IAS 2000 Annual Meeting IEEE

8-12 October 2000

Rome (Italy)

 

 

A.Cavagnino, M.Cristino, M.Lazzari, F.Profumo, A.Tenconi

Politecnico di Torino

Dipartimento di Ingegneria Elettrica Industriale

Corso Duca degli Abruzzi, 24 - 10129 Torino, Italy

 

Abstract - The paper presents an algorithm to calculate the parameters of an Axial Flux Interior PM (AFIPM) synchronous motor. The proposed procedure is based on a suitable "airgap function". At each instant, this function describes an equivalent airgap flux lines length due to the presence of the stator slots and the rotor geometry. Assuming infinite iron permeability, the magnetic potential of the rotor poles generated by the magnets and/or by the stator currents can be calculated. Therefore, for a specific machine radius, it is possible to determine the airgap flux density waveform. The complex AFIPM prototype geometry requires to calculate the flux densities at different machine radii. The performed analyses point out that ten radii values are sufficient to get a good precision. In the paper, the complete calculus procedure is presented. The obtained results are compared with the experimental data, measured directly on the AFIPM motor prototype. The induced EMF waveform is in excellent agreement with the measured one. Also the values of the synchronous inductances in the d-axis and in the q-axis confirm the validity of the proposed method.

 

I. INTRODUCTION

Axial Flux PM (AFPM) synchronous machines have been proposed for low-speed and high-torque applications, i.e. for direct traction drive systems in light-weight electric vehicles [1, 2]. In [3-5] a novel Axial Flux Interior PM (AFIPM) synchronous motor has been presented. Since the permanent magnets are buried inside the rotor magnetic structure, the AFIPM motor is capable to operate in the constant power speed range, with flux weakening operation. The experimental investigations on the AFIPM prototype performed in [3], have underlined some discrepancies between the measured motor parameters and those calculated through the simplified design procedure reported in [4]. This procedure bases itself on the results obtained by a 2D Finite Element Method (FEM) referred to the middle stator radius. Three-dimensional FEM analysis are very complex, because of the anisotropic behavior of the motor under study. Furthermore, 3D FEM techniques are not often appropriate for motor parameters determination, because of the time and the cost involved in the calculation [6-10]. This paper shows a simple algorithm to predict the induced EMF waveform and the d-axis and the q-axis synchronous inductances values of the proposed AFIPM motor prototype. The new algorithm is suitable to calculate, under suitable hypothesis, the airgap flux density waveform, for any machine radius. The stator slots and the flux barriers in the rotor are taken in to account considering an "airgap function". This function describes, at each instant, an equivalent length of the airgap flux lines, in each point of the considered circumference. Neglecting the reluctance of all the parts in iron, knowing the type and the magnetization direction of the magnets and/or the value of the stator currents, it is possible to define a function that describes the magnetic potential waveform, onto rotor surface toward the airgap. The potential value selection for the main and the leakage poles represents the major difficulty in the proposed approach. This value has been found resolving a simplified magnetic equivalent circuit of the motor. Due to the skewing of the flux barriers, the flux density must be calculated at different machine radii. The performed analyses point out that ten radii are sufficient to obtain a good precision. The obtained results are compared with the experimental data, measured directly on the AFIPM motor prototype. The induced EMF waveform is in excellent agreement with the measured one. Also the evaluation of the synchronous inductances in the d-axis and in the q-axis confirm the validity of the proposed method.

 

II. AFIPM MOTOR GEOMETRY

The exploded view of the AFIPM motor magnetic structure is shown in Fig.1, whereas in Fig.2 a double polar pitch particular, for half machine, in axial direction, is depicted.

Fig.1: AFIPM motor exploded view: some main and leakage rotor poles have been removed to show the magnets.

Fig.2: Double pole pitch for half machine in the axial direction.

Fig.3: Rotor geometry sketch: magnet and main pole corner coordinates.

The machine is constituted by two external stators and by one inner rotor. The Permanent Magnets (PMs) are axially magnetized and they find accommodation between two main rotor poles. Each main pole is separated by the adjacent leakage rotor poles through two airgaps, named "flux barriers". In order to reduce the cogging torque, the flux barriers have been skewed. Because of manufactory constraints, the skewing of two consecutive flux barriers is different (Fig.3). This geometrical peculiarity complicates the AFIPM motor parameters calculation. Besides the motor dimensions reported in Fig.3, the following geometric/design AFIPM prototype data have to be considered: · Winding type: concentric, single layer winding with 55 conductors/slot; the two stator windings are connected in series · Inner stator diameter: 119 mm · Outer stator diameter: 191 mm · Magnet type: Neodimiun-Iron-Borum (NdFeB) · Slot number, for one stator: 48 · Poles number: 8 · Stator slot opening: 2.9 mm · Airgap axial length between one stator and the rotor: 1 mm · Flux barrier thickness: 2 mm · Magnet axial length: 8 mm · Rotor pole axial length: 10 mm.

 

III. AFIPM MOTOR MAGNETIC FIELD ANALYSIS

A. Basic Assumptions

The induced EMF waveform calculation and the d-axis and q-axis magnetization inductance evaluation require the magnetic flux density waveform determination. The flux barriers skewing determines that the airgap field analysis for the AFIPM motor is a 3D field problem. If the 2D unfolded surface, at average machine radius, is considered as design reference geometry, some errors in the motor parameters calculation can be introduced [3]. For this reason, the proposed algorithm is capable to evaluate the flux density waveform, at different machine radii and for any rotor position. In the magnetic field analysis presented in this section, the following assumptions are made:

1) The calculus is made only for a double polar pitch and only for half machine in the axial direction (Fig.2).

2) The iron permeability of any iron parts is considered infinite: for each radial direction, the airgap MMF correspond to the magnetic potential difference between stator and rotor.

3) The rotor poles magnetic potential produced by the PMs or by the stator currents is constant.

4) The magnets are axially magnetized.

5) The electric degrees are used.

6) The reference system for the a angular coordinate is fixed onto stator. The q angle represents the rotor position measured as the phase difference between the main pole middle point, at the average radius (d-axis direction) and the a = 0 direction.

B. Theory

The airgap magnetic field generated by one MMF distribution acting along the airgap results weakened in correspondence to the stator slots and the flux barriers. This effect can be studied considering two theoretical reference cases: unilateral, infinite deep slot with teeth at the same potential (Fig.4.a) and unilateral, infinite deep slot with teeth at the conjugate potentials (Fig.4.b). In these pictures the magnetic potential of the smooth surface is zero. For each of these cases, it is possible to calculate the perpendicular component of the magnetic field waveform onto smooth surface, using the corresponding Schwartz-Christoffel conformal transformations [11, 12]. In particular, assuming that:

x : [m], linear airgap coordinate (centered in the middle point of the slot opening),

ac : [m], slot opening,

g : [m], geometrical airgap length,

A, DA : [A], teeth magnetic potential,

w, w' : [p.u.], conformal transformations variables,

: [p.u.], slot influence ratio,

the H(x) distributions assume the following formulations:

Case 1 (Fig.4.a)

...... (1)

where:

and when .

Case 2 (Fig4.b)

..... (2)

where:

and when .

Using the Eq.(1), the field weakening effect in correspondence of the slot can be studied through the function shown in Eq.(3). This function describes the equivalent length of the airgap flux lines. For convenience, in the AFIPM parameters calculation, the linear coordinate x is substitute by an electrical angular coordinate , where p is the machine pole pair number and R is the considered airgap radius.

..... (3)

Fig.4: Unilateral, infinite slot reference cases: (a) teeth at the same magnetic potential, (b) teeth at the conjugate magnetic potentials.

If the iron permeability is infinite, it is immediate to observe that, superimposing the solutions for the two previous theoretical cases, it is possible to calculate the airgap H(x) waveform in correspondence of a slot that produces a MMF step equal to DA (fig.5.a). In Fig.5.b the magnetic potential AMean is generated by the actual currents in the previous slots. It is possible to demonstrate that the same H(x) waveform can be evaluated considering the shown in Eq.(3) and considering that the slot MMF rises in a sinusoidal manner onto geometrical slot opening ( in Fig.5.b). It is important to remark that the fictitious distribution must be only used in the H(x) determination: it will be not right to use this function also for other calculation (i.e. the progressive phase conductors number and the linkage flux with the windings).

C. The "Airgap Function" (leq(a,q))

Repeating the Eq.(3) for all the stator slots, the stator airgap function shown in Fig.6 can be found. In a similar way, it is possible to model the flux barriers effect (Fig.7). In this case the equivalent length waveform, , depends on the rotor position. The slots and the flux barriers global effect onto the airgap flux lines length is evaluated by:

..... (4)

D. The Phase Conductors Progressive Number Function

The phase conductors progressive number function (NPhase(a)) is the integral of the airgap phase conductors density. By an other point of view, the NPhase(a) represents the airgap MMF waveform generated by one unitary direct current present in one phase winding. In order to define this function two hypotheses are necessary:

1) the conductors are distributed in the air gap at the middle point of the slot openings,

2) the conductors in the going coil side are assumed positive, whereas the return coil side is considered negative.

Fig.5: MMF slot effect: (a) sketch of the airgap magnetic field lines, (b) approximation of the MMF rise onto slot opening.

Fig.6: Equivalent airgap length due to the stator slots.

Fig.7: Equivalent airgap length due to the rotor flux barriers.

Fig.8: Phase conductors progressive number waveform.

Fig.9: Rotor magnetic potential waveform generated by the d-axis stator current of 1 APeak, at the machine average radius.

A NPhase(a) example is shown in Fig.8. The phase conductors progressive number function aids to evaluate the phase linkage flux. To evaluate the real airgap field waveform due to the stator sinusoidal currents three-phase systems, the fictitious function shown in Eq.(5) must be used.

..... (5)

In Eq.(5), are shifted by 120° electric degree and they are derived by the corresponding NPhase(a) distribution, substituting the vertical lines with sinusoidal arcs on the slot geometrical opening, in accordance with that shown in the section B. The angle is equal to at the synchronism.

E. The Rotor Magnet Potential Function (MMFR(a,q))

When any stator current is zero, the main pole assumes a magnetic potential due to the permanent magnet presence, whereas the leakage pole magnetic potential remains at zero for symmetry reasons. Also the stator surface is at zero potential. Assuming inactive the magnets, the d-axis stator current produces the same airgap rotor magnetic potential waveform created by the magnets, but the main pole potential value is different.

Fig.10: Rotor magnetic potential waveform generated by the q-axis stator currents of 1 APeak, at the machine average radius.

In Fig.9 an example of the rotor magnet potential waveform is shown. Because of manufactory constraints, each main pole shown in Fig.1 is constituted by two pieces closed together. For this reason, a small airgap for the q-axis reaction field lines must be considered. This parasitic airgap complicates the q-axis magnetic equivalent circuit and it determines the rotor magnet potential waveform depicted in Fig.10, when the q-axis stator current is supplied. The measured value of this airgap is equal to 0.2 mm. Independently by the magnetic excitation source, the magnetic potential determination of the rotor poles represents the critical aspect of the proposed method. In fact, the AFIPM motor parameters depend directly by these values. The rotor poles potential values are found resolving a simplified magnetic equivalent circuit of the motor. In particular, the reluctances of the air parts shown in Fig.11.a have been evaluated. To obtain accurate results, care must be taken in the definition of the sections for the magnetic flux passage, as at the inner and the outer stator radius. Due to the stator slot presence, the well known Carter factor is taken in to account for the evaluation of the stator-rotor airgap reluctance. In Eq.(6), the used Carter factor expression is reported, where ts is the slot pitch. Since in the AFIPM motor the slot opening is constant along the stator radius, whereas the slot pitch is proportional to the radius, a mean Carter factor has been used. The principal flux barriers effect onto Rg, RFB, Rleakage reluctances consists in the increasing of the useful sections for the magnetic flux path (Fig.11). These additional areas can be evaluated introducing the Dx [m] increment shown in Fig.11.b. In the case in study, the Dx value is determined through Eq.(2).

..... (6)

When the d-axis current is injected, the typical steps stator MMF waveform over the main pole is averaged, before resolving the equivalent magnetic circuit of the considered sector. When the q-axis current is considered, besides the mean value of the stator MMF waveform presents over the leakage rotor pole, also the mean value of the stator MMF waveform presents over half main pole must be considered, due to the parasitic airgap in the equivalent magnetic circuit.

F. The Airgap Flux Density Waveform (Bg(a,q))

Using the previous functions, it is possible to evaluate the flux density distribution through Eq.(7), when the machine radius and the rotor position are fixed.

Fig.11: AFIPM motor unfolded section for a stator radius: (a) equivalent magnetic circuit for the d-axis excitation, (b) sketch of the useful sections for the magnetic flux.

..... (7)

 

IV. INDUCED EMF WAVEFORM PREDICTION

A precise no-load EMF waveform prediction is obtained subdividing the AFIPM prototype machine in ten sectors in the radial direction. For each sector, the Eq.(7) is evaluated at the sector average radius. In the case in study, the stator MMF is zero along the whole periphery, because there are not current in the stator windings. Therefore, it is possible to calculate the flux linkage with a stator phase winding, as:

..... (8)

where Ro [m] and Ri [m] are respectively the outer and the inner radius of the considered machine sector. A 0.1 electric degree integration step is used. In Eq.ts (7) and (8), the phase displacement between the different sectors have to be carefully evaluated in order to obtain accurate results. Since the windings on the two stators are connected in series, the induced EMF can be determined using Eq.(9).

..... (9)

The obtained waveform is in excellent agreement with the measured one (Fig.12). The proposed algorithm is capable to take into account very well the stator slots presence and the flux barrier skewing effects. In fact, in the calculated EMF waveform, the rise and the descent fronts are different, because of the different skewing of two consecutive flux barrier. Since any damping effects onto the magnetic field variations due to the iron presence are modelled in the followed approach, the calculated EMF waveform results slightly more distorted that the measured ones.

Fig.12: Calculated and measured phase EMF waveform (comparison at 450 rpm).

 

V. MAGNETIZATION INDUCTANCES CALCULATIONS

In this case the PMs are considered inactive and the stator currents produce a MMFs(a,q) waveform aligned, at any instant, with the d-axis or with the q-axis. Always considering ten machine sector in radial direction, the d-axis and q-axis magnetization inductances (Lmd and Lmq) calculation is more complicated than the EMF evaluation procedure because, moreover the relative rotor sectors shifting, a precise synchronization between the stator MMF and the rotor position is requested. The stator-rotor synchronization is right when the calculated lPhase(q) waveform is exactly aligned to the corresponding sinusoidal phase current (Fig.13.a and Fig.14.a). Due to the rotor anisotropy, the waveform calculated by the Eq.(10) and depicted in Fig.13.b (in Fig.14.b for the q-axis), is not constant for different rotor position, but a ripple with periodicity appear. To describe in the correct way the electromagnetic conversion phenomena, it is convenient to define the magnetizing inductance as the ratio between the fundamental component of the phase linked flux and the amplitude of the symmetrical stator currents. It is important to remark that the rotor anisotropy, the stator slots effect and the stator MMF harmonics are correctly considered, if the lPhase(q) fundamental component is evaluated through the Eq.(8).

Fig.13: d-axis inductance calculation: (a) 1 APeak sinusoidal phase current and phase linked flux with one stator, (b) d-axis inductance function.

Fig.14: q-axis inductance calculation: (a) 1 APeak sinusoidal phase current and phase linked flux with one stator, (b) q-axis inductance function.

 

VI. CONCLUSIONS

To take in account the effective three-dimensional behaviors of the AFIPM prototype magnetic circuit, a simple method to predict the induced EMF waveform and the d-axis and the q-axis magnetization inductance is proposed. The algorithm is based on suitable functions and it requires the knowledge of the machine geometry. The basic hypothesis and the calculus procedure are presented in detail. The obtained results are compared with the experimental data measured directly on the AFIPM motor prototype. The induced EMF waveform is in excellent agreement with the measured one. Also the evaluation of the d-axis and the q-axis synchronous inductances confirm the validity of the proposed method. For these reason, this approach can be used for predict the airgap magnetic field distribution in any kind of PM synchronous motors.

 

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