application of the finite element method in design and … · motor magnetization curve. results of...

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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

1Department of Electronic Engineering, Faculty of Engineering, University of Isfahan, Isfahan 2Faculty of Electrical Engineering, Abhar Islamic Azad University, Abhar, Ghazwin

3Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan IRAN

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, permanent-magnet 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 permanent-magnet (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 series-exited 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 two-dimensional 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 permanent-magnet 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

FINITE ELEMENTS published by WSEAS Press 138

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instantaneous torque of a brushless permanent-magnet 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 non-trapezoidal 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 d-axis 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].

2 Characteristics of a PMDC and a FWDC Starter Motor

The influence of magnetic saturation on electromagnetic field distribution in both permanent-magnet 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.

Then, the magnetic field is modeled based on a two-dimensional 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]:

{ }11 −− −+= k

iki

ki

ki

ki µµλµµ

(1)

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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Fig. 2. Magnetic equivalent circuit of the motors

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

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

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

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.

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

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].

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


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

Table 1 includes the results of comparison of the

FEM analysis of the FWDC motor and the two-dimensional field distribution analytical method. In this analysis, the average value of the rotor angular speed, rotor output shaft torque, flux per pole, mid-airgap 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 previously-developed 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. 3 Brushless Permanent Magnet Motors 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 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

solutions and also winding distribution in the stator slots. Table 1. Comparison of FEM and analytical method for the FWDC motor

FEM Analysis

Analytical Method

No-load mω [RPM] 5828.84 5263

eT [Nm] 0.033 0.030

PΦ [Wb] 96.57e-5 100.2e-5

aigapmidB − [T] 0.7 0.7025

YokeB [T] 1.5 1.4

AI a 100=

mω [RPM] 1985.5 1966.1

eT [Nm] 0.3368 0.33065

PΦ [Wb] 105.84e-5 103.88e-5

aigapmidB − [T] 1.064 1.0421

YokeB [T] 1.622 1.5

AI a 150=

mω [RPM] 1747.9 1768

eT [Nm] 0.533 0.529396

PΦ [Wb] 111.648e-5 110.88e-5

aigapmidB − [T] 1.13 1.1722

YokeB [T] 1.727 1.6212

AI a 300=

mω [RPM] 0 0

eT [Nm] 1.2644 1.28

PΦ [Wb] 132.432e-5 35.0e-5

aigapmidB − [T] 1.596 1.733

YokeB [T] 2.08 2.12 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


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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

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

Fig. 13. Flux density distribution: radial magnetization

A comparison between lumped–parameter and

distributed-parameter flux calculations , i.e., using

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) 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 two-dimensional 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 Radial

B1(MEC) -------- 0.8200 B3(2DP) 0.8262 0.8385 B4(2DS) 0.8138 0.8260 B5(FEM) 0.7818 0.7953

The flux densities are in Tesla.

4 Interior Permanent Magnet Synchronous Motor 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.

Fig. 15. Operating limits for prototype IPM synchronous motor: family of maximum torque-per-ampere 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

1

)(i

sisirk kNAA

(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:

MA ×∇=×∇×∇ νν (2)

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

tAJA s ∂∂

+−=×∇×∇ σν

(3)

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

tAA∂∂

=×∇×∇ σν

(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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bbaabaab IRIRdtdV ][][)]([][ −+−= λλ

(5)

ccbbcbbc IRIRdtdV ][][)]([][ −+−= λλ

(6)

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

Lerr

eq TTBdt

dJ −=+ ωω )(

(7)

redtd ωωδ

−=

(8)

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

Calculation of electromagnetic field

Calculation of flux linkages, three phase Currents and instantaneous torque

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

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

airgap length

t > tmax

Stop

ttt ∆+=

Yes

No

Fig. 16. Transient Finite Element Method At last, convergence of the procedure is checked

by the velocity, magnetic vector potential and back-EMF errors. Fig. 17 (at the end of chapter) shows the

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].

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

The vector of design variables is indicated by:

TX ] r r r [ 3213213 21 θθθρρρ= (9) T

nXXXX ]...[ 321=Χ . (10)

Where parameters iii r θρ ,, are shown in Fig.18. The design variables are subject to constraint with upper and lower limits, that is:

iii XXX ≤≤ 9,,...,3,2,1 == nni

(11)

1,...,2,1)( migg ii =≤Χ

2,...,2,1)( mihh ii =Χ≤

3,...,2,1)( miwww iii =≤Χ≤

(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:

⎟⎠

⎞⎜⎝

⎛Ρ+Ρ+Ρ

+⎟⎠

⎞⎜⎝

⎛Ρ+⎟⎟

⎠

⎞⎜⎜⎝

⎛=Χ

∑∑∑

∑

===

=

3

1

2

1

1

1

10

)()()(*

)(),(

m

iiw

m

iih

m

iig

n

iiX

whgq

XffqQ

(13)

And for each optimization iteration )( j , a search direction vector, d , is calculated. The next iteration

)1( +j is obtained from the following equation [18,19]:

)()()1( jj

jj ds+Χ=Χ +

(14)

jS is the line search step size and )( jd is given by:

)1(1

)()( ),( −−+Χ−∇= j

jkjj drqQd (15)

Where, [ ] .

),(

),(),(),(2)1(

)()1()(

1

kj

kjT

kj

kj

jqQ

qQqQqQr−

−

−Χ∇

Χ∇Χ∇−Χ∇=

(16)

In this paper, the objective function is defined as:

avgTTT minmax −=Ψ

(17)

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, )(* tiA , )(* tiB and )(* tiC will be applied to the FEM model, according to Figs. 19,20. For example, )(* tiA can be estimated as:

( ))sin()cos(32)( *** θθ qdA iiti +−=

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

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

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

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

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. 24. Interior permanent-magnet synchronous-induction motor


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

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 synchrous-induction machine.

6 Conclusion In this research, different DC and AC permanent-

magnet 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.

At last, a new topology, i.e., Interior Permanent-Magnet 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 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.

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[28] 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.

[29] 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.

[30] M. Moallem, A. Kiyoumarsi and M. R. Hassanzadeh ," A novel technique on the analytical calculation of open-circuit flux density distribution in brushless permanent-magnet motor", International Journal of Engineering, pp. 51-58, IRAN, April 2004.

[31] 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.

[32] A. Kiyoumarsi, M. Moallem and M. Hassanzadeh, "The Torque ripples of brushless


12

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.

[33] 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.

[34] 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.

[35] 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.

[36] M. Hassanzadeh, A. Kiyoumarsi and M. Moallem, "Accurate methods for calculation of magnetizing inductances in interior permanent-magnet synchronous motors," presented in ICEM 2004, Appeared in CD Conference Proceedings, Paper No. 768, International Conference on Electrical Machines, Poland, September 2004.

[37] 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.

[38] 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 27-29, 2005.

[39] 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,

Paper No. DS1/9, Lausanne, Switzerland, September 27-29, 2005.

[40] A. Kiyoumarsi and M. Hassanzadeh, "Startup and steady-state performance of interior-permanent 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

Bmagnet Flux density due to magnet at stator inside surface

Bocb Open-circuit flux density; Boules’ method Boco Open-circuit flux density; Only assuming 0

~λ Bocn Open-circuit flux density; New method

Br Radial component of flux density; obtained by FEM analysis

Bsum Absolute resultant open-circuit flux b0 Stator slot-opening P Number of poles Qs Stator slot number

gr Mid-airgap radius

Sw Equivalent stator slot-opening

Tw Stator tooth width

α Angular displacement between the stator mmf and rotor mmf

mα Magnet arc angle

pα Pole-shoe arc angle

pα′ Pole arc angle

α sa Angular displacement between the stator slot axis and the axis of the coils of phase A

λ~

Relative permeance function

Λref Reference permeance


13

APPENDIX I PM and field-winding DC motor parameters

PMDC Motor Data Stator

mmDsi 0.59= , mmDso 0.79= ,

mmhm 7= ,

mechm °= 19.35α ,

TBr 45.0= , mKAH c 300= ,

P=6

Rotor

mmLr 60= , mmDri 6.38= ,

mmDro 5.58= , mmslot 7.2=ω ,

mmDshaft 6.16= , turnsNa .2= Field-Winding DC Motor Data: Stator

mmhp 6.8= , mmDso 5.89= , mmDsi 2.78= mechp °= 68α , mechp °=′ 37α , 8=fN .5,

P=4,

2

. .1*5.4 mmAcond = Rotor

mmLr 48= , mmDri 5.60= , mmDro 0.39= mmDshaft 45.17= , mmslot 7.2=ω ,

mmbrush 5=ω , mmlbrush .10= ,

turnsNa 2= , mmDcond 1.2. =

Stator lamination material M36,26 Gage Rotor lamination material M19,26 Gage

APPENDIX II Brushless PM motor parameters Stator outer radius (rso) 97.1 mm Stator yoke depth (hy) 17.4 mm Stator tooth depth (ht) 17.2 mm Stator slot opening (bo) 5.1 mm Stator Lamination material M36,26 Gage Rotor Lamination material M19,26 Gage Magnet arc angle )2( mα 60o

mechanical Magnet material Nd-Fe-B, N33 Remanence (Bres) 1.1 Tesla Relative recoil permeability (µr) 1.05

APPENDIX III IPM synchronous motor parameters Stator outer radius (rso) 97.1 mm Stator yoke depth (hy) 17.4 mm Stator tooth depth (ht) 17.2 mm Stator slot opening (bo) 5.1 mm Magnet arc angle )2( mα 2×37

mechanical degrees Stator lamination material M36, 26 Gage Rotor lamination material M19, 26 Gage Permanent-magnet material N33 Nd-Fe-B, Remanence (Bres) 1.1 T

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.

∫ ∫ ∫ ⋅=⋅×∇=⋅=ΦS S C

dlAdsAdstrBt )(),,()( θ (App.IV-1)

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

)(12

)( tNc

Pt swind Φ∗∗∗=λ

(App.IV-2)

And the instantaneous value of the induced no-load voltage is given by:

)()( tdtdte windλ=

(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


15

t=7.0 ms t=8.0 ms

t=9.0 ms t=10.0 ms

t=11.0 ms t=12.0 ms

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


application of the finite element method in design and … · motor magnetization curve. results of...

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