1. Introduction
Line-start permanent magnet synchronous motors (LSPMSMs) have the advantage of a high power factor, high efficiency, wide economic operation range, and high power density by using a permanent magnet (PM) to generate a strong magnetic field [
1,
2,
3,
4,
5]. At the time, brake torque occurs and causes a major drawback for the starting period by decreasing the start torque [
6,
7,
8]. In addition, the LSPMSM keeps synchronous speed in a steady state, and the ability of the rotor to pull in synchronization is also a key point for LSPMSMs [
9,
10,
11]. Therefore, more attention has been focused on the above issues, and the steady-state performances of LSPMSMs is easily neglected. The LSPMSM is different from a permanent magnet synchronous motor (PMSM) or an induction motor (IM), because of its special rotor structure. The rotor has both permanent magnets and cages; thus, the motor parameters have different influences on the steady-state performance, to some extent. There has been some literature on the study of steady-state performance, such as the influence of rotor bar depth, the frames of rotor, the stator winding connection mode, the winding structure, and the capacitor for the steady-state performances, invested respectively in [
12,
13,
14,
15], and so on. However, most studies have been exclusively concentrated on rated load power and neglected the influence on another load power. In fact, the motors not only work on rated load power, but have a range for different operational conditions. Therefore, the study of steady-state performance of line-start permanent magnet synchronous motors under different load powers is significant.
In this paper, the influence of the stator winding turns on the steady-state performance of LSPMSMs is studied. The winding turns have a direct effect on the induced electromotive force (EMF), stator resistance, end leakage reactance, synchronous reactance, etc. The performance of the motors will then be changed to some extent. It is convenient to change the performance by changing the stator winding turn value. Under normal conditions, the initial stator winding turn value is designed by the conventional calculation of the electromagnetic and thermal loads—besides, some experience is needed. This is not an accurate value. Changing the stator winding turns to some extent under the initial value is feasible.
The prototype is single-layer winding and the number of parallel branches is one. In order to simplify the analysis, the stator winding turns in particular represent the numbers of winding turns at a pitch called Ns throughout this study. The prototype operates in three-phase power of 380 V and 50 Hz. Based on the two-dimensional (2-D) finite element method (FEM), combined with the steady-state phasor diagram and torque angle characteristic, variation in the current, power factor, and maximum torque with the different stator winding turns is studied under different load powers for a 11 kW, LSPMSM-adopted, V-shape PM.
2. Two-Dimensional Finite Element Analysis Model and Parameters
The two-dimensional (2-D) finite element analysis (FEA) model was built, and the 2-D cross-section of the motor perpendicular to the axial direction was selected as the analysis model, as shown in
Figure 1. In this paper, the electromagnetic field of the motor does not take into account the influence of displacement current, and belongs to a quasi-steady field. In the process of finite element calculation, we consider the linear B–H curve. The vector magnetic potential
A and current density
J only has the
z-component. The transient 2-D electromagnetic field calculation equation could be described as
where
μ is the permeability,
A is the magnet vector potential,
Jz is the source current density,
n is the normal unit vector of the external permanent magnet,
Js is the equivalent current density of the permanent magnets,
σ is conductivity,
t is time,
l is the boundary between the equivalent surface current layer of permanent magnet material and other materials, Γ
1 is the first-type boundary, and Γ
2 is the second-type boundary.
In the numerical calculation of the 2-D electromagnetic field of an LSPMSM, the equivalent surface current is used to simulate the permanent magnet, and the equivalent surface current can be represented by surface current density
Js:
Here, where
n is the normal unit vector of permanent magnet,
Js is the equivalent current density of the permanent magnets,
Mr is the remanent magnetization,
μr is the relative recovery permeability.
When the magnetic vector potential is used to describe the field, at the boundary line between the equivalent surface current layer of the simulated permanent magnet and other media, the following boundary conditions are used:
where
l is the boundary between the equivalent surface current layer of permanent magnet material and other materials, the ν is the reluctivity,
A is the magnet vector potential,
n is the normal unit vector of the external permanent magnet,
Js is the equivalent current density of the permanent magnets.
When the FEM is used to solve the motor electromagnetic field, the range of solution areas should be minimized as much as possible. Because the permeability of ferromagnetic materials is far greater than the air permeability, the outer surface of the motor is used as the boundary surface. In other words, the magnetic force lines are closed along the outer surface of the motor, which belongs to the first-type boundary condition (Γ1). The second-type boundary condition (Γ2) is mainly applied to the interface of different materials in the motor, such as the interface between the motor cores and the air. It can be considered that the magnetic force lines are vertically crossed, and the variation of the magnetic potential along the geometric neutral line of the magnetic poles is 0—that is, .
The fundamental parameters of the prototype are shown in
Table 1. Based on the FEM, the numerical simulation of the model was obtained, and the result was verified by comparing it with the experimental data, as shown in
Table 2. The testing platform of the prototype is shown in
Figure 2. The masses of prototype materials are shown in
Table 3. Considering the losses of stator-core and rotor-eddy current in this study, the space harmonics should be calculated more carefully, so that the precise numerical simulation of the losses can be obtained. The
Ns is the number of turns. Some measures are needed, as follows.
The parts of the stator tooth, the edge between stator tooth and yoke, and the cages and permanent magnets of the model are more refined during triangulation, as shown in
Figure 1.
The time step is equal to one-eighth of the time when the rotor moves a slot pitch, and it can be described as the equation [
16]
where
f is the line frequency,
P is the number of pole pairs, and
N is the number of stator slots. Using the equation, the time step Δ
t is 1.38 × 10
−4 s.
It can be seen from
Table 3 that the number of turns has no effect on the masses of the stator rotor core, permanent magnet, or the squirrel cage. When the number of turns is changed, the wire diameter of winding should also be changed, so as to keep the copper filling factor constant.
3. The Influence of NS on the Current, the Power Factor, and Maximum Torque
It is obvious that the winding turns have a direct effect on the induced EMF, as well as on the no-load, back-induced EMF (called
E0 in next analysis); stator resistance; end leakage reactance; and synchronous reactance. Then the performance of the motors will be changed, to some extent. In this paper, the process used for the study of the stator winding turns can be presented in
Figure 3.
The initial
NS value is designed by the conventional calculation of the electromagnetics load, the thermal load, and some experience. As determined through the procedure shown in
Figure 3, the range of the
NS is from 27 to 33. Considering that the range is not large, and the initial stator resistance and end leakage values are little, they could therefore be ignored to simplify the calculation and analysis.
In the study, the steady-state phasor diagram under different loads can be utilized. Two factors are needed for this method. One is
E0, which can be obtained precisely by FEM. The
E0 with a different
NS, obtained through FEM, is shown in
Table 4.
A combination of the finite element method and the analytical calculation method is used to solve
Xd and
Xq.
Xd and
Xq are the quadrature axis reactance and direct axis reactance. The finite element method is used to calculate the voltage; power angle; power factor angle; no-load, back-induced EMF; resistance; and other parameters. The motor vector diagram is used to solve for
Id and
Iq. Finally,
Xd and
Xq are obtained by the analytical method. The calculation formula is as follows:
where
E0 is the no-load, back-induced EMF;
Id and
Iq are the quadrature axis current and direct axis current, respectively;
R1 is the resistance;
θ is the power angle; and
U is the voltage.
The other factor when using this method is the torque angle under different load powers, which can be estimated by the torque angle characteristic and the equation for the torque (neglecting the stator resistance), stated below as [
3]
where
Tem is electromagnetic torque;
m is the number of voltage phase;
U is the line voltage;
ω is the angular velocity of the rotor;
Xq and
Xd are the quadrature and direct axis reactance, respectively;
p is the number of pole pairs, and
θ is the torque-angle. Generally speaking,
E0 is directly proportional to
NS. In addition, the reactance is directly proportional to the winding turns squared. Assume that
k is equal to the ratio of
N2 to
N1, where
N1 and
N2 represent different values of
NS. When the value of the
NS is
N1, the
Tem is as shown in Formula 3. Then, when the value of
NS is
N2, the formula can be described as
When
N2 is less than
N1, then
k is less than 1. It is clear that the maximum value of
Tem is less than the maximum value of
T*em, and the conclusion can be gained that the maximum electromagnetic torque decreases as the value of
NS increases. However, estimating
Xq and
Xd involves saturation and the cross-coupling effect [
17]. The accurate values are difficult to obtain. By 2-D FEM, the clear results are obtained, as shown in
Figure 4. The
Tem is approximately consistent when the
θ changes within the range of 0° to 40°, and with the increase of
θ, the change of
Tem gradually become distinct. The maximum is reached near 100°, and the result is consistent with the above analysis.
The average torque of the LSPMSM is an important index to measure the performance of the motor in steady-state operation, which directly determines the ability of the motor to drag its load. The torque waveform and its Fourier decomposition is shown in
Figure 5.
When the number of turns is 29, the torque waveform and Fourier decomposition at no-load and rated-load operation are as shown in
Figure 5. The torque ripple of the motor at no-load is smaller than that at rated-load. The third, fifth, and seventh harmonic content of the torque is larger at a rated load than in the no-load state. With the increase of motor load torque, the motor harmonic content also increases. In addition, the cogging torque of the motor also causes the torque ripple of the motor. The LSPMSM torque ripple at rated-load operation is not more than 4% [
18]. However, the prototype is a first-generation motor. The torque ripple of the LSPMSM will be optimized in the follow-up work. The influence of stator winding turns on the steady-state performance of an LSPMSM is studied emphatically in this paper.
Neglecting the stator resistance and end leakage, the steady-state phasor diagram can now be depicted by
where
U is the phase voltage, which is equal to the line voltage, due to the delta connection; and
is is the phase current. The steady-state phasor diagram under different loads is shown in
Figure 6.
In
Figure 6, the A, B, and C represent the phasor diagram under different loads, which can be described as three states of light load, near-rated load, and overload, respectively. Under a light load, the
E0 has a key effect on the stator current and PF change, and the effect is nonlinear. The PF is the power factor. Whether or not
E0 is high or low enough, the
d-axis current component will get larger, which causes the stator current to increase and PF to decrease.
With the load torque from A to B, the torque angle gradually increases, and the influence of
E0 gradually decreases. The stator current and PF with different
NS values are approximately consistent. When the load torque continues to go up, as shown in
Figure 4, the torque angle with different
NS values become distinct, and its influence on the stator current and PF is gradually getting stronger, corresponding to the
Figure 6C.
The specific changes of the line current and PF, calculated by FEM, are shown in
Figure 7 and
Figure 8, respectively.
As is shown in
Figure 7 and
Figure 8, the variation of the current and PF with
NS are consistent with the above analysis. In
Figure 6, A, B, and C approximately correspond to the area of A, B, and C in
Figure 7. When
NS is 31, the value of
E0/U is 1.01, and the current is at its minimum of 2.07 A at no-load operation. Then the current increases or decreases with the value of
NS, as it goes up or down. Defining the load rate is a ratio of the fact load and the rated load. When the load rate is from 0 to 0.7, the current and PF difference with a different
NS is clear, and tends to be small. When the load rate is from 0.7 to 1.8, the differences between the stator currents and PF with different
NS are almost consistent. With the load rate continues to go up, the clear difference occurs again, and the value of the current increases as the value of
NS decreases. Take
NS equaling 27 and 33 as an example—the variation can be seen clearly in
Figure 9.
According to the demands of the prototype, the PF value of 0.94 was chosen was an index, so the influence of different
NS values on the current, the power factor, and maximum torque can be described in
Figure 10.
As is shown in
Figure 10, the range of the PF value, which is greater than or equal to 0.94, increases first, and then decreases with as the
NS value increases; the range is at its maximum when
NS equals 30 or 31. The change of range mainly appears at the light load, indicating that the change of
E0 value with
NS dominates the variation of stator current and PF. Besides, the maximum load decreases as the
NS value increases nearly 1 kW per turn. This can be seen from
Figure 4. This is because the synchronous reactance decreases as the
NS value increases, causing the load angle to be smaller than in a fixed load.
6. Conclusions
In this paper, the influence of NS on the steady-state performance of LSPMSMs, including stator current, power factor, stator core loss, and eddy current loss were obtained. It is noticeable that the analysis about the losses is not embedded, and in the next work these will be specially studied in-depth. The conclusions can be summarized as follows:
1. The influence of the stator winding turns for the current and PF can mainly be illustrated by two aspects. At light load, the approximate load rate is less than 0.7; in this paper, the main effect of the change of current and PF is the change of E0 due to the change of NS. However, with the load power increasing, the effect of E0 gradually weakens, and the effect of the change of synchronous reactance becomes stronger, so the stator current and PF tend to be consistent. Even when the load is large enough, the stator winding turns is higher, and the stator current is smaller. Finally, the capacity of overload improves as the NS value decreases.
2. For the stator core losses, when the armature reaction is not taken into account, the stator core losses will decrease with an increasing NS value. In contrast, as the load power increases, the rate of the stator core losses increase rises as NS value increases. What is more, this phenomenon is clear when the load power reaches the overload state.
3. The change of eddy current losses caused by an armature reaction at different NS values. In addition, the rate of the eddy current losses rise as the NS value increases.