Analytic and FEM evaluation of power density for various types of double-sided axial flux slotted PM motors.

Gholamian, S.A. ; Ardebili, M. ; Abbaszadeh, K. 等

Introduction

AFPMs (commonly called disc machines) are synchronous machines. In conventional machines, the gap flux density has normally radial direction; in AFPMs, the air gap flux density presents mainly axial direction. In general, AFPMs exhibit an axial length much smaller than the length of a conventional motor of the same rating [1].

There are two topologies for slotted double-sided AFPM motors. These topologies are axial flux slotted one-stator-two-rotor (TORUS) and two-stator-one-rotor (AFIR) type PM motors. Two AFPM motors and their acronyms are selected TORUS-S (Axial flux slotted external rotor internal stator PM stator) and AFIR-S (Axial flux slotted internal rotor external stator PM motor) for detailed analysis. The stator cores of the machine are formed by tape wound core with a lap and short-pitched polyphase AC winding located in punched stator slots. The rotor structure is formed by the axially magnetized NdFeB magnets [3-4].

The topologies used in the study are illustrated in Figure1.

[FIGURE 1 OMITTED]

Flux directions of both AFIR and TORUS slotted topologies at the average diameter in 2D are also shown in Figure.2a and 2b.

[FIGURE 2 OMITTED]

Selecting a double-sided AFPM motors with high power density is an important parameter, especially in electrical vehicle applications. So, comparison of power density between different topologies of double-sided AFPM motors seems to be necessary.

Increasing the air gap length, maximum power density will change in AFPM motors. These changes are not the same in different topologies. Maximum power density of TORUS-S is higher than AFIR-S in large air gap length.

In Section2, the generalized sizing approach for TORUS-S and AFIR-S types PM motors is briefly discussed. Then, some results of comparisons of the TORUS-S and AFIR-S topologies in terms of power density are illustrated in Section3. In Section4, Field analyses of both Topologies of slotted motors are investigated using Finite Element method (FEM) by MAXWELL10 software. The conclusions are given in Section 5.

Sizing Equation of Afpm Motors

In general, if stator leakage inductance and resistance are neglected, the output power for any electrical machine can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

where

e(t) and [E.sub.pk] are phase air gap EMF and its peak value, i(t) and [I.sub.pk] are phase current and the peak phase current, [eta] is machine efficiency, m is number of phases of the machine and T is period of one cycle of the EMF[2-4].

The quantity [K.sub.p] is termed the electrical power waveform factor and defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

where

[f.sub.e](t)=e(t)/ [E.sub.pk] and [f.sub.i](t)=i(t)/ [I.sub.pk] are the expressions for the normalized EMF and current waveforms. In order to indicate the effect of the current waveform, a definition for current waveform factor, [K.sub.i], is also useful,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

where

[I.sub.rms] is the rms value of the phase current. The peak value of the phase air gap EMF for AFPM in (1) is given by:

[E.sub.pk] = [K.sub.e] [N.sub.ph] [B.sub.g] * f/p * (1 - [[lambda].sup.2])[D.sup.2.sub.o] (4)

where

[K.sub.e] is the EMF factor which incorporates the winding distribution factor [K.sub.w] and the per unit portion of the total air gap area spanned by the salient poles of the machine (if any), [N.sub.ph] is the number of turn per phase, Bg is the flux density in the air gap, f is the converter frequency, p is the machine pole pairs, [lambda] is the diameter ratio for AFPM defined as [D.sub.i]/[D.sub.o], [D.sub.o] is the diameter of the machine outer surface, [D.sub.i] is the diameter of the machine inner surface. The peak phase current in (1) is given by:

[I.sub.pk] = A[pi][K.sub.i] 1 + [lambda]/2 * [D.sub.o]/[2m.sub.1] [N.sub.ph] (5)

where

[m.sub.1] is number of phases of each stator and A is the electrical loading.

Combining (1) through (5), the general purpose sizing equations take the following form for AFPM.

[P.sub.out] = m/[m.sub.1] [pi]/2 [K.sub.e][K.sub.p][K.sub.i]A[B.sub.g] [eta] f/p(1 - [[lambda].sup.2])(1 + [lambda]/2)[D.sup.3.sub.o] (6)

The machine power density for the total volume can be defined as

[P.sub.den] = [p.sub.out]/[pi]/4 [D.sup.2.sub.tot] [L.sub.tot] (7)

where

Dtot is the total machine outer diameter including the stack outer diameter and the protrusion of the end winding from the iron stack in the radial direction, [L.sub.tot] is the total length of the machine including the stack length and the protrusion of the end winding from the iron stack in the axial direction [2-4].

Sizing equations for the TORUS-S

The generalized sizing equation approach can easily be applied to axial flux permanent magnet TORUS type motor [4].

The outer surface diameter [D.sup.o] can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

The machine total outer diameter [D.sub.tot] for the TORUS-S motor is given by

[D.sub.tot] = [D.sub.o] + 2[W.sub.cu] (9)

where

[W.sub.cu] is the protrusion of the end winding from the iron stack in the radial direction. For the back-to-back wrapped winding, protrusions exist toward the axis of the machine as well as towards the outsides and can be calculated as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

where

[D.sub.g] is the average diameter of the machine, [J.sub.s] is the current density and [K.sub.cu] is the copper fill factor.

Note for the slotted topology machines the depth of the stator slot for slotted motors is

[L.sub.ss] = [W.sub.cu].

The axial length of the machine [L.sub.e] is given by

[L.sub.e] = [L.sub.s] + 2[L.sub.r] + 2g (11)

where

[L.sub.s] is axial length of the stator, [L.sub.r] is axial length of the rotor and g is the air gap length. The axial length of the stator [L.sub.s] is

[L.sub.s] = [L.sub.cs] + 2[L.sub.ss] (12)

The axial length of the stator core [L.sub.cs] can be written as

[L.sub.cs] = [B.sub.g][pi][alpha]p[D.sub.o](1 + [lambda])/4p [B.sub.cs] (13)

where

[B.sub.cs] is the flux density in the stator core and [[alpha].sub.p] is the ratio of average air gap flux density to peak air gap flux density.

The axial length of rotor [L.sub.r] becomes

[L.sub.r] = [L.sub.cr] + [L.sub.PM] (14)

Also, the axial length of the rotor core [L.sub.cr] is

[L.sub.cr] = [B.sub.u][pi][D.sub.o](1 + [lambda])/8p [B.sub.cr] (15)

where

[B.sub.cr] is the flux density in the rotor disc core, and [B.sub.u] is the attainable flux density on the surface of the PM.

The PM length [L.sub.PM] can be calculated as

[L.sub.PM] = [[mu].sub.r][B.sub.g]/[B.sub.r]-([K.sub.f]/[K.sub.d] [B.sub.g]) [K.sub.c]g (16)

where

[[mu].sub.r] is the recoil relative permeability of the magnet, [B.sub.r] is the residual flux density of the PM material, [K.sub.d] is the leakage flux factor, [K.sub.c] is the Carter factor, [K.sub.f] = [B.sub.gpk]/[B.sub.g] is the peak value corrected factor of air gap flux density in radial direction of the AFPM motor. These factors can be obtained using FEM analysis [4].

Sizing equations for the AFIR-S

The concept of Double-sided Axial Flux two-stator-one-rotor (AFIR) type PM motors was presented in [2-3].

The outer surface diameter [D.sub.o] is obtained from (6).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

The machine total outer diameter [D.sub.tot] for the AFIR type machines is given as

[D.sub.tot] = [D.sub.o] + 2[W.sub.cu] (18)

where

[W.sub.cu] is the protrusion of the end winding from the iron stack in the radial direction and can be calculated as

[W.sub.cu] = (0.49 - 0.62)[D.sub.o]/p (19)

The axial length of the machine [L.sub.e] is

[L.sub.e] = [L.sub.r] + 2[L.sub.s] + 2g (20)

where

[L.sub.s] is axial length of the stator, [L.sub.r] is axial length of the rotor and g is the air gap length. The axial length of a stator [L.sub.s] is

[L.sub.ss] = [.sub.Lcs] + [d.sub.ss] (21)

where

[L.sub.cs] is the axial length of the stator core, and the depth of the stator slot for slotted machines [d.sub.ss] is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)

where

[[alpha].sub.s] is the ratio of stator teeth portion to the stator pole. The axial length of the stator core [L.sub.cs] can be written as

[L.sub.cs] = [B.sub.g][pi][alpha][D.sub.o](1 + [lambda])/8p [B.sub.cr] (23)

Since there is no rotor core in rotor PM topologies, the axial length of rotor [L.sub.r] is

[L.sub.r] = [L.sub.PM] (24)

The PM length [L.sub.PM] can be calculated as

[L.sub.PM] = 2[[mu].sub.r][B.sub.g]/[B.sub.r]-([K.sub.f]/[K.sub.d] [B.sub.g]) [K.sub.c]g (25)

Comparoson of Torus-S And Afir-S

Comparison of two different Double-sided axial flux slotted PM motors in terms of power density is accomplished for 10KW output power, 4 poles and 60Hz drive. In this comparison, other constant parameters of motors are tabulated in table 1.

In AFPM motors, the air gap flux density and diameter ratio are the two important design parameters which have significant effect on the motor characteristics. Therefore, in order to optimize the motor performance, the diameter ratio and the air gap flux density must be chosen carefully. Figure. 3 shows the power density variation as a function of air gap flux density and the diameter ratio for the AFIR-S and TORUS-S motors.

[FIGURE 3 OMITTED]

As can be seen from Fig.3b, the maximum power density occurs at [B.sub.g]=0.528 (T) and [lambda] = 0.261. In various air gap length, the maximum power density occurs in different [B.sub.g] and [lambda]. Table 2 shows maximum power density with corresponding [B.sub.g] and [lambda].

Figure. 4 shows the maximum power density variation as a function of air gap length for the AFIR-S and TORUS-S motors for A=20000 (A/m), [J.sub.s]=6000000 (A/[m.sup.2]).

[FIGURE 4 OMITTED]

N special air gap length (this air gap length is called Gs) maximum power density of AFIR-S and TORUS-S motors will be the same. Considering Figure4, it can be concluded that in large air gap length, slotted TORUS motor has high power density.

2d Finite Element Analysis Of Field

In order to analyze the magnetic circuit and power density, 2D Finite Element Analysis was used for both TORUS-S and AFIR-S type motors. The purpose of the FEM is to get the overall picture of the saturation levels in various parts of the machine, to compare the flux densities obtained from FEM and sizing analysis.

FEM of the AFIR-S Motor

The motor parameters and important design dimensions used for the AFIR-S model are shown in Table 3.

Figure. 5 shows the flux distribution over one pole pair using FEM.

Figure. 6 shows the air gap Flux density over one pole at the average diameter ([D.sub.g]) using FEM. This curve shows that the flux density on the edge of the Slots is about 13% lower than the flux density on the center of the PM because of the magnet leakage flux.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

A flux density comparison between the FEM results and sizing analysis results on various parts of the slotted AFIR motor at no load is tabulated in Table4. The comparison table shows that the FEM results are consistent with the results obtained from the sizing analysis.

FEM of the TORUS-S Motor

The parameters and optimized TORUS-S motor dimensions used in the design which are calculated using sizing equations are shown in Table 5.

Figure 7: shows the flux distribution over one pole pair using FEM. The air gap flux density at the average diameter ([D.sub.g]) over one pole using FEM was obtained and is shown in Figure 8.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

A comparison of the flux densities between the FEM results and sizing analysis results for different parts of the machine at no load is tabulated in Table 6.

From the no load flux density plots, it is seen that the results are again consistent with the results obtained from the sizing analysis, the maximum flux density values on the rotor and stator came out almost the same. Also, the maximum and average airgap flux densities obtained from the FEM and sizing analysis agree well.

Effect of Electrical Loading and Current Density

The considerable point is that the value of Gs will vary when the electrical loading 'A' and current density 'Js' changes. Fig.9 shows the variation of the maximum power density as a function of air gap length in A=25000 (A/m), [J.sub.s]=6000000 (A/[m.sup.2]) for the AFIR-S and TORUS-S motors. Fig.10 shows the variation of the maximum power density as a function of air gap length in A=20000 (A/m), [J.sub.s]=7000000 (A/[m.sup.2]) for the AFIR-S and TORUS-S motors also.

[FIGURE 9 OMITTED]

According to Figure.9 it can be concluded that point Gs is shifted to larger air gaps and this means that in smaller air gaps AFIR-S motor has higher maximum power density. According to Figure.10 it can be concluded that point Gs is shifted to smaller air gaps and this means that in higher air gaps TORUS-S motor has higher maximum power density. Other value of Gs for various A and [J.sub.s] are tabulated in table 7.

Conclusions

Selecting an AFPM motors with higher power density is an important parameter in applications. The main goal of this paper has been introduce to double-Sided Axial Flux Slotted PM Motors with maximum power density. There are two topologies for slotted double-sided AFPM motors.

The maximum power density is changed by different value of the air gap, electrical loading and current density. TORUS-S topology has high power density in high current density and low electrical loading. But, AFIR-S topology has high power density in low current density and high electrical loading.

A flux density comparison between the various parts of the slotted AFIR-S and TORUS-S motors obtained from the FEM and sizing analysis at no load agree well.

References

[1] Jacek F. Gieras, Rong-Jie Wang and Maarten J. Kamper, "Axial Flux Permanent Magnet Brushless Machines",Publisher: Springer; 1 edition (January 4, 2005).

[2] S. Huang, J. Luo, F. Leonardi and T. A. Lipo, "A Comparison of Power Density for Axial Flux Machines Based on the General Purpose Sizing Equation", IEEE Trans. on Energy Conversion, Vol.14, No.2 June 1999, pp. 185-192.

[3] Aydin, M.; Huang, S.; Lipo, T.A.; "Optimum design and 3D finite element analysis of nonslotted and slotted internal rotor type axial flux PM disc Machines", Power Engineering Society Summer Meeting, 2001. IEEE Volume 3, 15-19 July 2001 Page(s):1409-1416 vol.3.

[4] Aydin, M.; Surong Huang; Lipo, T.A.; "Design and 3D electromagnetic field analysis of non-slotted and slotted TORUS type axial flux surface mounted permanent magnet disc machines", Electric Machines and Drives Conference, 2001. IEMDC 2001. IEEE International2001 Page(s): 645-651.

S.A. Gholamian, M. Ardebili and K. Abbaszadeh

Electrical Engineering Department of K.N. Toosi

University of Technology Tehran, Iran

Phone: (9821) 8846-9084

Fax: (9821) 8846-2066

Email: asghar_gholamian@ee.kntu.ac.ir

Table 1: Constant parameters of motors in comparison

Number of phases           3
Slot fill factor           0.8
Pole arc ratio             0.75
Slot per Pole per Phase    1
flux density in stator     1.5 T
flux density in rotor      1.5 T
Efficiency                 90%
Residual flux density of   1.1 T
PM

Table 2: Maximum power density with corresponding Bg and [lambda]

Type      g      Bg      [lambda]   Maximum
          (mm)   (T)                power
                                    density
                                    (W/[cm.sup.3])

TORUS-S   1      0.579   0.289      0.925
          1.5    0.569   0.271      0.907
          2      0.565   0.271      0.89
          2.5    0.569   0.271      0.878

AFIR-S    1      0.528   0.261      0.925
          1.5    0.518   0.261      0.902
          2      0.507   0.26       0.884
          2.5    0.518   0.251      0.87

Table 3: Parameters and dimensions of slotted AFIR-S motor

Air gap length                  1 mm
Slot depth 9 mm
Pole-arc-ratio                  0.75
Axial length of stator core    16 mm
Axial length of rotor core     40 mm
Axial length of PM              2 mm
Outer diameter                367 mm
Inner diameter                 95.5 mm

Table 4. Flux density comparison of slotless AFIR-S motor

         Rotor                 Air gap            Stator
         [B.sub.cr]   [B.sub.max]   [B.sub.avg]   [B.sub.cs]

FEM      1.5          0.82          0.55          1.45
Sizing   1.5 T        0.8           0.53          1.5
Eq.

Table 5: Parameters and dimensions of slotted TORUS-S motor.

Air gap length             1 mm
Slot depth                10 mm
Pole-arc-ratio             0.75
Axial length of stator    42 mm
core
Axial length of rotor     25 mm
core
Axial length of PM         2 mm
Outer diameter           356 mm
Inner diameter           103 mm

Table 6: Flux density comparison of slotless TORUS-S motor

         Rotor        Air           gap           Stator
         [B.sub.cr]   [B.sub.max]   [B.sub.avg]   [B.sub.cs]

FEM      1.52         0.85          0.6           1.44
Sizing   1.5 T        0.8           0.58          1.5
Eq.

Table 7: Other value of Gs for Various A and Js

A       Js              Gs
(A/m)   (A/[m.sup.2])   (mm)

20000   6000000         1.02
22000   6000000         1.2
25000   6000000         1.5
30000   6000000         2.23
20000   6500000         0.89
20000   7000000         0.83
20000   8000000         0.7
20000   9000000         0.58