathematical Problems in Engineering, vol. 2015, p. e195410, Oct. 2015.
[4] Z. Q. Zhu, L. J. Wu, and Z. P. Xia, “An accurate subdomain model for magnetic field computation in slotted surface-mounted permanent-magnet machines,” IEEE Transactions on Magnetics, vol. 46, pp. 1100–1115, April 2010.
[5] T. Lubin, S. Mezani, and A. Rezzoug, “2-d exact analytical model for surface-mounted permanent-magnet motors with semi-closed slots,” IEEE Transactions on Magnetics, vol. 47, pp. 479–492, Feb 2011.
[6] Z. J. Liu and J. T. Li, “Analytical solution of air-gap field in permanent- magnet motors taking into account the effect of pole transition over slots,” IEEE Transactions on Magnetics, vol. 43, pp. 3872–3883, Oct 2007.
[7] K. Ramakrishnan, M. Curti, D. Zarko, G. Mastinu, J. J. H. Paulides, and E. A. Lomonova, “Comparative analysis of various methods for modelling surface permanent magnet machines,” IET Electric Power Applications, vol. 11, no. 4, pp. 540–547, 2017.
[8] H. V. Xuan, D. Lahaye, H. Polinder, and J. A. Ferreira, “Influence of stator slotting on the performance of permanent-magnet machines with concentrated windings,” IEEE Transactions on Magnetics, vol. 49, pp. 929–938, Feb 2013.
[9] B. Hannon, P. Sergeant, and L. Dupre, “2d analytical torque study of slotless and slotted pmsm topologies at high-speed operation,” in Proceedings of ElectrIMACS 2014, pp. 248–231, Universitat Politecnica de Valencia, 2014.
[10] P. D. Pfister, X. Yin, and Y. Fang, “Slotted permanent-magnet machines: General analytical model of magnetic fields, torque, eddy currents, and permanent-magnet power losses including the diffusion effect,” IEEE Transactions on Magnetics, vol. 52, pp. 1–13, May 2016.
[11] F. Zhou, J. Shen, W. Fei, and R. Lin, “Study of retaining sleeve and conductive shield and their influence on rotor loss in high-speed pm bldc motors,” IEEE Transactions on Magnetics, vol. 42, pp. 3398–3400, Oct 2006.
[12] D. J. Griffiths, Introduction to Electrodynamics. Pearson Education, 2007. [13] B. Hanon, P. Sergeant, and L. Dupre, “2d analytical torque study of slotless
and slotted pmsm topologies at high-speed operation.” none. [14] B. Hannon, P. Sergeant, and L. Dupre, “2-d analytical subdomain model of
a slotted pmsm with shielding cylinder,” IEEE Transactions on Magnetics, vol. 50, pp. 1–10, July 2014.
[15] B. Hanon, P. Sergeant, and L. Dupre, “Torque and torque components in high-speed permanent-magnet synchronous machines with a shielding cylinder.” none.
[16] A. Tenconi, S. Vaschetto, and A. Vigliani, “Electrical machines for high- speed applications: Design considerations and tradeoffs,” IEEE Transac- tions on Industrial Electronics, vol. 61, pp. 3022–3029, June 2014.
[17] B. Hanon, P. Sergeant, and L. Dupre, “Two-dimensional fourier-based modeling of electric machines.” none.
[18] Z. Tian, C. Zhang, and S. Zhang, “Analytical calculation of magnetic field distribution and stator iron losses for surface-mounted permanent magnet synchronous machines,” Energies, 2017.
[19] J. J. Lee, W. H. Kim, J. S. Yu, S. Y. Yun, S. M. Kim, J. J. Lee, and J. Lee, “Comparison between concentrated and distributed winding in ipmsm for traction application,” in 2010 International Conference on Electrical Ma- chines and Systems, pp. 1172–1174, Oct 2010.
[20] M. Merdzan, J. J. H. Paulides, and E. A. Lomonova, “Comparative analysis of rotor losses in high-speed permanent magnet machines with different winding configurations considering the influence of the inverter pwm,” in 2015 Tenth International Conference on Ecological Vehicles and Renewa- ble Energies (EVER), pp. 1–8, March 2015.
[21] S. Niu, S. L. Ho, W. N. Fu, and J. Zhu, “Eddy current reduction in high-speed machines and eddy current loss analysis with multislice time-stepping finite-element method,” IEEE Transactions on Magnetics, vol. 48, pp. 1007–1010, Feb 2012.
[22] D. A. Co Huynh, Liping Zheng, “Losses in high speed permanent magnet machines used in microturbine applications,” J. Eng. Gas Turbines Power 131(2), 022301 (Dec 23, 2008) (6 pages), p. 6, 2008.
Contents
2.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Torque calculation and validation . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.1 Torque calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3.2 Torque components . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Eddy currents 19
5 Eddy-current losses 25
6.3.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
8.3 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.5.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 radial flux density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Harmonic content Br . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 machine topology torque inner rotor - open slots - shielding cylinder . . . 16
3.4 Validation total torque inner rotor - open slots - SC - no load . . . . . . . 18
4.1 Eddy current density in the PM . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Eddy currents comparison ANA (a) vs FEM (b) in the SC . . . . . . . . . 21
4.3 Eddy current density in the SC . . . . . . . . . . . . . . . . . . . . . . . . 22
4.4 Eddy current density in r3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.6 Eddy current density in the PM . . . . . . . . . . . . . . . . . . . . . . . . 24
6.1 Machine topology parameter study torque . . . . . . . . . . . . . . . . . . 31
6.2 Harmonic content of the studied machine Br . . . . . . . . . . . . . . . . . 32
6.3 Harmonic content of the studied machine Bφ . . . . . . . . . . . . . . . . . 33
6.4 Average torque as a function of the conductivity of the SC . . . . . . . . . 34
6.5 Illustration of the interaction between synchronous- and asynchronous field components at the slot opening . . . . . . . . . . . . . . . . . . . . . 34
6.6 Torque in the PM as a function of the conductivity of the SC . . . . . . . . 35
6.7 Torque ripple in the PM as a function of the conductivity of the SC . . . . 38
6.8 Average torque as a function of the thickness of the SC . . . . . . . . . . . 39
6.9 Torque ripple as a function of the thickness of the SC . . . . . . . . . . . . 40
xvii
6.10 machine topology parameter study Eddy current losses . . . . . . . . . . . 41
6.11 harmonic content of the studied machine Br . . . . . . . . . . . . . . . . . 42
6.12 harmonic content of the studied machine Bφ . . . . . . . . . . . . . . . . . 43
6.13 Eddy current loss as a function of the conductivity of the SC . . . . . . . . 44
6.14 Eddy current loss as a function of the conductivity of the SC and the penetration depth of the dominant asynchronous field components . . . . . 45
6.15 Induced eddy currents as a function of the conductivity of the SC . . . . . 46
6.16 the square of the induced eddy currents as a function of the conductivity of the SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.17 Resistivity of the SC as a function of the conductivity of the SC . . . . . . 47
6.18 Eddy current losses as a function of the thickness of the SC . . . . . . . . . 48
6.19 Eddy current losses as a function of the thickness of the SC . . . . . . . . . 48
xviii
xix
Symbol Definition δi,j Kronecker delta λ
(ν) k Eigenvalue related to the spatial-aspect of the solution λ(ν) n Eigenvalue related to the time-aspect of the solution µ2 Permeability of the magnets µ3 Permeability of the sleeve φm Angular span of a magnet φms Start angle of a magnet ρ Charge density σ Conductivity σ2 Conductivity of the magnets σ3 Conductivity of the shielding cylinder σPM Conductivity of the magnets σSC Conductivity of the shielding cylinder Jeddy Eddy current density Jext Externally imposed current density A Magnetic vector potential B Magnetic flux density Bm Residual flux density of the magnets D Electric flux density E Electric field strength EC Eddy currents F Total magnetic force f Force per unit volume fnom Nominal frequency H Magnetic field strengh Inom Current density J Current density ls Stack length
Symbol Definition m Number of phases MQS Magnetoquasistatic MV P Magnetic vector potential N Number of windings per slot nnom Nominal rotational speed Ns Number of slots p Number of pole pairs PM Permanent magnets PMSM Permanent magnet synchronous machine r1 Outer rotor iron radius r2 Outer PM radius r3 Outer sleeve radius r4 Outer airgap radius r5 Outer tooth tip radius r6 Outer winding radius r7 Outer stator iron radius rac Radius at the center of the airgap rmc Radius at the center of the magnet S Surface S Poynting vector Sms Surface area of a single magnet SC Shielding cylinder SM Surface-mounted T Maxwell stress tensor T Torque TPM Torque produced in the permanent magnets TSC Torque produced in the shielding cylinder Tt Time periodicity V Volume wτ tooth-tip width wt Tooth width
Chapter 1
Introduction
Designing high-speed electrical machines is a significantly more challenging task than traditional machines. Due to the high operating speed of the machines, rigorous mechanical, thermal and electromagnetic conditions need to be withstood. In the field of high-speed electrical machines the two most represented types of machines are induction machines and Permanent Magnet Synchronous Machines(PMSMs). Because of the higher power density achievable with PMSM, especially at very high speed, the focus nowadays leans towards PMSMs. These machine type will be the main focus of this work, exclusively the Surface-Mounted (SM) PMSMs with a retaining sleeve. This work will focus on the electromagnetic aspect of these machines such as eddy currents, eddy current losses and torque. Moreover a study of the effect of the so called Shielding Cylinder on these properties will be conducted. A Fourier based modeling technique is used to implement and study the mentioned properties.
This work consists of three parts. First, the modeling technique is shortly explained. The second part illustrates the implementation of the studied physical properties in the Fourier based model. The last part is the study itself where the considered properties are studied in regard to varying parameters.
1
Chapter 2
Fourier-Based Modeling
2.1 Introduction
The main goal of this work is to perform an analytical study of various physical properties of high speed machines. An analytical model is chosen because of various reasons. First, analytical models have proven to require less computational time [1], [2], although numerical approaches provide high accuracy results due to their ability to include complex geometries [3]. Secondly, when using readily available numerical software, less machine insight is required while developing the model. Developing an analytical model requires a more profound knowledge of the machine’s operation and the underlaying physics. Thus using an analytical model is in the authors best interest in gaining knowledge on the subject. Thirdly, analytical models are easier to parametrize, which is a key feature in this work. This chapter summarizes some of the basics of the analytical model used. It is mainly based on the authors’ thesis counselor’s work [[4],[5],[6]].
2.2 Fourier-Based Modeling
The goal of this technique is to solve Maxwell’s equations in the entire geometry. To do so Maxwell’s equations are reformulated to a single partial-differential equation using a magnetic potential, that equation is referred to as the governing equation. The machine is divided into subdomains in which the governing equation is simplified and solved using the separation of variables technique. Using this technique implies that the result will contain integration constants and eigenvalues. These are determined by imposing boundary conditions, these conditions also ensure a coherent solution over the entire geometry. The modeled machines in this work are radial-flux rotational machines. A cylindrical coordinate system (r, φ, z) is used to describe the geometry, where the z-axis is placed along the machine’s axis.
3
2.2.1 Assumptions
In order to enable an analytical solution of the problem with limited mathematical complexity, multiple assumptions are made. First, the situation is assumed to be magnetoquasistatic (MQS), this implies two aspects. The magneto aspect dictates that the magnetic field is dominant in regard to the electric field. The quasistatic aspect implies that time-dependent variations are propa- gated instantly throughout the entire geometry. Secondly the problem is assumed invariant along the z-axis, only two dimensions are considered. Thirdly, the boundaries of the chosen subdomains are either radial, i.e a constant φ, or circumferential, i.e a constant r. This assumption is required to solve the problem analytically, boundaries which aren’t radial or circumferential need to be simplified. Fourthly, it is assumed that all movements occur along the tangential direction. This means that all moving parts rotate around the same axis. This axis has to coincide with the longitudinal direction. Fifthly, all of the materials are assumed linear and isotropic. Furthermore soft-magnetic material are assumed to have an infinite magnetic permeability. Sixthly, the conductivity (σ) of the magnet region is assumed zero, so that no eddy current can be induced in the magnets. Finally, the machine is assumed to be operating in steady-state, this implies that tran- sient effects are not regarded, thus only time periodic components are present. This assumption is required because the FBM is based on periodicities in the magnetic field.
2.2.2 Potential formulation
The goal of Fourier-based modeling is to solve Maxwell’s equations, These can be written as follow in the MQS situation:
∇× E = −∂B ∂t
∇×H = J (2.1b) ∇ ·B = 0 (2.1c)
Note that the influence of the electric charges is neglected, because of the dominance of the magnetic field. Due to the difficulty in solving these equations analytically, the problem is reformulated using a magnetic potential; the Magnetic Vector Potential (MVP). The MVP, which is a vector is defined by its curl:
∇×A = B (2.2)
CHAPTER 2. FOURIER-BASED MODELING
For a complete description of MQS problems, the MVP has to be combined with the electric scalar potential V, which is defined by its gradient.
∇V = − (
) (2.3)
The following partial-differential equation for Maxwell’s equations (2.1) can be obtained from the above definitions and the constitutive relations:
∇2A−∇ ·A− µσ∂A ∂t
+ µσ(v× (∇×A)) = µσ∇V −∇×B0 (2.4)
It should be noted that the MVP is not uniquely defined,so gauge fixing is necessary. the Coulomb gauge (∇ ·A = 0) is chosen as is common in quasi static approximations.
Equation (2.4) can be physically interpreted by considering two current densities, the eddy current density (Jeddy) and the externally imposed current density (Jext). The eddy current density is accounted for by the time-derivative and speed-dependent terms of (2.4). The externally imposed current densities are accounted for by the µσ∇V term. This implies that each of the machine’s conductors should be modeled separately. As this would result in an augmented computational burden, the externally imposed current density is directly accounted for. This is done by substituting Jext for σ∇V in (2.4). Resulting from these changes, a governing equation for the MVP is obtained:
∇2A− µσ∂A ∂t
2.2.3 Subdomain technique
The governing equation (2.5) is too complex to be solved in the entire geometry. In order to avoid that complexity, the geometry is divided in regions called subdomains. In each chosen subdomain the problem is simplified and can therefore be solved. To ensure a physically correct behavior of the magnetic vector potential, boundary conditions are imposed. In choosing subdomains, two aspects should be considered. First, the governing equation should be simplified as much as possible and secondly, boundary conditions should be relatively easy to impose. These aspects are further discussed in this section.
Subdomains
In order to simplify (2.5), dividing the geometry in subdomains will reduce the num- bers of terms in the equation. Three types of terms can be recognized in the governing equation (2.5). First, the Laplacian term ∇2A is a general term that is present in every subdomain.
5
Secondly, the time-derivate and speed-dependent terms account for the eddy-currents. These terms are only present in conductive subdomains where eddy-currents are non- negligible. Last are the terms on the right hand side, these are sources terms. Jext accounts for externally imposed current densities and B0 accounts for the permanent magnets. The equation is also greatly simplified if σ and µ are constant, subdomain are chosen taking this in account. The model used in this work does not consider sudo- mains with multiple source terms or subdomains where both a source term and eddy- currents are present. This implies that only the following governing equations are considered:
∇2A = 0 (2.6a) ∇2A = µJext (2.6b) ∇2A = −∇×B0 (2.6c)
∇2A = ∂A ∂t
+ µσ(v× (∇×A)) (2.6d)
Where the simplest equation (2.6a) applies to subdomains with no source terms and no eddy currents. Equations (2.6b) and (2.6c) applies to subdomains where source terms are present. Finally (2.6d) applies to subdomains where the eddy-currents are non- negligible.
Boundary conditions
The boundary conditions ensure a physically correct behavior of the MVP. Initially a set of boundary conditions for B,D,H and E are considered. Because the situation is assumed MQS and the focus is thus on the magnetic field, only the conditions for B and H are regarded. These are the following:
n · ( B(ν) −B(ν+1)
) = Js
(2.7)
where n is the normal vector from subdomain ν to ν + 1 and Js is the current density on the boundary. In a 2-D model, Js is a line current density which is not considered in this model, thus Js is always zero. Using the magnetic vector potential definition (2.2) and the constitutive relations in the MQS situation (see appendix 8.2) the boundary condition can be written as:
A(ν)(rν , φ, t)−A(ν+1)(rν , φ, t) = 0 (2.8a)( ∂A(ν)(r, φ, t)
µ(ν)∂r − ∂A(ν+1)(r, φ, t)
µ(ν+1)∂r
Where (2.8a) imposes the continuity of A at the circumferential boundary between subdomains ν and ν + 1. Condition (2.8b) expresses the continuous behavior of the tangential component of the magnetic field strength. As stated in assumption (2.2.1), a subdomain is assumed to have four boundaries. Two of which are constant along the normal r direction and two along the φ periodic direction. A subdomain is referred as periodic or non-periodic, depending on whether or not they span the entire geometry along the periodic direction. In the spherical coordinate system a periodic subdomain is one which spans 2π radians.
2.2.4 Form of the solution
Equations (2.6) are solved using the separation of variables technique. The result contains both eigenvalues and integration constants. The eigenvalues, which determine the periodicity of the solution are found by imposing the periodic boundary conditions. The circumferential boundary conditions of all the boundaries form a set of equations which determines the integration constants. The solution in subdomain ν is written as:
Aν(r, φ, t) = ∑ λ
(ν) k
(r) is the r-dependent Fourier coefficient of the series, this coefficient contains the integration constants. λ(ν)
n is the eigenvalue related to the time-aspect of the solution. λ(ν)
k the eigenvalue related to the spatial-aspect of the solution. As stated above the eigenvalues are determined by the subdomain’s periodicity. The time periodicity (Tt) is the time the machine needs to perform one revolution and is equal for all the subdomains. The spatial periodicity of a subdomain is T (ν)
s radians. This periodicity is either 2π, for periodic subdomains, or 2βν , for non-periodic subdomains with soft magnetic boundaries. The eigenvalues can then be written as:
λ(ν) n = 2nπ
Tt = nω (2.10a)
(2.10b)
where ω is the angular speed of the machine, n and k are the time- and spatial harmonic order respectively. The solution for a periodic subdomain can be written as:
Aν(r, φ, t) = ∑ λ
and for a non periodic subdomain with soft-magnetic boundaries as:
Aν(r, φ, t) = ∑ λ
j( kπβν φ−nωt) (2.12)
As the equation is a summation over every harmonic combination, the governing equation can be considered for every time- and spatial-harmonic combination separately. This allows simplification of the time-derivative and speed-dependent terms of the governing equations (2.6). From the final form of solution (2.9), the time derivate can easily be calculated for every time- and spatial-harmonic combination:
∂A(ν) n,k
∂t = jnωA(ν)
n,k (2.13)
The time-derivate of A(ν) n,k can thus be written as the product of a constant and A(ν)
n,k
itself. Another important term of the governing equation is the speed-dependent term. As assumed in section (2.2.1), the speed of subdomain υ only has a φ-dependent term, i.e. v(υ) = r(υ)eφ. The speed-dependent term can then also be calculated for every harmonic combination based on (2.9):
v(ν) × ( ∇×Aν
(ν)A(ν) n,k (2.14)
Similar to the time-derivate, the speed-dependent term can be written as the product of a constant and A(ν)
n,k itself. Equations (2.6) can now be rewritten for an arbitrary harmonic combination (n, k) as:
∂2A(ν) n,k
) A(ν) n,k (2.15d)
The governing equations (2.6) can now be solved using the separation of variables technique. As an example the first equation (2.15a) is briefly discussed. Equation (2.15a) is a Laplace equation, it is applicable to regions where there is no magnetic field and the eddy-current density is negligible. These conditions are met in the air gap region, which is a periodic subdomain so Ts = 2π. The solution of the equation follows the form given in (2.9) and is written as:
Aν(r, φ, t) = ∞∑
8
C (ν) n,kr
where C(ν) n,k and D(ν)
n,k are the integration constants. The other solutions can be found in [5].
2.2.5 Conclusion
This chapter covered the basics of the Fourier-based model that is used in this work. The strategy of the technique, the necessary assumptions and the solutions for this model were briefly discussed. Note that there are many aspects which have not been discussed for the sake of simplicity. Some additional information can be found in the appendix. The authors made use of an existing model provided by faculty researcher and thesis counselor ing. Bert Hannon. The interested reader can find more in depth information regarding analytical modeling technique in [[4],[6],[7],[8], [9]].
9
3.1 Introduction
In this chapter an analytical method for torque calculation is introduced. The calculation is based on the Maxwell’s stress tensor, which is briefly described in this chapter. Secondly the implementation in the Fourier-based model is presented and the two considered torque components are described. Finally a validation is provided by comparing the analytical results with finite element calculations. This torque calculation method is later used in the parameter study.
3.2 Maxwell stress tensor
The calculation of the Maxwell’s stress tensor is based on the total magnetic force F on a rigid body with volume V and surface S which is placed in a magnetic field. The derivation starts with a calculation of the total force due to electromagnetic fields on the charges and current within a volume V . Lorentz force law states:
F = ˆ V
ρ(E + v×B)dV
(ρE + J×B)dV (3.1)
The integrand can be seen as a force per unit volume f in a region with magnetic flux density B and current density J.
f = ρE + J x B (3.2)
11
CHAPTER 3. TORQUE
By using Maxwell’s equations, Faraday’s law and the product rule of differential calculus, f can be found as:
f = ε0[(∇ · E)E + (E · ∇)E] + 1 µ0
[(∇ ·B)B + (B · ∇)B]
(3.3)
The Maxwell stress tensor T can now be introduced, which is a 3 × 3 matrix whose components are defined as:
Ti,j = ε0(EiEj − 1 2δi,jE
where δi,j represents Kronecker delta.
δi,j = 1, if i = j,
0, if i 6= j.
The subscripts i and j indicate the ijth component of the matrix. Those components each represent a unit of force per unit of area. Which can be interpreted as the force along the ith axis experienced by a surface normal to the jth axis, per unit of area. The diagonal elements represent pressures and the non-diagonal elements are shears stresses. The above equation can be used for force calculations in a Cartesian coordinate system. However this study is conducted in a cylindrical coordinate system, the equation is therefore rewritten accordingly. Furthermore the field is mainly magnetic so that a number of terms can be dropped and the equation can be simplified to
Tr,t = 1 µ0
(BrBt − 1 2δr,tB
2) (3.5)
where r is the radial direction and t is the shear in the tangential direction. The tangential component is the one that expresses torque in a motor, it is that component that is focused on later. The total force on volume V can be calculated by integrating the force per unit of area for each element of the Maxwell stress tensor over a surface S:
F = S
TdS (3.6)
Te = S
3.3.1 Torque calculation
Using the Maxwell stress tensor, the electromagnetic torque of a rotating electrical motor can be calculated. As described in the previous section a surface integral must be chosen, the considered surface must surround the studied motor. In a 2D model the surface integral is reduced to a line integral along the air gap. If a circle with radius r is considered as the integration path, the torque can be calculated as:
Te = 1 µ0
0 r2BrBφdφ (3.8)
This result can now be implemented in the Fourier based model. Where the radial (Br) and tangential (Bφ ) components of the flux density are calculated by applying (2.2).
B (3i) r,k,n = 1
(3.9)
(3.10)
The flux components are calculated in the air gap region following the surface integral chosen for the torque calculations. Substituting (3.9) and (3.10) in (3.8) and accounting for the shaft length ls.
T (t) = lsr 2 ac
µ0
2πˆ
0
φ (rac, φ, t) dφ (3.11)
By using the periodicity of the MVP, (3.11) can be simplified to:
T (t) = 2π lsr 2 ac
µ0
∞∑ n=−∞
∞∑ s=−∞
Tn,se−j(n+s)t (3.12)
Where n is the time harmonic order related to tangential flux component (B(3i) φ ) and s
the time harmonic order related to the radial flux component (B(3i) φ ) and Tn,s is calcu-
lated as: Tn,s =
B (3i) r,k,n(rac)B(3i)
φ,−k,n(rac) (3.13)
Equation (3.12) is implemented in the Fourier based model and used to calculate the electromagnetic torque.
13
CHAPTER 3. TORQUE
3.3.2 Torque components
The total electromagnetic torque in a machine equipped with a shielding cylinder is a combination of two torque producing phenomena. First of which is the interaction of the permanent magnets and the magnetic field due to the alternating stator current. This interaction produces the dominant torque and is referred to as TPM , the torque produced in the magnets. TPM is calculated by considering the surface surrounding the rotor yoke and the magnet ring. The other torque component is produced by the interaction of eddy currents in the shielding cylinder and the permanent magnet’s magnetic field. These eddy currents are introduced by two phenomena. First, the magnetic flux density in the air gap fluctuates as it passes from a point opposite to a slot, to a point opposite to a stator tooth. This because of the difference in permeability between the stator iron (high permeability) and the air in the slot opening (low permeability). This is known as the slotting effect.
0 1 2 3 4 5 6 −0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
Figure 3.1: radial flux density
These fluctuations can be observed on Figure 3.1, where the magnetic field density is plotted in function of the angle φ. The dips at the top are partially due to the slotting effect. With the aid of Fourier analysis, the periodic wave of the flux density can be de- composed in multiple harmonic components. this is shown in Figure 3.2. It can be seen that beside the fundamental harmonics, a number of asynchronous field components are presents, these represent the effect of the slotting effect.
14
Figure 3.2: Harmonic content Br
These changes in magnetic flux density induces eddy currents in the shielding cylinder, which in interaction with the magnets produces torque. Secondly, eddy currents are induced due to the alternating currents in the stator, this is similar to the principle of a squirrel cage rotor in an induction machine. The resulting torque component from both effects is referred to as TSC , the torque produced in the shielding cylinder. It can be calculated as the difference of the total torque T and the torque in the magnets TPM :
TSC(t) = T (t)− TPM(t) (3.14)
It should be noted that TPM and TSC are fictitious components, they do not occur separately but are regarded as such to provide a better understanding of the machine’s physics.
15
3.3.3 Finite element torque validation
Now that an analytical implementation of the torque has been derived, it needs to be validated. The analytically calculated torque is compared with a finite element torque calculation. The software used to conduct the FEM analysis is COMSOL Multiphysics. The considered topology is a surface mounted magnets PMSM with open slots and a shielding cylinder, the machine topology is shown in Figure 3.3.
Figure 3.3: machine topology torque inner rotor - open slots - shielding cylinder
A FEM model of the depicted topology is created in COMSOL, along with the various machine and material parameters. An overview of the model parameters is listed in table 3.1.
16
Table 3.1: Parameters of the studied machines
Parameter Symbol Value Number of slots Ns 12 Number of pole pairs p 2 Number of phases m 3 Number of layers / 1 Residual flux density of the magnets Bm 1.2 T Angular span of a magnet φm 80.00 Current density Inom 5.106 A
m2
Number of windings per slot N 6 Nominal frequency fnom 1000 Hz Nominal speed nnom 30,000 rpm Outer rotor iron radius r1 22.50 mm Outer PM radius r2 25.00 mm Outer sleeve radius r3 26.50 mm Outer airgap radius r4 28.50 mm Outer winding radius r5 40.40 mm Outer stator iron radius r6 50.00 mm Stack length ls 200 mm Permeability of the magnets µ2 1.0µ0
H M
17
1
4
7
T PM
T SC
Figure 3.4: Validation total torque inner rotor - open slots - SC - no load
Figure 3.4 shows a comparison of the different torque components between the FEM and the analytical calculations. A good agreement can be observed for all the torque components (T , TPM and TSC). From these results, it can be concluded that the implemented torque calculation is valid. In Chapter 6, a parameter study of the torque is conducted using the above described calculation method.
18
Chapter 4
Eddy currents
4.1 Introduction
In this chapter an analytical eddy current density calculation is presented. Knowledge of the eddy currents in a machine is crucial for the calculation of eddy current losses, which is an important design factor for electrical machines. The eddy current density is calculated in the permanent magnets (PM) and the shielding cylinder (SC). A validation of these calculations is provided by comparing the analytical results with a finite element calculation.
4.2 Eddy current calculation
In a moving conductive material, the current density J is related to the electrical field E and the magnetic field B surrounding the conductor.
J = σ (E + v×B) (4.1)
No external currents are applied in the magnets and the shielding cylinder, so the electric field E can be written as:
E = ∂A ∂t
(4.2)
By using (4.2) and the definition of the magnetic vector potential A ( 2.2 ) the current density J can be written as:
J = σ
( ∂A ∂t
+ v× (∇×A) )
(4.3)
By using the properties of vector calculus and (2.2) for A , the following is obtained for the SC:
19
kA (3) n,k(r, φ)ej(kφ−nωt) (4.4)
For the permanent magnets, a constant (C(t)) needs to be introduced to ensure that the net current in the magnet region is zero. This because in a 2D approximation, it is possible for currents to be closed in another magnet. Current flowing from one magnet to the other is not physically correct. When the net current through one magnet is not zero, eddy currents from another magnet are closed through the regarded magnet or eddy currents from the regarded magnet are closed through another magnet. The constant C(t) is the sum of the current flowing through all the magnets. Subtracting this value from each magnet, ensures that the net current in the magnet region is zero. Because the shielding cylinder is made of one single piece and thus short-circuited, it is possible for current loops to be closed in an other region of the shielding cylinder without violating the law of conservation of energy. Thus it is not necessary to implement this spatially constant current density in the shielding cylinder.
Ct(t) = 1 Sms
ˆ φms(t)+φm
) rdrdφ (4.5)
Where φms(t) is the start angle of a magnet and φm is the magnet arc angle. Sms is the surface area of a single magnet. The current density J in the PM is given by:
J = jσ(2)ω (n− k) ∞∑
−6
−4
−2
0
2
4
6
20
(b)
Figure 4.2: Eddy currents comparison ANA (a) vs FEM (b) in the SC
21
CHAPTER 4. EDDY CURRENTS
4.2.1 Eddy current validation
Yet again, the analytical implementation needs to be validated. To validate the Eddy current density calculation, the same FE model is used as in Section 3.3.3. The machine used for the calculations is a three-phase permanent-magnet synchronous machine with surface-mounted magnets and a shielding cylinder. The machine topology is shown in Figure 3.3, an overview of the parameters is listed in Table 3.1. In Figure 4.2, the contour plot of the eddy currents in the SC calculated with the analytical model and with the FE model is shown. It can be seen that these results show a good agreement between the analytical and the FEM results. A visual comparison of only the contour plot is not a good validation. To provide a more reliable validation, the eddy current density is calculated in a single point and compared with the analytical result in that same point. These results are shown in figures below. In the middle of the SC, the analytical eddy current density has an almost identical evolution as the one calculated using FEM. At the boundaries, the two models have the same waveform but the amplitude is different. This is because the FE model has difficulties at calculations at boundaries. Figure 4.4 shows that if we calculate the eddy current density a posteriori using the electrical field (E) of the FE model, the analytical calculation is indeed confirmed. The same is noticed for the eddy current density in de PM. Note that the eddy current density calculated with the FEM is zero because one of the assumptions made is that the conductivity of the PM is zero, so the validation occurs with the electrical field of the FE model.
0 0.05 0.1 0.15 0.2 0.25 0.3 −8
−6
−4
−2
0
2
4
6
22
−1
−0.5
0
0.5
1
1.5
0 0.05 0.1 0.15 0.2 0.25 0.3 −5
−4
−3
−2
−1
0
1
2
3
4
23
−3
−2
−1
0
1
2
3
4
−3
−2
−1
0
1
2
3
24
Chapter 5
Eddy-current losses
5.1 Introduction
To compute the eddy-current losses in the SC and the PM, two different methods are used. The losses in the PM can be calculated from Joule’s formula. Joule’s formula is not effective at computing the losses in the SC because the r-dependent part of J (3)(r, φ, t) contains Bessel functions, those functions can not be integrated analytically. To cope with this problem, the Poynting vector (S) can be used.
5.2 Shielding cylinder
A generalized form of Poynting’s theorem is derived in the appendix. This section uses the obtained results to calculate the power losses in the SC. The Poynting vector (S) is written as:
S = E×H (5.1)
The Poynting vector represents the rate per unit area at which energy crosses a surface. The integration of S over a closed surface determines the energy flowing through that surface. In this case, this implies that the total power P transmitted from the stator to the rotor can be calculated by integrating the Poynting vector over the surface of a cylinder that includes the rotor.
25
Pn,se −j(n+s)ωt (5.4)
where n is the time-harmonic order related to E(4) z and s is the time-harmonic order
related to H(4) φ . Pn,s is calculated as:
Pn,s = ∞∑
(5.5)
To find the losses in SC, the total power can now be divided in the mechanical power (Pm) and the eddy-current losses in the SC (P (3)
ec ). As the mechanical power can easily be derived from the torque, the eddy-current losses in the SC can be calculated as the difference of P and Pm:
P (3) ec (t) = P (t)− Pm(t) (5.6)
where
Pm(t) = ωT (t) (5.7)
The losses in the SC are validated using the same FE model as in Section 3.3.3. A deviation of 2.58% is obtained, this deviation is calculated as:
d = PdissSC,FEM − PdissSC,ANA PdissSC,FEM
5.3 Permanent magnets
The Poynting vector is calculated directly from the magnetic and electric fields. As the eddy-currents in the magnets are not directly included in the calculation of the magnets, their losses can’t be accounted for by the integration of S. So the losses in the magnets will be computed using Joule’s formula:
P (2) ec (t) =
dv (5.9)
This equation is used to calculate the eddy-current losses in the PM, for a 2D model it can then be rewritten as:
P (2) ec (t) =
J (2)(r, φ, t)J (2)(r, φ, t) σ(2) ds (5.10)
The losses in the PM are validated in the same way as in the SC, A deviation of 4.844% is obtained
27
Chapter 6
Parameter study
6.1 Introduction
Now that the analytical calculations have been validated, a theoretical study of the calculated properties can be conducted. One aspect of high-speed PMSMs that still requires a lot study is the effect of the Shielding Cylinder (SC) on the machine’s performance. This shielding cylinder is a conductive sleeve that is wrapped around the magnets. The goal of the SC can either be to keep the magnets in place at very high rotational speeds and/or to protect the magnets from overheating. The latter is due the fact that the resistance of the SC is much lower than that of the magnets. Eddy- currents, which will be induced in the SC instead of in the magnets, will therefore cause less losses. In the first section, the goal is to study the effect of the shielding cylinder on the electromagnetic torque production. Secondly, the effect of the SC on the dissipated power is discussed. To do so two properties of the shielding cylinder are studied, its thickness (tSC) and its conductivity (σSC).
29
6.2 Torque

(6.1)
6.2.1 Parameters
Table (6.1) shows the parameters of the studied machine. The two studied parameters thickness and conductivity although listed here are subject to change when studied. The machine geometry is depicted in Figure 6.1, the studied machine is a three-phase permanent-magnet synchronous machine with surface-mounted magnets and a shielding cylinder.
30
Table 6.1: Parameters of the studied machine
Parameter Symbol Value Number of slots Ns 12 Number of pole pairs p 2 Number of phases m 3 Number of layers / 1 Residual flux density of the magnets Bm 1.2 T Angular span of a magnet φm 80.00 Current density Inom 5.106 A
m2
Number of windings per slot N 6 Nominal frequency fnom 1000 Hz Nominal speed nnom 30,000 rpm Outer rotor iron radius r1 22.50 mm Outer PM radius r2 25.00 mm Outer sleeve radius r3 26.50 mm Outer airgap radius r4 28.50 mm Outer winding radius r5 40.40 mm Outer stator iron radius r6 50.00 mm Stack length ls 200 mm Permeability of the magnets µ2 1.0µ0
H M
Figure 6.1: Machine topology parameter study torque
31
6.2.2 Harmonic content
In order to gain a better understanding of the evolution of the studied properties, an overview of the harmonic content of the machine’s magnetic field is given. It can be observed that the dominant synchronous fields components are (n=2,k=2) and (n=6,k=6). Beside these synchronous components, a couple of asynchronous field components are mildly outspoken, these are (n=2,k=14), (n=2,k=-10),(n=2,k=26) and (n=2,k=-22). As discussed in Section 3.3.2, the produced torque can be explained by regarding the harmonic content of the magnetic field.
−50 −42
−34 −26
−18 −10
6.2.3 Conductivity
In this section, a constant shielding cylinder thickness of 1.5 mm is maintained. The parameter that is varied is the shielding cylinder’s conductivity. The goal of this section is to identify trends rather than to model a real machine, so the studied range contains
32
Figure 6.3: Harmonic content of the studied machine Bφ
extreme conductivities, which are not realistic. The range in which σSC is regarded is [0,108].
Figure 6.4 shows the torque and its components as a function of σSC . As expected, TPM shows little dependency of σSC . This is because TPM is mainly caused by the interaction of synchronous field components and the expectation is that these components are not affected by the shielding cylinder’s conductivity. Although when plotted separately, it can be seen that there is a variation on TPM . Figure 6.6 shows an initial decrease followed by a small increase and ultimately a continuous decrease. The reason for these small variations is that a part of TPM is produced by the interaction between asynchronous field components. At the boundary between the air gap and the slotting region, one space harmonic of the air gap (k) has an effect on all the space harmonics in the slotting region (m) with the same time harmonic (n) as shown in figure 6.5. Be- cause of this interaction, each asynchronous field component has an effect on the synchronous field components.
33
x 10 7
T or
qu e
(N m
T SC
Figure 6.4: Average torque as a function of the conductivity of the SC
Figure 6.5: Illustration of the interaction between synchronous- and asynchronous field components at the slot opening
A variation of the asynchronous field components can thus influence the synchronous field components. The evolution of TPM in Figure 6.6 is mostly due to synchronous field components (n=2,k=2) and (n=6,k=6). The synchronous field component (n=2,k=2) produces a positive torque for all considered conductivities. The other significant syn-
34
CHAPTER 6. PARAMETER STUDY
chronous field component (n=6,k=6) produces a negative torque for all the considered conductivities. These components are visualized in Figure 6.2. In each region the evolution is caused by the synchronous field components which varies the most. The initial decrease is caused by field component (n=2,k=2) which variation is dominant in this region. Component (n=2,k=2) has a positive effect on the torque so a decrease of (n=2,k=2) implies a decrease of TPM . The small increase that is observed in region B is caused by a decrease of field component (n=6,k=6). That decrease implies less negative torque produced by (n=6,k=6) and thus an increase of the produced torque. In region C, the decrease of field component (n=2,k=2) becomes greater than the decrease in (n=6,k=6) again, thus a decrease in TPM is observed.
0 2 4 6 8 10
x 10 7
T or
qu e
(N m
(A) (B) (C)
Figure 6.6: Torque in the PM as a function of the conductivity of the SC
The evolution of TSC with increasing σSC is plotted in Figure 6.4. TSC is negative which implies that, in the studied machine, the asynchronous components with a slower rotational speed than the synchronous are dominant. In what follows the absolute value of TSC,net will be discussed. The evolution of TSC,net is very similar to the speed-torque characteristic of an induction machine. Initially an almost linear increase is observed. However after reaching its maximum, TSC,net becomes inversely proportional to √σSC . As mentioned in the beginning of this chapter, the shielding cylinder can be regarded as the squirrel cage of an induction machine. Thus the evolution of TSC can be explained in a similar way as the speed-torque characteristic. This means that for a given component of the magnetic field (n, k), TSC depends on the resistance of the SC (RSC), the stator resistance (Rs), the rotor leakage inductance (Llr), the stator leakage inductance (Lls), the magnetization inductance (Lm), the slip (sn,k) and the pulsation (ωn,k) of the harmonic combination (n, k):
TSC,net,n,k ∼ RSC(
Following Pouillet’s law for low σSC , RSC is inversely proportional to σSC :
RSC = ls σSCSSC
(6.3)
with ls the stack length and SSC the surface of the SC. At high conductivities this is not longer true because the skin effect is no longer negligible. The threshold value is the conductivity at which the skin dept (δ) equals the thickness of the SC. If the skin dept is smaller than the thickness of the shielding cylinder, the conductive surface of the SC is no longer constant. Equation (6.4) shows that the conductive Surface is inversely proportional to √σSC and according to Pouillet’s law, RSC will then be inversely proportional to √σSC .
δn,k = √
(6.4)
At low conductivities of the shielding cylinder, RSC will be high and will dominate the denominator of (6.2).
RSC
TSC,net,n,k ∼ RSC((
∼ { σSC if δ > tSC√ σSC if δ < tSC
In this study, the area in which TSC,net increases is situated before the point at which δ = tSC . This confirms the linear behavior at low σSC . At high σSC , RSC will be low and thus the RSC term is negligible in the denominator of (6.2).
RSC
TSC,net,n,k ∼ RSC
if δ < tSC
This shows that until δ > tSC , TSC,net will decrease proportionally to 1 σSC
and after the threshold value TSC,net becomes proportional to 1√
σSC . Note that the conductivity for
which δ = tSC is located in the transition area between an increasing and a decreasing TSC,net. Which implies that only the decrease inversely proportional to √σSC is visible. A final remark is that the maximum of TSC,net, as (6.2) shows, depends on the ratio between the resistances and the inductances and thus on the entire machine geometry.
In Figure 6.7, the percentage torque ripple as a function of σSC is shown. It can be seen that there is a shift of the ripple from TPM to TSC when σSC is increased. This is explained by a stronger shielding effect. The total torque ripple is almost constant. The small variation of T% is explained by a similar variation of T (σSC). The evolution of TSC%(σSC) is as expected. Increasing σSC implies a better shielding effect. A better shielding effect translates in an increasing torque ripple. When δ becomes smaller than tSC , the SC already blocks most of the asynchronous content and the increase of TSC% stagnates. Because the ripple is defined as a percentage of T in (6.1), TSC% slightly decreases for higher σSC because the total torque T increases for higher σSC .
37
x 10 7
T or
qu e
rip pl
SC
Figure 6.7: Torque ripple in the PM as a function of the conductivity of the SC
6.2.4 Thickness
The second parameter to be studied is the thickness of the shielding cylinder and it’s effect on the different torque components. This is done by varying the outer radius of the SC (r4) and keeping the outer radius of the PM (r3) fixed. The width of the elements beyond r4 are kept constant to ensure a fair comparison. A first expectation of the effect of an increasing tSC would suggest a decrease in the net torque. As stated in 3.3.2, the net torque is mainly associated with the torque component TPM . TPM is produced by the interaction between the PM magnetic field and the stator current. The amount of non-magnetic material between the stator and the rotor is referred to as the effective air gap (tAG,eff = r4−r2). Increasing the effective air gap tAG,eff reduces the interaction which produces TPM and thereby decreasing the net torque. On the other hand, TSC is determined by two effects in regard to an increasing tSC . First, regarding the shielding cylinder as an electrical conductor, and applying pouillet’s law (6.3), which dictates that the SC resistance’s decreases as tSC increases. This decrease in resistance results in a higher eddy current density and thus a rise of TSC . Secondly, TSC is influenced by the effective air gap width (tAG). By increasing tAG, a smaller amount of eddy currents are induced in the shielding cylinder, resulting in a decrease of TSC .
Figure 6.8 shows the evolution of TPM , TSC and the total torque T in regard to a varying tSC . It shows an initial increase in TSC . Which means that the decreasing resistance
38
of the SC dominates the effect of a larger effective air gap. At a certain tSC , the increase of the effective air gap width dominates the evolution of TSC , which explains the continuous decrease at the end. The amplitude and tSC at which the maximum occurs depends on numerous factors, such as electrical frequency, machine geometry and num- bers of pole pairs.
0 1 2 3 4 5
x 10 −3
T SC
Figure 6.8: Average torque as a function of the thickness of the SC
Figure 6.9 shows the torque ripple with respect to Tnet as described in (6.1). As tSC widen, the torque ripple on TPM rapidly decreases. An increasing tSC also increases tAG, therefore less asynchronous field can interact with the permanent magnets. As the torque ripple is mainly caused by the slotting effect, this explains why TPM% decreases. Past a certain tAG, no more asynchronous field components reach the magnet region and therefore TPM% decreases to zero. The evolution of TSC% is similarly influenced by two same effects as TSC , the increasing surface area of the SC and the increasing tAG. As tSC increases more eddy currents are produced in the SC. When tSC becomes larger than the penetration depth of the asynchronous fields, no more additional eddy currents are induced. As tSC further increases, this results in a decrease of TSC .
39
x 10 −3
SC
Figure 6.9: Torque ripple as a function of the thickness of the SC
6.3 Eddy current losses
In this section a study of the shielding cylinder’s effect on the eddy current losses is conveyed. Eddy currents are induced due to asynchronous field components in the air gap’s magnetic field, which are a consequence of the slotting effect (see 6.5) and harmonics in the stator currents. These eddy currents add losses to the machine in the form of Joule losses (i2R), these losses generate a temperature rise which can possibly cause demagnetization of the permanent magnets. In order to reduce the total losses and risk of demagnetization, a shielding cylinder is added to the machine.
6.3.1 Parameters
To study the effect of the SC on the eddy current losses, two aspects of the SC, the thickness and conductivity, are regarded. Table 6.2 shows the parameters of the studied machine. A machine with concentrated windings is studied because these machines are more affected by eddy current losses in the PM at high speeds [10]. Figure 6.10 shows the studied machine geometry.
40
Table 6.2: Parameters of the studied machine
Parameter Symbol Value Number of slots Ns 6 Number of pole pairs p 1 Number of phases m 3 Number of layers / 2 Residual flux density of the magnets Bm 1.2 T Angular span of a magnet φm 180 Current density Inom 5.106 A
m2
Number of windings per slot N 2 Nominal frequency fnom 2500 Hz Nominal speed nnom 150,000 rpm Outer rotor iron radius r1 7.00 mm Outer PM radius r2 10.00 mm Outer sleeve radius r3 12.00 mm Outer airgap radius r4 15.00 mm Outer tooth tip radius r5 17.00 mm Outer winding radius r5 27.00 mm Outer stator iron radius r6 32.00 mm Stack length ls 36 mm Permeability of the magnets µ2 1.0µ0
H M
Figure 6.10: machine topology parameter study Eddy current losses
41
6.3.2 harmonic content
As with the previous machine an overview of the harmonic content of the magnetic field is given to provide better understanding. It can be observed that the primarily present synchronous fields are (n=1,k=1), (n=3,k=3) a,d (n=5,k=5). Although less outspoken the they are asynchronous fields components, these are (n=1,k=7) (n=1,k=-5) (n=1,k=13) (n=1,k=19) (n=1,k=-11) (n=1,k=-17) and (n=1,k=-23).
Figure 6.11: harmonic content of the studied machine Br
42
6.3.3 Conductivity
The first parameter to be studied is the conductivity of the SC. For this study a constant shielding cylinder thickness of 1.5 mm is considered. The range of the regarded σSC is [0− 13 · 106], as in the previous study (6.2.3), the goal is to analyze trends rather than model a real machine. Figure 6.13 shows the evolution of the eddy current losses as a function of σSC . As the conductivity increases, the shielding effect increases so less asynchronous field components reach the magnet region. This implies that less eddy currents are induced. This explains the parabolic decrease of Pdiss,PM as the joule losses are proportional to I2. The evolution of Pdiss,SC is plotted in Figure 6.14. As the conductivity increases two aspects can be regarded, the increase in the amount of eddy currents induced, and the decrease of the resistivity. These changes are shown in Figure 6.15, 6.16 and 6.17. First, the conductivity is zero, thus there are no induced eddy currents. As the conductivity increases, there is an increase in the amount of eddy currents induced. The percentage change of these induced eddy currents is bigger than the percentage change of the resistivity so there is an increase of the dissipated power. When the conductivity increases beyond 2.5 · 105, there is a decrease of the percentage change of the induced eddy currents and the percentage change of the resistivity becomes dominant. This results in a
43
decrease of the dissipated power. RSC is inversely proportional to σSC as shown in Fig- ure 6.17. As σSC further increases, the percentage change of the resistivity de

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