Analytical and FEM Methods for Line Start Permanent Magnet Synchronous Motor of 2.2kW

2.1 Magnetic flux density in air gaps

The performance of LSPMSM motors is actually dependent on permanent magnets. Thus, the modeling of magnets plays an important role in the design and analysis of local fields for these motors. The permanent magnets are often modeled in two forms [3], which are the magnetic vector modeling and equivalent current circuit modeling. To determine the working point on the magnetization characteristic line of a permanent magnet, it is necessary to consider the main magnetic flux line in the motor as presented in Figure 1.

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Figure 1. Magnetic flux lines

where, L_ab is the stator yoke length (L_sy), L_ah and L_bc are the stator tooth height (L_st), L_hg and L_cd are the air gap, L_gf and L_de are the rotor tooth height (L_tr), L_fe is the rotort tooth length (L_ry), L_m is the magnet thickness, W_m is the magnet width and W_f is the floating bridge distance. The details of the LSPMSM of 2.2 kW, with 36 stator slots and 28 rotor bars is given in Table 1. The geometry parameters of LS-PMSM can be investigated by two main parts (i.e., stator, rotor permanent magnet design) as given in Table 2. The rotor topology is determined during the design process by the design of stator windings. Finally, the placement of the required permanent magnets to the rotor core, makes the design process very practical.

Table 1. Main parameters of LSPMSM

Table 2. Gemometry parameters of the LSPMSM

This magnetic flux starts from the north pole of the magnet through the air gap, through the teeth and stator, then closes through the air gap through the teeth and cavities of the rotor. In this process, the word passes twice the magnet, twice the stator and rotor tooth length, twice the air gap and once the stator and rotor. In Figure 1, the magnetic circuit with the magnetic flux source is considered as a permanent magnet. By using Ampere's law, the nonlinear magnetic circuit is calculated through the dynamic magnetic forces falling on the main magnetic flux.

This process requires an additional magnetomotive force (MMF) falling throughout the circuit. It means that the EMFs of the magnet, stator, rotor, and air gap are enclosed in a loop circuit. Based on the B-H curve of electrical engineering steel, the flux density of air gap can be identified as:

$\oint H d l=2 H_m L_m+2 F_{t r}+2 F_{t s}+2 F_g+F_{s y}+F_{r y}=0$            (1)

By transforming from the Eq. 1, one gets [12, 19, 20]:

$2 H_m L_m+2 H_g g_e+2 H_{t r}\left(B_{t r}\right) l_{t r}+2 H_{t s}\left(B_{t s}\right) l_{t s}+H_{s y}\left(B_{s y}\right) l_{s y}+H_{r y}\left(B_{r y}\right) l_{r y}=0$          (2)

In addtion, from Figure 1, the magnetism of the permanent magnet is approximately the same as air, so the actual rotor length is only defined as $l_{r y}-2 L_m$. For that, one gets [19]:

$A=2 H_{t r}\left(B_{t r}\right) l_{t r}+2 H_{t s}\left(B_{t s}\right) l_{t s}+H_{s y}\left(B_{s y}\right) l_{s y}+H_{r y}\left(B_{r y}\right)\left(l_{r y}-2 L_m\right)$          (3)

where:

H_m: Magnetic field strength at the working point of the magnet (A/m);

H_tr: Rotor tooth magnetic field strength (A/m);

H_ts: Intensity from the head that stator (A/m);

H_sy: Magnetic field strength (A/m);

H_ry: Rotor magnetic field strength (A/m);

B_ts: Magnetic flux density of stator (T);

B_tr: Rotorr tooth magnetic flux density (T);

B_sy: Magnetic flux density (T);

B_ry: Rotorr magnetic flux density (T);

F_tr: Rotorr magnetic energy (A.T);

F_ts: Magnetic stator (A.T);

F_sy: Magnetic stator (A.T);

F_ry: Rotorr magnetic power (A.T);

F_g: Air gap dynamic magnetism (A.T).

Based on the Eq. (2) and Eq. (3), the magnetic flux density in air gap can be defined as [8-11]:

$B_g=-\frac{\mu_0}{2 g_e}\left(2 H_m L_m+A\right)=-\frac{\mu_0 H_m L_m}{g_e}-\frac{\mu_0 A}{2 g_e}$.          (4)

The effective air gap (g_e) is [11]:

$g_e=K_c g$

for $K_c=\left[1-\frac{b_{o s}}{\tau_s}+\frac{4 g}{\pi \tau_s} \ln \left(1+\frac{\pi b_{o s}}{4 g}\right)\right]^{-1}$           (5)

where, $\mu_0$ is the permeability of air gap: $\mu_0=4 \pi . 10^{-7} \mathrm{Tm} / \mathrm{A}$ and $t_s$ is the dental steps (m). Otherwise, one has

$\Phi_m=K_{l m} \Phi_g$.          (6)

The leakage flux coefficient $\left(K_{l m}\right)$ in (6) is defined

$K_{l m}=1+\mu_{r m} \frac{g_e}{L_m} \frac{W_m}{W_m+g_e}(2 \eta+4 \lambda)$

$\eta=\frac{L_m}{\pi \mu_{r m} W_m} \ln \left(1+\frac{\pi g_e}{L_m}\right)$

$\lambda=\frac{L_m}{\pi \mu_{r m} W_m} \ln \left(1+\frac{\pi g_e}{W_f}\right)$          (7)

From the Eq. (7), one gets:

$B_m S_m=K_{l m} B_g S_g$           (8)

where, $S_m$ is the permanent magnet area and $S_g$ is the air gap area under one pole step.

By substituting the term of B_g in Eq. (4) and the leakage flux coefficient $K_{l m}$ in Eq. (7) into the equation Eq. (8), the magnet magnetic flux density of the working point is determined as follows.

$B_m=-\frac{\mu_0 L_m K_{l m} S_g}{S_m g_e} H_m-\frac{\mu_0 K_{l m} S_g A}{2 S_m g_e}$          (9)

The demagnetization characteristic line can be written as follows

$B=B_r+\mu_0 \mu_{r m} H$          (10)

where, $\mu_{r m}$ is the relative magnetic range of magnet. It should be noted that the working point of the permanent magnet is the intersection of the air gap line with the demagnetization line. For that, the magnetic field strength at the working point of the magnet is defined.

$H_m=\frac{B_m-B_r}{\mu_0 \mu_{r m}}$          (11)

By substituting equation (11) into equations (from (3) to (10)) with some transformations, one gets

$c B_m=d-e A \rightarrow B_m=\frac{d-e A}{c}$

for $c=\left(1+\frac{L_m K_{l m} \quad \alpha_i \tau}{W m g_e \mu_{r m}}\right)$

$d=\frac{L_m K_{l m} \quad \alpha_i \tau B_r}{W m g_e \mu_{r m}}, e=\frac{\mu_0 K_{l m} \quad \alpha_i \tau}{2 W m g_e}$           (12)

where, τ is the polar embrace, $\alpha_i$ is the magnetic flux density shape coefficient. According to some electrical machine design documents, the coefficient is usually determined empirically. In the case of LSPMSM motors, this coefficient depends on the degree of saturation of the electrical engineering steel and the shape of the density distribution of the air gap magnetic flux $\alpha_i$.

2.2 Flux density of rotor and stator tooth

The magnetic flux densities of rotor and stator parts under one pole embrace of the stator and rotor are enclosed through the stator tooth and the rotor tooth. The flux density is then calculated as follows:

$B_g=\frac{B_m S_m}{K_{l m} S_g}=\frac{B_m W_m}{K_{l m} \quad \alpha_i \tau}$          (13)

In the case of stators, it gets $\Phi_g^{\tau_s}=\Phi_{t s}$. Thus, the term of B_st is then defined as

$B_{t s}=\frac{B_g \tau_s L_s}{b_{t s} k_{F e} L_s}=\frac{B_g \tau_s}{b_{t s} k_{F e}}$,           (14)

where, L_s is the stator effect length (m), $k_{F e}$ is the steel foil fastening coefficient and $b_{t s}$ is the stator tooth width (m).

By replacing the equation (13) with the equation (14), one has

$B_{t s}=\frac{W_m \tau_s}{\alpha_i \tau K_{l m} \quad b_{t s} k_{F e}} B_m \rightarrow B_{t s}=k_{s m} B_m$           (15)

where, the factor $k_{s m}$ is determined

$k_{s m}=\frac{W_m \tau_s}{\alpha_i \tau K_{l m}  \quad b_{t s} k_{F e}}$            (16)

In the case of rotor, it gets $\Phi_g^{\tau_s}=\Phi_{t r}$. Thus, the term of B_tr is then defined as

 $B_{t r}=\frac{B_g \tau_r L_s}{b_{t r} k_{F e} L_s}=\frac{B_g \tau_r}{b_{t r} k_{F e}}$          (17)

where, $b_{t r}$ is the rotor tooth width (m).

By replacing the Eq. (16) with the Eq. (17), one gets

$B_{t r}=\frac{W_m \tau_s}{\alpha_i \tau K_{l m} \quad b_{t s} k_{F e}} B_m \rightarrow B_{t r}=k_{r m} B_m$           (18)

where, the factor $k_{r m}$ is determined

$k_{r m}=\frac{W_m \quad \tau_r}{\alpha_i \tau K_{l m} \quad b_{t r} k_{F e}}$           (19)

Thetically, the half of air gap magnetic flux under per one pole through the stator and rotor can be considered in two cases of stator and rotor to find out the stator yoke fator ($k_{s y m}$) and the rotor yoke factor ($k_{r y m}$). In the case of stators, one gets $\Phi_g^\tau=\Phi_{s y}$.

$B_{s y} S_{s y}=\frac{1}{2} B_g \tau L_s \rightarrow B_{s y} h_{s y} L_s k_{F e}=\frac{1}{2} B_g \tau L_s \rightarrow B_{s y} h_{s y} k_{F e}=\frac{1}{2} B_g \tau$          (20)

where, $S_{s y}$ is the stator area and $h_{s y}$ is the stator height.

By replacing the expression (19) with the expression (20), one has

$B_{s y}=\frac{W_m}{2 K_{l m} \alpha_i h_{s y} \quad k_{F e}} B_m \rightarrow B_{s y}=k_{s y m} B_m$          (21)

where, the fator $k_{s y m}$ is expressed as

$k_{s y m}=\frac{W_m}{2 K_{l m} \alpha_i h_{s y} \quad k_{F e}}$           (22)

In the case of rotor, it gets: $\Phi_g^\tau=\Phi_{r y}$. For that one gets

$B_{r y} S_{r y}=\frac{1}{2} B_g \tau L_s \rightarrow B_{r y} h_{r y} L_s k_{F e}=\frac{1}{2} B_g \tau L_s \rightarrow B_{r y} h_{r y} k_{F e}=\frac{1}{2} B_g \tau$           (23)

where, $S_{r y}$ is the rotor area and  is the rotor height.

By substituting (22) into (23), it can be defined as

$B_{r y}=\frac{W_m}{2 K_{l m} \quad \alpha_i h_{r y}  \quad k_{F e}} B_m \rightarrow B_{r y}=k_{r y m} B_m$           (24)

where, the factor $k_{r y m}$ can be defined as

$k_{r y m}=\frac{W_m}{2 K_{l m} \quad \alpha_i h_{r y} \quad k_{F e}}$          (25)

These design factors of k_rm, k_sm, k_sy and k_ry are relationship of stator, rotor teeth and yokes with magnetic flux density at operation points, respectively. Those parameters have built from many electromagnetic calculations.

Model of the designed motor is shown in Figure 7. An experimental test bench has built to evaluate back electromagnetic force at different speeds.

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

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Figure 8. Model of motor test bench

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Figure 9. Back EMF measured by Osiloscope with 20V/div and 5ms/div (top) and back EMF simulation and measurement comparison (bottom)

The no load voltage of three stator windings is measured at the different speeds of 500 and 750 rpm. The back 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.

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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)

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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 torque characteristics. In compared with other induction motors, the LSPMS motor can improve efficiency by eleminating the rotor cage loss at the synchronous speed.