Journal Européen des Systèmes Automatisés

Vol. 55, No. 6, December, 2022, pp. 715-721

Journal homepage: http://iieta.org/journals/jesa

Analytical and FEM Methods for Line Start Permanent Magnet Synchronous Motor of 2.2kW
Hung Bui Duc, Dinh Bui Minh, Vuong Dang Quoc*

School of Electrical and Electronic Engineering, Hanoi University of Science and Technology, Hanoi 10999, Vietnam

Corresponding Author Email: vuong.dangquoc@hust.edu.vn

https://doi.org/10.18280/jesa.550603 ABSTRACT

Received: 26 November 2022 This paper has developed electromagnetic equations based an analytical model to design
Accepted: 17 December 2022 a three-phase line start permanent magnet synchronous motor (LSPMSM) of 2.2 kW. In
order to enhance the torque and efficiency of this machine, an optimized slot shape of rotor
Keywords: permanent magnet is proposed. The analytical model is first presented to evaluate the
line start permanent magnet synchronous magnetic characteristics, and then analytical expressions are derived under the open circuit
motor, magnetic flux density, leakage flux, condition. The influence of the magnet size variables on the analytical model is considered
analytic method, finite element method. to investigate the magnetic leakage flux and flux density distribution in air gaps. Moreover,
the variation of design has a significant impact on the steady-state performance
characteristics of this motor. In addition, in order to check the output results from the
analytical model, a finite element method is developed to evaluate these solutions. Finally,
experimental and simulation results have been compared and analyzed to validate the
development of method.

1. INTRODUCTION 2. COMPUTATION OF MAGNETIC FLUX DENSITY

As we have known, the line-start (LS) permanent magnet 2.1 Magnetic flux density in air gaps
synchronous motor (PMSM) is typically designed with a
squirrel cage inserted into the rotor of a PMSM [1-3]. The The performance of LSPMSM motors is actually dependent
magnetic flux density distribution in the air gap is determined on permanent magnets. Thus, the modeling of magnets plays
via the working point magnetic flux density of permanent an important role in the design and analysis of local fields for
magnets (PMs) [3-12]. However, there are some papers that these motors. The permanent magnets are often modeled in
have not considered yet the effect of the magnetization curve two forms [3], which are the magnetic vector modeling and
(B-H) of electrical engineering steels [3, 6, 7]. Hence, in order equivalent current circuit modeling. To determine the working
to simplify these documents, the aesthetics of electrical point on the magnetization characteristic line of a permanent
engineering steels are considered to be extremely large and magnet, it is necessary to consider the main magnetic flux line
only the relationship between permanent magnets and air gaps. in the motor as presented in Figure 1.
In reference [6], the magnetization curve (B-H) of electrical
engineering steels is taken into account. However, the width
of the stator and rotor teeth is not specifically mentioned in the
definite formula. In the research of Els et al. [7], the magnetic
flux density at the air gap through magnetic coefficients is also
determined, but it also makes no mention of tooth sizes and
stators and rotors. In addition, many different rotor designs for
LSPMSM have been also presented [13-18].
In order to serve the study of LSPMSM design, this paper
offers a computational research method and finds an algorithm
to determine the flux density and find out the analytic
relationship between the magnetic flux density and the tooth
magnetic flux density, the stator and the rotor.
The paper has developed electromagnetic equations based
the analytical model to design a three-phase LSPMSM of 2.2
kW. Then, an optimized slot shape of rotor permanent magnet
has been proposed to enhance the torque and efficiency of this Figure 1. Magnetic flux lines.
machine. The obtained results from this study can also apply
to bigger motors such as motors of 5kW and 7.5kW, with where, Lab is the stator yoke length (Lsy), Lah and Lbc are the
similar designs. stator tooth height (Lst), Lhg and Lcd are the air gap, Lgf and Lde
are the rotor tooth height (Ltr), Lfe is the rotort tooth length (Lry),
Lm is the magnet thickness, Wm is the magnet width and Wf is
the floating bridge distance. The details of the LSPMSM of 2.2

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kW, with 36 stator slots and 28 rotor bars is given in Table 1. of the magnet, stator, rotor, and air gap are enclosed in a loop
The geometry parameters of LS-PMSM can be investigated by circuit. Based on the B-H curve of electrical engineering steel,
two main parts (i.e., stator, rotor permanent magnet design) as the flux density of air gap can be identified as:
given in Table 2. The rotor topology is determined during the
design process by the design of stator windings. Finally, the ∮ 𝐻𝑑𝑙 = 2𝐻𝑚 𝐿𝑚 + 2𝐹𝑡𝑟 + 2𝐹𝑡𝑠 + 2𝐹𝑔 + 𝐹𝑠𝑦 + 𝐹𝑟𝑦 = 0 (1)
placement of the required permanent magnets to the rotor core,
makes the design process very practical.
By transforming from the Eq. 1, one gets [12, 19, 20]:
Table 1. Main parameters of LSPMSM
2𝐻𝑚 𝐿𝑚 + 2𝐻𝑔 𝑔𝑒 + 2𝐻𝑡𝑟 (𝐵𝑡𝑟 )𝑙𝑡𝑟 + 2𝐻𝑡𝑠 (𝐵𝑡𝑠 )𝑙𝑡𝑠
Main Parameters Materials Weight (kg) (2)
+𝐻𝑠𝑦 (𝐵𝑠𝑦 )𝑙𝑠𝑦 + 𝐻𝑟𝑦 (𝐵𝑟𝑦 )𝑙𝑟𝑦 = 0
Stator Lam (Back Iron) M800-50A 12.92
Stator Lam (Tooth) M800-50A 5.523
In addtion, from Figure 1, the magnetism of the permanent
Stator Lamination [Total] 18.44
Armature Winding magnet is approximately the same as air, so the actual rotor
Copper (Pure) 2.67 length is only defined as 𝑙𝑟𝑦 − 2𝐿𝑚 . For that, one gets [19]:
[Active]
Armature EWdg [Front] Copper (Pure) 1.475
Armature EWdg [Rear] Copper (Pure) 1.475 𝐴 = 2𝐻𝑡𝑟 (𝐵𝑡𝑟 )𝑙𝑡𝑟 + 2𝐻𝑡𝑠 (𝐵𝑡𝑠 )𝑙𝑡𝑠 + 𝐻𝑠𝑦 (𝐵𝑠𝑦 )𝑙𝑠𝑦
Armature Winding [Total] 5.62 (3)
+𝐻𝑟𝑦 (𝐵𝑟𝑦 )(𝑙𝑟𝑦 − 2𝐿𝑚 )
Magnet 0.56
Rotor Lam (Back Iron) M800-50A 4.326
Rotor Inter Lam (Back where:
1.97E-05
Iron) 𝐻𝑚 : Magnetic field strength at the working point of the
Rotor Lam (Tooth) M800-50A 4.468 magnet (A/m);
Rotor Inter Lam (Tooth) 2.04E-05 𝐻𝑡𝑟 : Rotor tooth magnetic field strength (A/m);
Rotor Lamination [Total] 8.794 𝐻𝑡𝑠 : Intensity from the head that stator (A/m);
Rotor Cage Top Bar Aluminium (Cast) 1.302
𝐻𝑠𝑦 : Magnetic field strength (A/m);
Rotor Cage Top Bar
Aluminium (Cast) 0.01513 𝐻𝑟𝑦 : Rotor magnetic field strength (A/m);
Opening
Rotor Cage (Front End) Aluminium (Cast) 0.2962 𝐵𝑡𝑠 : Magnetic flux density of stator (T);
Rotor Cage (Rear End) Aluminium (Cast) 0.2962 𝐵𝑡𝑟 : Rotorr tooth magnetic flux density (T);
Rotor Cage [Total] 1.91 𝐵𝑠𝑦 : Magnetic flux density (T);
Shaft [Active] Iron (Pure) 1.662
𝐵𝑟𝑦 : Rotorr magnetic flux density (T);
Flange Mounted Plate 7.158
𝐹𝑡𝑟 : Rotorr magnetic energy (A.T);
Table 2. Gemometry parameters of the LSPMSM 𝐹𝑡𝑠 : Magnetic stator (A.T);
𝐹𝑠𝑦 : Magnetic stator (A.T);
Sator Rotor 𝐹𝑟𝑦 : Rotorr magnetic power (A.T);
Quantity Quantity
parameters Parameters 𝐹𝑔 : Air gap dynamic magnetism (A.T).
Slot Number 36 Rotor Bars 40 Based on the Eq. (2) and Eq. (3), the magnetic flux density
Stator Lam in air gap can be defined as [8-11]:
200 Pole Number 4
Diamention
Stator Bore 122 Bar Opening 1 𝜇0 𝜇0 𝐻𝑚 𝐿𝑚 𝜇0 𝐴
Bar Opening 𝐵𝑔 = − (2𝐻𝑚 𝐿𝑚 + 𝐴) = − − . (4)
Tooth Width 6 0.8 2𝑔𝑒 𝑔𝑒 2𝑔𝑒
Depth
Rotor Tooth
Slot Depth 19.85
Width
4.2 The effective air gap (𝑔𝑒 ) is [11]:
Slot Corner
3.7 Bar Depth 20 𝑔𝑒 = 𝐾𝑐 𝑔
Radius
Tooth Tip Depth 1 Airgap 0.45 𝑏𝑜𝑠 4𝑔 𝜋𝑏𝑜𝑠 −1 (5)
Banding for 𝐾𝑐 = [1 − + 𝑙𝑛 (1 + )]
Slot Opening 2.5 0.1 𝜏𝑠 𝜋𝜏𝑠 4𝑔
Thickness
Tooth Tip Angle 30 Shaft Dia 41
Shaft Hole where, 𝜇0 is the permeability of air gap: 𝜇0 = 4𝜋. 10−7 𝑇𝑚/𝐴
Sleeve Thickness 0.1 32 and 𝑡𝑠 is the dental steps (m). Otherwise, one has
Diameter

This magnetic flux starts from the north pole of the magnet 𝛷𝑚 = 𝐾𝑙 𝑚 𝛷𝑔 . (6)
through the air gap, through the teeth and stator, then closes
through the air gap through the teeth and cavities of the rotor. The leakage flux coefficient (𝐾𝑙𝑚 ) in (6) is defined
In this process, the word passes twice the magnet, twice the
stator and rotor tooth length, twice the air gap and once the 𝑔𝑒 𝑊𝑚
𝐾𝑙𝑚 = 1 + 𝜇𝑟𝑚 (2𝜂 + 4𝜆)
stator and rotor. In Figure 1, the magnetic circuit with the 𝐿𝑚 𝑊𝑚 + 𝑔𝑒
magnetic flux source is considered as a permanent magnet. By 𝐿𝑚 𝜋𝑔𝑒
using Ampere's law, the nonlinear magnetic circuit is 𝜂= 𝑙𝑛 (1 + ) (7)
𝜋𝜇𝑟𝑚 𝑊𝑚 𝐿𝑚
calculated through the dynamic magnetic forces falling on the 𝐿𝑚 𝜋𝑔𝑒
main magnetic flux. 𝜆= 𝑙𝑛 (1 + )
𝜋𝜇𝑟𝑚 𝑊𝑚 𝑊𝑓
This process requires an additional magnetomotive force
(MMF) falling throughout the circuit. It means that the EMFs
From the Eq. (7), one gets:

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 𝐵𝑚 𝑆𝑚 = 𝐾𝑙𝑚 𝐵𝑔 𝑆𝑔 (8) has

where, 𝑆𝑚 is the permanent magnet area and 𝑆𝑔 is the air gap 𝑊𝑚 𝜏𝑠

𝐵𝑡𝑠 = 𝐵 → 𝐵𝑡𝑠 = 𝑘𝑠𝑚 𝐵𝑚 (15)
area under one pole step. 𝛼𝑖 𝜏𝐾𝑙𝑚 𝑏𝑡𝑠 𝑘𝐹𝑒 𝑚
By substituting the term of Bg in Eq. (4) and the leakage flux
coefficient (𝐾𝑙𝑚 ) in Eq. (7) into the equation Eq. (8), the where, the factor 𝑘𝑠𝑚 is determined
magnet magnetic flux density of the working point is
determined as follows. 𝑊𝑚 𝜏𝑠
𝑘𝑠𝑚 = (16)
𝛼𝑖 𝜏𝐾𝑙𝑚 𝑏𝑡𝑠 𝑘𝐹𝑒
𝜇0 𝐿𝑚 𝐾𝑙𝑚 𝑆𝑔 𝜇0 𝐾𝑙𝑚 𝑆𝑔 𝐴
𝐵𝑚 = − 𝐻𝑚 − (9) 𝜏
𝑆𝑚 𝑔𝑒 2𝑆𝑚 𝑔𝑒 In the case of rotor, it gets 𝛷𝑔 𝑠 = 𝛷𝑡𝑟 . Thus, the term of Btr
is then defined as
The demagnetization characteristic line can be written as
follows 𝐵𝑔 𝜏𝑟 𝐿𝑠 𝐵𝑔 𝜏𝑟
𝐵𝑡𝑟 = = (17)
𝑏𝑡𝑟 𝑘𝐹𝑒 𝐿𝑠 𝑏𝑡𝑟 𝑘𝐹𝑒
𝐵 = 𝐵𝑟 + 𝜇0 𝜇𝑟𝑚 𝐻 (10)
where, 𝑏𝑡𝑟 is the rotor tooth width (m).
where, 𝜇𝑟𝑚 is the relative magnetic range of magnet. It should By replacing the Eq. (16) with the Eq. (17), one gets
be noted that the working point of the permanent magnet is the
intersection of the air gap line with the demagnetization line. 𝑊𝑚 𝜏𝑠
For that, the magnetic field strength at the working point of the 𝐵𝑡𝑟 = 𝐵 → 𝐵𝑡𝑟 = 𝑘𝑟𝑚 𝐵𝑚 (18)
𝛼𝑖 𝜏𝐾𝑙𝑚 𝑏𝑡𝑠 𝑘𝐹𝑒 𝑚
magnet is defined.
where, the factor 𝑘𝑟𝑚 is determined
𝐵𝑚 − 𝐵𝑟
𝐻𝑚 = (11)
𝜇0 𝜇𝑟𝑚 𝑊𝑚 𝜏𝑟
𝑘𝑟𝑚 = (19)
𝛼𝑖 𝜏𝐾𝑙𝑚 𝑏𝑡𝑟 𝑘𝐹𝑒
By substituting equation (11) into equations (from (3) to
(10)) with some transformations, one gets Thetically, the half of air gap magnetic flux under per one
pole through the stator and rotor can be considered in two
𝑑 − 𝑒𝐴 cases of stator and rotor to find out the stator yoke fator (𝑘𝑠𝑦𝑚 )
𝑐𝐵𝑚 = 𝑑 − 𝑒𝐴 → 𝐵𝑚 =
𝑐 and the rotor yoke factor (𝑘𝑟𝑦𝑚 ). In the case of stators, one gets
𝐿𝑚 𝐾𝑙𝑚 𝛼𝑖 𝜏
for 𝑐 = (1 + ) (12) 𝛷𝑔𝜏 = 𝛷𝑠𝑦 .
𝑊𝑚𝑔𝑒 𝜇𝑟𝑚
𝐿𝑚 𝐾𝑙𝑚 𝛼𝑖 𝜏𝐵𝑟 𝜇0 𝐾𝑙𝑚 𝛼𝑖 𝜏
𝑑= ,𝑒 = 1
𝑊𝑚𝑔𝑒 𝜇𝑟𝑚 2𝑊𝑚𝑔𝑒 𝐵𝑠𝑦 𝑆𝑠𝑦 = 𝐵𝑔 𝜏𝐿𝑠 → 𝐵𝑠𝑦 ℎ𝑠𝑦 𝐿𝑠 𝑘𝐹𝑒
2 (20)
1 1
where, 𝜏 is the polar embrace, 𝛼𝑖 is the magnetic flux density = 𝐵𝑔 𝜏𝐿𝑠 → 𝐵𝑠𝑦 ℎ𝑠𝑦 𝑘𝐹𝑒 = 𝐵𝑔 𝜏
2 2
shape coefficient. According to some electrical machine
design documents, the coefficient is usually determined where, 𝑆𝑠𝑦 is the stator area and ℎ𝑠𝑦 is the stator height.
empirically. In the case of LSPMSM motors, this coefficient
By replacing the expression (19) with the expression (20),
depends on the degree of saturation of the electrical
one has
engineering steel and the shape of the density distribution of
the air gap magnetic flux 𝛼𝑖 . 𝑊𝑚
𝐵𝑠𝑦 = 𝐵 → 𝐵𝑠𝑦 = 𝑘𝑠𝑦𝑚 𝐵𝑚 (21)
2.2 Flux density of rotor and stator tooth 2𝐾𝑙𝑚 𝛼𝑖 ℎ𝑠𝑦 𝑘𝐹𝑒 𝑚

The magnetic flux densities of rotor and stator parts under where, the fator 𝑘𝑠𝑦𝑚 is expressed as
one pole embrace of the stator and rotor are enclosed through
the stator tooth and the rotor tooth. The flux density is then 𝑊𝑚
𝑘𝑠𝑦𝑚 = (22)
calculated as follows: 2𝐾𝑙𝑚 𝛼𝑖 ℎ𝑠𝑦 𝑘𝐹𝑒

𝐵𝑚 𝑆𝑚 𝐵𝑚 𝑊𝑚 In the case of rotor, it gets: 𝛷𝑔𝜏 = 𝛷𝑟𝑦 . For that one gets
𝐵𝑔 = = (13)
𝐾𝑙𝑚 𝑆𝑔 𝐾𝑙𝑚 𝛼𝑖 𝜏
1
𝜏 𝐵𝑟𝑦 𝑆𝑟𝑦 = 𝐵𝑔 𝜏𝐿𝑠 → 𝐵𝑟𝑦 ℎ𝑟𝑦 𝐿𝑠 𝑘𝐹𝑒
In the case of stators, it gets 𝛷𝑔 𝑠 = 𝛷𝑡𝑠 . Thus, the term of 2 (23)
1 1
Bst is then defined as = 𝐵𝑔 𝜏𝐿𝑠 → 𝐵𝑟𝑦 ℎ𝑟𝑦 𝑘𝐹𝑒 = 𝐵𝑔 𝜏
2 2
𝐵𝑔 𝜏𝑠 𝐿𝑠 𝐵𝑔 𝜏𝑠
𝐵𝑡𝑠 = = , (14) where, 𝑆𝑟𝑦 is the rotor area and ℎ𝑟𝑦 is the rotor height.
𝑏𝑡𝑠 𝑘𝐹𝑒 𝐿𝑠 𝑏𝑡𝑠 𝑘𝐹𝑒
By substituting (22) into (23), it can be defined as
where, 𝐿𝑠 is the stator effect length (m), 𝑘𝐹𝑒 is the steel foil 𝑊𝑚
fastening coefficient and 𝑏𝑡𝑠 is the stator tooth width (m). 𝐵𝑟𝑦 = 𝐵 → 𝐵𝑟𝑦 = 𝑘𝑟𝑦𝑚 𝐵𝑚 (24)
By replacing the equation (13) with the equation (14), one 2𝐾𝑙𝑚 𝛼𝑖 ℎ𝑟𝑦 𝑘𝐹𝑒 𝑚

717
where, the factor 𝑘𝑟𝑦𝑚 can be defined as It can be seen that the analytical result is very similar to the
result of finite eleement method (FEM) [21]. The error of the
𝑊𝑚 air gap flux density Bg in both analytical and FEM methods is
𝑘𝑟𝑦𝑚 = (25) 0.618%. This means that the obtained results from the analytic
2𝐾𝑙𝑚 𝛼𝑖 ℎ𝑟𝑦 𝑘𝐹𝑒
method is quite accurate. For that, t is useful to design and
These design factors of krm, ksm, ksy and kry are relationship optimize the electromagnetic parameters of LSPMSM. The
of stator, rotor teeth and yokes with magnetic flux density at flux density distribution for case 1 and 2 are pointed out in
operation points, respectively. Those parameters have built Figure 4.
from many electromagnetic calculations.

3. NUMERICAL TEST

Two cases of LSPMSM of 2.2kW are considered. Each case

proceeds to change the width Wm and length Lm of the
permanent magnet. Figure 2 shows the rotor groove structure
(left), stator groove (right) of 2.2kW motor.

Figure 4. Magnetic flux density in air gaps for case 1 (top)

and case 2 (bottom)

In order to evaluate the air gap flux density Bg, the magnetic
thickness Lm is changed from 2.8 mm to 4.8mm. For that, each
case is 1mm increasing.
Figure 2. Rotor groove structure (left), stator groove (right) The magnetic flux density Bg is increased from 0.1415 T to
of LSPMSM of 2.2kW 0.1428 T, while the magnet thickness is from 2.8 mm to
4.8mm, but the waveforms of magnetic flux density are the
where, the dimensions the rotor groove are given as: ho=0.4 similar profiles as presented in Figure 5. The waveforms of
mm, h1=1.4 mm, dl=3.89 mm, d2=1.99 mm, hr=12 mm. The magnet thickness of 2.8 mm, 3.8mm and 4.8mm are same
dimensions of the stator groove are: hos=1 mm, h12=12.4 mm, characteristics or shapes. Table 4 shows the flux density
bos=2.9 mm, dl=3.71 mm, d2=5.3 mm, and hs=13.6 mm. results with different thicknesses (Lm).
-For case 1: Lm=2 mm; Wm=54mm. The flux density Bg is increased with the magnetic width Wm
-For case 2: Lm=3 mm; Wm=51mm. with each step of 1mm. The air gap flux density distribution
Analysis results of the two cases are illustrated in Figure 3. with different magnetic widths is presented in Figure 6.
The results of the comparison between the analytic and
FEM methods are illustrated in Table 3.

Figure 5. Air gap flux density with different magnetic

Figure 3. Case structure 1 (left), case structure 2 (right) thickness

Table 3. Results of comparison between the two methods Table 4. Magnetic flux density results with different
thicknesses (Lm)
Bm Bg
Different methods
Case 1 Case 2 Case 1 Case 2 Lm (mm) Bg (T)
Analytical 0.891 0.952 0.485 0.496 2.8 0.1415
FEM 0.925 0.990 0.490 0.499 3.8 0.1425
Error (%) 3.479 3.466 0.618 0.639 4.8 0.1428

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 The flux density Bg is increased with the magnetic width Wm The no load voltage of three stator windings is measured at
with each step of 1mm. The air gap flux density distribution the different speeds of 500 and 750 rpm. The back
with different magnetic widths is presented in Figure 6. electromagnetic force (EMF) waveform are displayed in
Ossiloscope with DAI-Labvol interation as shown in Figure 8.
The back EFM waveforms of simulation and experiment result
are quite good agreement as presented in Figure 9.
The current and voltage waveforms were compared together
at rated speed by measurement and FEM simulation methods
in Figure 10. The torque, efficiency and power have been
compared with different methods (analytic, FEM and
experimental methods). The obtained results are shown in
Figures (from Figure 10 and Figure 11). It can be seen that the
measured results are checked to be very close to the torque,
EMF, current and power obtained by the FEM method.

Figure 6. Air gap flux density distribution with different

magnetic widths

4. EXPERIMENTAL RESULTS

Model of the designed motor is shown in Figure 7. An

experimental test bench has built to evaluate back
electromagnetic force at different speeds.

Figure 10. Back EMF measured by Osiloscope with 100V/div-

2A/div and 5ms/div (top) and back EMF and current I simulated by
FEM (bottom)

Figure 7. Stator (a), Rotor (b), Rotor drawing (c) and Rotor
assembly (d)

Figure 8. Model of motor test bench

Figure 11. Comparison of torque and power simulations and

measurement

The LSPMSM in this paper is consideered with I magnet

shape and curve flux barrier. The flux density structure of this
machine is developed from a convention induction motor. The
total torque is combined of magnetic torque and rotor cage
Figure 9. Back EMF measured by Osiloscope with 20V/div torque characteristics. In compared with other induction
and 5ms/div (top) and back EMF simulation and motors, the LSPMS motor can improve efficiency by
measurement comparison (bottom) eleminating the rotor cage loss at the synchronous speed.

719
5. CONCLUSIONS Machines and Drives (PEMD 2014), pp. 1-6.
https://doi.org/10.1049/cp.2014.0281
This study has found out an analytical calculation of the flux [9] Stumberger, B., Marcic, T., Hadziselimovic, M. (2012).
density distribution of air gap, stator and rotor poles, yokes Direct comparison of induction motor and line-start ipm
with very high accuracy. To validate the performed analytical synchronous motor characteristics for semi-hermetic
design, the designed motor is modeled by using the FEM in compressor drives. In IEEE Transactions on Industry
Maxwell software. The motor behavior and characteristics are Applications, 48(6): 2310-2321.
investigated in steady state using magnetostatic analysis as https://doi.org/10.1109/TIA.2012.2227094
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