2011_PhDCourse_SpecialElectricalMachines

Scuola di Dottorato in Ingegneria Industriale Attività didattica 2011 in Ingegneria Elettrotecnica

Special electric machines Dr. A. Tortella Laboratory of Electric Machines

Dipartimento Dipartimento di di Ingegneria Ingegneria Elettrica Elettrica

Summary • Introduction (motor classification and characteristics) • Magnetic materials (permanent magnets, SMC) • Small electric motors o Line-start single-phase induction and synchronous motors o Single-phase PM brushless motors o DC servomotors • Single-phase self-excited alternators (low rate) • Step motors (reluctance, PM and hybrid types) • Switched reluctance motors • Linear machines o Differences with rotating electrical machines o Induction and synchronous machines o Industry and transport applications

2

Medium and high rated motors

T.J.E. Miller : “Brushless Permanent-Magnet and Reluctance Motor Drives”

3

• Conventional motors (Ist row) o o o DC commutator (conventional excitation)

3-phase synchronous 3-phase induction (conventional excitation)

• Motors for electric drives (IInd e IIIrd rows) o o

o PM DC commutator

3-phase hybrid-PM synchronous

Normally operated using a power converter with a suitable control Favorable operating and manufacturing features with respect to the conventional motors Possibility of high speed operation (reluctance type machines)

• All the motors suitable for variable speed drives o o o

DC or sinusoidal brushless (PM excitation)

Normal operation if supplied directly by the mains Self-starting without adopting auxiliary devices Constant steady-state torque

Switched reluctance

Energy saving (maximum process efficiency, lower power for cooling) Position/speed control Improvement of transient phenomena (limitation of electric and mechanical stresses, suitability for start/stop processes)

Small electric motors • Single-phase or DC supply generally requested for both industrial and home appliances (HVAC, portable tools, washing machines, …) • Rated power ranging from some W to several hundreds of W • Requested performances often different from the high rated machines o Reduced weight and volume o Reliability (application and working cycles not defined in advance) o Reduced costs and maintenance o Low EMC and acoustic noise emissions • Design and manufacturing issues to obtain self-starting capability (AC) o 2-phase stator winding (main and auxiliary) with cage-type rotors o Pole air-gap shaping (PM machines) • Commutation concerns because of the low number of slots, involving current and torque ripple (DC) • Pulsating component (backward field) and harmonics in the main field (AC) o Efficiency and power factor lower than 3-phase machines o Significant torque ripple (especially 2nd harmonic)

4

Permanent magnets • Replacement of the conventional excitation in DC and AC synchronous machines Efficiency improvement and volume reduction Problems with flux control and operating temperature • High range of applications ⇒ from some tens of W (ferrites) to MW machines (rare earths) • Hard magnetic materials (Brinell hardness values as high as 690) Wide hysteresis cycle (high amount of magnetizing and demagnetizing energy) High coercivity with respect to the soft magnetic materials (operation in the II° quadrant of B-H curve) Low permeability at the normal operating point • Main materials (solid often sintered form, bonded or molded) Ceramics (strontium and barium ferrites) Alnico (alloy of aluminum, nickel and cobalt) Rare earths (samarium-cobalt, neodymium-iron)

5

Examples of PM machines Small DC motors

Traction motor (IPM)

High speed rotor

Small and high rated generators for wind turbines (axial and radial flux)

6

B-H characteristics • B = µ0H + J ⇒ Normal hysteresis loop • J-H curve ⇒ Intrinsic loop (domain orientation) • Experimental determination o Increasing H field in the virgin material o Domain orientation (J=Js) o H zero setting (B=Br≈Js) o H inversion ⇒ demagnetizing curve o Cancellation of B (H=Hc) • Influence of the magnetic knee position (quadrant II or III) o |H| reduction above the knee ⇒ B → Br o |H| reduction below the knee ⇒ B → B’r < Br o Recoil line based on µrec

Yeadon: ‘Handbook of small electric motors’

Intrinsic (J, Hi)

J

B=µ0H+J

Magnetization curves

7

PM typical properties • Remanence Br: defines the PM section needed to obtain a given magnetic flux • Operating remanence Bd: B value after removing the magnetic load o Linear curve ⇒ Bd≡Br o Non-linear curve ⇒ Bd depends on µrec related to the linear part • Coercivity Hc: defines the maximum allowable electric load without the material demagnetization • Maximum specific energy or grade BHmax: defines the minimum PM volume to obtain a given (air-gap) energy o Optimal operating point to minimize costs (important for design purpose) o Constraint on the torque density (Bm → φ/Am , Hm → NI/lm) • Temperature coefficients TC(Br), TC(Hc): define the BH curve modification when the operating temperature changes o TC(Br)=(dBr/dT)/Br·100 – TC(Hc)=(dHc/dT)/Hc·100 o Reversible during cooling only if the curve remains linear (condition fixed by the maximum temperature Tmax), otherwise a new magnetization is needed • Curie temperature TC: defines the temperature limit after which the magnetic domains lose their orientation ⇒ complete and irreversible demagnetization

8

Alnico

9 Dexter Magnet Technology: “Permanent Magnet Catalog”

• High temperature stability (operation up to 550 ) and relatively high remanence • Non- linear B-H curve with low Hc (long and thin shapes, use of magnetic shunts) • Production with casting processes (for complex shapes) or by sintering • Troublesome machining because of the hardness and brittleness of the material • Isotropic (un-oriented particles which can be magnetized with any pattern) or anisotropic (particles oriented according to the magnetization direction) property

10

Ferrites

• High coercivity (demagnetization robustness), resistance to oxidation and low electric conductivity • Cheap material widespread for low rated PM machines (nowadays considered also for medium sized machines because of the cost) • Ceramics ferrites troublesome to machine because of the hardness and brittleness of the material (cut effectively only with diamond tools) • Flexible ferrites (combined with rubber) to obtain complex shapes or direct incorporation with shaft

Grade 1: anisotropic (not oriented) Magnetic knee

Grade 5: readily available and very inexpensive Grade 7: B-H curve knee below the H axis (high level of resistance to demagnetization) Grade 8 (and various subgrades): more powerful, useful for new design of ferrite permanent-magnet motors and actuators

11

Neodymium--Iron Neodymium Iron--Boron • Highest magnetic performances (remanence and grade) • Low temperature and oxidation resistance (protection coating made of zinc, nickel or polymers), electric conductivity (shielding requires), troublesome production and machining (brittleness, toxic materials, dangerous to handle, damage of devices sensitive to high magnetic fields) • Production by direct particle sintering (sintered magnets) or covering them by polymers as nylon or epoxy resins (bonded magnets → lower performances, easier production and shaping, low conductivity) Define operating temperature

Samarium cobalt

12

• Common compositions Sm1Co5 and Sm2Co17 • Less powerful and more expensive than neodymium-iron, very brittle (small pieces), very good temperature (250 C), linear curve and corrosion resistance • Production by sintering or by bonding with polymer binders (needed also in case of large assemblies, lower operating temperature)

Sm1Co5

Sm2Co17

Bonded magnets • Precision: superior mechanical tolerances because of the elimination of the sintering operation, finish machining not required (more cost-effective) • Isotropic behavior: multiple magnetization patterns including axial, diametric, radial and multi-pole are possible • Form: compacted to the net shape through a die (elimination of subsequent machining, greater consistency) • Negligible eddy currents: insulation due to the polymer bonding Temperature dependence Magnetic properties (rare earth)

13

14

Magnetization for radial flux machines

http://www.mqitechnology.com

• Three basic orientations with bonded magnets 1) Straight: Flux lines are parallel and unconstrained by magnet geometry 2) Radial: Flux enters and exits the ring along a radial vector

3) Halbach: Flux orientation is continuously rotating with respect to the magnet (only one side is magnetized) Implications regarding the flux density profile and the backiron design DC brushless machine

Sinusoidal machine

15

Magnetization skewing

http://www.mqitechnology.com

• Adopted to reduce cogging or noise in a motor without skewing armature laminations (too complex and expensive) • Reduction of the magnetic flux harmonic content according to the well-known skewing coefficient • h: harmonic order Example of fixtures f (ξ ) = sin (hp ξ 2 ) (hp ξ 2 ) • p: pole pairs sk , h

sk

sk

sk

•

ξsk: skewing angle

o Total amplitude reduction o Shape modification (important when cogging is used for the motor starting) o Proper choice to avoid excessive decrease of the output torque

18° skewing

Steel plates

16

Commercial bonded magnets Rare earths

Top (°C) =110=110-150 µrec= 1.101.10-1.20

Ring PM axial Halbach magnetization Ferrites

Interp. (I-III harm.)

interpolation 0.2 B [T]

Measured

0.15

Top (°C) =80=80-120 µrec= 1.3

0.1 0.05 0 -0.05 -0.1

C

C

-0.15 -0.2

[°] 0

20

40

60

80

100

120

140

160

180

Permeance coefficient: calculation example Brushless motor with surface magnets φt/2

At

hm

PM section: Air-gap section: Air-gap reluctance: PM permeance: Rotor permeance:

φt

ℜt

φt/2

φm ℘r0 2

φr

℘m0

℘r0 2

Am r1

t

Am = 2π 3 ⋅ (r1 − t − hm 2 ) ⋅ Lm

At = [2π 3 ⋅ (r1 − t 2) + 2t ] ⋅ (Lm + 2t ) ℜt =

Am Cφ = At

PM flux concentration factor

kc ⋅ t µ 0 At

℘m 0 =

µ0 µrec Am hm

℘m = ℘m 0 +℘r = ℘m 0 ⋅ (1 + pr 0 )

pr0 = (0.05÷0.2)

Rotor leakage coefficient

17

Permeance coefficient: calculation example Cφ

Air-gap flux density:

1 φt = φr 1 + ℘mℜ t

Bt =

PM flux density:

1 φt = φm 1+℘rℜt

1 +℘r ℜ t Bm = Br 1 +℘m ℜ t

Permeance coefficient: H m = −

Br − Bm µ 0 ⋅ µ rec

PM characteristic

B Br

PC

P Hm

Magnetic circuit characteristic

Bm 1 + ℘r ℜ t = ⋅ µ rec = µ 0 ⋅ H m ℘m 0ℜ t 1 + pr 0 µ rec Cφ k c t Lm Lm = ≅ Cφ k c t Lm Cφ ⋅ k c t PC =

• By substituting PC in the magnetic circuit characteristic

Bm

Hc

1 + ℘mℜt

Br

H

Bm ≅ 0.85 Br

Bm PC = Br PC + µ rec PC ≈ (5÷6)

18

Parametric variations Air-gap variation (linear motors, eccentricity problems)

External m.m.f. (no-load to load condition, shortcircuit, …)

19

Summary of PM characteristics

Reference sizes • Distance point P 5 mm • Flux density BP 100 mT

Very interesting as far as cost/energy ratio is concerned NdFe

SmCo

20

Neodymium magnet cost (2010)

21 http://www.ndmagnets.com

• Price determined by three categories o manufacturing process o supply of its raw materials o required performance • Sintered Neo: anisotropic material whose alignment is imposed during the pressing operation

Other factors affecting the price Prices of certain rare earth elements (such as dysprosium or terbium) employed to enhance the magnet ability to withstand more extreme operating or environmental conditions (availability only in some regions) Improvement in densification of the magnet material and/or with better orientation of the magnetic powder (anisotropic sintered Neo is very favorable)

• Isotropic bonded Neo magnets: made from isotropic powder magnetized after molding ⇒ simpler and more economic process, though methods which develop greater densification consequently produce higher magnetic remanence and hence better price performance • Anisotropic bonded Neo magnets: highest $/kg because their fine powder is quite unstable and has to be handled in a batch process, which must also incorporate the magnetic aligning field; but this orientation produces far superior magnetic properties compared to isotropic bonded magnets.

Soft magnetic composites (SMC)

• Innovative material adopted to produce magnetic cores of DC and AC electric machines with isotropic magnetic properties • Iron particle powder covered by an insulating material (organic resin, polymers) thermally and mechanically processed to obtain unconventional 3D shapes • Main features o Realization of complex magnetic geometries with 3D flux patterns (axial or transverse flux machines, …) using suitable moulds o Low eddy current losses ⇒ high frequency (speed) applications o Manufacturing automation (final form obtained by combining two or three moulds, easy mounting of the winding coils) o Easy to recycle (crumbling and separation from the winding) o Temperature stability of the magnetic properties

22

Comparison with laminations Radial flux machines

Poles for axial flux machines

Linear tubular machines (stator assemblies)

23

24

Comparison with laminations

W/kg

f = 50 Hz

f = 100 Hz

GKN – Ancor. Lam.35 GKN – Ancor.

f = 200 Hz

f = 400 Hz

Lam.35

GKN – Ancor.

Lam.35

GKN – Ancor.

Lam.35

B=0.5 T

1.85

0.55

3.81

1.6

8.0

2.9

17.4

7.0

B=1.0 T

6.08

1.6

12.5

4.0

26.5

10.0

58.7

24.0

more than quadratic

more than double

• Moreover: o o o o

Lower mechanical resistance and thermal conductivity Unsuited for reluctance machines (too high magnetizing current) High production costs Difficult efficiency prediction from prototypes obtained from sample machining (loss of particle electrical insulation)

1

Single--phase induction motors Single • Main winding directly supplied by the mains ⇒ presence of a pulsating field which can be decomposed in two rotating fields F+ and F-, with forward (+) and backward direction (-) F1 (θ , t ) = FM cos ωt ⋅ cos pθ =

FM F cos( pθ − ωt ) + M cos( pθ + ωt ) 2 2

F+

F

F− ω/p ω/p F+

F−

• Induced e.m.f. E+ and E- related to the rotating field components which represents the rotor reaction due to the eddy current in the cage bars R1

Electromagnetic torque

X1 I 12+

I1 E+

Xm 2

V

E−

Xm 2 I 12−

R 12 2s

C+ C

X 12 2 R 12 2 2−s X 12 2

Absence of a starting torque due to the balanced field action (s=1)

-0.5

1 C−

0

0.5

Null torque for s>0 (n ω0 a V=cost. oppure abbassamento di V con ω=cost. • Potenza P = E·I erogata verso l’alimentazione ⇒ ricarica batterie Corrente circa doppia di quella allo spunto Brake operating mode Applicazione di una coppia di carico in opposizione ⇒ I = (V + | E|)/(Ra+Rsp)

Solutions to improve current commutation • Riduzione campo nei pressi dell’asse neutro o Scelta opportuna della larghezza del magnete o Sagomatura del traferro ai bordi del magnete • Riduzione dell’induttanza dell’avvolgimento di armatura o Cave meno profonde o Scelta opportuna della larghezza dell’apertura di cava o Riduzione del numero di spire • Spostamento del piano di commutazione in anticipo rispetto alla posizione naturale o Spazzole arretrate rispetto al senso del moto per f.e.m. che aiutano l’inversione della corrente o Valido per carichi praticamente costanti (I poco variabile) • Uso di spazzole in elettrografite o Alta caduta di tensione che compensa la f.e.m. indotta nella matassa in commutazione

5

6

Example Ampere

12.5

Current

Rated power 140 W Analysis at n = 7000 rpm)

10

Load Flux 7.5

(E-3) Weber s.

3

(E-3) N.m 2

150

1

Torque

0

-1

-2

-3

s. 2.5E-3

0.005

125

100

s.

7.5E-3 0.003

0.004

0.005

0.006

0.007

0.008

7

Characterization from measurements Speed characteristic ω=

V − (Rsp + Ra ) ⋅ I

ω = α1 ⋅ I + α 2

K ⋅φ

Calcolo α1 e α2 dalla caratteristica elettromeccanica della velocità

Shaft torque characteristic CL = K φ I − b ω Ra + Rsp   V  I − b =  K φ + b Kφ  Kφ 

CL = α 3 ⋅ I + α 4

Derived quantities V φ= α2 K

Rsp = −α1 K φ − Ra

b=−

α4 K φ

Ra + Rsp    α 3 ⋅ 100 ∆α 3 = α 3 − K φ − b Kφ  

V Utilizzata per verifica

Calcolo α3 e α4 dalla caratteristica elettromeccanica della coppia all’asse

8

Experimental data interpolation ω = α1 ⋅ I + α 2

CL = α 3 ⋅ I + α 4

C0=α4 Supplied current I

Shaft torque CL

Angular speed ω

ω0=α2

Motor general performances • Determinazione sperimentale delle caratteristiche elettromeccaniche ω(I) e CL(I) • Grandezze derivate dalle caratteristiche elettromeccaniche o Flusso φ o Resistenza equivalente delle spazzole Rsp o Coefficiente di attrito b • Altre grandezze derivate o Potenza assorbita Pa = V·I o Potenza resa PL= CL·ω o Perdite ohmiche Pj= (Ra + Rsp) ·I2 • Grandezze calcolate o Coppia elettromagnetica C = K·φ·I o Potenza convertita P = C·ω o Perdite meccaniche Pm= b·ω2 o Perdite nel ferro(+addizionali) PFe=Pa- Pm - Pj - PL

9

Configuration with Alnico PMs polo involucro

polo

10

• Uso per motori DC ad alte prestazioni • Configurazioni generalmente con 2 o 4 poli (minor flusso per polo)

(a)

nuclei

(b)

• Magnetizzazione nel senso della lunghezza per resistere agli effetti della reazione d’indotto • Strutture che si differenziano in base alla funzione dell’involucro esterno (a): materiale non magnetico con funzione di solo contenimento

(c)

(d)

(c): materiale magnetico (acciaio dolce) per ottenere la richiusura del flusso • Uso anche di magneti di tipo anisotropo per micromotori (e)

(e)

Demagnetization due to the armature m.m.f.

11

Z Z: zona del magnete più sensibile alla smagnetizzazione

A vuoto

Solo armatura

sovracorrente

Br

P P”

r Hc

poli

• A causa di una sovracorrente (es. inversione

della V per decelerare il motore) ⇒ P → P” BP BP” • Forte riduzione del flusso e quindi della coppia per l’abbassamento della retta di recupero (verifica dalla misura della velocità a vuoto)

P’

r’

Z A carico

HP HP”

Magnete

• Contromisure ⇒ Uso di espansioni polari (miglioramento distribuzione di flusso, incremento costante tempo elettrica) che sono solide solo se il rotore è privo di cave ⇒ Traferro incrementato ai bordi del magnete

Configuration with Ferrite PMs nucleo

nucleo magnete polo

(a)

(b)

12

• Magneti con elevato campo coercitivo ⇒ spessore ridotto con ampia area per incrementare il flusso • Possibilità di utilizzo di espansioni polari (b) per ridurre ancora lo spessore del magnete e migliorare la concentrazione del flusso • Nucleo con funzioni magnetiche coincidente con l’involucro del motore (d) (spessore del nucleo e quindi peso molto ridotto)

nucleo

• Magnete sempre più lungo del pacco rotorico per aumentare il flusso e quindi la coppia (c)

(d)

• Uso anche di magneti di tipo anisotropo per micromotori (c)

13

Comparison with rarerare-earth PMs magnete polo

polo

magnete

• Uso per motori ad alte prestazioni (alto campo coercitivo e induzione residua) • Minore spessore rispetto ad Alnico con più ampie espansioni polari (flusso meglio distribuito) • A parità di area e di lunghezza,

Samario-cobalto

Alnico

flusso doppio rispetto a quello prodotto da una ferrite

Comparison for a given motor size

(A): SmCo

Torque

Max power

CA/ CB

Pmax,A/ Pmax,B

1.5

2.0

(B): Ferrite

Mechanical time constant

Electrical time constant

Tm,A/ Tm,B

Te,A/ Te,B

0.5

0.7

14

Rotor slots and winding • Conduttori in filo smaltato inseriti in cave di tipo semichiuso • Cave inclinate e possibilmente in

numero elevato per ridurre il ‘cogging’ e quindi la rumorosità • Numero di cave dispari per ridurre il ‘cogging’, ma più difficile da costruire ⇒ in genere si sceglie un numero pari • A parità di coppia ⇒ NIa↑,φ↓ copper motor (a) oppure NIa↓,φ↑ iron motor (b) (a)

(b)

(a): uso con ferriti per il basso valore di flusso (denti sottili, molti conduttori) (b): uso con Alnico (denti larghi per non portarli in saturazione) • Sistemazione dei conduttori sul fondo cava per applicazioni ad alta dinamica

(c)

(basso sfruttamento del motore)

15

Alternative structures Motori slotless nucleo

• Rotore con conduttori fissati al nucleo in ferro senza usare le cave (‘slotless motor’) • Bassa inerzia e assenza di ‘cogging’ • Fissaggio conduttori problematico anche a causa dell’azione diretta esercitata dalla forza elettromagnetica • Utilizzo di magneti a terre rare o Alnico per avere un flusso accettabile

Motori moving-coil Fibra di vetro

• Rotore formato da un cilindro cavo in fibra di vetro su cui sono fissati i conduttori inserito tra due nuclei magnetici fissi • Bassissima inerzia, velocità ed accelerazioni molto elevate e assenza di ‘cogging’ • Costruzione complessa, traferro elevato

Nucleo interno

(uso di Alnico), problemi di raffreddamento

Motor comparison Motori con cave

x

Motori slotless

x

16

1

Single--phase self Single self--excited alternator Stator windings • Main winding (1) connected to the load • Auxiliary winding (2) connected to a capacitor (huge backward field component to enable selfexcitation) Rotor windings

1 1 1

d

θ

2

2

2 2

b2b6

1

b5

b1

1

1 3

1

3

1 1

1

• Field winding (3) connected to a diode (rectifying the induced e.m.fs) • Separate damping cages (b1b2-b3-b4 e b5-b6-b7-b8) to reduce voltage harmonic distortion without weakening the backward field

3

1

3 b8

b

1 2

b3 b7

4

2

2

2

1 1

1 1

q

2

Self--excitation process Self • Residual magnetism ⇒ e.m.f induced in the stator windings • Backward rotating field due to the stator currents ⇒ e.m.fs induced in the field winding (II harmonic order components) • Non-zero mean flux in the field winding due to the rectified e.m.f.s ⇒ flux and current increase in the stator winding • Final working point dependent on the magnetic saturation and on the terminal impedances 3

7

i3 [A]

6

λ3 [Wb]

2.5

5

2

4 1.5 3 1 2 0.5

1

t [s]

0

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

3

I2 growth during self self--excitation (no no--load load)) Xσ

Xm(I2)

Equivalent circuit of the auxiliary winding

I2

E2

Xc

Series of instants at constant excitation flux (ϕ3=cost.) (Xm,1+Xσ+Xc)I2

(Xm,2+Xσ+Xc)I2 E2,1

ϕ3,1 I2,1

ϕ3,3> ϕ3,2

E2,2

(Xm,3+Xσ+Xc)I2

ϕ3,2> ϕ3,1 I2

I2,2

E2,3

I2

I2 increases ⇒ ϕ3 increases⇒ both E2 and magnetic saturation increase (Xm ↓) ⇒ I2 increases further

I2,3 I 2

4

Steady--state operation Steady • Auxiliary current I2 90 leading the e.m.f. E2 phasor • Ohmic load (current I1 in phase with the e.m.f. E1) • E1 90 leading the e.m.f. E2 phasor (windings displaced by 90 ) Low phase displacement between I1 and I2 ⇒ High backward field component ⇒ Magnetic axes of the main field rotated of less than 45° with respect to the d axis

Load flux lines

Load flux density map

Linee di Flusso

Mappa Magnitudine Induzione

Main current

Auxiliary current

Output characteristic 280

• Pn=2.2 kVA, Vn=240 V

240

• Magnetizing effect due to the combined action of I1 and I2 at light loads

200

V0

[V]

160

• Quick reduction at high currents because of voltage drop, saturation effects (lower flux) and machine heating

120 80 40

Icc

In

V1[V]

00

10

20

30

40

voltage stability

• Pn=1 kVA, Vn=100 V, without damping cages

V1 I2

[A] • Icc ≈ 4 In to guarantee adequate

I2 [A] • I2 increase with C leading to higher V0 • Choice of C in order to have V1≈ cost.

I1 [A]

• C smaller ⇒ V1(I1) increase because of the series magnetization produced by the load current and the lower saturation with reduced I2 values

5

6

Effects of the damping cages Self-excitation process

Nuova Saccardo Motori documentation

Magnetic saturation

E22 + E32 + E42 + E52 + E62 + L+ En2 ⋅100 THD = Reduction of the harmonic distortion E1 With damping cages

Harmonic spectrum

1 3 5 7

Harmonic order

Without damping cages

Harmonic spectrum

Harmonic order

7

Machine model Main problems • High harmonic content in the air-gap m.m.f. • Complex rotor configuration • Magnetic saturation in the polar shoes (cross-coupling between d and q axes)

Analytical methods

o Difficult to obtain a general formulation o Approach limited to analyze the steady state conditions

FEM transient

o Multiple solutions for each configuration o High number of simulations o Elaboration time

New method (FEM magnetostatic module) o Definition of a general d-q model o Integration procedure to solve the dynamical equations

8

Electric equations cage equivalent winding

Main winding

di1 v1 = Rci1 + Lc dt Auxiliary winding

2

v5=0 i5

5

i3 = I 0 e

v3 ηVt

)

Damping cages vk=0

1

θ

v1

−1

4

3

i1

Field winding

v4=0

i4

i2

dv2 i2 = dt C

(

v2

i3

v3

d

ψ

i6

cage equivalent winding

q

6 v6=0

k=4, 5, 6

Solution of the matrix equation

− p[λ ] = [v] + [R ] ⋅ [i ] Non-linear set of equations (Lapp

[λ ] = [Lapp ]⋅ [i ]

dependent on θ and [i]) to solve numerically

9

Numerical solution 1

d ϕ1 di − = R1i1 + v1 = R1i1 + RL i1 + LL 1 dt dt

ϕ1′ = ϕ1 + LL i1

2

d ϕ2 − = R2i2 + v2 dt

d 2ϕ2 d i2 1 − 2 = R2 + i2 dt dt C

3

−

i2 = C ⋅ dv2 dt

d ϕ3 = R3i3 + v3 = R3i3 + rd i3 dt

R1′ = R1 + RL

rd = R f +

Rr − R f 1 + e i3 / I 0

−

−

d ϕ '1 = R '1 i1 dt

d ϕ3 = R'3 i3 dt

Step-by-step integration (step ∆t = tk - tk-1) ⇒ algebraic equations −

− −

ϕ1′,k − ϕ1′,k −1 ∆t

i1,k + i1,k −1 = R1′ 2

ϕ 2′,k − 2ϕ 2′ ,k −1 + ϕ 2′,k −2 ∆t

ϕ ′j ,k − ϕ ′j ,k −1 ∆t

= R′j

i j ,k

• Non-linear system of equations derived by applying the trapezoidal rule

i2,k + i2,k −1 = R2′ (i2,k − i2,k −1 ) + ∆t • Solution by an iterative 2C method adopting an adequate + i j ,k −1 relaxation parameter to avoid ( j = 3,...,6) 2 numerical instability

Semplified approach

10

Saturation model • Definition of two equivalent m.m.f. distributions having amplitudes Md along d axis and Mq along q axis, dependant on the position, on the winding currents and on the geometric configurations by suitable shape factors Ki (harmonic analysis of the air-gap m.m.f. waveform) • Calculation of the d and q permeances Λd and Λq, using the characteristics ϕd(Md) and ϕq(Mq) obtained by FEM analyses Mutual permeance Λij between i-th and j-th windings

Λ ij (θ i ) = K i K j Λ d cos θ i cos θ j + K i K j Λ q sin θ i sin θ j

Self-inductance

Lii = N (Λ ii (θ i ) + Λ 0i ) 2 i

Mutual inductance

Lij = N i N j Λ ij (θi ,θ j )

Simplified approach

11

Flux along d and q axes (no cross-coupling) Values in linear condition

ϕ d (M d ) = Λ d ⋅ M d = Λ d 0 K sd (M d ) ⋅ M d 80

ϕ q (M q ) = Λ q ⋅ M q = Λ q 0 K sq (M q ) ⋅ M q • Ksd, Ksq: reduction factors because of the magnetic saturation (values ≤1)

[mWb/m]

70 60

• Interpolation of the FEM values by analytical functions

50 40 30

ϕd

20

ϕq

• In linear condition Λd0≈1.5 Λq0

FEM

10

• Saturation effect similar 5000 for the two axes

[A] 0

0

500 1000 1500 2000 2500 3000 3500 4000 4500

12

Simplified approach 1.6

15

i3 [A]

1.4

i1 [A]

10

1.2 1

5

0.8 0

0.6 0.4

-5

0.2 -10

0 0 2.5

0.02

0.04

0.06

0.08

0.1

0.12

[s] 0.14

i3 [A]

Experimental Simulated

-15 6 i [A] 2

Experimental Simulated

4

2.0 2 1.5 0 1.0

-2

0.5 0.0 t0

t0+10

Experimental Simulated [ms] t0+2 0

-4 -6

t0

t0+10

t0+20

t0+30

[ms] t0+40

13

Self and mutual inductance Representation by analytical functions

 2π  ( θ + ξ j, h , k ({i })) l h , k (θ , {i }) = ∑ L j, h , k ({i })cos  j j= 0  T h ,k  n h ,k

h , k = 1,.., 6

Determination of Lj,h,k and ξj,h,k 1. Reproduction of the magnetic saturation by 2D FEM magnetostatic analyses (air-gap current sheet ⇒ total air-gap f.m.m.) 2. Calculation of the inductance matrix [Lapp*], independently from θ, related to suitable elementary circuits derived from the machine windings 3. Definition of a connection matrix [C] for [Lapp] calculation so that [Lapp] = [C]t ⋅ [Lapp*] ⋅ [C] 4. Interpolation of the inductance values calculated for different positions by a Fourier series expansion

Dependance on current Definition of an equivalent air-gap current distribution ρ(ψ) which reproduces the resultant m.m.f. distribution Ampereturns

Mh mh , j (ψ ) = ν h , j cos( j (ψ − θ h )) 2 Position of Winding coefficient

ρ h , j (ψ ) = − jν h , j ρ (ψ ) =

nd

∑r

d, j

j =1 j odd

j-th m.m.f harmonic (h-th winding)

magnetic axis

Mh 2R

cos jψ + ρd (ψ)

sin ( j (ψ − θ h ))

j-th current harmonic (h-th winding)

Mean airgap radius nq

∑r

q, j

j =1 j odd

sin jψ ρq (ψ)

Resultant d-q current distribution (by elaborating the previous equation)

rd,j e rq,j new state variables (nd=3,nq=1 and h≤3 ensures good accuracy and acceptable computational time)

14

15

Elementary circuits d2 d1

d3

4 d4

Full pitch coil

6

5 Elementary stator coil

Elementary rotor coils

Equivalent cage coils

• Connection matrix [C] to transform the elementary circuits (independent on position → same model also for meshing) to the actual windings • Slot pitch rotation simulated by sliding the [C] coefficients, for a given saturation condition • Cage bars connection reproduced by three equivalent windings

16

Complete circuit model Cage windings circuit Rb12 ⁄ 2 Rb1 2

i4

Rb23 ⁄ 2

Rb2 2

i5

Rb12 ⁄ 2

Rb3 2

Rb23 ⁄ 2

Rb34 ⁄ 2

i6 Rb34 ⁄ 2

Rb4 2

R '4 − R b2 2 0   i4  λ d  4   − λ5 = − R b2 2 R '5 − R b3 2  ⋅  i5     dt  λ   0 − R 2 R ' b3 6  6    i6  λ′ λ  R ' 0 0   i4   d  4  d  −1  4    4 − λ′5 = −  [ρ] ⋅ λ5  = 0 R '5 0  ⋅  i5        dt  λ′  dt  λ 0 0 R ' 6   i6   6  6   

Diagonalization matrix

Connection matrix

[L ] = [C ] ⋅ [L ]⋅ [C ] t

app

* app

[C s ] ns ×2 [C ] =   [ 0 ] 5×2

Position θ = 0°

0 0 0 1 1 1 1 1 1 1 1 0   [Cs ] =  1 1 0 0 0 0 0 0 0 0 − 1 − 1  

[ 0 ] n s ×4   [Cr ] 5×4 

1 [Cr ] =   [0]  1×4

[0] 3×1   [Cd ] 

−1 1 0 0    Indepen[Cd ]t =  0 0 1 − 1 dant    1 1 1 1  from θ  

Procedure for the inductance calculation Position θ Currents {i}

Database of the elementary inductances (2D FEM analyses)

Preliminary step

Air-gap current sheet parameters Matrix of the elementary inductances Set of inductance matrixes for one slot pitch rotations

Step 1

Step 2

Coefficients of the Fourier series expansion

 2π  (θ+ ξj,h,k ({i})) l h,k (θ,{i})= ∑Lj,h,k ({i}) cos j j=0  Th,k  Actual self and mutual h,k=1,..,6 nh,k

17

inductances

Step 3

Verification on commercial machines Main ratings and electrical parameters P=2.2 kVA

V=230 V

f=50 Hz

n=3000 rpm

RL=24.04 Ω

C=13.5 mF

R1=2.07 Ω

R2=9.24 Ω

R3=5.36 Ω

R4=0.78 mΩ

R5=1 mΩ

R6=0.78 mΩ

Rr=6 kΩ

I0=3.33 mA

Rf=1 mΩ

Nuova Saccardo Motori srl

Current source (without cages)

18

19

Simulation at steadysteady-state (rated load) 400

1000

v1 [V]

Experimental Simulated

300

500

100

250

0

0

-500

V1,rms = 221.6 V THD = 4.4 %

-300 -400 0.56

8

-250

V1,rms = 217.1 V THD = 4.6 %

-200

0.565

-750 t [s] 0.57

0.575

i3 [A]

V2,rms = 479.0 V V2,fund = 643.4 V V2,rms = 466.8 V V2,fund = 624.5 V

-1000 0.58 0.56

Experimental Simulated

7 6 5 4

0.565

t [s] 0.57

0.575

0.58

• Very good concordance as regard load voltage, V2,rms and 〈i3〉 • Reduced THD for v1

3 2

Mean value

1 0 0.56

Experimental Simulated

750

200

-100

v2 [V]

0.565

0.57

0.575

• Problems: saturation probably underestimated, t [s] 0.58 auxiliary modelization

Test bench Alternator ratings P1=5 kVA

V0=230 V

f=50 Hz

n=3000 rpm

Capacitor C=30 µF

20

21

Output characteristic Compound effect due to the main winding 8

V1

250

7 6

200 V1 [V]

P1

150

4 Measured

100

3

d-q model 2

FEM transient

50

Doubled calculation times than d-q model 0

0

5

10

15 I1 [A]

20

25

1 0 30

P1 [kW]

5

22

Magnetic saturation reduction B [T] 2.8 2.4 2.0 1.6 1.2 0.8 0.4 0.0

Initial configuration

Rotor modified configuration

Conf.

C [mF]

∆V1 [V]

∆v% [%]

THD [%]

I1 [A]

I2 [A]

I3 [A]

φ3m [mWb]

Pd [W]

Initial

32.8

18.4

9.9

4.5

23.6

7.93

5.66

7.7

1419

Modified

29.2

13.6

8.2

4.3

23.1

7.13

5.72

7.8

1314

Diff. [%]

-11.0

-26.5

-

-

-2.3

-10.1

1.1

2.2

-7.3

C adjusted during the parametric analysis to obtain the same rated no-load voltage V0

23

Modification of the bar connections Comparison of different connections using an objective function to be minimized

weighted average of the performance indexes

fob = α1

Penalties introduced if constraints are not fulfilled (THD, current densities,…)

∆V1 ∆v % Pd3 Pd + α + α + α 2 3 * 4 * * ∆v % Pd3 Pd ∆V1*

250

8

V1

Optimized connection

7

6

7

200

Optimized configuration

8

6

Initial configuration 5

V1 [V]

150

P1

4

100

3 2

50

8’

7’

6’ 5’ 4’ 3’

2’

1’

1 0

Initial connection

0

5

10

15

I1 [A]

20

25

0

P1 [kW]

1

4 5 2 3

Stepper motors Electromechanical converters operated to obtain an incremental (not continuous) motion ⇒ a current pulse produces a fixed rotation depending on the stator/rotor poles Benefits • Open control loop operation (no sensors are needed) • Suitable for digital control (no current modulation • Economic manufacturing (simple magnetic configurations) • • • •

Drawbacks Low efficiency Fixed (discrete) angular step (problematic for fine rotations) Oscillations around the standstill position with high inertial loads Positioning errors with high frictional loads Motor types (based on rotor configuration)

• Variable reluctance (VR) • Permanent magnet (PM) → polarity-dependant torque • Hybrid PM-reluctance

1

Applications

2

Variable reluctance stepper motors (VR) -

+ phase A A B C

C

B' C'

N

A C

S

B C

• Salient stator and rotor magnetic circuits (low rotor cost and inertia → high acceleration)

B

S N

A

A

C A' 6/2 (m=3 phases)

A 12/8 (m=3 phases)

• Torque related only to the reluctance variation linear condition

B

B

3

∂ Wec′ (F ,θ ) 1 2 ∂Λ C= = F ∂θ 2 ∂θ

C (12/8 12/8 motor) motor

• Unipolar current ⇒ simplification of supply converter topology

typical operation

step angle

• Step angle ⇒ ε= 2π π /(m⋅⋅Nr) o m: number of phase o Nr: rotor teeth (high to reduce ε) o np=m Nr : n.steps/rev 0

phase A

phase B

phase C

15

30

45

60

o displacement between the single-phase torques

Configuration for very low step angles

4

1

1

• High number of steps without increasing too much the number of phases (m≤8) 2

4

2

3

3

3

3

• Each stator poles subdivided in multiple teeth having the same pitch of the rotor ones (lower stator pole saturation ) • Condition to enable a regular motion:

4

2 2

1

1

4 stator poles

2π π/Nr

2π π/Ns

• Example

ε

• Verification

2π πq/Nr maximum n.teeth for each stator pole

4

Multiple--stack VR stepper motor Multiple phases

5

Teeth of each stator module (“stack”) displaced by a step angle with respect to the adjacent one (in the figure, 1/3 of the single rotor step angle) Same effect by displacing the rotor teeth instead of the stator ones

flux lines paths benefits: high number of steps, simple winding structure drawbacks: high inertia (3 rotors), use of unconventional laminations (see flux lines placed in the transverse plane)

6

Stepper motors with PM rotor A+

B+

B–

IA IB IA

A–

• Torque due to the interaction between the supplied winding field and PMs (Nr coincident with the rotor poles) • Bipolar current operation complicating converter topology or winding structure o Wave drive (conventional) o Full step (higher torque and current) o Half step drive (higher number of steps) • Presence of a detent torque with no supply which holds the rotor in position • Generally lower number of steps (higher step angle) than VR motors because of the more complicated manufacturing

IB IA IB

VR

PM

Frequency

1200 imp/s

400 imp/s

Step angle

1.8° – 15°

15° – 90°

Torque production (single phase supply supply)) Linear condition ∂W'ec ( Fi ,Fm ,θ ) 1 m 2 ∂ Λi 1 2 ∂ Λm m ∂Ψ im C= = ∑ Fi + Fm + ∑ Fi ∂θ 2 i =1 ∂θ 2 ∂ θ i =1 ∂θ Rotor reluctance Stator reluctance torque torque

θ

Cylindrical torque

7

Fi, Λi: m.m.f. and permeance related to the i-th phase self-inductance (i=1,2,...,m) Fm, Λm: m.m.f. and permeance related to the PM flux Ψim: flux generated by the PM and linked with i-th phase

• Λi independent on θ (magnet isotropic behavior and µr≈1)⇒ ∂Λi/∂θ≈0 (null rotor reluctance torque) cylindrical torque

full-step supply

resultant

IB 0°

+ B

– A

I

– B

A

+ A

45° 90° 135° 180° 225° 270° 315° 360°

reluctance torque

• Ψim fundamental varies according the function cos(½Nr(θ-2π(i-1)/Ns)) • Λm fundamental varies according to the function cos(mNrθ) → at every step the PM is always positioned in the same way with respect to the stator teeth (m·Nr is the number of steps/rev) • Fm costant and ∂Λm/∂θ has null mean value ⇒ stator reluctance torque with null mean value, but generates a significant torque ripple worsening the dynamic behavior

8

Bipolar supply circuits Bifilar windings (2 switches/phase)

Unifilar windings (4 switches /phase)

S1 S2

phase

S3 S4

S1

S2

tightly coupled coils

Current suppression tecniques Free-wheeling to avoid overvoltage on the turning off switch Current fall dependent on the circuit time constant τe=L/R

Half unipolar switches → cheap supply converter

Branches in parallel to the winding: see solutions 1,2,3

1

Wound on the same pole to decrease inductance during the simultaneous conduction Bulky and expensive windings, utilized only for a half of the conducting period

1

2

3

2 3

9

Hybrid stepper motor Back rotor

Motor exploded view

Front rotor

+A +B -B -A

• Rotor divided in two modules with both saliencies (teeth) and permanent magnets (axially magnetized)

• Half slot pitch displacement between the rotor modules Half pitch to double the active poles displacement (90° electrical) (number of steps/rev 2·m·Nr) • Λm now independent on rotor position because of the teeth displacement (improvement of dynamic performance) Magnet

-B +B +A +A +B

• Supply sequence (example with m=2)

+A -A

Phase supply sequence (final state indicated)

+A +

S N S N S N S

+B

+B

S N S N S N S

-A +

-A

N S N S N S N

-B

-B

N S N S N S N

10

Comparison between stepper motor configurations VR

PM

Hybrid

Torque/mass

Low

High

High

Steps/rev

High

Low

high

n° switch/phase

1

4 (2 if bifilar)

4 (2 if bifilar)

Efficiency

Low

High

High

Dynamic performance (torque/inertia)

Low

High

High

Manufacturing complexity/cost

Low

Medium-high (1)

High

(1):

depending on the PM poles

11

Torque characteristic • Torque which can be produced without losing the step as a function of frequency

o Performance decrease with increasing frequency (less time to drive the load) o Different curves according to the dynamic operation (pull-in and pull-out)

Cm

coppia di trattenuta holding torque

coppia di agganciamento pull-in torque

coppia di sganciamento pull-out torque

campo di risposta start-stop region

campo di funzionamento continuo slew range

f (n.steps/s)

12

Torque characteristic pull-in torque • upper bound of the start-stop region (dynamic operation)

• torque-frequency values that can be applied in dynamic condition without losing the step (for instance, typical sequence of starting, stopping and reversing rotation)

pull-out torque • upper bound of the slew range (continuous operation) • Maximum torque-frequency values that can be applied at constant frequency operation (without accelerating) f

Cm coppia di trattenimento coppia di agganciamento holding torque pull-in torque coppia di sganciamento pull-out torque

f

t campo di risposta start-stop region

campo di funzionamento continuo slew range

t f (n°passi/s)

13

Torque characteristic Holding torque

• Maximum torque with locked rotor which can be produced by supplying the phase with constant current • With no supply ⇒ detent torque: maximum torque due to the interaction between the magnets and the salient stator poles (rotor locked without current, presence of a torque ripple at load) holding torque +B detent torque Stable standstill points (without supply)

–A

–B

+A

Torque profile in dynamic condition Hyp.: constant torque as θ varies (mean value), initial speed=0 electromagnetic torque frictional torque load torque (effective value)

Torque equation

J

d2θ 2

dt

= Cem - Cfr - Cm

Dynamic condition ⇒ constant acceleration

α=

d2θ dt2

=

Cem - Cfr - Cm J

⇒

1 2∆θ 2J ∆θ α ∆t2 = ∆θ ⇒ ∆t = = 2 α Cem - Cfr - Cm Step angle

• ∆t is the minimum interval needed to cover the step angle and then the waiting time before supplying the next phase • The supplying frequency f must be therefore lower than 1/∆t: start-stop region

Cem - Cfr - Cm 1 f< = ∆t 2J ∆θ

⇒ Cm < Cem - Cfr -2J ∆θ f2

14

Torque profile at steadysteady-state Steady-state (pull-out torque) ⇒ f=const. ⇒

d2 θ dt

2

15

=0

Torque equation

Cem - Cfr -Cm = 0 ⇒ Cm = Cem - Cfr For a given Cem (same frequency and supply current),Cm is higher than in dynamic condition because of the lack of the inertial component

Cm < Cem - Cfr -2J ∆θ f2

When frequency increases: • increase of the frictional torque Cfr; • Cem decreases because of the current is decreasing as stated by the voltage equation

v = Ri +

∂ϕ ∂ϕ di dϕ ⇒ i= = Ri + ω + × dt ∂θ ∂i dt

v-ω

dϕ ∂ϕ di × dθ ∂i dt R

• f.c.e.m. increase with ω (f) ⇒ current decrease for a given voltage ⇒ Cem decrease

16

Switched reluctance motors (SRM) +V

• Doubly salient motor with number of rotor teeth Nr different form the stator one Ns • Torque generated only by the rotor tendency to assume a minimum reluctance positionS • Supply by unipolar switches with frequency inversely proportional to the step angle • Quite different from the stepper VRM (speed control, presence of the position sensor , possibly continuous and smooth torque,

1

0

2 m=3

βr

3 efficiency) m=4

m=4

βs

6/4

8/6

24/18

Main characteristics

Benefits

+V

0

17

1

2

• Simple rotor configuration with low inertia • Stator windings ease to manufacture • Losses mainly located in the stator, easier to cool than the rotor • Torque independent on the current polarity (simple converter topology) 3 • Generator operation very simple to obtain • Higher operating temperature than PM motors • Very high starting torque and maximum speed • Rotating direction reversed only modifying the switching sequence

Drawbacks • • • •

High torque ripple and radial forces (source of the motor noise) Very low air-gap length to maximize the torque production High current ripple (need of a capacitive filter) High supply frequency for a given winding utilization with respect to 3-phase motors because of the pulsed supply (vernier effect)

18

Applications (1)

http://www.srdrives.co.uk/

Applications (2) Electric motorbike (Lectra 24)

19

Applications (3)

20

Operating principle (linear condition condition))

21

• Trapezoidal inductance profile 1 R2

1 R1

1

R2

R3

1 R2

R3

• Useful zone to produce torque restricted to βs (dL/dθ>0: motor – dL/dθ>0: generator)

R2

L

• Favorable conditions: βr≈ βs and low unaligned inductance βs

βr-βs

L1

L2

βs

2π/Nr-βr-βs

θ

• Current waveform affected by both the inductance and the back-emf variation (especially at high speed)

L3

Ideal supply scheme to obtain a constant torque operation θ I1

I2

I3 θ

Consideration on the torque production

22

• Sign determined by the inductance derivative (position sensor is needed) • Motor design must emphasize the ratio Lmax/Lmin • Significant torque ripple because: dL/dθ θ≠const. (magnetic saturation , pole shapes) i≠ ≠const. (chopping at low speed, presence of a back-emf at high speed) • Phase supply Nr times per revolution to have continuous torque ⇒ switching frequency higher than a conventional AC machine (increased core losses, lower flux per pulse) Switching frequency

ω=

∆θ 2π 2π = fs ⋅ = ⋅n ∆t N r 60

Nr fs = ⋅n 60

Synchronous (p=1): f 0 =

p n ⋅n ≡ 60 60

SRM 6/4: f s = N r ⋅ f 0

• Step angle: rotation angle for each torque pulse

2π ε= m ⋅ Nr Phase number

• mNr : pulses/rev. • ε must be lower than βs to have continuous torque

Actual apparent inductance (8:6) Lapp [mH] 100 4A 8A

i

90 10A

80 12A

70 14A

60 16A 18A

50 40 30 20 10

0

Aligned position

-5

-10

-15

θ[°]

-20

-25

-30 Un-aligned position

23

Actual e.m. torque (I=const I=const., ., 8:6) Cem [Nm] 35 Steep decrease with high saturation (deviation fron the rectangular profile)

i

30

18A 16A

25

14A 20 12A 15 10A 10

8A

5 4A 0

0

Aligned position

-5

-10

-15

θ [°]

-20

-25

-30 Un-aligned position

24

25

FEM simulations (8:6) Rated power/speed: 4 kW/1500 rpm Length: 153.5 mm 3

11

20

Flux lines

10

7

Flux density map 20.2° 21.7° 28 96.8

26

Single phase supply • Low speed operation (current modulation) • Supply of the next phase which produces the maximum torque (step angle interval) • Energy balance examination on the ϕ – i characteristic Wm, W’m: converted mechanical energy between (-30≤θ≤0°) and (θ”≤θ≤θ’) respectively W’f: stored magnetic energy (θ=θ’) θ θ • Wc=W’m+W’f: supplied energy by the converter • ER=η=W’m/Wc: conversion efficiency

resultant torque T(θ) single-phase torque

Tav

θ’=-α α 15

ϕ' ϕ”

θ”=-(α α+ε)

0

-15

-30 θ [°]

0°

Wm

ϕ

θ’

W'f

θ” −30°

W’m O

in

i

Design consideration βs

2π β s + β r = ___ Νr

βs = βr

1

2

B

C

• 1: stator poles width lower than the rotor slot width to avoid a magnetic short-circuit between two adjacent rotor poles in the unaligned position • 2: stator poles narrower than the rotor ones because the winding mounting

3

βs= ε

27

• 3: angle βs higher than the step angle to avoid null torque zones

A βr

• Vertex A: higher room for the winding, but remarkable effect of the flux fringing at the pole edges (increase of the minimum inductance) • Vertex B: high minimum inductance value and smaller volume available for the winding • Vertex C: higher efficiency and power density, but significant increase of the torque ripple

Choice of the pole number • Most common combinations 6/4, 8/6, 12/10 (2 poles/phase) – 12/8,16/12 (4 poles/phase) • High rotor poles o High commutation frequency (core and switching losses) o High importance of the current rise and fall intervals (higher ohmic losses, conduction overlap) o Lower torque ripple with high harmonic order • Adoption of many poles/phase vs 2 poles/phase o Higher cost and winding manufacturing o Lower filling factor (insulation, spacers) ⇒ lower power density o Lower pole amperturns and then lower iron flux density for a given air-gap length (poor utilization of the magnetic material) o Reduced flux lines length and unidirectional stator flux (limited core losses, higher efficiency) • Adoption of slotted stator poles (see VR stepper motor) in case of high number of phases

28

Influence of the geometric parameters

Mean torque [Nm]

• Most convenient pole arc/pitch ratio 4045% with βr=βs (higher values lead to room and weight problems) • Stator pole arc/pole pitch more sensitive as regards the mean torque

Pole arc ————— (rotor) Pole pitch

Pole arc ————— (stator) Pole pitch

Aligned inductance [H]

Pole arc ————— (βs= βr) Pole pitch

Mean torque [Nm Nm]

29

Pole arc ————— (rotor) Pole pitch

βs / βr

Performances of a 8:6 motor 30.0

30 B 50

Favorable zone

27.5

Single phase supply

48

A

25.0

Parametric analysis varying the stator pole width βs and the rotor one βr

46

22.5 P 20.0

Conversion efficiency

44

17.5

15.0

92

B

30.0

D 20

25

βr [°]

30

35

B Favorable zone

27.5 88

A

25.0

βs [°]

[Nm/m]

40

Torque ripple 30.0

Favorable zone

27.5

42

C 15

Mean torque

[%]

βs [°]

20

A

25.0

[%]

22.5

84

βs [°] 22.5

P

15

P

20.0

20.0 80

17.5

15.0

17.5

C 15

D 20

βr [°]

25

30

35

15.0 76

10

C

D 15

20

25

30

35

5

31

Dynamic analysis Voltage equation

∂λ di ∂λ di dλ v = Ri + = Ri + ⋅ +ω = linc (i,θ )⋅ + kω (i,θ )⋅ ω dt ∂i θ =cost dt ∂θ i =cost dt Incremental inductance

Torque equation

dω Cem (i, θ) = Cm + J + Cf dt

Back-emf coefficient

dθ ω= dt

Numerical integration

ik − ik −1 1 ik + ik −1 ωk + ωk −1  Vk + Vk −1 = ⋅ − R⋅ − kω (ik , θk ) ⋅  ∆t linc (ik , θk )  2 2 2 ωk − ωk −1 1 = [Cem (ik , θk ) − Cm − C f ] J ∆t θk − θk −1 ωk + ωk −1 Algebraic non-linear system of equations to be solved = iteratively for each k-th step 2 ∆t

32

Flux linkage 0

-10

θ[°] -20

-30 0.8 0.6

λ [Wb]

0.4 0.2 0 15 10

i [A]

5 0

33

Static torque

30 20

Cem [Nm]

10 0

0 15 -10 10

i [A]

-20 5

0 -30

θ [°]

Incremental inductance 0

θ [°]

-10

-20

-30 100 80

l inc [mH]

60 40 20 0 15 10

i [A]

5

34

35

Back--emf coefficient Back 15

2

k ω [Wb/rad]

i [A] 10

1 0

5

0 -10 -20 -30

θ[°]

Current and torque waveform (1 phase supply supply)) [A]

Low speed (a)

[Nm]

(a)

Useful interval as concerns the torque production

[A]

High speed (b)

Advanced and longer conduction angle

[Nm]

(b)

36

Typical torque speed characteristic Hysteresis control Mean torque

Conduction angle

Lower conduction time

Profile suitable for transport application (electric vehicles)

Operation with increased voltage

Switching frequency limitation

ωb

(2÷3) ωb

ω

Angular speed

37

Comments Low speed • Very low back-emf ⇒ current controlled by chopping the supply voltage • Possibility to operate with increased voltage ⇒ current increase ⇒ saturation increase ⇒ higher converted energy ⇒ reduction of the conducting interval • Frequency limitation for the switch ⇒ current reduction to limit th switching losses • Base speed ωb: highest speed value for which i ≤ imax only by voltage commutation (the conduction angle θD and the maximum voltage VMAX are fixed)

High speed • Increase of θD by advancing the phase turn on to enable a faster current rising • Current increase limited by Linc and Cω • Advanced turn off to avoid the operation in generator mode (dL/dθ