Optimal design of rotor geometry in interior permanent magnet machine

Abstract

Interior permanent magnet machine has been widely used in electric vehicles due to its high-power density and good flux weakening performance. This paper firstly introduces the principle of flux-weakening control, which are the foundation to describe the effect of d-axis inductance and q-axis inductance on the speed range of motor. Secondly, regarding an interior permanent magnet synchronous motor with V -type permanent magnet structure as the research object, an improved rotor structure has been proposed which concludes asymmetric segmentations of permanent magnets and protuberances from centers of every magnetic poles toward airgap. Furthermore, the Taguchi method is utilized to optimize the proposed improved structure. Finally, the optimized rotor structure which meet the requirement of d-axis inductance, q-axis inductance and electromagnetic torque has been obtained. Compared with the initial structure, the optimized rotor structure can increase d-axis inductance effectively, and hence broaden the speed range with constant power operation.

Keywords

Speed range interior permanent magnet synchronous machine inductance Taguchi method

1. Introduction

At present, the problems about resource consumption and environmental pollution have become more and more serious. To remit these problems, traditional diesel locomotives are expected to be replaced by electric vehicles which can utilize electricity supplied by renewable resources. Electric vehicles (EVs) have a broad prospect which can overcome the inherent defects of traditional diesel locomotives. Driving motors of EVs need to meet the requirements of wide speed range with constant power operation and smooth operation, because EVs need to start and stop frequently and withstand a large acceleration and deceleration in the operating process. The interior permanent magnet machine (IPMSM) has a larger torque density due to the existence of reluctance torque, resulting from the difference between their d-axis inductances and q-axis inductances. What’s more, compared with surface permanent magnet machine (SPMSM), IPMSM have a wider speed range with constant power operation and better field-weakening ability because of the bigger d-axis inductance. These characteristics make IPMSM suitable for the vehicles’ requirements of wide speed range and high reliability. However, compared with the machines excited by electricity, the magnetic field produced by the permanent magnet (PM) limits the speed range of the IPMSM to some extent [1]. In order to achieve a higher speed and extend the speed range of EVs, it is necessary to rational design the motor structure [2].

In order to improve speed range of IPMSM, a large number of scholars all over the world have done a lot of research and analyzed the mechanism of the flux-weakening ability of IPMSM. On this basis, a variety of rotor structures which can effectively extend the speed range of the motor are proposed. In [3] and [4], the conception called variable reluctance loop is put forward, in which some movable regions in the rotor can be pushed by the centrifugal force when the motor operates under high speed. Therefore, the flow of magnetic flux can be hindered, thereby the speed range of motor is widened. In [5], adding ring soft irons in the rotor is proposed. Then, the flux path produced by PM can be changed by the d-axis current under high speed, and thus the flux from the PMs to armature windings is reduced. But it is difficult to control this kind of motor and select the sizes of soft irons. In [6], q-axis magnetic flux can be hindered by inserting the magnetic barriers into rotor. This method can reduce the total flux through the airgap with the prerequisite that d-axis flux is not influenced, and thus the speed range is increased. But the processing craft of this kind of structure is difficult and costly. It is proposed in [7] and [8] to drill holes in rotor to reduce useful flux in order to enhance the speed range of motor. In [9–11], it is suggested that the multi-layer permanent magnet structure (double layers, three layers and four layers structure) can effectively widen the speed range. However, the prototype with multi-layer permanent magnet structure is difficult to process and not suitable for mass production. The segment structure of PM is studied in [12–15], which can increase the flux leakage coefficient and decrease the flux linkage 𝜓_f produced by PMs. At the same time, d-axis inductance can be effectively improved, and further the flux-weakening ability is improved. Meanwhile, the amount of PMs should be kept constant in order to ensure that the output torque of the motor barely reduces. The memory motor proposed by [16–18] as well as the hybrid rotor structure proposed by [19] can both change the magnetic direction and the intensity of the PMs by applying stator current, which can achieve better flux-weakening performance. In addition, several kinds of motors with different rotor magnetic circuit structures are compared in [20] and [21], including radial-type, tangential-type, V -type, U-type and W-type. Although the speed range of W-type PM structure is the widest, its structure is complex and its mechanical strength is low. Besides, the torque produced by V -type permanent magnet structure is larger, and its flux-weakening ability is also quite strong.

Furthermore, in order to perfect the improved rotor structure, it is necessary to optimize the improved structure of the motor. Taguchi method is one of the most widely used methods. It can obtain the optimal combination of design variables quickly by the least amount of experiment [22]. It has been widely applied in the surface permanent magnet synchronous motor [23,24] and interior permanent magnet motor [25].

In this paper, the principle of flux-weakening control is firstly introduced. On this basis, the effect of the d-axis inductance and q-axis inductance on expanding the speed range is further explained. Secondly, an improved rotor structure for V -type interior permanent magnet motor is proposed in order to widen the flux-weakening speed range The improved rotor structure includes asymmetric segmentations of PMs and protuberances from the centers of magnetic poles toward airgap. Finally, Taguchi method is used to optimize the improved rotor structure so as to increase the d-axis inductance and expand speed range effectively. Meanwhile, it is ensured that the electromagnetic torque barely decreases, and the output ability of the motor is guaranteed.

Table 1
The main parameters of the machine

Parameter Symbol Value Unit

Outer radius of stator D _os 200 mm

Outer radius of rotor D _or 128 mm

Airgap length δ 1 mm

Lamination length of iron core L 140 mm

Rated speed n 3000 r/min

Rated torque T _N 64 Nm

Width of PM b _PM 39 mm

Relative permeability of PM 𝜇_r 1.010 –

Remanence B _r 1.125 T

Maximum supply voltage u _max 320 V

Maximum current for peak loads i _max 150 A

Maximum speed n _max 7600 r/min

Parameter	Symbol	Value	Unit
Outer radius of stator	D _os	200	mm
Outer radius of rotor	D _or	128	mm
Airgap length	δ	1	mm
Lamination length of iron core	L	140	mm
Rated speed	n	3000	r/min
Rated torque	T _N	64	Nm
Width of PM	b _PM	39	mm
Relative permeability of PM	𝜇_r	1.010	–
Remanence	B _r	1.125	T
Maximum supply voltage	u _max	320	V
Maximum current for peak loads	i _max	150	A
Maximum speed	n _max	7600	r/min

2. Principle of flux-weakening

An assumption that the influences of hysteresis loss, eddy current loss and temperature change of the motor are ignored in the analysis process is done [26]. According to the double reaction theory of the motor, the steady mathematical model of IPMSM in d-q coordinate system can be obtained as follows $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{d}=Ri_{d}-{\omega}L_{q}i_{q}\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{q}=Ri_{q}+{\omega}{\psi}_{f}+{\omega}L_{d}i_{d}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle T_{em}=\displaystyle \frac{3}{2}p[{\psi}_{f}i_{q}+(L_{d}-L_{q})i_{d}i_{q}]\end{eqnarray}$ (3) where u _d and u _q are the d-axis and q-axis voltages, respectively. i _d and i _q are the d-axis and q-axis currents, respectively. R is the phase resistance. ω is electric angular velocity. 𝜓_f is the permanent magnet flux linkage. L _d and L _q are the d-axis and q-axis inductances, respectively. T _em is the electromagnetic torque. p is the number of pole-pairs.

Fig. 1.

The speed regulation mode of IPMSM.

When the motor operates under higher speed, the resistance is much smaller than the reactance, the voltage drop of the resistance is negligible. Therefore, the terminal voltage of the motor can be obtained by combining (1) and (2) $\begin{eqnarray}\displaystyle u=\sqrt{(-{\omega}L_{q}i_{q})^{2}+({\omega}L_{d}i_{d}+{\omega}{\psi}_{f})^{2}}. & & \displaystyle\end{eqnarray}$ (4) The input voltage and current of the motor should meet the following constraints $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle u_{d}^{2}+u_{q}^{2}\leq u_{\lim }^{2}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle i_{d}^{2}+i_{q}^{2}\leq i_{\lim }^{2}\end{eqnarray}$ (6) where u _lim and i _lim are the limit values of phase voltage and phase current, respectively.

Fig. 2.

Original IPMSM machine.

Fig. 3.

Improved rotor geometry and optimization variables diagram.

Combining (4) and (5), the speed of motor can be obtained as follows $\begin{eqnarray}\displaystyle {\omega}=\displaystyle \frac{u_{\lim }}{\sqrt{({\psi}_{f}+L_{d}i_{d})^{2}+(L_{q}i_{q})^{2}}}. & & \displaystyle\end{eqnarray}$ (7)

In fact, when motor operates in the constant torque region, the voltage increases with the increase of the speed of the motor. When the base speed is reached, the terminal voltage and current reach the maximum values, as shown in Fig. 1. Thereafter, if the control mode is unchanged, the motor terminal voltage will exceed the maximum value with the speed of motor further increasing. Therefore, the flux-weakening control is needed above the base speed. The flux-weakening control essentially changes the distribution of the d-axis current and the q-axis current, so that the negative d-axis current increases and the positive q-axis current decreases. In this process, the terminal voltage and current of the motor are maintained at the maximum, and the torque decreases gradually.

The ideal maximum speed of the motor can be obtained from (7), as follows $\begin{eqnarray}\displaystyle {\omega}_{\max }=\displaystyle \frac{u_{\lim }}{{\psi}_{f}+L_{d}i_{\lim }}. & & \displaystyle\end{eqnarray}$ (8)

According to (8), because u _lim and i _lim are constant values, there are two methods to increase the maximum speed: decreasing the flux linkage 𝜓_f of PMs or increasing the d-axis inductance L _d

What’s more, it can be seen from (3) that electromagnetic torque consists of permanent-magnet torque and reluctance torque. As to permanent-magnet torque, with the decrease of 𝜓_f, the electromagnetic torque will decrease, and the load capacity of the motor will decrease too. Therefore, reducing 𝜓_f to widen the speed range does not meet the actual requirement.

As to reluctance torque, because i _d < 0 and L _d < L _q under flux-weakening control in IPMSM, if the way of increasing d-axis inductance is adopted to widen the speed range, the q-axis inductance needs to correspondingly increase to ensure the reluctance torque as well, thereby the output torque of the motor is ensured. Therefore, it is necessary to improve the rotor structure.

3. Basic structure of motor before and after improvement

3.1. Basic structure of the original motor

This paper adopts the IPMSM with the interior V -type permanent magnet cavity geometry as the research object. The structure of the IPMSM is shown in Fig. 2. The machine with this kind of PM structure has bigger reluctance torque and higher efficiency compared with other structures having the same amount of PMs [21]. The motor in Fig. 2 has 8 poles and 48 slots with double-layers distributed winding. The main parameters of the motor are shown in Table 1.

3.2. Improved rotor structure

The improved rotor structure of the IPMSM is shown in Fig. 3. In the improved rotor structure, the PMs are divided into four sections, which can increase d-axis inductance. Meanwhile, the amount of PMs are kept constant before and after the improvement. What’s more, outer circle of rotor core protrudes from centers of every magnetic poles toward airgap, which can decrease the reluctance of d-axis magnetic path, and further increase the d-axis inductance and widen the speed range.

Table 2
Levels and design area of design variables

Factors

Optimization variables

A B C D E F

Levels (mm) (°) (mm) (mm) (mm) (mm)

I 0.12 4 0.2 2.5 2.5 2.5

II 0.24 7 0.4 3.0 3.0 3.0

III 0.36 10 0.6 3.5 3.5 3.5

IV 0.48 13 0.8 4.0 4.0 4.0

V 0.60 16 1.0 4.5 4.5 4.5

	Factors
I	0.12	4	0.2	2.5	2.5	2.5
II	0.24	7	0.4	3.0	3.0	3.0
III	0.36	10	0.6	3.5	3.5	3.5
IV	0.48	13	0.8	4.0	4.0	4.0
V	0.60	16	1.0	4.5	4.5	4.5

Table 3

L ₂₅ (5⁶) orthogonal table

Experiment number	A	B	C	D	E	F
1	I	I	I	I	I	I
2	I	II	II	II	II	II
3	I	III	III	III	III	III
4	I	IV	IV	IV	IV	IV
5	I	V	V	V	V	V
6	II	I	II	III	IV	V
7	II	II	III	IV	V	I
8	II	III	IV	V	I	II
9	II	IV	V	I	II	III
10	II	V	I	II	III	IV
11	III	I	III	V	II	IV
12	III	II	IV	I	III	V
13	III	III	V	II	IV	I
14	III	IV	I	III	V	II
15	III	V	II	IV	I	III
16	IV	I	IV	II	V	III
17	IV	II	V	III	I	IV
18	IV	III	I	IV	II	V
19	IV	IV	II	V	III	I
20	IV	V	III	I	IV	II
21	V	I	V	IV	III	II
22	V	II	I	V	IV	III
23	V	III	II	I	V	IV
24	V	IV	III	II	I	V
25	V	V	IV	III	II	I

Table 4

Simulation results

Experiment number	Optimization objective			Experiment number	Optimization objective
	L _d (uH)	L _q(uH)	T _em(Nm)		L _d(uH)	L _q(uH)	T _em(Nm)
1	212.4	432.1	60.93	14	232.3	449.7	63.73
2	231.4	436.4	58.17	15	257.5	467.9	61.01
3	250.8	442.3	55.22	16	283	445.4	52.79
4	270.1	449.3	52.12	17	319.6	455.1	50.49
5	289.3	458.6	48.91	18	237.9	445.3	64.34
6	233.2	436.1	58.17	19	267.6	463.2	62.1
7	256.3	442.2	55.55	20	298.7	488.1	59.42
8	279.7	450.6	52.69	21	314.3	450.1	50.14
9	302.6	460.9	49.64	22	241.1	442	64.33
10	225	450.8	62.66	23	275	456.2	62.63
11	256.7	441.1	55.58	24	312.5	479.7	60.36
12	285.4	448.1	53.14	25	354.7	515.1	58.48
13	314.3	459.2	50.33

There are magnetic barriers between the segments of permanent magnets in the improved structure, which makes the leakage flux of the improved structure higher than that of the original structure. Due to the increase of leakage flux in the improved structure, the flux linkage of PMs decreases, and then the permanent-magnet torque descends. But at the same time, the inhomogeneous air-gap in the improved structure makes the magnetic energy of the motor increased, so the dropped electromagnetic torque caused by leakage flux is offset to some extent. Therefore, it is necessary to optimize the improved rotor structure, in order to increase the d-axis inductance and q-axis inductance of the improved structure as much as possible. Meanwhile, the output torque of the motor is ensured.

Table 5

The total average of the experimental results

L _d (μH)	L _q (μH)	T _em (Nm)
272.056	454.620	56.9172

Table 6

Average results under each level of each factor

Factor	Level	The average value			Factor	Level	The average value
		L _d (μH)	L _q (μH)	T _em (Nm)			L _d (μH)	L _q (μH)	T _em (Nm)
	I	250.80	443.74	55.070		I	274.82	457.08	57.152
	II	259.36	448.12	55.742		II	273.24	454.30	56.862
A	III	269.24	453.20	56.758	D	III	278.12	459.66	57.218
	IV	281.36	459.42	57.828		IV	267.22	450.96	56.632
	V	299.52	468.62	59.188		V	266.88	451.10	56.722
B	I	259.92	440.96	55.522		I	276.34	457.08	57.096
	II	266.76	444.76	56.336		II	276.66	459.76	57.242
	III	271.54	450.72	57.042	E	III	268.62	450.90	56.652
	IV	277.02	460.56	57.790		IV	271.48	454.94	56.874
	V	285.04	476.10	58.096		V	267.18	450.42	56.722
	I	229.74	443.98	63.198		I	281.06	462.36	57.478
	II	252.94	451.96	60.416		II	271.28	454.98	56.830
C	III	275.00	458.68	57.226	F	III	267.00	451.70	56.598
	IV	294.58	461.70	53.844		IV	269.28	450.50	56.696
	V	308.02	456.78	49.902		V	271.66	453.56	56.984

Table 7

Effect of all factors on electromagnetic performance

Factor	S _Ld	S _Lq	S _Tem
A	292.354304	76.33616	2.16093536
B	73.764384	159.13984	0.89026016
C	763.094464	38.31216	22.08859936
D	19.198304	11.46832	0.05362016
E	15.092544	12.8104	0.04955296
F	23.020064	17.30712	0.09547616

4. Optimization design

At present, there are many optimization methods for motor design, which can be divided into global optimization and local optimization according to the effective range of the final optimized structure. Among them, the local optimization calculation is simple and easy to understand. In particular, the Taguchi method proposed by Taguchi gen’ichi, a famous Japanese statistician, can realize the optimal design of multi-objective function. By establishing the orthogonal table related to the number of experiments, the final optimal solution can be obtained by using the least number of experiments. It is a scientific and effective robust design method [27,28]. In order to maximize the speed range, the improved rotor structure is optimized based on Taguchi method.

Table 8
The ending value of optimization variables

Variable A B C D E F

(mm) (°) (mm) (mm) (mm) (mm)

Value 0.60 16 0.4 3.5 3 2.5

Variable	A	B	C	D	E	F
Value	0.60	16	0.4	3.5	3	2.5

Fig. 4.

Comparisons of L _d and L _q. (a) L _d (b) L _q.

4.1. Optimization objective and variable selection

The optimization objectives are to maximize the d-axis inductance L _d and the q-axis inductance L _q. Constraint condition is that the reduction of the electromagnetic torque is not more than 3%. The selected optimization variables are represented as A, B, C, D, E and F, respectively, as shown in Fig. 2. It can be illustrated from Fig. 2 that A represents the thickness of the protuberance from the center of each pole of rotor to airgap. B represents the angle between the d-axis and the line which is from original point to the end of arc of the protuberance. Crepresents the width of magnetic barrier bridge between adjacent segmented PMs. D, E, and F represent the width of three segmented PMs to determine the position of magnetic barrier bridges, respectively.

Table 9
Comparison of the initial and final motor design indexes

Initial motor Final motor

Rated point Maximum speed point Rated point Maximum speed point

Torque 64 Nm 25.2 Nm 64 Nm 20 Nm

Stator current 91.23 A 91.2 A 91.4 A 91.39 A

Stator voltage 79.84 V 130.64 V 84.58 V 130.64 V

Power factor 0.953 0.567 0.90 0.57

The contribution of the reluctance torque to the total torque 10.02% 26.2% 9.48% 27.8%

Speed 3000 r/min 7600 r/min 3000 r/min 9600 r/min

	Initial motor	Final motor
Torque	64 Nm	25.2 Nm	64 Nm	20 Nm
Stator current	91.23 A	91.2 A	91.4 A	91.39 A
Stator voltage	79.84 V	130.64 V	84.58 V	130.64 V
Power factor	0.953	0.567	0.90	0.57
The contribution of the reluctance torque to the total torque	10.02%	26.2%	9.48%	27.8%
Speed	3000 r/min	7600 r/min	3000 r/min	9600 r/min

The values of the above six optimization variables are determined according to the geometry parameters of the machine. The limit value of A is 1 mm, that is, the value of A cannot exceed the airgap length and needs to leave sufficient margin for air gap length. In this paper, the value of A is selected from 0.12 mm to 0.60 mm. The limit value of B is 22.5° which is the angle of half the pole pitch. In order to keep sufficient margin, B is selected from 4° to 16°. In this paper, the value of C is selected from 0.2 mm to 1.0 mm (the width of magnetic barrier bridge is generally not more than 1 mm, otherwise the leakage flux is too much). What’s more, since the length of a single PM is 19.5 mm, in order to make DE and F have the same value range, the above three values are selected from 2.5 mm to 4.5 mm. According to the value ranges of the above six variables, five levels can be set up evenly for each variable. The factor-level table can be established, as shown in Table 2.

Fig. 5.

Comparisons of the flux-weakening ability. (a) T _em − n curve. (b) P − n curve. (c) Voltage-speed curve.

Fig. 6.

Electromagnetic torque comparisons of motor under original structure and improved structure.

Fig. 7.

Airgap flux density waveforms under no load condition (a) original structure (b) improved structure.

Fig. 8.

Airgap flux density waveforms under load condition (a) original structure (b) improved structure.

Fig. 9.

The stress analysis diagram of the final rotor structure at the maximum speed.

4.2. Orthogonal experiment

According to the data shown in Table 2, the orthogonal table L ₂₅ (5⁶) can be established, as shown in Table 3. Then, the finite element method is used to simulate the motor, and the results of orthogonal test are obtained, as shown in Table 4.

4.3. Data processing and analysis

4.3.1. The analysis of average value

In the Taguchi method, in order to analyze the influence of optimization variables on L _d, L _q and the constraint condition T _em, the average value of the experimental data needs to be analyzed. The average value analysis is divided into the total average analysis and the average analysis under each level of each factor.

The calculation formula of the total average value of the experimental data is shown as follows $\begin{eqnarray}\displaystyle \overline{m}=\displaystyle \frac{1}{N}\mathop{\sum }_{i=1}^{N}m_{i} & & \displaystyle\end{eqnarray}$ (9) where m is the average value of experimental results in certain column of Table 3; m _i is the value of certain optimization objectives of ith experiment in Table 4, in which N is the number of orthogonal experiments, and N is 25 in the paper. Therefore, the total average analysis of experimental results in each column of Table 4 can be obtained, as shown in Table 5.

The average value of q-axis inductance under level (I) of factor B is set as an example to explain the average analysis under each level of each factor, as follows $\begin{eqnarray}\displaystyle \overline{L}_{q∼BI}=\frac{1}{5}(L_{q1}+L_{q6}+L_{q11}+L_{q16}+L_{q21})=440.96. & & \displaystyle\end{eqnarray}$ (10) Therefore, the average values of d-axis inductance, q-axis inductance and electromagnetic torque under each level of each factor are shown in Table 6.

Then, some conclusions can be obtained from Table 6 that d-axis inductance, q-axis inductance and electromagnetic torque change with the change of levels of different factors. To be specific, with the increase of the value of the optimization variable A, the d-axis inductance, the q-axis inductance, and the electromagnetic torque T _em increase. With the increase of the value of variable B, the d-axis inductance, the q-axis inductance and the electromagnetic torque increase. With the increase of the value of the variable C, the d-axis inductance increases and the electromagnetic torque decreases. With the increase of the value of the variable F the d-axis inductance, the q-axis inductance and the electromagnetic torque increase.

As can be seen from Table 6 the combination of the level taken by each factor that maximize the d-axis inductance is A (V)B (V)C (V)D (III)E (II)F (I). The combination of the level taken by each factor that maximize the q-axis inductance is A (V)B (V)C (IV)D (III)E (II)F (I). The combination of the level taken by each factor that maximize the electromagnetic torque is A (V)B (V)C (I)D (III)E (II)F (I). It can be seen from the above analysis results that the combinations of the level taken by each factor that make the d-axis inductance, q-axis inductance, and the decrement of the electromagnetic torque smallest, respectively, are not the same. To be specific, in the three combinations, the levels taken by factors A, B, D, E and F are the same, but the levels taken by factors C is different. Therefore, it is necessary to analyze the variance of the experimental results to obtain the relative importance of the influence of each optimization variable on the d-axis inductance, q-axis inductance and electromagnetic torque. Then, the optimized scheme can be obtained.

4.3.2. Analysis of variance

The variance analysis of the experimental results is needed to obtain the relative importance of the different levels of variables on the optimization objectives. The variance expression is as follows (take factor B as an example) $\begin{eqnarray}\displaystyle S_{B}=\displaystyle \frac{1}{Q}\mathop{\sum }_{i=I}^{V}[\overline{M_{Bi}}-\overline{M}]^{2} & & \displaystyle\end{eqnarray}$ (11) where M is the total average value of certain optimization objective; M _Bi is the average value under level i of factor B of certain optimization objective. Q is the number of levels taken by each factor, and Q = 5 in this paper.

Therefore, the variances of L _d, L _q and T _em under each optimization variable can be obtained, and the results are shown in Table 7.

It can be seen from Table 7 that variable C has the greatest influence on L _d and T _em, and variable B has the greatest influence on L _q. The influence of variable A on L _d and L _q is also great. What’s more, in these variables which represent the segment position (D, E, F), the variable F has the greatest influence on L _d and L _q

For the d-axis inductance, the variance ratio of variable C to the sum of each factor is 64.313%. For the q-axis inductance, the variance ratio of variable C to the sum of each factor is 12.148%. So that, the effect of the variable C on the d-axis inductance is greater than that on the q-axis inductance. If the level of variable C is chosen as the level C (V), which makes the d-axis inductance maximum, the electromagnetic torque is 54.42 Nm and the dropping amplitude is 14.969%, which does not meet the requirements. At the same time, for the electromagnetic torque, the variance ratio of variable C to the sum of each factor is 87.174%. Therefore, for the variable C, the influence on the electromagnetic torque is greater than that on the d-axis inductance. However, the trends are reverse about increasing d-axis inductance and increasing electromagnetic torque. Therefore, in order to maximize d-axis inductance and meet the requirement of the constraint, the value of variable C should be adjusted. By the finite element analysis, if the variable C is selected as the level C (II), the electromagnetic torque is 63.96 Nm, and the dropping range is only 0.0625%, which meets the requirement of design. If the level is selected as C (III), the electromagnetic torque is 60.97 Nm, and the dropping range is 4.73%, which does not meet the requirements. Because the level of C (III) does not meet the requirement, C (IV) will not need to be investigated. As a result, the level of variable C is chosen as C(II). Finally, the optimization scheme is obtained, and the value of each optimized variable is shown in Table 8.

4.4. Optimization results and analysis

In order to cover most of the operating points of the motor, the range of d-axis current is selected as −150 ∼ 0A and the range of q-axis current is selected as 0 ∼ 150 A. Then, the d-axis inductances and q-axis inductances in the whole operation range achieved by optimized rotor geometry are compared with those achieved by initial rotor geometry, as shown in Fig. 4.

As can be seen from Fig. 4, compared with the original structure of the rotor, L _d of the final optimized structure are greatly increased and L _d are also obviously increased under different i _d and i _q conditions. Then operating speed range of the motor can be widened effectively.

Furthermore, the relationship diagram between the electromagnetic torque T _em and the motor speed n can be obtained, as shown in Fig. 5(a). The relationship diagram between the output power P and the motor speed n can be obtained, as shown in Fig. 5(b). As can be seen from Fig. 5(a), the maximum speed of the motor is expanded from 8200 r/min to 10240 r/min after adopting the optimized improved rotor structure. In Fig. 5(b), the speed range with constant power operation is defined as the speed range from the rated speed to the speed that the rated power is reached again, that is, the range indicated by the solid and dashed arrows in the diagram, respectively. As can be seen that the optimized motor structure has a wider speed range than the original motor structure. For the initial and final design at the rated and maximum speed, the design indexes are compared respectively, as shown in Table 9.

At the same time, by improving the rotor structure, the electromagnetic torque ripple can be reduced to a certain extent, as shown in Fig. 6. The operating stability of the motor can be effectively improved. In addition, the airgap magnetic density waveforms of the motor under no load and rated load conditions before and after adopting the improved rotor structure are shown in Fig. 7 and Fig. 8, respectively. It can be seen that the airgap magnetic density waveforms of the motor under no load and rated load conditions are more sinusoidal. Meanwhile, the 3rd, 5th and 7th harmonics of the waveforms are effective weakened. Finally, considering the mechanical constraints of the motor, the stress of the final rotor structure is simulated at the maximum speed, as shown in Fig. 9. The maximum stress point of the rotor in the region close to the rotor outer diameter is 39.878 MPa. Meanwhile, the maximum stress point outside the rotor shaft is 43.304 MPa. The yield strength of the silicon steel sheet is 260 MPa, so the stress at the maximum speed can meet the design requirements.

5. Conclusion

This paper presents an improved rotor structure for interior permanent magnet synchronous motor (IPMSM) with V -type PM to widen the speed range with constant power operation. Then, the improved rotor structure parameters are optimized based on the Taguchi method. Meanwhile, orthogonal experiment, average value analysis and variance analysis are adopted. Then, the influence of each optimization variable on the d-axis inductance, q-axis inductance and electromagnetic torque and the relative importance of the influence is obtained. Finally, the optimization scheme of rotor structure is obtained from the analysis results. Compared with the original rotor geometry, by adopting the optimized improved rotor structure, the d-axis inductance and q-axis inductance are effectively increased. Thereby the speed range is broadened. Meanwhile, the electromagnetic torque decreases only by 0.0625% after adopting the optimized improved rotor structure, which meets the requirements of constraints. In addition, by adopting the optimized improved rotor structure, the airgap magnetic density waveforms are improved, in which the harmonic content and the torque ripple are reduced. Then the stability of the motor is enhanced.

Footnotes

Acknowledgements

This work was supported in part by the Major Program of National Natural Science Foundation of China under Grant 51690183, and in part by the National Natural Science Foundation of China under Grant 51507111.

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	Factors
	Optimization variables
	A	B	C	D	E	F
Levels	(mm)	(°)	(mm)	(mm)	(mm)	(mm)
I	0.12	4	0.2	2.5	2.5	2.5
II	0.24	7	0.4	3.0	3.0	3.0
III	0.36	10	0.6	3.5	3.5	3.5
IV	0.48	13	0.8	4.0	4.0	4.0
V	0.60	16	1.0	4.5	4.5	4.5

	Initial motor		Final motor
	Rated point	Maximum speed point	Rated point	Maximum speed point
Torque	64 Nm	25.2 Nm	64 Nm	20 Nm
Stator current	91.23 A	91.2 A	91.4 A	91.39 A
Stator voltage	79.84 V	130.64 V	84.58 V	130.64 V
Power factor	0.953	0.567	0.90	0.57
The contribution of the reluctance torque to the total torque	10.02%	26.2%	9.48%	27.8%
Speed	3000 r/min	7600 r/min	3000 r/min	9600 r/min

Optimal design of rotor geometry in interior permanent magnet machine

Abstract

Keywords

1. Introduction

3.1. Basic structure of the original motor

3.2. Improved rotor structure

Table 2 Levels and design area of design variables Factors Optimization variables A B C D E F Levels (mm) (°) (mm) (mm) (mm) (mm) I 0.12 4 0.2 2.5 2.5 2.5 II 0.24 7 0.4 3.0 3.0 3.0 III 0.36 10 0.6 3.5 3.5 3.5 IV 0.48 13 0.8 4.0 4.0 4.0 V 0.60 16 1.0 4.5 4.5 4.5

Table 8 The ending value of optimization variables Variable A B C D E F (mm) (°) (mm) (mm) (mm) (mm) Value 0.60 16 0.4 3.5 3 2.5

4.3. Data processing and analysis

4.3.1. The analysis of average value

5. Conclusion

Footnotes

Acknowledgements

References

Table 2
Levels and design area of design variables

Factors

Optimization variables

A B C D E F

Levels (mm) (°) (mm) (mm) (mm) (mm)

I 0.12 4 0.2 2.5 2.5 2.5

II 0.24 7 0.4 3.0 3.0 3.0

III 0.36 10 0.6 3.5 3.5 3.5

IV 0.48 13 0.8 4.0 4.0 4.0

V 0.60 16 1.0 4.5 4.5 4.5

Table 8
The ending value of optimization variables

Variable A B C D E F

(mm) (°) (mm) (mm) (mm) (mm)

Value 0.60 16 0.4 3.5 3 2.5