Optimum design of axial flux permanent magnet slotted TORUS motor using genetic algorithm.

Gholamian, S. Asghar ; Hemmati, S. ; Nasiri, Reza 等

Introduction

Now days, axial flux PM motors are used in industry because of high efficiency and low volume [1, 2 and 5].

These Configurations of motors have many various motors that divided two groups Axial flux PM Motor (AFPM) and Radial flux PM Motor (RFPM). Axial flux PM Motor are different because of direction of flux movement with electrical usual machine because in these motors flux move parallel with machine shaft [6, 7].

The required electric motors in vehicles should have the features like, proper shape, low volume, high torque and power density, high reliability and high efficiency. In comparison that accomplished between axial flux motors and electric machines, the axial flux motors can be the best choice to be used in electrical vehicles [9].

For this cause many design of this motors built by researcher. In 1988 Spooner suggested axial flux permanent magnet slotless for application in automobile. The purpose of using this kind of axial flux motor is the simple configuration and high efficiency [10]. In 1991, Jensen designed an axial flux machine with permanent magnetic, and without any slot, to be used in automobile [11].

Motors similar to the above motors will be observed in other researches that they have some difference in the way of action and design, but the substructure of their work can be assumed unchanged. Typically refer to these references: [3, 12, 13, 14, 15 and 17].

Among the different configurations of axial flux motors, double-sides configuration, has the best and the most application. Thus axial flux motor are designed with slotted stators and without slot, that the slot configuration has the higher strength and power density that without one [1], [6].

AFPM Motors and Sizing Equation

Axial flux permanent magnetic motors have single-sided, double-sided and multisided structures. Figure 1 show different structures of Axial flux motors.

[FIGURE 1 OMITTED]

An easiest and cheapest structure of axial flux motors is single-sided. But because of producing low moment and bearing problem that result from high magnetism in air levels that causes closing two parts to each other using from this Configuration is not usual. We can defuse high magnetism between Rotor and Stator by using from second Stator or rotor that set up by first symmetrical. This structure called double-sided. Double-sided motors are the best and used more than others. Despite this fact that double-sided are suitable but any Configuration of several sides is the best solution for special application. For example, in cases that we need to high power with much limitation on outside diameter, increasing the number of disks is a good suggestion.

Slotted AFPM motors can be used as a stimulant part of electric vehicle because have featured like: low volume, high configuration, high torque, power density and high efficiency. A typical configuration of a slotted double-sided axial flux motor with internal stator is presented in figure 2. More details of this configuration are discussed in [1, 5 and 7].

[FIGURE 2 OMITTED]

In this structure stator armature winding is as permanent magnet synchronous motors and permanent magnetic are placed surface of the rotor.

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

[P.sub.out] = [eta] m/t [??] e(t) x i(t)dt = m[K.sub.p][eta][E.sub.pk][I.sub.pk] (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[1, 7].

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] x f/p x (1 - [[lambda].sup.2]) [D.sup.2.sub.0] (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 x [K.sub.i] 1 + [lambda]/2 x [D.sub.o]/2[m.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] x/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.0] (6)

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

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

Where [D.sub.tot] 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 [1, 7].

The outer surface diameter [sup.[D.sub.0]] can be written as

[D.sub.0] = [([P.sub.out]/[??]m/2[m.sub.1] [K.sub.e] [K.sub.p] [K.sub.i]A[B.sub.g] [eta] f/p (1 - [[lambda].sup.2])(1 + [lambda]/2)).sup.1/3] (8)

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

[D.sub.tot] = [D.sub.0] + 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

[W.sub.cu] = [D.sub.i] - [square root of ([D.sup.2.sub.i] - (2A[D.sub.g]/[K.sub.cu] [J.sub.S])/2 (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] and The axial length of the stator [L.sub.s] are given By

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

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

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 core [L.sub.cs] can be written as

[L.sub.cs] = [B.sub.g] [xxx.sub.p] [D.sub.0] (1 + [lambda])/4p [B.sub.cs] (13)

Where [B.sub.cs] is the flux density in the stator core and [sup.[alpha]] 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] and The PM length [L.sub.PM] can be calculated as

[L.sub.cr] = [B.sub.u] x [D.sub.0] (1 + [lambda]/8p [B.sub.cr] (15)

[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 [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. [sup.[mu]]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 [1, 5].

Genetic algorithm and optimization

Genetic algorithm (GA) is used a powerful optimization tool, in many different optimization problems GA is a search algorithm that is based on natural selection mechanisms and natural genetics. For using GA, parameters are coded to arrays with specific length in solution area. Each array has a definite fitness that depends on the application. Then GA search to find the optimal solution. GA includes chromosome representation of solution, initializing the first generation, cross over and mutation operators that considering the problem to be optimized, these parameters are defined. According to the illustrations at the beginning of this section, parameters of GA are coded as below [2, 18].

Chromosome representation

Each chromosome in proposed GA is an [sup.1 x 5] array as shown in figure 3.

In which g, [N.sub.ph], A, [lambda] and [B.sub.g] are air gap length, number of turn per phase, electrical loading, the diameter ratio and flux density of air gap, respectively.

Cross over

For instance, considering one point cross over in figure4. There is not the possibility of mutation in the proposed algorithm. In this paper, the method of tournament is used as the selection operator.

Fitness function

As mentioned at the beginning of this paper, the proposed algorithm will be used to find the optimum power density. The motor power density for the total volume can be defined as

[P.sub.den] = [P.sub.out]/[??]/4 [D.sup.2.sub.tot][L.sub.e] (W/[m.sup.3]) (17)

where, [D.sub.tot] 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.e] 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

Result of optimizing design by using genetic algorithm

In this section, the result of designing double-sides slotted axial flux motor is presented by genetic algorithm. Nominal design parameters of this motor and related restriction are presented in table1.

By choosing generation 1400, and after executing program several times, algorithm approaches to the optimal point, variations of fitness-function (power density) for the best chromosome in every generation are presented in figure 5.

[FIGURE 5 OMITTED]

By executing program and its convergence after 313 generations, the related consequences to the optimal chromosome is shown in table2 that the first line shows the best choosing chromosome. Dimensions of optimal double-sides slotted axial flux motor is tabulated in Table3.

Simulation by 2D FEM

In order to analyze the magnetic circuit and power density, 2D Finite Element Analysis was used for double-sides slotted axial flux motor. 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 [1, 6 and 7].

FEM of the slotted AFPM Motor

The motor parameters and important design dimensions used for the double-sides slotted axial flux motor model are shown in Table4. Figure6 shows the flux distribution over tow pole pair using FEM.

[FIGURE 6 OMITTED]

Fig.7 shows the air gap Flux density over one pole at the average diameter (Dg) using FEM and Fig 8. show the stator yoke Flux density over one pole pair at the average diameter, too.

Average flux density comparison between the FEM results and sizing analysis results on various parts of the slotted AFPM motor at no load is tabulated in Table5. The comparison table shows that the FEM results are consistent with the results obtained from the sizing analysis.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

Conclusions

Selecting an AFPM motors with higher power density is an important parameter in applications especially electric vehicles. The main goal of this paper has been optimal design of double-Sided Axial Flux Slotted PM Motors with maximum power density.

The relations of design and dimensions of this configuration of motors were investigated and among optimization methods, a genetic algorithm was used for optimization.

This paper has designed a 1kW, 48 V TORUS slotted axial motor aimed to be integrated a drive system for EVs.

A flux density comparison between the various parts of the optimal motor and obtained from the FEM and sizing analysis at no load agree well.

References

[1] S.A. Gholamian, M. Ardebil, K. Abbaszadeh and S. Mahmodi charati; "Optimum Design of 1kW Axial Flux Permanent Magnet Slotted TORUS Motor", European Journal of Scientific Research, ISSN 1450-216X, Vol.21 No.3 (2008), pp.488-499.

[2] S.A. Gholamian, M. Ardebili and K. Abbaszadeh; " Analytic and FEM Evaluation of Power Density for Various Types of Double-Sided Axial Flux Slotted PM Motors", International Journal of Applied Engineering Research, ISSN 0973-4562, Vol.3, No.6 (2008), pp. 749-762.

[3] N. A. Rahim, Hew Wooi Ping and M Tadjuddin, "Design of Axial Flux Permanent Magnet Brushless DC Motor for Direct Drive of Electric Vehicle", PES2007, IEEE.

[4] Parag R. Upadhyay and K. R. Rajagopal; "FE Analysis and Computer-Aided Design of a Sandwiched Axial-Flux Permanent Magnet Brushless DC Motor", IEEE TRANSACTIONS ON MAGNETICS, VOL. 42, NO. 10, OCTOBER 2006

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

[6] 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.

[7] 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.

[8] Caricchi, F.; Capponi, F.G.; Crescimbini, F.; Solero, L.; "Experimental study on reducing cogging torque and core power loss in axial-flux permanent-magnet machines with slotted winding", Industry Applications Conference, 2002. 37th IAS Annual Meeting. Conference Record of the Volume 2, 13-18 Oct. 2002 Page(s):1295-1302 vol.2.

[9] 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.

[10] E. Spooner and B. J. Chalmers, "Toroidally-wound, slotless, axial flux, permanent magnet, brushless-DC motor," in Conf. Rec. ICEM-88, Pisa, Italy, 1988, pp. 81-86.

[11] C. C. Jensen, F. Profumo, and T. A. Lipo, "A low loss permanent magnet brushless DC motor utilizing tape wound amorphous iron," IEEE Trans. Ind. Applicat., vol. 28, pp. 646-651, May/June 1992.

[12] F. Caricchi, F. Crescimbini, E. Fedeli, and G. Noia, "Design and construction of a wheel directly coupled axial flux PM prototype for EV's," in Conf. Rec. IEEE-IAS '94, Denver, CO, 1994, vol. 1, pp. 254-261.

[13] Z. Zhang, F. Profumo, and A. Tenconi, "Axial flux interior PM synchronous motors for electric vehicle drives," J. Electromotion, vol. 1, no. 1, pp. 23-29, 1994.

[14] Federico Caricchi, Fabio Crescimbini, Eugenio Fedeli and Giuseppe Noia, " Design and Construction of a Wheel-Directly-Coupled Axial-Flux PM Motor Prototype for EVs", 1994 IEEE.

[15] Upadhyay, P.R.; Rajagopal, K.R.; Singh, B.P.; " Design of a compact winding for an axial-flux permanent-magnet brushless DC motor used in an electric twowheeler", Magnetics, IEEE Transactions on Volume 40, Issue 4, Part 2, July 2004 Page(s):2026-2028.

[16] Kartik Sitapati and R. Krishnan;"Performance Comparisons of Radial and Axial Field, Permanent-Magnet, Brushless Machines", IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 37, NO. 5, SEPTEMBER/OCTOBER 2001.

[17] Profumo, F.; Zheng Zhang; Tenconi, A.; "Axial flux machines drives: a new viable solution for electric cars", Industrial Electronics, IEEE Transactions on Volume 44, Issue 1, Feb. 1997 Page(s):39-45.

[18] Uler, G.F.; Mohammed, O.A.; Chang-Seop Koh: Design optimization of electrical machines using genetic algorithms.--Magnetics, IEEE Transactions on, Volume: 31 Issue: 3, 5-7 Jul 1994, Page(s): 2008-2011.

(1) S. Asghar Gholamian, (2) S. Hemmati, (3) Reza Nasiri (2) S. A. Saied and (4) S. Esmaeili Jafarabadi

(1) Electrical Engineering Department of Noushirvani University of Technology Babol, Iran.

(2) Electrical Engineering Department of K.N. Toosi University of Technology Tehran, Iran.

(3) Electrical Engineering Department of Tafresh Amir Kabir University Tehranl, Iran

(4) Department of Electrical Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Email: asghar_gholamian@ee.kntu.ac.ir

Table1: Nominal design parameters of motor

design parameters              value

Voltage                        48  V
Out put power                   1 Kw
Number of poles                    4
Number of phases                   3
number of slots                   15
Slot fill factor                 0.8
Pole arc ratio                  0.65
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 PM    1.1 T

Table 2: Optimal chromosomes

            [P.sub.den]    [N.sub.ph]   A      g
            W/[cm.sup.3]   Turn         A/m    mm

Chromosom   0.35           74           1593   1
e1                                      0
Chromosom   0.35           82           1799   1.06
e2                                      0
Chromosom   0.35           74           1541   1.1
e3                                      0
Chromosom   0.35           74           1516   1.09
e4                                      0
Chromosom   0.35           84           1745   1.01
e5                                      0

           [B.sub.g]   [lambda]
            T

Chromosom   0.5         0.5
e1          3
Chromosom   0.5         0.5
e2          3           2
Chromosom   0.4         0.4
e3          6           5
Chromosom   0.4         0.5
e4          4           2
Chromosom   0.5         0.5
e5          1           5

Table 3: Dimensions of optimal double-sides slotted AFPM motor

design parameters                     value

Voltage                                48 V
Out put power                          1 Kw
Number of poles                           4
Number of phases                          3
number of slots                          15
Power density                   W / cm 0.37
Outer diameter                     150.8 mm
inner diameter                        75 mm
PM length                              4 mm
axial length of rotor               14.9 mm
axial length of stator              15.5 mm
protrusion of the end winding      17   mm
Slot depth                            20 mm
Teeth width                          7.9 mm

Table 4: Parameters and dimensions of slotted double-sides AFPM motor

Air gap length                 1 mm
Slot depth                    17 mm
Pole-arc-ratio                 0.65
Axial length of stator core   15 mm
Axial length of rotor core    16 mm
Axial length of PM             5 mm
Outer diameter               150 mm
Inner diameter                77 mm

Table 5: Flux density comparison

             Stator       Air gap      Stator
             [B.sub.cr]   [B.sub.ag]   [B.sub.cs]

FEM          1.42 T       0.52 T       1.45 T
Sizing Eq.   1.5 T        0.5 T        1.5 T

Figure.3: Chromosome representation.

g   [N.sub.ph]   A   [B.sub.g]   [lambda]

Figure 4: one point crossover.

g   [N.sub.ph]   A   [B.sub.g]   [lambda]   Parent 1
g   [N.sub.ph]   A   [B.sub.g]   [lambda]   Parent 2

g   [N.sub.ph]   A   [B.sub.g]   [lambda]   Children 1
g   [N.sub.ph]   A   [B.sub.g]   [lambda]   Children 2