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Application of the Finite Element Method in Design and Analysis of Permanent-Magnet Motors ARASH KIYOUMARSI1, PAYMAN MOALLEM1, MOHAMMADREZA HASSANZADEH2 and MEHDI MOALLEM3 Department of Electronic Engineering, Faculty of Engineering, University of Isfahan, Isfahan 2 Faculty of Electrical Engineering, Abhar Islamic Azad University, Abhar, Ghazwin 3 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan IRAN 1

Abstract- In this research, the results of approximate analytical methods and Finite Element Method (FEM), those are used for prediction of airgap flux density distribution, are compared. In this comparison, permanentmagnet direct-current (PMDC) motors and brushless permanent magnet motors are considered. In addition, a coupled magnetic field, electrical circuit, and mechanical system program by which the FEM analysis is accomplished, is briefly discussed. Then, time stepping finite element method is used for the magnetic field analysis. At last, an example of shape design optimization, i.e., optimal shape design of an interior permanentmagnet (IPM) synchronous motor, is considered. Key-Words: The Finite Element Method, Brushless Permanent Magnet Motors, DC Motors.

1 Introduction Prior to the development of reliable high-power solid-state switching devices, the DC motor was the dominant electric machine for all variable-speed motor drive applications. The DC motor turns out to be the most economical choice in the automotive industries for cranking motors, wind shield wiper motors, blower motors and power window motors [1]. In this paper, first, the magnetic flux density of both a six-pole, 29-slot PMDC motor and a seriesexited four-pole, 21-slot field-winding DC motor are obtained based on an iterative analytical method. Then, the magnetic field is modeled based on a twodimensional field analysis method considering the effect of rotor slots [1-3]. The results of calculations are compared with finite element method results and predicted output characteristics of both motors are also compared with those obtained by real output measurements [4-7]. Brushless permanent magnet motors can be divided into the PM synchronous AC motor (PMSM) and PM brushless DC motor (PMBDCM). The former has sinusoidal airgap flux and back EMF, thus has to be supplied with sinusoidal current to produce constant torque. The PMBDCM has the trapezoidal back EMF, so the rectangular current waveform in its armature winding is required to obtain the low ripple torque. Generally, the magnets with parallel magnetization are used in the PMSM while the

magnets with radial magnetization are suitable for the BDCM [8-9]. Permanent-magnet synchronous motors (PMSM) have higher torque to weight ratio as compared to other AC motors. There are different rotor topologies that divide into two basic types, i.e., Exterior Permanent-Magnet motors (EPM) and Interior Permanent-Magnet motors (IPM). Surface–mounted permanent-magnet synchronous motor (SPMSM) and inset permanent-magnet motor, belong to former and buried or interior–type permanentmagnet synchronous motor (IPMSM) and flux concentration or spoke–type permanent-magnet synchronous motor, belong to the latter. Because of the mechanical and electromagnetic properties of each type, different topologies have different advantages and disadvantages used in high-speed applications [10, 11]. They have different control strategies and there is usually torques, speed, angular position and current-control loops in the control system. Interior permanent-magnet synchronous motor, has many advantages over other permanent-magnet synchronous motors. It has usually larger quadrature axis magnetizing reactance than direct axis magnetizing reactance. These unequal inductances in different axes, enable the motor to have both the properties of SPMSM and synchronous reluctance motor (SynchRel) [12]. The total resultant

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instantaneous torque of a brushless permanentmagnet motor has two components, a constant or useful average torque and a pulsating torque which causes torque ripple. There are three sources of torque pulsations. The first is field harmonic torque due to non-ideal distribution of flux density in the airgap, i.e., non-sinusoidal in the PMSM or nontrapezoidal in the PM brushless DC motor. The second is due to the cogging torque or detent torque caused by the slotted structure of the armature and the rotor permanent-magnet flux. The third is reluctance torque, produced due to unequal permeances of the d- and q-axis. This torque is produced by the self-inductance variations of phase windings when the magnetic circuits of direct- and quadrature-axis are unbalanced [13]. In IPM synchronous motor, the effective airgap length on the d-axis is large so the variation of the daxis magnetizing inductance, Lmd, due to magnetic saturation, is minimal. For the q-axis, there is an inverse condition, i.e., the effective airgap length on the q-axis is small and therefore the saturation effects are significant [14-16].

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Then, the magnetic field is modeled based on a twodimensional field analysis method considering the effect of rotor slots [1-3]. The results of calculations are compared with finite element method results and predicted output characteristics of both motors are also compared with those obtained by real output measurements.

2.1 Analytical Method Figs. 1-a and 1-b show the frame assembly and armature of two ideal motors. Fig. 1-c shows armature and field current densities applied to the FWDC motor in this study. Having determined the structure of the magnetic circuit (Fig. 2) and electric circuit of both motors, the node permeance matrix and the node magnetic flux source vector can be found and interactive calculation is used to solve the equation. The permeability of segment i, for the (k+1)th iteration, considering magnetic saturation, is then corrected and determined by the following expression [5]:

µ ik = µ ik −1 + λki {µ ik − µ ik −1 }

(1)

2 Characteristics of a PMDC and a FWDC Starter Motor The influence of magnetic saturation on electromagnetic field distribution in both permanentmagnet direct-current (PMDC) and field-winding (wound-field) direct-current (FWDC) motors with the same output mechanical power, have been studied. An approximate analytical method and FEM are used for prediction of airgap flux density distribution. No-load and rotor-lucked conditions, according to experimental measurements, and the FEM and analytical method studies of the motor, have been studied. A sensitivity analysis has also been done on the major design parameters that affect motor performance. At last, these two DC motors are compared, in spite of their differences, on the basis of measured output characteristics. Boules developed a two-dimensional field analysis technique by which the magnet and armature fields of a surface–mounted brushless synchronous machine can be predicted [1,2]. In a comprehensive proposed model, Zhu et al. presented an analytical solution for predicting the resultant instantaneous magnetic field in the radially-magnetized BDCM and PMDC under any load conditions and commutation strategies [3]. In this research, first, the magnetic flux density of both a six-pole, 29-slot PMDC motor and a series-exited four-pole, 21-slot FWDC motor are obtained based on an iterative analytical method.

Fig. 1-a

Fig. 1-b

Fig. 1-c Fig. 1. Frame of ideal DC motor and applied current densities to the prototype motor Fig. 3 shows results of this method for FWDC motor magnetization curve. Results of design sensitivity analysis are also evident on this figure for changing the number of turn of armature windings [1-3].

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The no-load flux distribution is also shown in Fig. 7 in details with magnifications. The direction and magnitude of the flux density distribution vector is also included in the Fig. 7.

Fig. 2. Magnetic equivalent circuit of the motors (b) (a) Fig. 7. (a) Flux lines, and (b) colored vector plot of vector field: no-load conditions; FWDC motor

Fig. 3. Flux per pole of series-exited DC motor vs. line current

2.2 Finite Element Method

Figs. 8 and 9 show the flux distribution and radial and tangential components of flux density curve at no-load without and with the magnetic holder, for the PMDC motor. Finally, Fig. 10 shows the flux distribution at full-load with the magnetic holder [6,7].

Figs. 4 and 5 show the flux distribution and radial and tangential components of flux density curves at no-load and rated load, respectively.

(a) (b) Fig. 8. (a), Frame of the prototype PMDC motor and rotor windings, (b), Equipotential lines for magnetic vector potential: no-load conditions; PMDC motor

Fig. 4. Flux lines: no-load conditions; FWDC motor

Fig. 5. Flux lines: rated-load conditions; FWDC motor Fig. 6 shows the equi-potential lines for magnetic vector potential and their color map by considering the effect of screw and bolt on the stator frame, for the FWDC motor.

(a) (b) Fig. 6. (a) Flux lines and (b) colored contour plot, effect of stator frame screw: no-load conditions; FWDC motor

Fig. 9. (a) Equipotential lines for magnetic vector potential: no-load conditions, PMDC motor, magnetic holder considered.

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solutions and also winding distribution in the stator slots. Table 1. Comparison of FEM and analytical method for the FWDC motor FEM Analytical Analysis Method No-load

ω m [RPM]

5828.84

5263

Te [Nm]

0.033

0.030

96.57e-5

100.2e-5

0.7

0.7025

1.5

1.4

ω m [RPM]

1985.5

1966.1

Te [Nm]

0.3368

0.33065

105.84e-5

103.88e-5

1.064

1.0421

1.622

1.5

ω m [RPM]

1747.9

1768

Te [Nm]

0.533

0.529396

111.648e-5

110.88e-5

1.13

1.1722

1.727

1.6212

ω m [RPM] Te [Nm]

0

0

1.2644

1.28

3 Brushless Permanent Magnet Motors

Φ P [Wb]

132.432e-5

35.0e-5

In a comprehensive proposed model, Zhu et al. presented an analytical solution for predicting the resultant instantaneous magnetic field in the radiallymagnetized BDCM and PMDC under any load condition and commutation strategy [3]. Recently, Zhu et al.[17] extended Rasmussen’s model[9] to predict magnetic field due to the armature reaction both in the three phase overlapping and non-overlapping stator windings. Finally, open–circuit field distribution and load condition field distribution can be expressed as relative permeance functions including the field

Bmid − aigap

1.596

1.733

2.08

2.12

Fig. 10. (a) Equipotential lines for magnetic vector potential: full-load conditions, PMDC motor, magnetic holder considered.

Φ P [Wb]

Bmid − aigap

Table 1 includes the results of comparison of the FEM analysis of the FWDC motor and the twodimensional field distribution analytical method. In this analysis, the average value of the rotor angular speed, rotor output shaft torque, flux per pole, midairgap flux density distribution waveform and yoke flux density distribution waveform are considered. Magnetic holders are devised to prevent the demagnetization of the PMs during the influence of a strong armature reaction field on the stator field. The FEM results have completely validated the operation of the holders during different load conditions. These two prototype direct-current motors, i.e., a PMDC and a FWDC are compared from the point of view of output characteristics. Results of an approximate analytical method, a previouslydeveloped two-dimensional field analysis technique and finite element method are compared for prediction of airgap flux density distribution and possible replacement of the FWDC motor with PMDC motor is briefly studied. The results of a few experimental measurements are also involved in this study and the results of all methods are also compared.

[T]

BYoke [T] I a = 100 A

Φ P [Wb]

Bmid − aigap

[T]

BYoke [T] I a = 150 A

Φ P [Wb]

Bmid − aigap

[T]

BYoke [T] I a = 300 A

BYoke [T]

[T]

Cross section of the 5 HP, 1500 rpm, 4-pole surface-mounted brushless permanent magnet motor [10], is shown in Figure 11. Figure 12 shows comparison of flux lines in radial magnetization configuration. Figure 13 shows comparison of analytical and numerical results. The results shows that the flux density curves of that new improved model follows the FE curves more closely, especially at the corners of the stator slots. As a result, the 4

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analytical model is a powerful tool for torque pulsation calculations, using magnetic flux density distribution in the airgap. Also average torque can be obtained using useful flux per pole and unit length of stator. The results show that the flux density curves of improved model follow the FE curves more closely, especially at the corners of the stator slots.

Fig. 11. Cross section of the machine

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the Magnetic Equivalent Circuit(MEC) model of machine, Fig. 14, which takes into account the non– linear characteristics of the iron in the machine and the FEM method is also accomplished and carried out.

Fig. 14. Magnetic Equivalent Circuit model (Non-Linear elements are in black)

Fig. 12. Equipotential lines: flux distribution (opencircuit condition) at time t=0 (Radial Magnetization)

The fundamental component of flux density distribution is obtained using MEC analysis (B1:MEC), the two-dimensional Cartesian-based coordinates method(B3:2DR)[8], the twodimensional polar-based coordinates method with stator slot effects (B4:2DS) [9], and FEM analysis (B5:FEM), are compared in Table 2. The figures show the peak of fundamental component of flux density distribution considering both the parallel and radial magnetization. The good agreement of the analytical method with FE results has proved validity of this method for fast calculation of back-EMF and torque ripples. Table 2. Comparison of different field analysis methods Parallel B1(MEC) -------B3(2DP) 0.8262 B4(2DS) 0.8138 B5(FEM) 0.7818 The flux densities are in Tesla.

Fig. 13. Flux magnetization

density

distribution:

radial

A comparison between lumped–parameter and distributed-parameter flux calculations , i.e., using

Radial 0.8200 0.8385 0.8260 0.7953

4 Interior Permanent Synchronous Motor

Magnet

Kim et al. [18,19], in a Recent comprehensive proposed method, have presented a shape design optimization method to reduce cogging torque of an IPM synchronous motor using the continuum

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sensitivity analysis combined with FEM. They used the finite element nodes on the rotor outer surface as control points and used the B-spline curves to relate design variables vector and control points vector to each other. There is only a slight difference between initial shape and final shape of rotor outer surface. They also presented an optimal shape design method for reducing the higher back-EMF harmonics generated in IPM synchronous motor; however, they did not evaluate the optimal design at different load conditions. The proposed optimal design method in this part is industrially applicable to the rotor and motor drive operation is considered too. The optimal shape is evaluated at different load conditions which shows good improvement in all loading conditions. Electrical circuits’ equations of different stator windings and rotor mechanical motion equations are coupled to magnetic field equations, to obtain a comprehensive model for the IPM motor drive. The optimal shape design is obtained based on addition of three circled-type holes that are drilled in the rotor iron. Motor-drive operation is also discussed.

4.1 Steady-State Operating Curves of IPM Synchronous Motors The main torque control strategies for the speeds lower than base speed operation are zero d-axis current, maximum torque per unit current (MTPC), maximum efficiency, unity power factor and constant mutual flux linkage. The main control strategies for the speeds higher than base speed operation are constant back-EMF and six-step voltage. The MTPC control strategy provides maximum torque for a given current. This minimizes copper losses for a given torque; however, it does not optimize the total losses. Maximum torque per current curve is shown by curve 6 in Fig.15. The operating point, A, shown in this figure is for below base speed. The path ABCD, shows the above speed operation. Shape design optimization of the motor-drive has been done in these five operating points, considering field weakening on path ABCD [15]. These are shown in Fig. 15 by curves 8 to 10. According to motor-drive limits given by these curves, two basic operating conditions can be identified: infinite-speed operation and finite-speed operation. Below and above the rated speed, there are three control modes, current limited, current-voltage and voltage limited regions. The motor-drive here is finite maximum speed drive because the infinite speed operating point lies outside the current-limited circle.

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Fig. 15. Operating limits for prototype IPM synchronous motor: family of maximum torque-perampere curves, constant torque curves, rated stator current curve and voltage-limit ellipses

4.2 Modeling of Motor-Drive System 4.2.1 Magnetic Field Model Rotor mesh and stator mesh are coupled together at a slip interface to allow for rotation which is a cylindrical slip surface in 3D and a circular slip path in two-dimension in the middle of the air gap. So, there is a weak boundary condition enforced on this interface. Using Lagrange multiplier and Kth rotor variable, Ark as magnetic vector potential at node k on rotor, rotor can be coupled to the corresponding stator variables Asi, as: 4

Ark = ∑ Asi N si (k )

(1)

i =1

The local virtual work method is used for torque calculation in FEM method. In the rotor PM, the magnetic field equation can be expressed as:

∇ ×ν∇ × A = ν∇ × M

(2) In the stator conductors, the magnetic field equation is given by:

∇ ×ν∇ × A = − J s + σ

∂A ∂t

(3)

And in the rotor and stator iron regions is expressed as:

∇ × ν∇ × A = σ

∂A ∂t

(4)

4.2.2 The Electrical Circuit Model [20-22] The voltage equations of the phase stator windings are given by:

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[Vab ] = [

[Vbc ] = [

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d (λ a − λb )] + [ Ra ]I a − [ Rb ]I b dt

(5)

d (λb − λ c )] + [ Rb ]I b − [ Rc ]I c dt

(6)

4.2.3 Mechanical System Model Two equations of the rotation of rotor are given as follows,

J eq (

dω r ) + Bω r = Te − TL dt

(7)

dδ = ωe − ωr dt

(8)

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results (contour plots of the resultant absolute value of flux density distribution) of the time stepping FEM for this IPM synchronous motor.

4.3 Shape Optimization Method To reduce the pulsation torque, which consists of cogging torque and torque ripples, it is necessary to redistribute the flux in rotor. For flux pattern optimization, it needs to change the iron and air combination in a practical approach. In this research, small holes have been drilled in the flux path at rotor surface (Fig. 18). The place and radius of these holes are found to minimize the torque pulsations [22-40].

The above-mentioned equations are coupled in the two-dimensional circuit-field-torque coupled time stepping finite element method. This model is used to optimize the shape of the rotor of motor as will be discussed in the following section. This analysis is done by the algorithm shown in Fig.16. Start Calculation of static magnetic field and initial mechanical angle of the rotor position Coupling of magnetic field equations and electric circuit equations

Fig. 18. The model used to optimize the IPM synchronous motor

Calculation of electromagnetic field

The vector of design variables is indicated by:

Calculation of flux linkages, three phase Currents and instantaneous torque t = t + ∆t

Solving the two motion equations, i.e., rotation of rotor equations(speed and torque angle equations) The rotation of rotor: Moving mesh, modification of element coordinate systems on each permanent magnet

X = [ ρ1 ρ 2 ρ 3 r1 r2 r3 θ1 θ 2 θ 3 ]T

(9)

Χ = [ X 1 X 2 X 3 ... X n ]T .

(10)

Where parameters ρ i , ri , θ i are shown in Fig.18. The design variables are subject to constraint with upper and lower limits, that is:

X i ≤ Xi ≤ X i i = 1,2,3,..., n, n = 9

Write related new constraint equations according to new node positions corresponding to mid airgap length

(11)

g i ( Χ) ≤ g i i = 1,2,..., m1 No

t > tmax

hi ≤ hi ( Χ) i = 1,2,..., m2

Yes

wi ≤ wi ( Χ) ≤ wi i = 1,2,..., m3

Stop

Fig. 16. Transient Finite Element Method At last, convergence of the procedure is checked by the velocity, magnetic vector potential and backEMF errors. Fig. 17 (at the end of chapter) shows the

(12)

A first order optimization method is used and this constrained problem is translated into an unconstrained one using penalty functions. Each iteration is composed of sub-iterations that include search direction and gradient (i.e., derivatives)

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computations. The unconstrained objective function is formulated as follows: ⎛ f ⎞ ⎛ n ⎞ Q ( Χ, q ) = ⎜⎜ ⎟⎟ + ⎜ ∑ ΡX ( X i ) ⎟ + ⎠ ⎝ f 0 ⎠ ⎝ i =1 (13) m1 m2 m3 ⎛ ⎞ q * ⎜ ∑ Ρg ( g i ) + ∑ Ρh (hi ) + ∑ Ρw ( wi ) ⎟ i =1 i =1 ⎝ i =1 ⎠ And for each optimization iteration ( j ) , a search direction vector, d , is calculated. The next iteration ( j + 1) is obtained from the following equation [18,19]: Χ ( j +1) = Χ ( j ) + s j d ( j ) (14)

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Fig. 19. Diagram for applying desired currents of the IPM synchronous motor [12]

S j is the line search step size and d ( j ) is given by:

d ( j ) = −∇Q ( Χ ( j ) , qk ) + rj −1d ( j −1) Where, rj −1 =

[∇Q(Χ

( j)

(15)

]

T

, qk ) − ∇Q( Χ ( j −1) , qk ) ∇Q( Χ ( j ) , qk ) ∇Q( Χ ( j −1) , qk )

2

.

(16)

In this paper, the objective function is defined as:

Ψ=

Tmax − Tmin Tavg

(17) Fig. 20. Cross section of the IPM synchronous motor based on different phase mmf axes

4.4 Motor-Drive Control Strategy The d-q components of the reference currents that were calculated according to the method described in section 4-1, are used for estimation of phase currents. So the reference currents, i *A (t ) , i B* (t ) and iC* (t ) will be applied to the FEM model, according to Figs. 19,20. For example, i *A (t ) can be estimated as:

i *A (t ) =

(

2 * − id cos(θ ) + iq* sin(θ ) 3

)

(18)

Fig.21 shows equi-potential lines for the magnetic vector potential solutions at a time obtained by time stepped FEM. Fig. 22 presents different wave forms of motor torque, corresponding to different rotor shapes, for the operational point A. Fig. 23 shows the spectrum of curves shown in Fig. 22. In this Fig., curve 1 is the motor torque and the rotor has no holes, curve 2 presents the output torque of motor when three equal-area circles are created on the rotor and curve 3 shows the rotor torque when three optimized circles are drilled into the rotor.

Fig. 21. Equipotential lines: flux distribution (full load condition) at time t=0.556 S, after creating optimal circular holes It can be seen that there is a valuable improvement both in performance index, torque pulsations and saliency ratio. This new shape of the rotor with optimized holes, has the advantage of increasing the maximum speed for the field weakening region from almost 1.83 p.u. to 1.96 p.u. above the base speed. Optimal shape design of rotor has large effect on reduction of torque pulsations of IPM synchronous motor.

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Fig. 22. Comparison of electromagnetic torque calculated by FEM for different rotor structures in point A, curve 1: no holes on the rotor, curve 2: same holes on the rotor and curve 3: optimized holes on the rotor

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In this part, the shape design optimization is carried out by drilling internal circular holes of optimal radius in the flux path at rotor surface. The torque curves of the optimized motor show lower pulsating torque and higher average torque. Another advantage is that the field weakening region has been extended for optimized motor. Although the shape optimization is done at nominal operation point, the performance evaluation of optimized motor at other operation conditions shows improvement too. The new shapes are easily applicable in the factory by drilling the holes of different radius at predetermined positions.

5 Interior Permanent-Magnet Synchronous-Induction Motor In this part, a synchronous-induction motor has been considered and a rotor cage for self starting of the IPM synchronous motor is included in the rotor. Fig. 24 shows this new topology. Time stepping FEM is used to simulate this new machine. Fig. 25 shows the equipotential lines for this motor. Figs. 26 and 27 show the results obtained for the motor at startup from standing using a three-phase 50Hz voltage source under 15 N.m. load torque obtained by time stepped finite element method and d-q model respectively. In these waveforms it is evident that the results of two dimensional filed modeling, i.e., time stepping FEM has contained the rotor and stator slots and rotor saliency affected by permanent magnet shapes. Using a cage on the rotor of a permanent magnet motor has this advantage that the motor can be started directly as an induction motor and it needs not to have a frequency control process for starting [23-24].

Fig. 23. Spectrum of three curves shown in Fig.22, curve 1: no holes on the rotor, curve 2: same holes on the rotor and curve 3: optimized holes on the rotor

Fig. 24. Interior permanent-magnet synchronousinduction motor

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At last, a new topology, i.e., Interior PermanentMagnet Synchronous-Induction Motor has also been completely studied. Among these different rotor topologies for these PM machines, it can be seen that the last one is the best from the point of view of efficiency and pulsating torque.

Acknowledgment Fig. 25. Equipotential lines for magnetic vector potential: full-load conditions, interior permanentmagnet synchronous induction motor

The authors would like to really appreciate decent considerations of all people who engaged on different parts of this work. Unless they had accompanied and had helped us, the work could not have been finished and finalized. At last, we would say that the work presents the results of almost 8 years of interpretations and main conclusions of different research and educational projects in the field of FEM analysis of PM machines. References N. Boules, "Prediction of no-load flux density distribution in permanent magnet machines", IEEE Transactions on Industry Applications, Vol. IA-21, pp. 633-643, 1985. [2] N. Boules, "Two-dimensional field analysis of cylindrical machines with permanent magnet excitation", IEEE Transactions on Industry Applications, Vol.IA-20, pp. 1267-1277, September-October 1984. [3] Z. Q. Zhu, D. Howe, "Instantaneous magnetic field distribution in brushless permanent magnet DC motors, part III: Effect of stator slotting", IEEE Transactions on Magnetics, Vol.29, No.1, pp. 143-151, January 1993. [4] J. J. Cathey, Electric machines, analysis and design, applying MATLAB, chapter 5, MCGRAW-HILL, 2001. [5] M. Cheng, et al., "Nonlinear varying-network magnetic circuit analysis for doubly salient permanent-magnet motors", IEEE Transactions on Magnetics, Vol. 36, No. 1, pp. 339-348, January 2000. [6] US Steel Manual, Non-Oriented Electrical Steel Sheets, Pittsburgh, Pa. 15230. [7] ANSYS 5.6 Theory and APDL Reference Manual, 2003. [8] W. Cai, D. Fulton and K. Reichert, "Design of permanent magnet motors with low torque ripples: a review", ICEM 2000, ESPO, Finland, pp. 1384-1388, 2000. [9] K. F. Rasmussen, et al., "Analysis and numerical computation of air-gap magnetic fields in brushless motors with surface permanent magnets", IEEE Transactions on Industry Applications, Vol. 36, No.6, pp. 15471550, November-December 2000.

[1]

Fig. 26. Electromagnetic torque obtained by time stepping finite element method

Fig. 27. Electromagnetic torque obtained by equivalent d- and q-axes model of this synchrousinduction machine.

6 Conclusion In this research, different DC and AC permanentmagnet motors have been analyzed via both analytical and numerical methods. The time stepping Finite Element Method (FEM) has been used as a numerical method. The PM motors that are considered in this research include PMDC motor, brushless AC and DC permanent magnet motors and interior type permanent magnet synchronous motor.

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[10] T. L. Skvarenina, The Power Electronics Handbook, Purdue University, West Lafayette, Indiana, CRC Press, 2002. [11] Stephan Meier, Theoretical design of surfacemounted permanent-magnet motors with fieldweakening capability, Master Thesis, Royal Institute of Technology, Stockholm, 2002. [12] T. M. Jahns and W. L. Soong, "Interior permanent-magnet synchronous motors for adjustable-speed drives", IEEE Transactions on Industry Applications, Vol. IA-22, No. 4, pp.738-747, July-August, 1986. [13] I. Boldea, Reluctance Synchronous Machines & Drives, Oxford University Press, 1996. [14] C. Mademlis and N. Margaris, "Loss minimization in vector-controlled interior permanent-magnet synchronous motor drives", IEEE Transactions on Industrial Electronics, Vol. 49, No. 6, pp.1344-1347, December 2002. [15] W.L. Soong and T.J.E. Miller, "Fieldweakening performance of brushless synchronous AC motor drives", IEE Proceeding of Electrical Power Application, Vol. 141, No.6, pp.331-340, November 1994. [16] Stephan Meier, Theoretical design of surfacemounted permanent magnet motors with fieldweakening capability, a paper, Royal Institute of Technology, Department of Electrical Engineering, Electrical Machines and Power Electronics, Stockholm, 2000. [17] Z. Q. Zhu, D. Howe and C.C. Chan, "Improved analytical model for predicting the magnetic field distribution in brushless permanent– magnet machines", IEEE Transactions on Magnetics, Vol. 38, No.8, pp. 1500-1506, July 2002. [18] D. H. Kim, et al., "Optimal shape design of iron core to reduce cogging torque of IPM motor", IEEE Transactions on Magnetics, Vol.39, No.39, pp.1456-1459, May 2003. [19] J. H. Lee, et al., "Minimization of higher backEMF harmonics in permanent magnet motor using shape design sensitivity with B-spline parameterization", IEEE Transactions on Magnetics, Vol.39, No.3, pp.1269-1272, May 2003. [20] Y. Wang , K. T. Chau, C.C. Chan, and J. Z. Jiang, "Transient analysis of a new outer-rotor permanent-magnet brushless DC drive using circuit-field-torque coupled time-stepping finite-element method", IEEE Transactions on Magnetics, Vol. 38, No. 2, pp.1297-1300, March 2002. [21] A.B.J. Reece and T.W. Preston, Finite Element Methods in Electrical Power Engineering,

[22] [23] [24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

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Oxford University Press, chapters 11, 12 and 15-18, 2000. US Steel Manual, Non-Oriented Electrical Steel Sheets, Pittsburgh, Pa. 15230. P. C. Krause, Analysis of electric machinery, McGraw Hill, 1986. E. Nipp, "Parameter decoupling in permanent magnet motor including space harmonics and saturation", nternational Conference on Electrical Machines (ICEM 2004) proceeding, pp.69-71, 2005. A. Kioumarsi, M. Moallem, and B. Fahimi, "Mitigation of torque ripple in interior permanent magnet motors via optimal shape design", IEEE Transactions on Magnetics, Vol. 42, No. 11, pp.3706-3711, November 2006. A. Kiyoumarsi, M. Hassanzadeh and M. Moallem, "A new analytical method on field calculation for interior permanent-magnet synchronous motors", International Journal of Scientia Iranica, Vol. 13, No. 4, pp. 364-372, October 2006. B. Mirzaeian-Dehkordi, A. Kiyoumarsi, P. Moallem and M. Moallem, "Optimal design of switching-circuit parameters for switched reluctance motor drive based on genetic algorithms", Electromotion, Vol. 13, No. 3, pp.213-220, July - September 2006. A. Kiyoumarsi, M. Moallem and M. Hassanzadeh, "An accurate method for calculation of magnetizing inductances in interior permanent-magnet synchronous motors", Electromotion, Number 1, pp.3-8, January-March 2005. M. Moallem and A. Kiyoumarsi, "The new definitions for the power terms in distorted and unbalanced conditions and calculation of these terms for an electric arc furnace", Esteghlal Engineering Magazine (International and National Journal), pp.30-38, May 2005. M. Moallem, A. Kiyoumarsi and M. R. Hassanzadeh ," A novel technique on the analytical calculation of open-circuit flux density distribution in brushless permanentmagnet motor", International Journal of Engineering, pp. 51-58, IRAN, April 2004. M. Moallem, A. Kiyoumarsi and M. Hassanzadeh, "New methods on field calculation of brushless PM motors", presented in the 12th Symposium of Power Electronics– Ee 2003, Appeared in CD Conference Proceedings, Paper No. T3-1.2, Novisad, Serbia & Montenegro, November 5-7, 2003. A. Kiyoumarsi, M. Moallem and M. Hassanzadeh, "The Torque ripples of brushless

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[34]

[35]

[36]

[37]

[38]

[39]

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permanent-magnet motors", presented in the 12th Symposium of Power Electronics–Ee 2003, Appeared in CD Conference Proceedings, Paper No. T3-1.3, Novasid, Serbia & Montenegro, November 5-7, 2003. A. Kiyoumarsi, M. Hassanzadeh and M. Moallem, "Eccentric magnetic field analysis of electric machines", presented in ACEMP’2004, International Aegean Conference on Electrical Machines and Power Electronics, pp. 172-177, 26-28, Istanbul, Turkey, May 2004. A. Kiyoumarsi, M. Moallem, J. Soltani and M. Hassanzadeh, "Calculation of magnetizing inductances in interior permanent-magnet synchronous motors", presented in OPTIM‘04 International Conference on Electrical and Electronic Equipment, pp. 145-148, Poiana Brasov, Romania, May 20-23, 2004. A. Kiyoumarsi and M. Moallem, "New analytical methods on field calculation for interior permanent-magnet synchronous motors", presented in ICEM 2004, Appeared in CD Conference Proceedings, Paper No. 323, International Conference on Electrical Machines, Poland, September 2004. M. Hassanzadeh, A. Kiyoumarsi and M. Moallem, "Accurate methods for calculation of magnetizing inductances in interior permanentmagnet synchronous motors," presented in ICEM 2004, Appeared in CD Conference Proceedings, Paper No. 768, International Conference on Electrical Machines, Poland, September 2004. A. Kiyoumarsi and M. Moallem, "Optimal shape design of an interior permanent-magnet synchronous motor", Presented and published in International Electric Machines and Drives Conference, San Antonio, TX, May 15-18, 2005. B. Mirzaien-Dehkordi, A. Kiyoumarsi and M. Moallem, "Comparison of Output Characteristics of a Permanent-Magnet DC and a Field-winding DC Starter Motor", Presented in Electromotion 2005, 6th International Symposium on Advanced Electro-Mechanical Motion System, Appeared in CD Conference Proceedings, Paper No. OS1/5, , Lausanne, Switzerland, September 2729, 2005. A. Kiyoumarsi and B. Mirzaien-Dehkordi, "Rotor eccentricity of third kind in a rotating electric machine", Presented in Electromotion 2005, 6th International Symposium on Advanced Electro Mechanical Motion System, Appeared in CD Conference Proceedings,

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Paper No. DS1/9, Lausanne, Switzerland, September 27-29, 2005. [40] A. Kiyoumarsi and M. Hassanzadeh, "Startup and steady-state performance of interiorpermanent magnet synchronous-induction motors", 8th International Conference on Electrical Machines and Systems 2005 (ICEMS’2005), Appeared in CD Conference Proceedings, pp. 200-202, Nanjing, P. R. China, September 27-29, 2005.

LIST OF SYMBOLS Bocb

Flux density due to magnet at stator inside surface Open-circuit flux density; Boules’ method

Boco

Open-circuit flux density; Only assuming λ0

Bocn Bsum b0 P Qs

Open-circuit flux density; New method Radial component of flux density; obtained by FEM analysis Absolute resultant open-circuit flux Stator slot-opening Number of poles Stator slot number

rg

Mid-airgap radius

wS

Equivalent stator slot-opening

wT

Stator tooth width

α

Angular displacement between the stator mmf and rotor mmf Magnet arc angle

Bmagnet

Br

αm αp

Pole-shoe arc angle

α ′p

Pole arc angle

α sa λ

Angular displacement between the stator slot axis and the axis of the coils of phase A Relative permeance function

Λ ref

Reference permeance

~

12

~

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APPENDIX I

APPENDIX II

PM and field-winding DC motor parameters

Brushless PM motor parameters Stator outer radius (rso) Stator yoke depth (hy) Stator tooth depth (ht) Stator slot opening (bo) Stator Lamination material Rotor Lamination material Magnet arc angle ( 2α m )

97.1 mm 17.4 mm 17.2 mm 5.1 mm M36,26 Gage M19,26 Gage

Magnet material Remanence (Bres) Relative recoil permeability (µr)

Nd-Fe-B, N33 1.1 Tesla 1.05

PMDC Motor Data Stator

Dsi = 59.0mm , Dso = 79.0mm ,

hm = 7mm , α m = 35.19°mech , Br = 0.45T , H c = 300 KA m , P=6 Rotor

Lr = 60mm , Dri = 38.6mm ,

Dro = 58.5mm , ω slot = 2.7mm , Dshaft = 16.6mm , N a = 2.turns

o

60 mechanical

APPENDIX III IPM synchronous motor parameters

Field-Winding DC Motor Data: Stator

h p = 8.6mm , Dso = 89.5mm , Dsi = 78.2mm

α p = 68°mech , α ′p = 37°mech , N f = 8 .5,

P=4, Acond . = 4.5 * 1. mm 2

Stator outer radius (rso) Stator yoke depth (hy) Stator tooth depth (ht) Stator slot opening (bo) Magnet arc angle ( 2α m ) Stator lamination material Rotor lamination material Permanent-magnet material Nd-Fe-B, Remanence (Bres)

97.1 mm 17.4 mm 17.2 mm 5.1 mm 2×37 mechanical degrees M36, 26 Gage M19, 26 Gage N33 1.1 T

Rotor

Lr = 48mm , Dri = 60.5mm , Dro = 39.0mm

Dshaft = 17.45mm , ω slot = 2.7mm ,

ω brush = 5 mm , lbrush = 10. mm , N a = 2turns , Dcond . = 2.1mm Stator lamination material Rotor lamination material

M36,26 Gage M19,26 Gage

APPEND IV Back EMF calculations by FEM The instantaneous flux Φ trough a coil of the winding is given by integrating the flux density over the coil area, which is bounded by the contour C. Φ(t ) = ∫ B(r ,θ , t ) ⋅ ds = ∫ (∇ × A) ⋅ ds = ∫ A ⋅ dl (App.IV-1) S

S

C

Furthermore the instantaneous flux linkage of the winding is found as:

λ wind (t ) =

P 1 ∗ ∗ N s ∗ Φ (t ) 2 c

(App.IV-2)

And the instantaneous value of the induced noload voltage is given by:

e(t ) =

13

d λ wind (t ) dt

(App.IV-3)

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t=1.0 ms

t=2.0 ms

t=3.0 ms

t=4.0 ms

t=5.0 ms

t=6.0 ms

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t=7.0 ms

t=8.0 ms

t=9.0 ms

t=10.0 ms

t=11.0 ms

t=12.0 ms

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Fig. 17. Magnetic vector potential produced by means of time stepped finite element method

15