Design and comparison of PMaSynRM versus PMSM for pumping applications

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INOM

EXAMENSARBETE ENERGI OCH MILJÖ,

AVANCERAD NIVÅ, 30 HP STOCKHOLM SVERIGE 2018,

Design and comparison of PMaSynRM versus PMSM for pumping applications

VIKTOR BRIGGNER

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TRITA TRITA-EECS-EX-2018:496

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Design and comparison of PMaSynRM versus PMSM for pumping applications

VIKTOR BRIGGNER

Master of Science Thesis in Electrical Energy Conversion at the School of Electrical Engineering and Computer Science

KTH Royal Institute of Technology Stockholm, Sweden, August 2018.

Supervisors: Tanja Hedberg and Øystein Krogen Examiner: Oskar Wallmark

TRITA-EECS-EX-2018:496

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Design and comparison of PMaSynRM versus PMSM for pumping applications VIKTOR BRIGGNER

c VIKTOR BRIGGNER, 2018.

School of Electrical Engineering and Computer Science

Department of Electric Power Engineering and Energy Systems Kungliga Tekniska h¨ogskolan

SE–100 44 Stockholm Sweden

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Abstract

This master thesis aimed to design a permanent magnet assisted synchronous reluctance machine (PMaSynRM) rotor for pump applications which were to be implemented in an existing Induction Machine stator. The machine were to be compared with a similar permanent magnet synchronous machine (PMSM) with similar torque production in terms of cost and performance.

This thesis goes through the theory of the Synchronous Reluctance Machine and the Permanent Magnet assistance. A rotor was designed by utilizing existing design approaches and simulation of performance by use of finite element analysis. A demagnetization study was conducted on the added permanent magnets in order to investigate the feasiblity of the design.

The final design of the PMaSynRM was thereafter compared to the equivalent surface-mounted PMSM in terms of performance and cost. The performance parameters was torque production, torque ripple, efficiency and power factor. Due to the lower torque density of the PMaSynRM, for equal torque production the PMSM had a 40% shorter lamination stack than the PMaSynRM.

The economic evaluation resulted in that when utilizing ferrite magnets in the PMa- SynRM it became slightly cheaper than the PMSM, up to 20%. However, due to the fluc- tuating prices of NdFeB magnets, there exist breakpoints below which the PMaSynRM is in fact more expensive than the PMSM or where the price reduction of the PMaSynRM is not worth the extra length of the motor. However, it was shown that the PMaSynRM is very insensitive to magnet price fluctuations and thereby proved to be a more secure choice than the PMSM

Keywords: Demagnetization, economic evaluation, permanent magnet assistance, synchronous reluctance machine.

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Sammanfattning

Detta examensarbete avs˚ag att designa en rotor till en permanentmagnetsassisterad synkron reluktansmaskin (PMaSynRM) för pumpapplikationer, vilken skulle implement- eras i en befintlig asynkronmaskin (IM) stator. Maskinen jämfördes ekonomiskt och pre- standamässigt med en liknande synkronmaskin med permanentmagneter (PMSM) med jämförbar vridmomentsproduktion.

Uppsatsen avhandlar teorin bakom synkrona reluktansmaskiner och konceptet kring permanentmagnetassistans. Rotorn designades genom användandet av befintliga design- metoder och simulering genom finit elementanalys (FEA). En avmagnetiseringsstudie utfördes p˚a de adderade magneterna för att undersöka rimligheten kring designen

Den slutgiltiga designen av PMaSynRMen jämfördes därefter mot den jämlika PMSMen i termer om prestanda och kostnad. De undersökta prestandaparameterarna var vridmoment, vridmomentsrippel, verkningsgrad och effektfaktor. Eftersom vridmoments- densiteten i en PMaSynRM är lägre än hos en PMSM s˚a visade sig PMSMen ha en 40%

kortare lamineringskropp ¨an PMaSynRMen vid j¨amnlik vridmomentsproduktion.

Den ekonomiska utvärderingen resulterade i att vid användandet av ferritmagneter i PMaSynRMen s˚a blev den n˚agot billigare än PMSMen, upp till 20%. P˚a grund av fluk- tuerande priser hos NdFeB magneter, s˚a finns det brytpunkter där PMaSynRMen faktiskt blir dyrare än PMSMen eller d˚a kostnadsreduktionen för PMaSynRMen kan bedömas att vara för l˚ag med tanke p˚a den ökade längden och vridmomentsrippel. Däremot visades det att PMaSynRMen är väldigt okänslig för prisvariationer och därför visades vara ett kostnadsmässigt tryggare val än PMSMen.

Nyckelord: Avmagnetisering, ekonomisk utv¨ardering, permanentmagnetassistans, synkron reuktansmaskin.

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Acknowledgements

This master thesis has been carried out at the department of Research and Development for electrical motors at Xylem Water Solutions in Stockholm, Sweden.

I would like to thank Xylem Water Solutions for giving me the opportunity to do my master thesis for them and for the great experience that it has entailed. I would especially like to thank Tanja Hedberg and Øystein Krogen for their supervision and help throughout the duration of the project. Furthermore would I like to thank my co-workers at Xylem Water Solutions for making my stay there even more enjoyable with their company.

I would also like to express a special thanks to Associate Professor Oskar Wallmark for sparking my interest in electrical machines and for inspiring me to pursue this field of engineering. Additionally I would like to thank him for acting as my examiner for this thesis.

I also want to give thanks to all of my friends here in Stockholm who has made my years at KTH unforgettable to say the least. Thank you for all the memories and for your friendship. Even if we eventually find ourselves in different parts of the world, I know that we will always stay in touch.

Finally, I would like to give my deepest gratitude to my parents and my sister who always have supported me and helped me whenever I needed it. I would also like to especially thank my girlfriend, Saga Kubulenso, for her never-ending patience with me when my studies has gotten the best of me and for always being there for me no matter what. Thank you so much.

Viktor Briggner Stockholm, Sweden August 2018

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Acronyms

Notation Description Page

List

ALA axially laminated anisotropy 11

CSPR constant power speed range 11

FEA finite element analysis 7, 21

FFT fast fourier transform 38

FSCW fractional slot concentrated winding 40

IM induction machine 6

IPF internal power factor 15

IPMSM interior PM synchronous machine 11

LSPM line start PMSM 7

MOOA multi-objective optimization algorithms 73

MTPA maximum torque per ampere 16

MTPkVA maximum torque per kVA 15

NdFeB neodymium-iron-boron 6

PF power factor 15

PM permanent magnet 10

PMaSynRM permanent magnet assisted synchronous machine 7 PMSM permanent magnet synchronous machine 6

SP salient pole 11

SynRM synchronous reluctance machine 7, 9

TLA transversally laminated anisotropy 11

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Acronyms

List

VFD variable frequency drive 7

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Nomenclature

List

AF e Area of rotor segments 27

e Complex valued emf 12

vs Complex valued stator voltage 12

B_r Remanent flux density 30

Fc Centrifugal force acting on the rotor material 26 fd,i Average MMF seen by the i:th iron segment due

to sinusoidal d-axis MMF

34 fq,i Average MMF seen by the i:th iron segment due

to sinusoidal q-axis MMF

34 f_q,i Average q-axis MMF difference across the i:th

flux barrier

35

g Air-gap length 25

Hc Coercivity 30

Hc,b Normal coercivity 31

Hc,i Intrinsic coercivity 31

H_k Value of magnetic field at the knee of the intrinsic curve

31

ic Complex valued iron loss current 12

i Complex valued stator current vector after iron losses

12

id d-axis stator current 13

Im Magnetic polarization 31

iq q-axis stator current 13

Is Magnitude of current vector in dq-frame 16 is Complex valued stator current vector in dq-frame 12

kmag,a Magnet fill factor for center barrier 46

kmag,b Magnet fill factor for barrier arms 46

k_w,d Insulation ratio in d-axis 21

Ld d-axis inductance 14

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Nomenclature

List

Wk,q Height of barrier k in q-axis 21

Lq d-axis inductance 14

L_stk Stack length of the machine 26

nr Number of rotor barrier slots per pole 24

ns Number of stator slots per pole 24

p Number of poles 13

ps Stator slot pitch 23

R1 Rotor radius 35

R_c Equivalent iron loss resistance 12

RG Center of gravity of rotor segments 27

Rs Stator resistance 12

R1 Shaft radius 35

Sh,q Height of iron segment h in q-axis 21

✓b,k Rotor barrier end angle of barrier k 21

v_d d-axis stator voltage 13

vq d-axis stator voltage 13

kw,q Insulation ratio in q-axis 21

Wk,d Height of barrier k in d-axis 21

wma,i Width of barriers placed in center of flux barrier 28 wmb,i Width of barriers placed in arms of flux barrier 28

w_r,i Width of radial ribs 26

wt,i Width of tangential ribs 26

wtooth Stator tooth width 23

↵i Angle between flux barrier arm and center of flux barrier

21

↵_m Rotor slot pitch angle 33

Torque angle 13

s Rotor slot displacement angle 33

Load angle 13

⌘ Machine efficiency 38

Current angle from d-axis 13

⌫_rib Safety factor for dimensioning of radial barrier ribs

26

!e Electrical angular frequency 12

!m Mechanical angular rotor frequency 27

' Power factor angle 13

'i Internal power factor angle 13

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Nomenclature

List Complex valued flux linkage in dq-frame 12

d d-axis flux linkage 13

P M Permanent magnet flux linkage 18

q q-axis flux linkage 13

⇢lam Steel mass density 27

r Tensile strength of steel lamination 26

⌧em Electromagnetic torque 13

⌧P M PM torque 19

⌧_rel Reluctance torque 19

⇠ Saliency ratio 15

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Chapter 1 Introduction

1.1 Background and objectives

Electrical machines is an ever-present piece of equipment found in numerous industrial and household applications. In fact, it is estimated that the energy usage by electrical machines correspond to approximately 40% of the total electrical power consumption in industrialized countries and up to 65% of the industrial energy consumption. Addition- ally, in the EU approximately 22% of the energy consumed by electrical machines used in industry found its usage for pumping applications [1, 2]. Therefore, an increase in efficiency of these machines would turn out to be hugely important in the strive for reducing overall energy consumption in the world.

The vast majority of the electrical machines on the market today are induction machines (IMs). In Sweden it has been estimated that 90 % of all electrical machines of power ratings between 0.75-375 kW are IMs [3]. These machines have a relatively poor power factor and efficiency compared to permanent magnet synchronous machine (PMSM), keeping power rating and pole numbers equal. However, the main benefit of the IM is the sheer simplicity and reliability of the machine and its long history in industrial applications. It can be line-started without the need for any power electronic drives and doesn’t have any magnetic components which are expensive and are at risk of being demagnetized.

However, as the price of power electronics continue to decline [4], synchronous machine-based drives increase in popularity. This because synchronous machines are easier to control and generally has a higher or comparable torque density and higher efficiency compared to IMs [5]. Furthermore, in order to keep the power ratings, and thus costs, of the power electronic components down for these drives, the power factor plays a significant role when evaluating these kinds of machines.

High-efficiency PMSMs utilize rare-earth magnets such as neodymium-iron-boron (NdFeB) magnets which are not only relatively expensive compared to non rare-earth magnets but also pose with quite poor price-stability as shown over the last few years

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1.1. Background and objectives

(a) PMaSynRM (b) PMSM

Fig. 1.1: Example of PMaSynRM and surface-mounted PMSM

[6]. Furthermore, the extraction process of rare-earth magnets entails both a hazardous environment for workers and for the people living nearby the extraction and refinement plants, as well as the process used being very environmentally harmful [7].

synchronous reluctance machines (SynRMs), which utilize the anisotropy of the rotor to produce torque known as reluctance torque, has been shown to perform better than induction machines in terms of efficiency [8] while however falling short of equivalent PMSMs by a large margin both in terms of efficiency and power factor, but also torque density [9]. However, lately permanent magnet assisted synchronous machines (PMaSyn- RMs) has been a source of interest in order to find a feasible competitor to the PMSMs.

The permanent magnets utilized in a PMaSynRM are either of far lesser quantities of rare-earth magnets or alternatively of weaker, more abundant, and cheaper magnets such as ferrite magnets [9]. High-efficiency PMSMs also generally perform better than PMa- SynRM in terms of efficiency and torque density [5, 9], however this difference might be small enough that the economical benefit can outweigh the reduction in performance. In Fig. 1.1 a PMaSynRM and a surface-mounted PMSM is shown.

In the present thesis, a PMaSynRM will be designed and analyzed based on torque density, efficiency and power factor and thereafter compared to an equivalent high-efficiency surface-mounted PMSM. The machines are simulated and analyzed by means of finite element analysis (FEA). The work conducted in this thesis is in part based on the work conducted by Adrian Ortega Dulanto as a master thesis [10]. However, the investigated dimensions of the machines has been increased as to see if the PMaSynRM might display a relative increase in performance at greater dimensions as well as investigate the scalability of the design developed in [10].

Furthermore, the choice to compare the PMaSynRM to a PMSM can be argued to be a better comparison rather than to a line start PMSM (LSPM) or IM as both PMaSynRM and PMSM requires a variable frequency drive (VFD) to operate as opposed to the other

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1.2. Thesis outline two and therefore their applications are more similar.

1.2 Thesis outline

The thesis consist of 5 chapters and will hold the following structure

• Chapter 1: Introduction, background and justification of the thesis

• Chapter 2: Theoretical foundation of PMaSynRM design and analysis

• Chapter 3: Description of the analysis and design process

• Chapter 4: Results and comparison of the developed PMaSynRM and PMSM

• Chapter 5: Conclusions and discussions regarding further work

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Chapter 2 Synchronous Reluctance Machine and Permanent Magnet assistance

As was briefly stated in Chapter 1 the synchronous reluctance machine (SynRM) and PMaSynRM relied on the anisotropy of the rotor in order to produce torque. In this chapter the theory of the SynRM will be discussed and how utilizing magnets to further improve the operation of SynRM influence the operation, thus producing a PMaSynRM.

2.1 Concept of reluctance torque

Reluctance torque relies on, as the name would suggest, the reluctance of an object. More specifically, it relies on the difference in reluctance in different directions. This difference means that depending on the how the object is placed in a magnetic field relative to the direction of the field, different magnetic behaviour is displayed, i.e. the specimen is anisotropic. This anisotropy is easiest achieved by altering the geometry of a magnetic material specimen. The torque produced is dependent on the interaction of the specimen and an applied magnetic field. By defining an object-orientated coordinate system, with the direct (d) axis aligned with the lowest reluctance and the quadrature (q) axis along the path of highest reluctance we can begin to define the torque produced.

To illustrate this, assume that the anisotropic specimen is subjected to a homoge- neous magnetic field and there is an angle between the d-axis of the specimen and the magnetic field. This angle means that a distortion in the field is introduced and hence the curl of the field will be non-zero. This in turn creates a force which does not cancel out and hence a torque is produced. In Fig. 2.1 objecta) is completely isotropic and thus displays equal reluctance in all directions in the plane of the magnetic field B and therefore no torque is exerted on it and consequently it remains unaffected by the field. However, objectb) is anisotropic with an angle to the field and therefore experience a net torque.

The angle is known as the load angle and it is the angle which determines the magnitude of the torque since the system always aim to keep equal to zero [11].

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2.2. Synchronous reluctance machine

B

a) b) c)

F

F d

q

d q

Fig. 2.1: Conceptual description of reluctance torque. a) an isotropic object, b) an anisotropic object, c) a rotor-like anisotropic object.

Objectc) is added in Fig. 2.1 as to illustrate how the concept is transferred to rotating machines. Anisotropy is achieved by introducing air gaps, or flux barriers, in a rotor structure and a torque is produced. However, note that in a SynRM the magnetic field will be directed radially and rotating in order to produce a continuous torque, but the same concept applies.

From this qualitative description of the reluctance torque it can be derived that the torque production in a SynRM (and PMaSynRM) is dependent on and that there has to be an optimal angle if it is sought to maximize the torque. The main flux in these machines is induced by the current and therefore by controlling the current, the torque is controlled.

How the flux, current and reluctance is related is expanded on in the coming sections.

2.2 Synchronous reluctance machine

The theory behind torque production from salient pole machines has been well-established since the 1920s. Following the development of inverter technology in the 1970s the thought of commercializing these types of synchronous machines became a source of growing interest [9]. However, given the development of high-energy permanent magnets (PMs) such as NdFeB the interest for machines driven purely by reluctance torque faded as they couldn’t compete with the high-energy magnet machines.

The drawbacks of the SynRM is that inherently has a high torque ripple, lower torque density and low power factor compared to equivalent PMSMs. This can in part be remedied by the addition of permanent magnets, giving a higher torque production and power factor albeit still worse than the equivalent PMSM [5].

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2.2. Synchronous reluctance machine The main advantage of the SynRM when compared to a PMSM is generally the lower price range as it doesn’t utilize expensive rare-earth magnets. However, there are more advantages over the PMSM of the SynRM and PMaSynRM as outlined in [9] and [12]. To name a few we have

• The SynRM is not as vulnerable to short-circuit conditions as the lack of magnets means that no current is induced.

• The constant power speed range (CSPR) is very good for SynRM and especially PMaSynRM

• The rotor saliency provides with easy rotor position detection at stand-still

The design of the SynRM rotor as it looks today is still conceptually based on the work done by Kostko in 1923 where the rotor is divided into different segments with flux barriers in order to achieve a high saliency [9, 13] as seen in Fig. 2.2b and c. Salient pole machines can be constructed in a few different ways. First, there is the conventional salient pole (SP) rotor, the axially laminated anisotropy (ALA) rotor and the transversally laminated anisotropy (TLA) rotor [14] and these types can be seen in Fig. 2.2.

However, the SP design configuration has been shown to be sub-optimal for SynRM drives and is more suitable for wound rotor synchronous machines. The ALA is more theoretically appealing and is believed to be able to provide a higher saliency ratio than the TLA configuration. However, the TLA configuration is much easier to mass produce as it utilizes the same punching and assembly procedure as traditional electrical machines [15]

and therefore will be the focus of this thesis.

The geometry of a TLA SynRM is similar to that of objectc) in Fig. 2.1. As was explained in Section 2.1 the SynRM produces its torque by differences in reluctance around the rotor. Generally several flux barriers are introduced and it has been shown that the pole number should be kept as low as possible where four poles are held as the most suitable pole number. In [11] a thorough investigation regarding pole numbers are presented. Fig.

2.3 displays a common four-pole SynRM rotor design with 3 flux barriers.

2.2.1 Definition of axes

Due to the inherent ansiotropy of the SynRM rotor, an analysis in a stator reference frame is difficult. The inductance of the machine is not only current dependent, but also dependent on the position of the rotor. In order to derive useful expressions and to perform a proper analysis, a rotor-based (synchronous) reference frame needs to be established, this coordinate system can be seen in Fig. 2.3.

The direct axis (d) is defined along the path which displays the highest inductance and conversely the quadrature axis (q) axis is defined along the path which has the lowest inductance. Note that this definition differs from the convention of interior PM synchronous machines (IPMSMs) and surface-mounted PMSMs where the d-axis is defined

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Fig. 2.2: Different rotor designs for rotor saliency. a) Conventional salient pole. b) Axially laminated anisotropy. c) Transversally laminated anisotropy. From [14]

along the flux from the PM, this distinction will prove important when discussing permanent magnet assistance since the consequence will be that the axes are reversed in terms of permanent magnet flux.

2.2.2 Governing equations

A model over the SynRM can be formulated in accordance with the equivalent circuit seen in Fig. 2.4 [16], where all bold-faced quantities are complex-valued. is the flux linkage of the machine, vsis the stator voltage, isis the stator current, i is the current not contributing to the iron losses, icis the iron loss current, Rsis the stator resistance, Rc is the equivalent iron loss resistance and !e is the angular frequency. Here e represents the emf as

e = d

dt + j!e (2.1)

Neglecting iron losses, the dynamics of the synchronous machine in dq-frame can be described by

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q

d

Flux barriers

Iron segments

Fig. 2.3: Cross-section of SynRM with defined dq-reference frame.

vd= Rsid+d d

dt !e q (2.2a)

v_q = R_si_q+ d _q

dt + !_e _d (2.2b)

where vdand vq are the stator voltages, idand iq are the stator currents, dand q are the machine flux linkages [9].

Fig. 2.4 can be utilized to draw a phasor diagram in the dq-plane as seen in Fig. 2.5.

Here the is the load angle, is the torque angle, is the current angle from the d-axis, 'and 'iare the power factor angle and internal power factor angle respectively.

The torque, ⌧em, can be described as

⌧em = 3

4p( diq qid) (2.3)

where p is the number of poles [9].

The d- and q-axis flux linkages are both very dependent on the operating point and

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R

_s _i

s

R

_c

e

ic

+

j!

e

i

d L v

_s

dt

Fig. 2.4: Circuit diagram including iron losses.

q

d vs

e

iq i

id

is ic

Ldid

jL_qi_q Rsiq

j!eLdid

Rsid

!eLqiq

'i

'

Fig. 2.5: Phasor diagram for SynRM.

experience cross-coupling from currents in the adjacent axis [9, 11], i.e.

(

d = d(id, iq)

q = _q(i_d, i_q) (2.4)

This cross-coupling occurs since the q-axis current cause a flux component in the d-axis and vise verse. This not only contributes to the total flux in the respective axis but it also affect the saturation level of the iron in the respective axes. Hence, when rewriting the flux linkages as current times inductances it is very important to note that the inductances (Ld, Lq) are indeed not constant [9, 11]

(

d= L_d(i_d, i_q)i_d

q= Lq(id, iq)iq

(2.5)

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2.3. Saliency and performance Note that this definition of the inductances is a simplification and for a more thorough discussion see Chapter 4 in [9]. Note also that the inductances includes not only magnetizing inductances but also the leakage inductances, which are not significantly in- fluenced by the aforementioned cross-coupling and saturation [16].

2.3 Saliency and performance

The saliency ratio, ⇠ is of great significance for the performance of the SynRM. The saliency ratio is defined as

⇠ = Ld

L_q (2.6)

and in a SynRM, ⇠ generally do not exceed values much higher than 10 [9].

Consider the phasor diagram in Fig. 2.5. The following relationships hold for the defined angles

⇡

2 + = + 'i (2.7)

= + (2.8)

Utilizing these relationships we find that we can write the internal power factor (IPF) as

IPF = cos 'i = cos⇣⇡

2 + ⌘

= ⇠ 1

r

⇠² 1

sin² + 1 cos²

(2.9)

for the derivation of this expression refer to Appendix A.1. And thus, it becomes obvious that the saliency ratio influences the IPF heavily, as seen in Fig. 2.6 where the it is plotted for different values of ⇠ as a function of the current vector angle. It is important to note that IPF is not the same thing as power factor (PF) but they are, however, related and a high value of IPF leads to a high value of PF since the difference between these two are only governed by Rs and Rc which can be verified by looking at Fig. 2.5 and Fig.

2.4. Therefore, IPF is discussed here as it is quite straightforward to derive an analytical expression from current angle and saliency.

From equation (2.9) it is obvious that for any given value of ⇠ there exist a value of the current vector angle which allows for the optimal IPF. It can be shown that this occurs when tan =p

⇠and then the IPF is equal to ⇠ 1

⇠ + 1. This operating point is often called maximum torque per kVA (MTPkVA) [16] and thus correspond to the operating point when the least amount of reactive power is required by the supply.

With the definition of flux in equation (2.5) and dropping the parentheses for simplicity we find that equation (2.3) can be rewritten as

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2.3. Saliency and performance

0 10 20 30 40 50 60 70 80 90

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

[deg]

IPF

⇠=2

⇠=5

⇠=10

⇠=15

⇠=20

⇠=100

Fig. 2.6: IPF as function of current angle for different values of ⇠.

⌧em= 3

4p(Ld Lq)idiq (2.10)

Thus, the torque is proportional to the difference in inductance between the two axes. It is interesting to note that while the IPF is governed by the saliency ratio, the torque production is dependent on the saliency difference. Even though these parameters are related, it can prove difficult to maximize both parameters simultaneously when de- veloping a rotor design [10]. This is due to the non-linear dependency of d- and q-axis inductances on the rotor geometry [17].

Another conflict occurs when operating a SynRM and it becomes apparent when rewriting equation (2.10) utilizing the phasor quantities presented in Fig. 2.5 in steady state, i.e.

⌧em = 3

4p(Ld Lq)I_s²sin 2 (2.11)

where Isis the stator current magnitude in steady state. Here we see that for given (constant) inductances and current magnitude, the maximum torque is achieved for a current vector angle of 45 degrees. This operating point is referred to as maximum torque per ampere (MTPA). Again, looking at Fig. 2.6 we see that this current vector angle does not coincide with the angle which maximizes the IPF for moderately high values of ⇠.

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2.4. Iron saturation It should also be noted that saturation affect both the inductance difference and saliency ratio such that in reality ⇠ and inductance difference decrease with increasing current, which affect both the torque production and power factor. The effect of iron saturation will be expanded upon below.

2.4 Iron saturation

As has been stated above the performance of SynRMs and PMaSynRMs is affected by the saturation of the iron. The most prominent effect of this is that the inductances change as a function of current level. This in turn contributes to a different behaviour in terms of torque and PF between low and high levels of current. In Fig. 2.7 the two inductances are plotted as functions of the applied current modulus. Since the d-axis is defined along the path of lowest reluctance and therefore the path which contains the greatest amount of iron it is quite straightforward to understand that it is also affected by the saturation to a greater extent. In fact, to saturate the q-axis is desirable because it allows for ⇠ and the inductance difference to increase. However, the decreased values of Ldresults in undesirable effects.

As the torque is proportional to the inductance difference it becomes obvious that the torque does not vary with the square of the current level when considering saturation.

The saturation of the iron leads to a shift towards higher values of optimal current vector angle for maximum torque (MTPA) as the current increases. The same tendencies can be seen when analyzing the PF curve. Saturation implies that ⇠ decrease in value for greater current levels. Again, referring to Fig. 2.6 we see that as ⇠ decrease, so does the internal power factor and therefore the power factor. However, some saturation effects actually benefits the PF when operating in the maximum torque (MTPA) operating point as discussed in [18]. This was attributed to the fact that the current angle for maximized power factor (MTPkVA) and MTPA current angle values were shifted closer together and thereby resulting in a higher PF.

Moreover, as was stated previously the d- and q-axis inductances are not only dependent on the d- and q-axis currents respectively but cross-saturation also occurs to varying degrees. In Fig. 2.8 the flux is plotted against different values of idand iq. Here we see difference in flux when the opposite axis current is non-zero.

2.5 Permanent magnet assistance

Reviewing Fig. 2.6 we see that in order to for an ordinary SynRM to have a IPF greater than 0.9 a saliency ratio beyond 20 is needed, which is an unrealistic value [16]. There- fore, in order to make the SynRM feasible compared to PMSMs some modifications need to be done. Such a modification is to add magnets in the rotor barriers, thus making a PMaSynRM. This addition slightly alters the characteristics of the machine. The magnets

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2.5. Permanent magnet assistance

0 10 20 30 40 50 60 70 80 90 100

Current magnitude [A]

Inductance[mH]

Ld

Lq

Fig. 2.7: Ldand Lqas function of current modulus for machine with 26 A rated current (RMS).

add a flux linkage component in the q-axis and thus equation (2.5) becomes (

d = Ld(id, iq)id

q = Lq(id, iq)iq P M

(2.12) where P M is the permanent magnet flux linkage addition. As can be seen in Fig. 2.9 the phasor diagram is altered due to this addition. The effect of the PM flux is highlighted in red. The voltage phasor is rotated towards the current vector effectively increasing the power factor. Additionally, since the PM-flux is largely directed in the q-axis direction this flux also help saturate the iron in the q-axis which reduces the q-axis inductance and thus increases the saliency ratio [9].

To derive an analytical expression for the IPF as a function of the saliency ratio of a PMaSynRM is not as straightforward as it were for the SynRM due to it being dependent on the q-axis PM flux linkage aswell. However, when comparing the two phasor diagrams in Fig. 2.5 and Fig. 2.9 we notice that the difference is the aforementioned rotation of the voltage vector. This rotation occur because the added PM flux counteracts the q-axis stator flux linkage and thus shifts the flux vector away from the current vector. For both machines in the steady state, the emf (e) will be perpendicular to the flux vector ( ) and therefore the emf will be rotated towards the current vector which increases the IPF and hence PF.

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2.5. Permanent magnet assistance

100 80 60 40 20 0 20 40 60 80 100

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5

idor iq [A]

Fluxlinkage[Wb]

d(i_d, i_q = 0)

d(id, iq = 50)

d(id, iq = 100)

q(iq, id = 0)

q(iq, id = 50)

q(iq, id = 100)

Fig. 2.8: dand qas function of different values of idand iq, for machine with rated current of 26 A (RMS).

The added PM flux also significantly alter the expression for the torque production and equation (2.10) becomes

⌧em = 3

4p P Mid+ (Ld Lq)idiq (2.13) i.e. we get two torque components, the PM induced torque (⌧P M) and reluctance torque (⌧rel). Note that this equation is similar to that of a salient-pole PMSM, but the key differences being that the torque contribution of the PM-flux is lower compared to the the reluctance torque and that the reference frame is rotated.

Rewriting equation (2.13) in the same fashion as equation (2.11) we get

⌧em= 3

4p P MIscos + (Ld Lq)I_s²sin 2 (2.14) This implies that the PM torque and reluctance torque does not display coinciding maxima with respect to . Hence, the optimal value of depends on the ratio between the two but typically lies around 40 [9] when neglecting saturation.

In addition to the shifting of flux vectors and PM torque production, another feature of permanent magnet assistance benefits the operation of the PMaSynRM. The PM flux helps saturate the ribs within the rotor structure (which are further discussed in Section

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2.5. Permanent magnet assistance q

d vs

e

iq i

i_d is i_c

L_di_d

jLqiq

j P M

Rsiq

j!eLdid

Rsid

!eLqiq

j! P M

'i

'

Fig. 2.9: Phasor diagram for PMaSynRM including iron losses.

2.6.6). This hence helps reduce the flux between the iron segments in the rotor which further reduces the q-axis inductance [18].

2.5.1 PM flux magnitude

As was discussed previously adding PMs to an ordinary SynRM improves its performance in more than one way. However, the amount of added PM flux linkage needs to be determined in order to know the amount and type of magnet to utilize in the design. The limit between ”permanent magnet assisted machine” and ”permanent magnet machine” is not clearly defined and it can be discussed when the machine stops being assisted. In [18]

an analytical expression describing the PM flux linkage magnitude has been derived and corresponding FE analysis has been carried out. It was shown that for maximum torque production for lower saliency ratios the required PM flux linkage were far greater than 50% of the nominal flux and at that point it can be argued that the machine is no longer just PM assisted since such a large part of the flux is supplied by the magnets. Addition- ally this amount of flux linkage implies the use of high-energy rare-earth PMs and the economical benefits vanishes as it closes in on being a IPMSM. However, this shows that there are no real upper limit in terms of PM flux magnitude when utilizing ferrite PMs or similar lower energy magnets.

Additionally, another property shown in [18] was that the torque curve and PF curve as function of the PM flux linkage had a very flat maxima, meaning that even quite low values of PM flux linkage would generate acceptable levels of power factor and torque production and even a small PM flux linkage contributed to reverse the q-axis flux linkage.

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2.6. Geometry and performance of PMaSynRM In fact, it is stated in [18] that the PM flux should be in the vicinity of 25-35% of the nominal flux. This also implies that ferrite PMs might be sufficient in order to reach the desired operating point.

2.6 Geometry and performance of PMaSynRM

The geometry of a SynRM rotor is inherently complex and as the operation of this machine depends on the saliency of the rotor, it is of utmost importance that the geometry is well defined in terms of parameters to analyze. Due to the complexity and non-linearity of these machines, it is very difficult and of questionable use to derive an analytical model of the machine and optimize. Instead, it is better to combine analytical expressions which determines some key geometrical parameters and thereafter conduct the performance analysis utilizing computer-aided finite element analysis (FEA) as is done in [11], [19] and [10].

The parameterization in this thesis is based on the work conducted in [15] and [11] and its theoretical relevance is expanded upon in Section 2.8.

2.6.1 Parameterization of PMaSynRM

In order to properly describe the rotor of the PMaSynRM it is crucial to define the geometry through comprehensible parameters. In Fig. 2.10 a pole of a PMaSynRM rotor with 3 barriers is shown. This parameterization follows that conducted in papers such as [15], [19].

• The barrier height of barrier k is described by Wk,q in the q-direction and Wk,d in the d-direction.

• The height of the iron segment h between barriers is described in the q-direction as Sh,q

• The angle that the barrier k makes at the periphery of the rotor with the d-axis is defined as ✓b,k

• The angle between flux barrier arm and center of flux barrier is defined as ↵i

Often, the q-axis position of a rotor barrier along the q-axis is of interest. The dis- tance to the n:th barrier is defined as

D0,n = Xn

h=1

Sh,q+ Xn 1

k=1

Wk,q (2.15)

Furthermore, in order to give an indication as to how much air versus iron there is in the rotor in both the q- and d-direction the insulation ratios kw,q and kw,d are qualitatively

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2.6. Geometry and performance of PMaSynRM

R_sh

R₁

✓_b,1

✓_b,2

✓_b,3

S_1,q

S2,q

S_3,q

S_4,q

W1,q

W_2,q

W_3,q

W_1,d W2,d

W_3,d

↵₁

↵2

↵₃

Fig. 2.10: Parameterization of PMaSynRM rotor.

defined as

kw,q = Amount of air

Amount of iron _{q axis} (2.16)

k_w,d = Amount of air

Amount of iron _{d axis} (2.17)

which gives that a value below 1 of these ratios means that there is more iron than air in the respective direction and conversely a value above 1 means that there is more air than iron. Note that the path of calculation for the q-axis is easily defined along the axis. For the d-axis it is a somewhat more complicated. The expressions for the insulation ratios are given in Section 2.8.

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2.6.2 Insulation ratios

The aforementioned insulation ratios have a great influence on the torque production as well as the power factor. The reason of this is most easily explained by the fact that these ratios determine the reluctance of the machine. This, in turn, affects the flux linkage inside the machine and therefore the saturation level of the iron and thus the inductance is altered.

More air introduced in the q-axis direction lowers the q-axis inductance, thus allowing for increasing the inductance difference and saliency ratio. Therefore it is common to optimize the insulation ratio in terms of torque production. However, the relationship of the insulation ratios are not linear towards either ⇠ or inductance difference (for fixed values of and current). It can be shown that there exist an optimum in terms of kw,qand kw,d when it comes to inductance difference and saliency ratio [11, 20]. These optimum values does not necessarily coincide and therefore a design trade-off has to be made in terms of power factor and torque production. Furthermore, an upper limit of the rotor insulation ratios has been suggested, which is related to the stator insulation ratio. This value is described by

kw,s = ps wtooth

ps (2.18)

where psis the stator slot pitch and wtoothis the stator tooth width. It is desirable to choose the q-axis insulation ratio of the rotor to a value close to or below the value of kw,s[9,15].

This is true since the insulation ratios in large determines the flux density magnitude in the stator and rotor and hence the saturation levels in rotor and stator. Therefore, if kw,q <

kw,s the stator teeth experience a greater saturation flux than the rotor and conversely if k_w,q > k_w,sis true the opposite applies. This affects the flux linkage magnitudes at higher current levels and therefore the torque production. This was shown in [9] and there it was concluded that a lower value for kw,q was preferable to a higher in terms of flux linkage and torque production. Following the same reasoning, it can be concluded that the d-axis insulation ratio kw,d should be equal or less than the value of kw,q, i.e. the amount of iron in the d-axis should be higher than in the q-axis [15, 21].

2.6.3 Number of flux barriers

While it can be shown that the insulation ratios mostly govern the torque production the number of flux barriers also impacts the performance of the SynRM. Generally speaking, a greater amount of flux barriers has a positive impact on both torque production and power factor. However, when it comes to aspects such as torque ripple the interaction between stator and rotor is of great importance and therefore the number of barriers will have a great influence on these parameters [10,15,19,22], albeit there are no simple rules for how many are optimal.

The choice of number of barriers is non-trivial and a simple analytical expression is difficult to derive. However, in order to minimize the torque ripple a general rule was

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2.6. Geometry and performance of PMaSynRM presented in [23] as

nr = ns± 4 (2.19)

where nr is the number of rotor barrier slots per pole and nsis the number of stator slots per pole. Whether the equation should be treated with a plus or a minus is determined by the feasibility of the structure albeit it is stated that +4 generates better results.

In [22] a thorough analysis of the behavior of different numbers of stator slots and barriers is presented. There it was derived that different number of stator slots perform at its best for different number of rotor barriers For instance was it shown that for a 48 slot machine, torque production was maximized for 4 or 6 number of barriers whereas efficiency was maximized for 4 barriers and torque ripple was minimized for 6 barriers.

Whereas in [10] it was determined that for the 36 slot machine 3 barriers generated the overall best performance in terms of torque production and power factor.

2.6.4 Torque ripple and rotor slots

High torque ripple is an inherent problem that plagues the SynRM and other machines such as IPMSMs [24]. It has already been determined that the interaction between stator and rotor influences the performance of the machine and most importantly the torque ripple. Therefore it becomes obvious that the positioning of the rotor barrier ends, or rotor barrier slots, impacts the operation of the machine.

The ripple that occurs in a SynRM is due to the variation in reluctance that occur when a rotor slot passes a stator slot [19, 23–25]. Therefore, it is of interest to place the rotor slots such that the torque ripple is minimized. A great deal of research has been made in the field of reducing the torque ripple of these types of machines. Torque ripple reduction can be achieved in a variety of ways. The method discussed above presented in [23] focuses on equally distributed rotor slots along the circumference and focus more on number of barriers rather than the placement of the rotor slots, [25] discusses the possibility of asymmetrically placed stator slot openings as a means of combating torque ripple, and [24] proposes asymmetrically shifting the rotor slots between every other or several laminations. A rule of thumb is presented in [19], where it is stated that the rotor barrier should be placed such that when one of the two barrier slots enters below a stator slot, the other barrier slot should enter below a stator tooth. Fig. 2.11 illustrates how the stator and rotor slots can be aligned. In this case the innermost barrier might be properly aligned since the top slot it is about to pass below a stator slot while the right slot is about to pass below a tooth. However, the middle barrier is more problematic since both slots are situated below the middle of both teeth which cause discrepancies in reluctance and thereby torque ripple.

However, for pumping applications torque ripple is not a serious issue as the load torque typically is proportional to the square of the rotor speed, and therefore the torque

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Fig. 2.11: PMaSynRM stator and rotor interaction.

ripple won’t correspond to any major speed ripple. Hence, to optimize torque ripple is not the focus of this paper but measures in terms of altering the placement of the rotor slots will be made as long as it does not negatively impacts the torque production.

2.6.5 Air-gap length

It was shown in [17] that the air-gap length, g, of a SynRM greatly affects the d-axis inductance while leaving the q-axis inductance virtually unaffected. This thus decreases the inductance difference and saliency ratio and consequently the torque production and power factor. This is due to the air-gap being the only air that the d-axis inductance sees while in the q-axis the air-gap is a small fraction of the total amount of air for the q- axis inductance due to the rotor flux barriers [15]. Hence, it is preferable to maintain the air-gap length as low as possible in order to get as large torque production as possible.

However, there is a manufacturing gain of increasing the air-gap as the tolerances can be kept larger. Additionally, a bigger air-gap yields lower torque ripple and lower iron losses in the rotor and therefore the effect of the air-gap length is of interest.

2.6.6 Radial and tangential ribs

Due to the shape of the flux barriers of a SynRM rotor, structural weaknesses is something that needs to be addressed. This means that the rotational forces might compromise the structural integrity of the rotor as the radial forces can cause the structure to break down.

This can be remedied by adding radial ribs to some, or all, barriers as to reinforce the structure. [10, 15, 26]. Another structural issue is that of the flux barrier ends towards the air gap. Ideally, there would be no steel between the rotor barrier ends - or rotor slots - and the air gap. But as has been mentioned previously the chosen manufacturing process,

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w

r,1

w

r,2

w

r,3

w

t,1

w

t,2

w

t,3

Fig. 2.12: Illustration of radial and tangential ribs in SynRM.

TLA, means that the rotor is punched and therefore require a continuous sheet of metal.

The width of this rib is in part determined by the tolerance of the punching machine, but also by the expected tangential forces from torque ripple or load variations. However, the calculation of the thickness of the tangential ribs are determined to be outside of the scope of this project. In Fig. 2.12, these ribs are illustrated, and the parameters describing the widths defined, wr,i is the widths of the radial ribs and wt,i is the width of the tangential ribs.

Not all machines require radial ribs and it is rather a question of size of the rotor, radial positioning of the flux barriers, and maximum allowable speed of the machine which determines the need and widths of them. The width of the radial ribs can calculated by diving the rotor into i segments and calculate the rotational force exerted on each segment [26]. This is done as

wr,i = ⌫rib

F_c,i

rLstk

(2.20) where Fc is the centrifugal force acting on the rotor, r is the tensile strength of the material, Lstk is the total stack length of the rotor and ⌫ribis a safety factor usually in the

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D

0,1

D

_0,2

D

_0,3

1 2 3

Fig. 2.13: Areas for radial rib calculation.

range between 2 and 3 [26]. The centrifugal force can be calculated as

F_c,i = ⇢_lam!_m²R_G,iA_{F e,i}L_stk (2.21) where AF e is the area of the relevant rotor segment, RG is the center of gravity of the rotor segment, !m is the mechanical angular frequency of the rotor and ⇢lam is the mass density of the steel.

An overestimation of the radial ribs can be achieved by simplifying the geometry as shown in Fig. 2.13 and use these sections to calculate the force that each rib will experience. Note that each area contains all steel within its boundaries, i.e. Area 1 includes 2 and 3 and so on. This is an overestimation since the air in the rotor structure is neglected which means that the force in reality is lower provided that the center of gravity is not increased dramatically. This also holds when magnets are added to the flux barriers since the density of ferrite magnets is lower than the steel and NdFeB magnets are about comparable with the steel while they will likely not fill up the entirety of the flux barriers. For the equations to derive the center of gravity refer to Appendix A.2.

Introduction of radial ribs in the barrier structure leads to an unwanted flux path in the rotor which contributes to increase the q-axis inductance and therefore leads to a torque reduction. In Fig. 2.14, the influence of the ribs on the q-axis flux is visible when looking at the span -5 A to 5 A, the slope of the flux, and thus inductance, is significantly higher in that span than elsewhere. This increased inductance is due to the flux path pro-

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60 50 40 30 20 10 0 10 20 30 40 50 60

2.5 2 1.5 1 0.5 0 0.5 1 1.5 2 2.5

idor iq[A]

Fluxlinkage[Wb]

d(id, iq = 0)

q(iq, id= 0)

Fig. 2.14: Influence of radial ribs on flux, for machine with 26 A rated current.

vided by the radial ribs. However, it is also visible that for greater values of the current, the ribs saturate and start to behave as air. In [15] it was shown that this reduction is in the magnitude of a few percent of the nominal torque. In [27] an expression to estimate of the magnitude of the torque reduction is presented where the torque reduction is proportional to the number of poles in square times the width of the ribs assuming constant width of the ribs. In [11] further analysis of the influence and design of radial ribs is conducted.

2.6.7 Magnet dimensions and placement

When adding magnets to the SynRM design they will be placed in the flux barriers as shown in Fig. 2.15. The width of the magnets in the center barrier is given by the param- eter wma,iand the width of the magnets in the barrier arms are given by wmb,i.

The placement of the magnets is quite important with regards to the performance of the machine. [28] showed that when keeping the total volume of the magnets constant it was more suitable in terms of torque ripple, and to some extent torque production, to distribute the magnets in all of the center barriers instead of filling some barriers with magnets while leaving some empty. Likewise, in [29] it was shown that when placing magnets only in the barrier arms and leaving the center barriers empty the torque production increased but the torque ripple became much greater compared to distributed magnets

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w

_ma,1

w

ma,2

w

ma,3

w

mb,1

w

mb,2

w

mb,3

Fig. 2.15: Magnet variables and magnet placement in rotor.

in the center barriers.

Furthermore, to keep the majority of the magnet volume deep within the rotor also helps to protect against demagnetization due to stator flux [28], a property which is very desirable when utilizing weaker magnets such as ferrite.

2.6.8 Stator and rotor steel

The grade of the stator and rotor steel primarily determines the iron losses of the machine as the iron losses in a machine is dependent on the flux density variation in the material and the material specific hysteresis curves of the steel. For a through description of calculation of iron losses, refer to [30]. Therefore, it is important to take the steel grade into consideration as it can have a significant impact on the efficiency of the machine. How- ever, for synchronous machines the iron losses in the rotor are generally lower in the rotor than the stator as the rotor follows the fundamental of the magnetic field and therefore experience only flux variations in terms of harmonic content.

In [19], it was shown through simulations that lower-loss steel grades affected iron losses and output power of a SynRM greatly. In [19], when only varying the steel type

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2.7. Permanent magnets

Hc

Br

H B

Fig. 2.16: Typical B-H curve for a hard magnetic material.

for a machine in the range of 12 kW output power, the efficiency saw an increase of 9 percentage units between the lowest and highest loss steel grade. At the same time, the output power increased by 8 percent when using the low loss grade steel compared to the higher loss grade.

2.7 Permanent magnets

Magnetic materials are in general characterized by the hysteresis loop of theB-H curve which describes how the magnetic flux density varies when an external magnetic field is applied. A magnetic material is largely defined by the remanent flux density, Br, and the coercivity, Hc. Br defines the flux density in the material when no external H-field is applied and Hc describes theH-field required in order to bring the flux density inside the material to zero. Magnetic materials can be divided into two major groups, hard and soft magnetic materials. Hard magnetic materials are defined by large values of Br and Hcwhile soft magnetic materials have lower values [31,32]. Fig. 2.16 illustrates a typical curve for a hard magnetic material.

Permanent magnets are hard magnetic materials and are, as the name suggests, permanently magnetized. This magnetization,M, relates to the magnetic flux density as [32]

B = µ0(H + M) (2.22)

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2.7. Permanent magnets

Hc,b

Hc,i

Br

H B or Im

Fig. 2.17: Typical normal (dashed) and intrisic (line) curve for permanent magnet.

2.7.1 Demagnetization

Typically, permanent magnets do not lose its magnetization when the flux density is reduced to zero, i.e. when the coercivity is reached. Utilizing the above stated relationship one can define the magnetic polarization,Im, as

Im = µ0M = B µ0H (2.23)

Both Im and B can be plotted in the same graph as is done in Fig. 2.17. The Im-H plot is often referred to as the intrinsic curve and the B-H plot is called normal curve.

In these plots two coercivities appear, the intrinsic and normal coercivity Hc,i and Hc,b. Analogous to the definition of Hc, Hc,i is the value at which the magnetization is forced to zero and beyond this point the magnetization will start to shift polarity [31]. Note that Hc,b = Hc as defined previously.

Demagnetization of the permanent magnet occurs when the magnetic field intensity approaches the intrinsic coercivity. In fact, when the magnetic field passes the value close to the knee of the intrinsic curve partial demagnetization start to occur. For most practical situations in electrical machines, only the second quadrant of the hysteresis loops are of interest. One can define the magnetic field knee value in the second quadrant as Hk and when that value is exceeded and thereafter reduced to below that value again, the magnetization of the magnet will be reduced and therefore also the remanent flux density. It can be shown that when Hk is exceeded, the new intrinsic curve follows the so called recoil lines shown in Fig. 2.18. The slope of the recoil lines are similar to that of the slope of

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2.8. Theoretical foundation of design approach

Hk

Hc,i

B_k Br

Fig. 2.18: Demagnetization curve for permanent magnet with recoil lines.

the original intrinsic curve when the magnetic field is zero [33]. This reduction of magnetization is what is referred to as demagnetization of the magnets and the consequences of this is that the maximum energy product of the magnets is reduced which lowers the magnetic torque and saturation flux of the magnets and is therefore undesirable.

As with all materials, magnetic materials are temperature sensitive where the temperature of the magnet alter the magnetic characteristics. NdFeB magnets experience a reduction in both remanent flux and coercivity for higher temperatures [31], meaning that rated magnet values, which often are given at room-temperature, are slightly misleading since the operating temperature tends to be higher. This, in turn, means that the NdFeB magnets are more sensitive to demagnetization for higher temperatures. Ferrite magnets are also affected by temperature differences, but as opposed to NdFeB the remanent flux actually increases with decreasing temperature while the coercivity decreases [31]. This means that the ferrite magnets are the most sensitive to demagnetization at lower temperatures.

2.8 Theoretical foundation of design approach

In a previous section a range of geometric variables were defined. In this section the relationship between these are explained and the relevant design parameters are defined to allow for manageable design variables. The design approach is expanded on in [15] and

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2.8. Theoretical foundation of design approach it relies on a number of key assumptions which are

• Saturation effects are neglected

• Stator slotting effects are neglected

• Magnetic scalar potential drop in the iron is neglected

• The stator winding is assumed to be ideal

• Distribution effects of the MMF is disregarded

2.8.1 Rotor barrier end angles

The rotor barrier end angles are are distributed along the rotor periphery with the constant rotor slot pitch angle, ↵m, between them according to

✓_b,h = (2h 1)↵_m

2 (2.24)

This is done as this part of the design is based on the work conducted in [23] which advocates constant rotor slot pitch. However, in order to allow for a greater degree of free- dom in order to minimize the torque ripple and other unwanted side-effects of rotor/stator slotting as expanded upon in Section 2.6.4 the point(B) is introduced on the periphery of the rotor which allows for altering the position of the rotor slots. The outermost barrier slots is shifted further from the q-axis with the displacement angle s. Fig. 2.19 displays the situation in a machine with three flux barriers. The addition of the displacement angle entails that all rotor slots are shifted from each-other with equal angles except the slots closest to the q-axis.

Hence, the rotor slot pitch angle can be calculated as

↵m =

⇡

2p ^s

k + 1 2

(2.25)

where k is the number of rotor barriers and scan be regarded as a design variable.

2.8.2 d/q-axis MMF and barrier sizing

The applied magnetomotive force (MMF) is assumed to be sinusoidal in d- and q-axis.

When this MMF is applied in either the q- or d-axis direction it is possible to derive a step function where the steps has the value of the average MMF seen by the corresponding iron segment as seen in Fig. 2.20 [15, 23].

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2.8. Theoretical foundation of design approach

q

d

↵_m/2

↵m

↵_m

↵_m (C)

s(B)

Fig. 2.19: Rotor barrier angles in design approach.

Hence, assuming that there are k barriers the average values seen by the iron segments for a d- and q-axis MMF, fd,i and fq,i, represented by the steps in Fig. 2.20 can in per unit be expressed as

fd,i = 1

✓_b,i

✓Z_b,i+1

✓_b,i

cos ✓d✓ = sin ✓b,i+1 sin ✓b,i

✓_b,i+1 ✓_b,i i = 0, ..., k 1 (2.26a)

fq,i = 1

✓b,i

✓Zb,i+1

✓b,i

sin ✓d✓ = cos ✓b,i cos ✓b,i+1

✓b,i+1 ✓b,i

i = 0, ..., k 1 (2.26b)

Where the angles are defined as in Fig. 2.10. Note that ✓b,0=-✓b,1 since the iron segment aligned with the d-axis is shared with another pole.

It was shown in [15] that if the ratio of the permeances across each flux barrier were assumed to be constant for any barriers, the following relationship for the flux barrier

Design and comparison of PMaSynRM versus PMSM for pumping applications

Design and comparison of PMaSynRM versus PMSM for pumping applications

VIKTOR BRIGGNER

Design and comparison of PMaSynRM versus PMSM for pumping applications

Abstract

Sammanfattning

Acknowledgements

Contents

Acronyms

Nomenclature

Chapter 1 Introduction

1.1 Background and objectives

1.2 Thesis outline

Chapter 2

Synchronous Reluctance Machine and Permanent Magnet assistance

2.1 Concept of reluctance torque

B

2.2 Synchronous reluctance machine

2.2.1 Definition of axes

2.2.2 Governing equations

q

d

R

R

j!

d L v

dt

2.3 Saliency and performance

2.4 Iron saturation

2.5 Permanent magnet assistance

2.5.1 PM flux magnitude

2.6 Geometry and performance of PMaSynRM

2.6.1 Parameterization of PMaSynRM

2.6.2 Insulation ratios

2.6.3 Number of flux barriers

2.6.4 Torque ripple and rotor slots

2.6.5 Air-gap length

2.6.6 Radial and tangential ribs

w

w

w

w

w

w

D

D

D

1 2 3

2.6.7 Magnet dimensions and placement

w

w

w

w

w

w

2.6.8 Stator and rotor steel

2.7 Permanent magnets

2.7.1 Demagnetization

2.8 Theoretical foundation of design approach

2.8.1 Rotor barrier end angles

2.8.2 d/q-axis MMF and barrier sizing