Comparison of two 5-phase Permanent Magnet machine winding configurations. Application on naval propulsion specifications. Franck Scuiller, Jean-Fr´ed´eric Charpentier

Eric Semail, St´ephane Cl´enet

Institut de Recherche de l’Ecole Navale French Naval Academy BP 600, 29240 BREST-ARMEES, FRANCE {scuiller,charpentier}@ecole-navale.fr

Laboratoire d’Electrotechnique et d’Electronique de Puissance Ecole Nationale Sup´erieure d’Arts et M´etiers 8, bd Louis XIV, 59046 LILLE, FRANCE {eric.semail,stephane.clenet}@lille.ensam.fr

Abstract— This paper describes a design approach dedicated to multiphase Surface Mounted Permanent Magnet SMPM machines. Based on a vectorial multimachine modelling that splits the multiphase machine in a set of magnetically independent fictitious machines, this approach allows to determine a pertinent control topology in case of Pulse Width Modulation (PWM) Voltage Source Inverter (VSI) supply. Design rules to reduce the parasitic currents and the torque ripples can also be deduced. This method is applied to improve the adaptation of a naval propulsion PM 5-phase machine to its converter. From a classical initial design with fully-pitched concentrated winding, the machine is improved by performing a new fractional-slot winding which drastically decreases the cogging torque. To make this winding possible, only the pole number has been modified (from 16 to 14). Iron, magnet and copper volumes are unchanged. For the two machines, a vector control to have sinusoidal currents is considered. For a same average torque, a decrease of the copper losses (less 35%) is observed mainly due to the reduction of parasitic currents. Furthermore a significant diminution (less 36%) of the torque ripples is obtained as a consequence of weakened interactions between the harmonics of the electromotive force and current. In comparizon with the full-pitched concentrated winding, the fractional-slot winding improves significantly the torque quality (better density and lower ripples) and makes the current control easier without over-sizing the electronic components.

we make without changing the stator magnetic core a new 5-phase winding design that improves the compactness and reduces the torque ripples. In section I, a vectorial multimachine model dedicated to SMPM machines is described. By cancelling the numerous electromagnetical couplings inherent to the multiphase machine, this tool allows to predict the behavior of the machine from rules linked with spatial harmonic interactions. The section II shows that this multimachine modeling is an appropriate way to define systematic design and control rules for SMPM multiphase machine supplied with PWM VSI. In the section III, the multimachine design rules are used to improve the adaptation of a 5-phase machine to its converter.

I. I NTRODUCTION

In this paper, only the case of odd phase number is considered. The multimachine approach needs the usual following assumptions: • the N phases are identical and regularly shifted • the rotor is smooth • saturation and damper windings are neglected • the back electromotive force (back-EMF) in the stator windings is not disturbed by the stator currents. According to these hypothesis, the inductance matrix Mss that links the stator flux due to the stator with the voltage phase currents is symetric and circular. So it can be diagonalized by using a generalized Concordia transformation. This transformation makes F = (N + 1)/2 eigenspaces that are orthogonal each other appear: • the first eigenspace E0 , associated with eigenvalue λ0 , is a vectorial line

Multi-phase motors are widely used in electrical marine propulsion for reasons as reliability, smooth torque and partition of power [1], [2]. Among the different kinds of multiphase motors, the synchronou PM one appears as an attractive solution to improve the compactness of the propulsion system. These multi-phase motors can now be supplied by a PWM VSI which increases the flexibility of the control. In order to take advantage of this attractive topology, efficient vector control laws must be defined [3], [4] but also adapted designs must be etablished [5]. It is possible to work on the shapes and the repartition of the magnets and on the windings. After usual windings with one slot per pole and per phase, windings with a fractional number of slots per pole and per phase begin to be examined [3], [6]. In this paper, from a reference 5phase machine designed for a small-podded ship propeller,

II. M ULTI - MACHINE MODELLING OF A MULTI - PHASE MACHINE

A multiphase machine is difficult to study owing to its numerous inherent magnetical couplings. The multimachine modelling [7] enables a systematic study of this system with the particularity of taking into account the whole space harmonics. A. Property of the stator inductance matrix

• the other eigenspaces Eg , associated with eigenvalues λg , are vectorial planes The eigenvalues correspond to the cyclic inductance. The projection of the vecorial voltage equation of the N-phase machine onto the different eigenspaces allow to obtain a set magnetically independant vectorial voltage equation as shown by the following relation (where Rs is the stator resistance): → − → − → − → − v = Rs i + Mss di dt + e  − → → − → − → − 0  v0 = Rs i0 + λ0 di   dt + e0    ...  −→ (1) → − dig → − → − ⇐⇒ i = R + λ + e v g s g g g dt   ...    − −−− →  −−−→ −−→  −→ vF −1 = Rs iF −1 + λF −1 diFdt−1 + e−− F −1 → − → In this relation, the voltage vector − v , the current vector i and → the back-EMF vector − e have N components whereas the pro-

Fig. 1.

Multimachine decomposition of a 5-phase machine TABLE I

H ARMONIC CHARACTERISATION OF A 3- PHASE MACHINE

jected vectors onto the vectorial planes has two components: → an α-one and a β-one (and the projected vector − v0 onto the vectorial line has only one component). This set of uncoupled equations allow the introduction of multimachine concept.

Fictitious Machine 1-phase machine 2-phase machine

Eigenspace E0 E1

Odd harmonics family 3, 9, 15, 21, ... 3h 1, 5, 7, 11, ... 3h ± 1

B. Eigenspaces and fictitious machines Since the generalized Concordia transformation is orthonor→ − → mal, the dot product − v . i that determines the electrical power P of the N -phase machine is distributed onto each eigenspace: →  − → − − → → vg . ig = Pg P =− v.i = F

F

g=1

g=1

(2)

So each eigenspace provides a part of the total electrical power. By injecting relation (1) into (2), similar relation is obtained for the EM torque T : → − .− T =→ i =

F  g=1

→ − − → g . ig =

F 

In other words, each fictitious machine is characterized by a familly of particular harmonics that allows the calculation of its electrical quantities. Consequently each fictitious machine can be characterized by a specific familly of harmonics. The tables I and II gives the decomposition for respectively 3-phase and 5-phase machines. In any case, the machine sensitive to the harmonic 1 is called main machine; the 1-pase machine corresponds to the homopolar machine and the other 2-phase machines are nammed secondary machines. D. Pulsating torque analysis of a fictitious machine

Tg

(3)

g=1

→ In this relation (3), the vector −  is the back-EMF vector divided by the speed: this vector is called elementary backEMF. The equations (2) and (3) state that each eigenspace behaves as a real machine concerning the power balance. That is why an eigenspace can be considered as a fictitious machine. The phase number of the fictitious machine is equal to the dimension of its associated eigenspace. So the N-phase machine is split into (N − 1)/2 2-phase fictitious machines that are magnetically independant. The figure 1 shows the multimachine decomposition of a five-phase machine into one 1-phase fictitious machine and two 2-phase fictitious machines.

The pulsating torques result from the interaction between the current and back-EMF harmonics. By introducing the harmonic famillies, the multimachine approach allows to accurately determine the origin of the pulsating torques by discriminating the contribution of each fictitious machine. The table III shows the origins of the pulsating torques for a 5-phase machine by giving for each current and back-EMF harmonics interactions the pulsation of the corresponding EM torque (with ω, the current pulsation): • • • •

empty box if no interaction 0 for average torque generation 10ω for first harmonic pulsating torque 20ω for first harmonic pulsating torque...

C. Harmonic characterisation of fictitious machines The Fourier serie expansion of a N-phase vector and then its projections onto the different eigenspaces put forward the fact that the spatial harmonics are distributed among the fictitious machines: the projection of a vector associated with a given harmonic order number is not null only for one eigenspace [8].

TABLE II H ARMONIC CHARACTERISATION OF A 5- PHASE MACHINE Fictitious Machine 1-phase machine 2-phase machine 2-phase machine

Eigenspace E0 E1 E2

Odd harmonics family 5, 15, ... 5h 1, 9, 11, 19, ... 5h ± 1 3, 7, 13, 17, ... 5h ± 2

Back-EMF harmonic 1 3 5 7 9 11 13

1 0

3

Current harmonic 5 7

0 10ω 10ω

10ω 10ω

(0, 10ω)

10ω

9 10ω

11 10ω

0 20ω

20ω 0

0 20ω

13 10ω 20ω 0

TABLE III P ULSATING TORQUE GENERATION FOR A 5- PHASE MACHINE

For example, the first harmonic pulsating torque (10ω mark in the table III) has a contribution from the main fictitious machine (interaction between fundamental and harmonics 9 and 11) and another contribution from the secondary fictitious machine (interaction between harmonic 3 and harmonic 7 and 13). This statement is a consequence of the magnetical decoupling of the fictitious machines. This table also points out the inability of the homopolar fictitious machine to provide a constant torque. Consequently, a phase wye-coupled connection is required in order to make possible the achievement of a smooth EM torque. The existence of the harmonic families explain why a large back-EMF spectrum is less embarrassing with high phase order machine than with three-phase machine. Indeed, in case of classical sinusoidal supply, the first pulsating torque harmonic results from the interaction between the fundamental of the current and a back-EMF harmonic rank as higher as the phase number is high. For example (and as shown by the table III), for a 5-phase machine supplied with sinusoidal current, the first pulsating torque results from the interaction of fundamental current with the 9th back-EMF harmonic instead of the 5th harmonic with a 3-phase machine. So the design of multiphase machine with sinusoidal phase currents does not demand sinusoidal back-EMF. For a 5-phase machine, harmonic 5 in the back-EMF spectrum can be admitted. III. C ONTROL AND DESIGN RULES BASED ON THE MULTIMACHINE MODELLING

A multiphase machine machine is all the more difficult to supply with PWM VSI that its phase number is high. This phenomenon results from the high number of electrical constant times that characterises such a machine. Since each fictitious machine models a time constant system, the multimachine theory is an appropriate tool to get onto the multiphase PWM VSI control issue and to define design rules to adapt the machine to its converter. A. Analyse of the behaviors of the fictitious machines with PWM VSI topology With a classical sinusoidal control strategy, only the fundamental phase current is regulated. This statement means that only the main fictitious machine is controlled (the main fictitious machine is the one sensitive to the 1st harmonic). This strategy is not sufficient with a multiphase machine in

Fig. 2.

Regulation scheme of a fictitious machine

so far as the whole fictitious machines are subject to be unintentionally supplied. Indeed, according to the fictitious voltage equation (1), the fictitious machine current depends on two signals: • the fictitious voltage that is a pulse train signal resulting from the comparison between the modulating signal and the carrying signal of the Pulse Width Modulation system • the fictitious back-EMF that can be seen as a disturbance. This analysis points out the necessity to control the current in the other machines. Otherwise, because these fictitious machines are in open-loop current, the back-EMF harmonics that belong to the family harmonics will induce disturbed currents that will interact with the back-EMF to produce pulsating torques. Furthermore, if the Pulse Width Modulation frequency is not adapted to the whole fictitious machine electrical time constants, some pulse train voltage harmonics will not be enough filtered which will generate parasitic currents. These effects can’t be admitted because they create unpredicted copper losses and then unexpected temperature increasement. B. Multimachine control topology A possible regulation topology consists in controlling each fictitious machine by using the multi-reference frame strategy [4]. Each fictitious machine is associated with a rotating rotor frame that is obtained by referencing the first minus odd harmonic of the familly harmonic. Thus, for a 5-phase machine, the main fictitious machine is associated with a dqframe that rotates at ω speed and the secondary fictitious machine is associated with another dq-frame that rotates at 3ω speed. The figure 2 represents the current control loop of each fictitious machine. More precised descriptions are available in [8], [9] for a five-phase machine. The final goal of the control loop is to regulate a first-order system (the fictitious machine dynamic block on the figure 2) that is characterised by a static Rs . Proportionalgain 1/Rs and a cut-frequency fg = 2πλ g Integrator controllers are known to be able to regulate this kind of system. In other words, algorithms dedicated to 3phase machine can be used to perform the control of each fictitious current. Concerning the fictitious current references, the set of values depends on a choice of repartition of the total real machine torque. The more simple strategy only aims to supply the main machine: that is the sinusoidal strategy that suggests to regulate the other fictitious currents to zero. For these regulations, it is necessary to compensate the back-EMF disturbance with specific control laws. For example, the paper

[10] focuses on the control of a 5-phase brushless DC machine and uses a controller enabled to compensate the 7th EMF harmonics that is dramatically disordering for the regulation of the second two-phase machine (see table II for harmonic family description). C. Multiphase machine design to make easier the PWM VSI multimachine control Actually the absence of these undesired back-EMF harmonics would be the ideal situation. A multi-phase design machine that specifies back-EMF spectrum goals is an efficient mean to reduce these harmonics. An efficient design must lead to a sinuoidal back-EMF for the whole fictitious machine. These design goals favour a smooth torque. To perform such a design, the main freedom degrees are the magnets disposition on the rotor and the stator windings: on the one side, the magnets disposition determines the spectral composition of the flux density generated by the rotor; on the other side, the winding acts on the back-EMF spectrum by filtering the flux density generated by the rotor [11]. Some elements about systematic windings are available in [12], [13]. As underlined previously, the PWM carrier-frequency must be adapted to the highest fictitious machine cut frequency. Consequently, the design specification must guarantee sufficient fictitious inductance values for all the fictitious machine. By this way, on the one side, the requirements about electronic components stay reasonable; on the other side, the available time for computing operations stay acceptable. Since the number of fictitious machines increases with the phase number, this issue can become crucial. In other words, if the multi-phase machine design does not care about inductance values, a high number of fictitious machines must be controled in a time all the shorter as the phase number is high. IV. E XAMPLE OF A 5- PHASE MACHINE This section applies the described multi-phase design to a 5phase surface-mounted permanent magnets machine. An initial small-podded propeller motor is improved only by changing the stator windings and the pole number: 2D finite difference calculations and Matlab simulations predict significant peformance increasements. A. Initial design and control According to the multi-machine theory and the figure 1, a 5-phase machine can be divided into three fictitious machines: the 2-phase Main Machine that is sensitive to the fundamental, a 2-phase Secondary Machine whose the minus hamonic of the familly is the third and the homopolar machine that can be unsupplied in case of wye-coupled connection. The considered 5-phase machine is wye-coupled. So only the main and secondary machines are to be considered. Moreover, the chosen control strategy classicaly makes the Main Machine provide the whole torque; which means that the Secondary Machine current is regulated to zero. The table IV gives the main parameter set of the motor. The motor is supplied with a PWM VSI with a frequency fP W M = 2000 Hz. The objective

TABLE IV P ROPELLER PARAMETER S ET Number of phases Stator core thickness Air gap Angular teeth width Axial machine length Bore Diameter Slot depth Slot Number Conductors by slot Slot fill factor Required Average Torque Power (at 500 rpm) Magnet total volume Magnet material Magnet Magnetization

80 slots 16 poles 5 phases

(a) Initial full-pitched machine

N =5 Th = 6 mm e = 1 mm W t = 2.25 degrees L = 35 cm D = 166 mm Ds = 1 cm Ns = 80 (16 slots/phase) 10 0.5 60 N m (100 to 500 rpm) 3.1 kW 1042 cm3 (almost 6 kg) NeFeB 0.6 T

80 slots 14 poles 5 phases

(b) Modified fractional-pitched machine

Fig. 3. Comparison of the two 5-phase machines with the coil disposition of the first phase

of the approach is to improve the motor running by only changing stator winding and pole number without changing the electronics components characteristics. As shown on the figure 3-a, the initial situation of this 5-phase motor is the following: • the rotor consists of 16 poles, each pole made with a radial magnet that covers the whole pole pitch • the windings is fully-pitched (the winding step is equal to the pole pitch). The 2D Finite Difference Calculation Software Difimedi [14] allows the calculation of the inductance values. The stator resistance Rs is about 0.71 Ω. The table V summarizes these results and puts forward the ratio between the PWM frequency and the cut frequencies of the 2-phase fictitious machines. Obviously the PWM frequency is adapted to the cut-frequencies of the two two-phase fictitious machines. Nevertheless the ratio concerning the Secondary Machine is significantly lower than the one concerning the Main Machine. Consequently the current regulation of the Secondary Machine can not be as efficient as the current regulation of the main machine. This issue is all the trickier as the back-EMF spectrum that results from this initial design does not imply a sinusoidal back-EMF for the Secondary Machine. As shown on the Figure 4, the elementary back-EMF is almost squarewaved. Consequently

TABLE V

TABLE VI

F ICTITIOUS I NDUCTANCE VALUES WITH THE INITIAL FULLY- PITCHED

F ICTITIOUS I NDUCTANCE VALUES WITH THE FRACTIONAL - SLOTS

WINDINGS

WINDINGS

Inductance Value (mH) L1 = 3.1 L2 = 1.5

Machine Main Secondary

fP W M /fg 55.0 26.5

3

3

2

2

1

1

Back−EMF (V/rad/s)

Back−EMF (V/rad/s)

Machine Main Secondary

0

−1

fP W M /fg 65.5 60.2

0

−1

−2

−3

Inductance Value (mH) λ1 = 3.7 λ2 = 3.4

−2

0

30

60

90

120

150

180

210

240

270

300

330

360

Electrical Angle (deg)

−3

0

30

60

90

120

150

180

210

240

270

300

330

360

Electrical Angle (deg)

Fig. 4. Elementary back-EMF for the initial machine (with fully-pitched windings)

Fig. 5. Elementary back-EMF for the modified machine (with the fractionalslot windings)

this signal does not only contain harmonics 1 and 3. The ratio between the harmonics 7 and 3 is about 31% which means the current regulation of the Secondary Machines is strongly disturbed. To sumarize, significant parasitic current on the Secondary Machine can’t be avoided: unexpected copper losses and pulsating torques will appear.

It can be noticed that the two inductances λ1 and λ2 are almost equal. Consequently the PWM frequency is now correctly adapted for the two controlled fictitious machines. The figure 5 represents the new elementary back-EMF. In comparizon with the former signal (figure 4), the permeance wave has disappeared. That’s why a quasi-null cogging torque is predicted by Difimedi software. The figure 6 that compares the back-EMF of the two machines stigmatizes the improvements concerning the elementary back-EMF spectrum. The two fictitious machine back-EMF are more sinusoidal: the Main Machine can be considered absolutely sinusoidal and the Secondary Machine is significantly improved. All the undesired harmonics have been reduced with a very interresting drop for the 7th harmonic.

B. New winding to favour fictitious machine control without change power electronics specifications In order to make easier the current control of the Secondary Machine, the two following goals must be reached: • to increase the electrical time constant λ2 /Rs • to reduce the 7th, 13th and 17th harmonic of elementary back-EMF in comparizon with the 3rd harmonic As underlined in the subsection III-C, the winding stator is a pertinent design freedom degree to undertake this issue. In the intial machine, the slot number (80) is a multiple of the pole number (16): this kind of structure usually leads to an important cogging torque particularly embarrassing at low speed. A classical way to reduce this phenomenon is to increase the least common multiplier between slot number and pole number [15], [16]. That’s why it is chosen to prospect for a fractional slot winding enabled to improve Secondary Machine characteristics. By reducing the pole number from 16 to 14, a convenient winding is found. The figure 3-b presents the corresponding coils distribution for the new winding. To perform this winding with the same number of conductor by slot, the required copper volume is almost the same. So the stator resistance is unchanged (about 0.71 Ω). The Table VI gives the corresponding fictitious machine inductance values.

C. Performance improvements In order to put forward the improvements of the drivemachine association, the regulation system of the motor is simulated with Matlab-Simulink software. The load is sized to make the motor run at its nominal point 60 N m torque at 500 rpm speed. The figure 7 compares the electromagnetical torque produced by the initial machine and the modified one. The quality torque improvement is clear: the ripples level is reduced to about 36%. As shown on the figure 8, the current wave is more sinusoidal with the modified machine. The amplitude of the two signals are quitely equal. So, although the new back-EMF is less square-waved (see figures 4 and 5), the motor does not demand a higher current to reach the same running point. This is because the new motor is more adapted to the Main Machine regulation than the previous. With the

Initial Modified

Ratio betwenn harmonics 9 and 1 is reduced from 6,6 % to 0,8 %

Phase current (A)

1.6

1.4

20

15

15

10

10

5

0

−5

−10

1.2

Amplitude

20

Phase current (A)

1.8

Ratio betwenn harmonics 7 and 3 is reduced from 31,0 % to 14,3 %

1

5

0

−5

−10

−15

−15

−20

−20 1.43

1.44

1.45

1.46

1.47

1.48

1.49

1.5

1.42

Time (s)

Fig. 8.

0.6

1.44

1.45

1.46

1.47

1.48

1.49

1.5

Time (s)

(a) Initial Machine

0.8

1.43

(b) New Machine

Comparison of the parasitic currents in the two machines

0.4

R EFERENCES

0.2

0

1

3

5

7

9

11

13

15

17

19

21

23

25

Harmonic Rank

Back-EMF spectrum of the two machines (at 1 rad/s)

68

68

66

66

64

64

62

62

Torque (Nm)

Torque (Nm)

Fig. 6.

60

60

58

58

56

56

54

52 1.6

54

1.62

1.64

1.66

1.68

1.7

1.72

1.74

1.76

1.78

1.8

52 1.6

1.62

1.64

Time (s)

1.66

1.68

1.7

1.72

1.74

1.76

1.78

1.8

Time (s)

(a) Initial Machine

(b) Modified Machine

Fig. 7. Comparizon of the torque produced by the two 5-phase motors for the same torque-speed point: 60 Nm @ 500 rpm

new machine, the amplitude of the parasitic currents are less important (as shown on the figure 8) and then the copper losses are reduced to 35%. This improvement is explained by the two following points: the new back-EMF is less disturbing and the new electrical time constants are more adapted to the PWM frequency. V. C ONCLUSION The paper shows the torque quality improvements and copper losses reduction that can be expected if the PM multiphase machine is sized and controlled by considering the multimachine theory. In the case of a PWM VSI supply, the phase winding must be designed by taking into account the whole fictitious machine time constants in order to limit parasitic currents. As illustrated in this paper, the fractional slot windings are an efficient way to reach this goal without oversizing the electronic components. The expected results are all the more promising as they could be still better by an adaptation of the rotor geometry to make other fictitious machines able to provide torque. In the case of the 5-phase machine, if the back-EMF contains a significant part of harmonic 3, the torque density can be increased by supplying the second fictitious machine with third current harmonic injection.

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