Design and FEM Validation for an Axial Single Stator Dual Rotor PMSM

Lucian Nicolae Tutelea1 , Sorin Ioan Deaconu2 , Ion Boldea1

1 Electrical Engineering Department 2 Electrical Engineering and Industrial Informatics Department

Politehnica University of Timisoara sorin.deaconu@fih.upt.ro

Abstract-A novel synchronous machine-the single stator dual PM rotor brushless, axial-flux, concentrated double layer fractional tooth winding, single inverter with dual frequency PWM independent control for two shafts has been proven in previous papers of these authors to be able to improve the machine efficiency and boost the torque density. This paper will present the key design equations and design procedure of the Single Stator Dual Rotor PMSM by the equivalent magnetic circuit method , analyze skewed PM angle effects on machine performance and give design guidelines to achieve specific design objectives. A quasi-3D finite-element analysis with specialized software is given to prove the effectiveness of the design equations and find the main characteristics of the machine.

I. INTRODUCTION

The axial flux permanent magnet (AFPM) machine, also called the disc-type machine, is an attractive alternative due to its pancake shape, compact construction and high power density. AFPM motors are particularly suitable for electrical vehicles, pumps, fans, valve control, centrifuges, machine tools, robots etc [1], [2].

Axial flux machines appeared in the technical literature in the early 70s and trading of axial flux induction motors started few years later [3]-[6]. Nowadays, direct drive applications that require actuators or generators capable of operating at low speeds with large torques have revived the attention towards Axial Flux Machines, especially for the PM type, as they are capable of larger torque density and efficiency [3], [7]-[11].

However, AFPM Synchronous Machines become advantageous whenever a number of design prescriptions are fulfilled. Most notably, it is widely accepted that the number of pole pairs must be conveniently high [3], [12].

Fractional slot windings can be often realized in concentrated layouts: this happens when windings overhangs are not overlapped and the coils are wound individually around the stator teeth. Fractional Slot Concentrated Windings offer remarkable advantages both on the end user and to the manufacturer. In fact, they allow the physical separation of the phases and of the magnetic circuits of the phases, thus reducing the risk of phase-to-phase faults and minimizing the mutual inductance among the phases [3], [13].

The features of Single Stator Dual Rotor PMSM are summarized as the following: greatly shortened end windings,

high ratio of diameter to length, high efficiency, high torque density and low material costs [14], [15].

II. DESIGN EQUATIONS

This paper will derive the main design equation and a design procedure for these machines, but provided with two rotors. In addition, quasi 3D finite-element analysis is employed to prove the effectiveness of the design equations and the main machine characteristics. In fig. 1 a drawing of longitudinal section is shown. The single stator dual PM rotor axial synchronous machine has in centre the stator assembly (1) with the two three-phase windings (2) placed in open slots, fixed rigid in the casing (3), provided with two side covers (4), (5) in which the two ball bearings supports (6), (7), one radial and one axial, are introduced. The ball bearings allow the two shafts (8), (9) to rotate independently, each shaft having in the side towards the stator a disk of solid steel on which the permanent magnet poles are placed in circular and symmetric manner . The other end of the shaft is inserted into a half-coupling which is connected to the thermal engine (10), respectively, to the gears towards the drive wheels (11).

Many unknown parameters are involved in the design of the Single Stator Dual Rotor PMSM. As a result, it is necessary to assign some description to these parameters. They will be further explored in the design equations. Table I gives a list of the parameters used in the design approach.

8

2 3

1

10 6

4 5

7

11

9

Fig.1. Longitudinal section through the dual PM rotors machine.

TABLE I DESIGN PARAMETERS

Symbol Description fs tangential force per surface unit Rin inner radius Rout outer radius T1 torque of rotor 1 T2 torque of rotor 2 Hc cohercive field of PM hag air-gap height kc Carter factor for air-gap hpm PMs height 0 permeability of the air-gap PM permeability of the PM kyPM pitch factor for PM alpm x ratio between PM width and pole pitch pole pitch for rotors hcs equivalent machine length kw winding factor ky pitch factor kq zone factor for winding wst width of one slot hst slot height rho_copper stator winding resistivity at 200C alfa_copper temperature winding coefficient at 200C Tw winding temperature Scopper copper section Js current density hry tickness of the rotor discs on which the PM are fixed ry magnetic permeability in the rotor discs hs4 slot neck height Fez magnetic permeability in stator teeth wt teeth width Fesy magnetic permeability in the stator yoke hsy stator yoke width Is stator current bipm interpole PM width GmPM leakeage permeance of PMs beta skewing PM angle alpm ratio between PM width and pole pitch at beta=0 Rmed medium radius g air-gap PM PM linkage flux Uf phase voltage

TABLE II RATED PARAMETERS

Parameters Machine M1 Machine

M2 Base continuous power Pn [kW] 55 110 Base speed, nb [rot/min] 4200 2200 Maximum speed, nmax [rot/min] 5500 12000 Maximum voltage, Vn [V] 220 Number of phases, nphase 3 Pole pairs p 7 5 Number of stots Nst 12 12

In general to fulfill the imposed performance, the

dimensioning calculus has to be redone a few times. So the geometrical dimensioning is found iteratively even within a classical design. The number of iterations depends on the strategy chosen to change the electric/magnetic stresses after each verification calculus routine. To reduce the number of iterations, a correlation between electric/magnetic stress changes and geometrical parameter variations has to be

established. However, the number of iterations remains high, except for a highly experienced designer [16], [17], [18].

The main parameters used in the design and optimization of the axial PM rotor single stator machine are given in Table II. For an axial machine the torque is:

outin

R

R

2s drr2fT . (1)

If

out

inr R

Rk , (2)

3outs3r Rfk132T . (3) In many cases the design theme contains the maximum

power at a certain speed, wherefrom the maximum torque is extracted:

b

n

n260PT . (4)

Based on the equation (3), with the help of equation (4), there results the fundamental equation of dimensioning (5) is obtained:

3 3rs maxout k1f2 T3R , (5) Tmax = max(T1, T2), (6) Rin = kr Rout . (7)

For machines with surface permanent magnets, the magnetic gap (the mechanic gap and the equivalent thickness of the permanent magnets) is high enough , so, that for preliminary calculations of magnetic flux density in the gap , we may neglect the magnetic saturation (we consider the saturation factor as being unitary). The PM air-gap flux density peak value (8) and its fundamental magnitude (9) are:

PM

0

pm

cag

c0pmag

hkh

HB , (8)

yPMpmagpmag1 KB4B , (9)

where

2

alpmxsink yPM . (10)

The PM linkage flux without saturation (one turn/coil) is:

pmagwcsphase

st0pm Bkhn1N2 , (11)

where qyw kkk - is the winding factor. (12)

The phase mmf (peak value) yields:

0pm

nt p3

T2I , (13) while the stator main inductance (one turn per coil) is:

cag

2in

2out

2w0

phasesm kh

1RRkn

L . (14)

Slot geometric permeance

st

st4s w13/hhlambdac , (15)

end coil length stslotf wl , (16) coil turn length fcsturn lh2l , (17) end coil geometric permeance

ststst

in

stst

out

N21

whR74lg

whR74lglambdaendc

, (18)

are used to calculate slot leakeage inductance (one turn/coil)

lambdachn

N4L csphase

st0sigma , (19)

end coil leakeage inductance (one turn/coil)

lambdaendcln

N2L fphase

st0sigmaend , (20)

total phase inductance of machine (one turn/coil) sigmaendsigmasms LLLL . (21) Stator winding resistivity

copper_alpha2011

Tcopper_alpha1copper_0rh01rh w

, (22)

phase resistance

phase

st

copperturns n

NS

1l01rhR , (23) with

Fig. 2. Matlab diagram design and optimization program.

s

tcopper J2

IS (24) complete the picture of circuit parameters.

The program used for design and optimization was made in Matlab. Figure 2 presents the utilized subroutines, starting from the constants and input data and ending with the files in which the results are being saved, the characteristics are being drawn and the necessary data for validating the results using the finite element method is collected.

Besides electric circuit the equivalent magnetic circuit is crucial in the design. The equivalent magnetic circuit for one pole is presented in fig.3. Rmry represents the magnetic reluctance of the portion from the rotor disc (1 or 2) that corresponds to one pole:

csryry

cmry h

125,0hp6

RR

, (25) hcs = Rout Rin . (26)

With RmpM , the magnetic reluctance of a permanent magnet:

cspmPM

cpmmPM hb

khR

. (27)

Rmsz1 Rmag1

RmPM1 Rmsy

es1 Rmry1 RmPM1 emPM1

Rms1 Rms2

es2 Rmsz2 Rmag2 emPM2 RmPM2 Rmry2

RmPM2

Fig. 3. The equivalent magnetic circuit for one pole at Rc radius (between Rin and Rout) (explicit section).

hry1

hpm1

hag1

hs4

hst1

wst1

wt1 wt2

wst2

hst2

hag2

hpm2

hry2

hs4 hsy

The magnetomotive force (mmf) of a permanent magnet is: cpmPM Hhe . (28)

The magnetic reluctance of the air-gap Rmag, of one stator teeth Rmsz, of the portion from the stator yoke that corresponds to one pole Rmsy, the one pole armature mmf es, the magnetic leakage reluctance of the slot Rms and the leakage reluctance of the permanent magnet RmPM are:

cscy0

cagmag hRk

pkhR

, (29)

sty N

psink , (30)

cstFe

syst4smsz hw

4/hhhR

z

, (31)

cssystFe

cmsy hhN4

RRsy

, (32)

es = Is , (33)

cs4s

st0

stms

hh2

h2

wR

, (34)

csmPM

mPM hG1R

, (35)

where GmPM is: if agipm h2b

0pm

ag

mPM 2hh

1logG

, (36) else

0

ipm

ag

pm

ipm

mPM 2

2/1

bh

1

h2b

1log

G

. (37) The ratio between PM width and pole pitch at Rc radius and

optimum skewed PM angle is:

betaR

betasinRarcsinp

alpmalpmxc

med

, (38)

PMbalpm , (39)

p2

RR inout , (40)

2RRR outinmed

. (41) If beta 0 the conditions of existence are (for the radius

Rin we need to have at least a point of the magnet both in the upper and the lower side so that the magnets wont share positions):

0betaR

betasinRarcsinp

alpmin

med

, (42)

Fig. 4. Skewed PM angle.

p

betaR

betasinRarcsinp

alpmout

med

. (43)

If beta < 0 only one condition is required:

p

betaR

betasinRarcsinp

alpmout

med

. (44)

Then, the width of the PM is: cpm Ralpmxb , (45) and the width of teeth

stst

ct wN

R2w . (46) The slot pitch st is:

st

cst N

R2 , (47) with the equivalent magnetic air-gap:

PM

0pmagage hhh

, (48)

age

stage

st

hw5

1hwg , (49)

agest

stc hg

k . (50)

Few sample results of design (Table III) for a case study are given here (these all machines data will used in quasi 3-D FEM validation):

TABLE III PRELIMINARY DESIGN RESULTS

Parameters Machine M1 Machine

M2 Mechanical base speed wb [rad/s] 439.82 230.38 Rated torque rotor Mn [Nm] 125.05 477.46 Inner radius Rin [mm] 110 Outer radius Rout [mm] 197 Equivalent machine length hcs [mm] 87 Slot pitch tauslot [mm] 80.37

0

Rc

Rin

Rout

Rmed

PMs

bipm bpm

beta beta

/p

Parameters Machine M1 Machine

M2 Polar pitch for rotor tau [mm] 68.89 96.44 Slot height hst [mm] 29.5 65 Slot width wst [mm] 30.2 32.7 Average teeth width bt_med [mm] 50.12 47.61 Outer teeth width bt_out [mm] 72.91 70.41 Inner teeth width bt_inn [mm] 27.3 24.77 PM height hpm [mm] 3 6.3 Inner PM width bpm_in [mm] 31.44 61.49 Average PM width bpm_med [mm] 41.33 79.08 Outer PM width bpm_out [mm] 51.22 96.7 Carter factor for airgap kC 1.258 1.218

III. QUASI - 3D FEM VALIDATION

It is possible to divide the machine into a certain amount of computation planes: nlayer number of computation planes, ilayer current computation plane, nlayer,1ilayer 1nlayer

1RR1ilayerRR inoutinc . (51) The number of computation planes needed for computation

depends on the purpose (precision) of the computation. The finite element method (FEM) analysis may be

necessary to verify the analytical result and to calculate the flux distribution more accurately. Figure 10 shows the magnetic flux lines at no load in the q axis of machine 1.

The total distribution of the flux density at full load for the full scale machines considered here in d axis of rotor 1 is presented in fig. 11.

Figure 12 presents the flux density amplitude in an axial section in rotor discs, permanent magnets, air-gaps, teeth and stator yoke at five loads ( -2In, - In, 0, In, 2 In) and in the fig. 13 the flux density Y component at the same X coordinate at full load.

Fig. 10. A portion of magnetic field lines with rotor 1 in q axis at no load.

Fig. 11. The total distribution of the flux density at full load and d axis position of rotor 1.

Fig. 12. The values of flux density module in an axial section in rotor discs, permanent magnets, air-gaps, teeth and stator yoke at five loads ( -2In, - In, 0, In, 2 In)

Fig. 13. The values of flux density Y component in an axial section at full load

The results obtained with Matlab optimization were compared with results from quasi 3D FEM validation (Table III). Due to high saturation in rotor disk (yoke) and stator tooth corner, and reduced order of analytical model it a notable difference could be observed. The mesh has 68 regions, 12 symmetry pairs and 76430 elements.

The linkage PM flux per phase is quite sinusoidal (fig.14) despite of fractionary tooth wound windings. The largest harmonics is the third harmonic and it represents only 2.21% of fundamental for M1, respectively 0.7% of fundamental for M2. Classical vector control with PWM inverter could be used with good results in order to control the proposed machine. Figure 15 shows the d-q inductances for machines 1 and 2. The d axis inductances become larger than q axis inductances for deep flux weakening, leading an increase in speed range. The interaction torqueses T1, T2, (fig. 16) have been computed as a difference between total torque and cogging torque from FEM. The cogging torque is acceptably low for an automotive power drive application: 1.8% for generator (M1) and 2.9% for traction machine (M2). The torque and PM flux computed in FEM (Table IV) are larger than values from analytical model for M1 (about 10% for torque) and they are smaller than values from analytical model for M2 (about 8% for torque). These results are in relation with the saturation level on M1 respectively M2 rotor disk as noticed in fig. 17, 18.

The points 3 on this characteristic curves correspond to the rated current, and, because of saturation effect the inductances drop.

BY [T]

[mm]

Bmed [T]

[mm]

TABLE IV PARAMETERS COMPARISON

Parameters FEM quasi 3D MATLAB

design Torque of machine 1 [Nm] 138.8 125 Torque of machine 2 [Nm] 438.1 477.5 Inductance of machine 1 [mH] 0.5 0.567 Inductance of machine 2 [mH] 0.35 0.398 Linkage PM flux of machine 1 [mWb] 73.8 63.7 Linkage PM flux of machine 2 [mWb] 191.5 206.4

Fig. 14. Linkage PM flux for machines 1 and 2 and three phase.

Fig. 15. D-q inductances for machines 1 and 2 versus current.

0 50 100 150 200 250 300 350 400-500

-400

-300

-200

-100

0

100

200

300

400

500

Position (deg)

Torq

ue (N

m)

FEM Torques

T1cg

T2cg

T1

T2

Fig. 16. Interaction and cogging torques computed with FEM.

The points denoted with 1 on the characteristics Ld1 and Lq1 from the figure 15 correspond to a current I = -In. They are shown in the figures 17 and 18. Where a more significant drop for Lq1 with respect to Ld1, due to a higher saturation in this axis (figure 17 with respect to the figure 18) is to be noticed. When the current is zero (at the points denoted by 2) a higher saturation in the axis d1 appears, in comparison with axis q1; and this is why the inductance Ld1 is lower.

Fig. 17. The total distribution of the flux density at full load and q axis position of rotor 1

Fig. 18. The total distribution of the flux density at full load and d axis position of rotor 1.

IV. CONCLUSIONS

A novel machine family is analized in order to improve machine radial and axial dimensions, torque density and efficiency. The key design equations and procedure for the Single Stator Dual Rotor PMSM have been presented, some design guidelines to achieve the design objectives have been given and a quasi 3D FEM analysis was presented the influence of the saturation effect on machine inductances.

The proposed analytical model was validated by FEM with an accuracy of 8-10% for torque, and 13% for inductances. This is acceptable but it could be further reduced by a more complex analytical model. The maximum computed torque versus speed shows that both machines are able to produce the requested torques over the entire speed range.

This paper shows that it is possible to design an assembly of two machines (generator 55 kW/motor 110 kW) for a hybrid electric vehicle with a mass of active materials of only 133kg at 6000 rpm, an outer diameter 394 mm and an axial length of 188 mm.

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0 50 100 150 200 250 300 350 400-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Position (deg)

PM

flux

(Wb)

Linkage PM flux

a2 b2 c2

a1 b1 c1

-800 -600 -400 -200 0 200 400 600 8000.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55d-q inductances

Indu

ctan

ces

(mH

)

Current magnitude (A)

Ld2 Lq2

Ld1 Lq1

1

3

2

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