Optimal Design and Control of a Spoke-Type IPM Motor with Asymmetric Flux Barriers to Improve Torque Density

Abstract

This paper studies the optimal design and control scheme of a spoke-type interior permanent magnet motor (SIPM). An asymmetric rotor structure with flux barriers is designed to improve the torque density of SIPM. The design method improves the torque density by approximating the maximum value of the magnetic torque and the reluctance torque, wherein the torque components are separated by the frozen permeability method (FPM) to evaluate the contribution. This scheme does not increase the amount of permanent magnets or the motor size, and reduces motor weight while increasing motor torque output. Firstly, the asymmetric flux barriers are applied in a 27/4 SIPM to illustrate the design principle. Further, by optimizing the width of flux barriers, based on finite-element analyze (FEA), a higher torque density is obtained. Compared with the basic model, the output torque and the torque density of the optimal model are both increased. Based on the optimal model, an angle scanning method is proposed to orient the flux vector and dq-axis. Then, the mathematical model of the optimal model is established, and the maximum torque per ampere (MTPA) control system is designed. Compared with the conventional control system, the proposed control system has a higher torque per ampere (TPA), which shows that the designed control system can give full play to the advantages of the high torque density.

1. Introduction

Permanent magnet motors are used in various fields due to their high torque density, high efficiency, etc. [1,2,3,4] Most permanent magnet motors use high-performance rare earth permanent magnet materials [5]. However, the issues, such as the outbreak of the epidemic, have caused a shortage and price increase in rare-earth permanent magnet materials, which brings challenges to the application of mass production [6,7]. To solve this problem, people are constantly looking for alternatives. Ferrite permanent magnets have proven to be an effective alternative, but their magnetic density per unit volume is small, where the improved design is required to enhance motor performance.

There are many schemes which can improve motor performance. Among them, the motors with asymmetric rotor structures have been proved to be an effective scheme to improve torque density [8,9,10,11]. This type of motor can superimpose the maximum value of reluctance torque and magnetic torque at the same or similar current phase angles, thereby increasing the torque density of the motor. This type of motor can improve utilization of magnetic torque and reluctance torque, and increase the torque without increasing the amount of permanent magnets and motor size [11,12], which can effectively reduce the adverse effects of ferrite permanent magnet. Among them, unlike some schemes that make the structure too complicated [8,9,10], the reference [11] only uses the flux barriers to improve the torque output while reducing the rotor weight, which is advantageous in some applications where the motor needs to follow the mechanical motion (such as electric vehicles, ships, aviation, etc.), which can reduce energy consumption [13,14,15]. However, the basic model in [11] adopts a “V”-shaped permanent magnet structure, which is still more complex and expensive than the spoke-type interior permanent magnet motor (SIPM) [16]. Based on this, a novel SIPM is proposed, where asymmetric flux barriers are applied to obtain high torque density and low weight.

In addition, in the field of motor control, scholars have proposed many motor control methods [17,18]. Among them, the maximum torque per ampere (MTPA) control scheme is a mainstream efficient control scheme [19,20,21]. However, due to the asymmetric structure of the rotor, the direction of the flux vector is shifted, and the law of the torque generated becomes more complicated. At the same time, due to the change of the magnetic circuit of the dq-axis, the dq-axis oriented by the geometric centerline has a serious cross-saturation effect. If the conventional method in [20,21] is directly applied, the control system would have a poor effect. A mathematical model is established in [22] to solve these problems. However, the modeling process is complex and only considers the case where the two torque component maximums are summed at the same current phase angle. To sum up, a new set of orientation schemes is needed to establish the torque equation, achieve the decoupling of the dq-axis magnetic circuit, and achieve the design of the control system.

In this paper, a novel SIPM and its control system are proposed to improve torque density, which utilizes asymmetric flux barriers. The effect of torque density improvement is achieved by approximating the maximum values of magnetic torque and reluctance torque. The design does not increase the amount of permanent magnets or motor size, and increases the output torque while reducing motor weight. To solve the problem of the flux linkage offset caused by the utilization of asymmetric flux barriers, a scanning angle method is proposed to establish the mathematical model of the SIPM. Then, the MTPA control curve is derived to establish the control system. The remainder of this paper is organized as follows. Section 2 analyzes the impact of the utilization of asymmetric flux barriers on the performance of SIPM, and optimizes the asymmetric flux barriers. Section 3 designs the control system of the novel SIPM and compares its control effect with the conventional control scheme. Finally, Section 4 presents the conclusion of this paper.

2. Motor Modeling

2.1. Basic Model and Toque Performance

To evaluate the effect of the asymmetric rotor structures on motor performance, a traditional SIPM is shown in Figure 1a as basic model. The stator has twenty-seven half-open slots with distributed windings. The rotor is inserted with spoke-type ferrite permanent magnets. The specific parameters of the traditional SIPM motor are listed in

Table 1. Specific parameters of the basic SIPM motor.

Item	Unit	Value
Magnet poles/Stator slots	-	4/27
Stator outer diameter	mm	130
Rotor outer diameter	mm	74
Motor axial length	mm	75
Airgap length	mm	0.5
Remanence of magnet	T	0.4
Rated speed	rpm	1000
Rated peak current	A	4

.

In the dq coordinate system, the torque of the SIPM motor can be expressed as:

$$T=\frac{3p}{2}[\frac{1}{2}(L_q-L_d)I_{\mathrm{a}}^2\sin 2\beta +{\lambda }_{\mathrm{PM}}{I}_{\mathrm{a}}\cos \beta ]$$

(1)

where p is the number of pole pairs, L_d and L_q are the dq-axis inductances, I_a represents the peak value of the phase current, β means the angle of the current phase under the dq-axis, and λ_PM represents the peak fundamental value of the magnet flux linkage. The first term of the torque component is the reluctance torque and the second one is the magnetic torque.

The frozen permeability method (FPM) is used to separate magnetic torque (T_PM) and reluctance torque (T_re). Firstly, the total torque (T_e) generated by the permanent magnet and the rated current was calculated by finite element analysis (FEA). Based on this, the T_re generated only by the rated current excitation was obtained by the FPM. Finally, the T_PM was obtained by subtracting the T_re from the T_e. A plot of typical per-unit torque performance for SIPM is shown in Figure 1b. This figure shows that the peaks of the two torque components are usually not at the same current phase angle, which leads to the T_PM and T_re of the SIPM are not being fully utilized.

2.2. Proposed Model and Comparison

A proposed SIPM with the design variable selected as the open angle of the assisted barriers θ is shown in Figure 2, where θ is 46°. For a reasonable comparison, the size, rated value, and the amount of permanent magnet material of the proposed model are consistent with the basic model.

FEA under no-load conditions is firstly carried out. Figure 3 shows the magnetic flux line distribution of the investigated models. Differing from that of the basic model in Figure 3a, the magnetic flux lines of the proposed models exhibit asymmetric flux distributions as shown in Figure 3b.

Through FPM results in Figure 4, it can be seen that the difference between T_{PM_max} and T_{re_max} of the basic and the proposed model is 60° and 30°, respectively, which improved output torque. Therefore, the effect of increasing torque while reducing weight is achieved. At the same time, the motor efficiency also increases due to the increase in torque. The efficiency is defined as

$$\eta =\frac{P_{\mathrm{out}}}{P_{\mathrm{out}}+P_{\mathrm{copper}}+P_{\mathrm{iron}}}$$

(2)

where P_copper is the copper loss, P_iron is the iron loss in the stator and rotor core, and P_out is the output power.

A comparison of electromagnetic torque is shown in Figure 5. The torque ripple factor is defined as the ratio of the peak-to-peak value to the average value of torque, i.e.

$$K_t=\frac{T_{\mathrm{e}\_\max }-T_{\mathrm{e}\_\min }}{T_{\mathrm{e}\_\mathrm{ave}}}$$

(3)

The torque density is defined as the ratio of torque-to-weight, i.e.

$$TPW=\frac{T_{\mathrm{e}\_\mathrm{ave}}}{m}$$

(4)

The detailed data of the basic and proposed model are listed in

Table 2. Comparison of performance data of the basic and proposed models.

Item	Unit	Basic Model	Proposed Model
Rotor weight	kg	2.152	1.945
Max. torque (Average)	Nm	6.646	6.881
Torque ripple	%	11.659	9.991
Torque density	Nm/kg	3.088	3.538
Output power	W	696.074	720.652
Efficiency	%	93.936	94.092

.

2.3. Optimal Design

Based on the proposed model, an optimization process was implemented to obtain a larger torque density. The optimal variable was selected as the assisted barriers span angle θ in Figure 2, and its right boundary is fixed. Considering that the magnetic flux linkage offset effect is more obvious when the assisted barriers cross the centerline of the rotor core, the lower limit of θ was set to 34°. At the same time, if the assisted barriers are too wide, a large amount of magnetic flux leakage will occur, so the upper limit of θ was set to 66°, which is the value when the rotor is just symmetrical. To sum up, the range of θ is shown in Equation (5).

Design variables: 34° ≤ θ ≤ 66°

(5)

FEA results of the optimization process are shown in Figure 6a. It can be seen that the torque density of the motor is the largest when θ = 54°. At this time, the torque characteristics are shown in Figure 6b. It can be seen that the T_{PM_max} and the T_{re_max} are superimposed at almost the same current phase angle. Figure 7 is a comparison figure of the electromotive force (EMF), air gap flux density and maximum output torque. As can be seen from Figure 7a,b, the RMS value of the optimized model EMF decreases and the harmonic content increases, which is due to the change in the air gap flux density caused by the change of the asymmetric flux barriers as shown in Figure 7c,d, where the total harmonics distortion (THD) of the proposed model and the optimal model are 78.43% and 97.36%, respectively. Meanwhile, the motor output torque is reduced by 0.0188 Nm as shown in Figure 7e, but it is only 0.27% of the torque output of the proposed model, which is negligible. This is because the torque superposition effect cancels out the output torque reduction caused by flux leakage. The comparison of detailed performance data between the proposed model and the optimized model is shown in

Table 3. Comparison of performance data of the proposed and optimal models.

Item	Unit	Proposed Model	Optimal Model
Rotor weight	kg	1.945	1.909
Max. torque (Average)	Nm	6.881	6.862
Torque density	Nm/kg	3.538	3.595
EMF (RMS)	V	39.581	36.241

.

3. Control System Design

3.1. Mathematical Modeling of the Optimal Model

Before establishing the mathematical model of the motor, make the following assumptions:

• Ignore the effect of magnetic saturation;
• Ignore the magnetic field harmonics.

3.1.1. Orientation of the Main Flux Linkage Vector

A scanning electrical angle method is proposed to identify the position of the main flux linkage vector, which achieves the transformation of the flux linkage in the three-phase coordinate system shown in Figure 8a into the dq_f coordinate system shown in Figure 8b. Firstly, the objective function was identified as shown in Formula (6), which can minimize the projection of the main flux linkage vector on the q_f-axis and meet the requirements of the traditional flux linkage orientation theory. Then, the accuracy of θ_f was taken as two decimal places, and the final solution result is θ_f = 189.77° (electrical degree).

$$\{\begin{array}{l}{\psi }_{d\_\mathrm{f}}>0\\ \min \left|average({\psi }_{q\_\mathrm{f}})-0\right|\end{array}$$

(6)

where

$$\left[\begin{array}{l}{\psi }_{d\_\mathrm{f}}\\ {\psi }_{q\_\mathrm{f}}\\ {\psi }_{0\_\mathrm{f}}\end{array}\right]=\mathit{P}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{f}})\left[\begin{array}{c}{\psi }_A\\ {\psi }_B\\ {\psi }_C\end{array}\right]=\frac{2}{3}\left[\begin{array}{ccc}\cos ({\theta }_{\mathrm{e}}+{\theta }_{\mathrm{f}})& \cos ({\theta }_{\mathrm{e}}+{\theta }_{\mathrm{f}}-120°)& \cos ({\theta }_{\mathrm{e}}+{\theta }_{\mathrm{f}}+120°)\\ -\sin ({\theta }_{\mathrm{e}}+{\theta }_{\mathrm{f}})& -\sin ({\theta }_{\mathrm{e}}+{\theta }_{\mathrm{f}}-120°)& -\sin ({\theta }_{\mathrm{e}}+{\theta }_{\mathrm{f}}+120°)\\ 1/2& 1/2& 1/2\end{array}\right]\left[\begin{array}{c}{\psi }_A\\ {\psi }_B\\ {\psi }_C\end{array}\right]$$

where ψ_A, ψ_B, ψ_C are the there-phase flux, ψ_{d_f}, ψ_{q_f}, ψ_{0_f} are the dq0_f-axis flux; θ_f is the transform angle for flux; and θ_e is the electrical angle of rotor position.

It can be seen from Figure 8 that the irregular flux linkage waveform can be transformed into relatively constant direct axis flux linkage and quadrature axis flux linkage by selecting a suitable θ_f, and the quadrature-axis flux linkage ψ_{q_f} is approximately 0. The average value of the direct axis flux linkage ψ_{d_f} is 0.2259 Wb. It is worth mentioning that, due to the star connection of motor windings, the zero-axis flux linkage ψ_{0_f} does not play the role of electromechanical energy conversion, so its influence is ignored here.

3.1.2. Orientation of the Inductance

According to the traditional dq-axis orientation theory, the purpose of the inductance orientation is to achieve the decoupling of the inductance, that is, the mutual inductance is 0. According to this idea, the inductance orientation was carried out according to the principle of the smallest mutual inductance of dq_L-axis. And the objective function can be expressed as:

$$\{\begin{array}{l}\min ({\displaystyle \sum \left|L_{dq\_\mathrm{L}}^n\right|}+\left|L_{qd\_\mathrm{L}}^n\right|)\\ \min (\left|{\theta }_{\mathrm{L}}-{\theta }_{\mathrm{f}}\right|)\end{array}$$

(7)

where

$$\left[\begin{array}{ccc}{L}_{d\_\mathrm{L}}& M_{dq\_\mathrm{L}}& M_{d0\_\mathrm{L}}\\ M_{qd\_\mathrm{L}}& L_{q\_\mathrm{L}}& M_{q0\_\mathrm{L}}\\ M_{0d\_\mathrm{L}}& M_{0q\_\mathrm{L}}& L_{0\_\mathrm{L}}\end{array}\right]=\mathit{P}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{L}})\left[\begin{array}{ccc}{L}_{AA}& M_{AB}& M_{AC}\\ M_{AB}& L_{BB}& M_{BC}\\ M_{AC}& M_{BC}& L_{CC}\end{array}\right]{\mathit{P}}^{-1}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{L}})$$

where L_ii is the self-inductance of winding i; M_ij is the mutual inductance of winding i and j, and θ_L is the transform angle for inductance.

It is worth mentioning that in (7), due to the symmetry of the magnetic circuit, there are four θ_L that satisfy the first equation. Thus the second formula is added as a further constraint. Meanwhile, considering that the formula of the induced voltage is u = L(di/dt), that is, the induced voltage is proportional to the first power of the inductance, so the decoupling effect is the best when n = 1 in (7).

After the scan, θ_L was identified to 173.66° (electrical degree). The transformation effect is shown in Figure 9.

As shown in Figure 9, a transformation of the irregular three-phase inductance into a relatively constant dq-axis inductance is achieved by an appropriate selection of θ_L. Among them, the average value of the direct axis inductance L_{d_L} is 0.0845 H, while the quadrature axis inductance L_{q_L} is 0.237 H, and the mutual inductance of the quadrature axis and the direct axis is approximately 0.

3.1.3. Mathematical Model Equation

To sum up, the orientation position of each electrical quantity are shown in Figure 10b, which is compared with the basic model in Figure 10a. As can be seen from Figure 10, the flux linkage of the optimal model is shifted to the q-axis, which is in line with the design concept described in Section 2.2.

In the SIPM, the voltage equation of the three-phase winding can be expressed as

$$\left[\begin{array}{l}{u}_A\\ u_B\\ u_C\end{array}\right]=R\left[\begin{array}{l}{i}_A\\ i_B\\ i_C\end{array}\right]+\frac{\mathrm{d}}{\mathrm{d}t}\left\{\left[\begin{array}{ccc}{L}_{AA}& M_{AB}& M_{AC}\\ M_{BA}& L_{BB}& M_{BC}\\ M_{CA}& M_{CB}& L_{CC}\end{array}\right]\left[\begin{array}{l}{i}_A\\ i_B\\ i_C\end{array}\right]+\left[\begin{array}{l}{\psi }_A\\ {\psi }_B\\ {\psi }_C\end{array}\right]\right\}$$

(8)

where u_A, u_B, u_C are the three-phase voltage of motor; R is phase resistance, and i_A, i_B, i_C are the three-phase current.

After ignoring the fluctuations of flux and inductance in the dq_f or dq_L coordinate system, the differential term on the right-hand side of (8) is transformed using the Park transform to

$$\begin{array}{l}\mathit{P}({\theta }_{\mathrm{e}},{\theta }_L)\left\{\left[\begin{array}{ccc}{L}_{AA}& M_{AB}& M_{AC}\\ M_{BA}& L_{BB}& M_{BC}\\ M_{CA}& M_{CB}& L_{CC}\end{array}\right]\left[\begin{array}{l}{i}_A\\ i_B\\ i_C\end{array}\right]+\left[\begin{array}{l}{\psi }_A\\ {\psi }_B\\ {\psi }_C\end{array}\right]\right\}\\ =\mathit{P}({\theta }_{\mathrm{e}},{\theta }_L)\left[\begin{array}{ccc}{L}_{AA}& M_{AB}& M_{AC}\\ M_{BA}& L_{BB}& M_{BC}\\ M_{CA}& M_{CB}& L_{CC}\end{array}\right]{\mathit{P}}^{-1}({\theta }_{\mathrm{e}},{\theta }_L)\mathit{P}({\theta }_{\mathrm{e}},{\theta }_L)\left[\begin{array}{l}{i}_A\\ i_B\\ i_C\end{array}\right]+\left[\begin{array}{ccc}\cos ({\theta }_L-{\theta }_f)& -\sin ({\theta }_L-{\theta }_f)& 0\\ \sin ({\theta }_L-{\theta }_f)& \cos ({\theta }_L-{\theta }_f)& 0\\ 0& 0& 1\end{array}\right]\mathit{P}({\theta }_{\mathrm{e}},{\theta }_f)\left[\begin{array}{l}{\psi }_A\\ {\psi }_B\\ {\psi }_C\end{array}\right]\\ =\left[\begin{array}{ccc}{L}_{d\_\mathrm{L}}& & \\ & L_{q\_\mathrm{L}}& \\ & & L_{0\_\mathrm{L}}\end{array}\right]\left[\begin{array}{l}{i}_{d\_\mathrm{L}}\\ i_{q\_\mathrm{L}}\\ i_{0\_\mathrm{L}}\end{array}\right]+\left[\begin{array}{c}{\psi }_{d\_\mathrm{f}}\cos ({\theta }_L-{\theta }_f)\\ {\psi }_{d\_\mathrm{f}}\sin ({\theta }_L-{\theta }_f)\\ {\psi }_{0\_\mathrm{f}}\end{array}\right]\end{array}$$

(9)

Therefore, the flux linkage equation of the optimal model is obtained as:

$$\{\begin{array}{l}{\psi }_{d\_\mathrm{L}}=L_{d\_\mathrm{L}}{i}_{d\_\mathrm{L}}+{\psi }_{d\_\mathrm{f}}\cos {\theta }_{\mathrm{s}}\\ {\psi }_{q\_\mathrm{L}}=L_{q\_\mathrm{L}}{i}_{q\_\mathrm{L}}+{\psi }_{d\_\mathrm{f}}\sin {\theta }_{\mathrm{s}}\end{array}$$

(10)

where θ_s = θ_f − θ_m, ψ_{d_L} and ψ_{q_L} are the total flux linkages in the d_L and q_L directions, respectively.

(8) is transformed using the Park transform to

$$\begin{array}{l}\left[\begin{array}{l}{u}_d\\ u_q\\ u_0\end{array}\right]=\mathit{P}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{L}})\left[\begin{array}{l}{u}_A\\ u_B\\ u_C\end{array}\right]=R\mathit{P}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{L}})\left[\begin{array}{l}{i}_A\\ i_B\\ i_C\end{array}\right]+\mathit{P}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{L}})\frac{\mathrm{d}}{\mathrm{d}t}\left\{{\mathit{P}}^{-1}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{L}})\mathit{P}({\theta }_{\mathrm{e}},{\theta }_{\mathrm{L}})\left(\left[\begin{array}{ccc}{L}_{AA}& M_{AB}& M_{AC}\\ M_{BA}& L_{BB}& M_{BC}\\ M_{CA}& M_{CB}& L_{CC}\end{array}\right]\left[\begin{array}{l}{i}_A\\ i_B\\ i_C\end{array}\right]+\left[\begin{array}{l}{\psi }_A\\ {\psi }_B\\ {\psi }_C\end{array}\right]\right)\right\}\\ =R\left[\begin{array}{l}{i}_{d\_\mathrm{L}}\\ i_{q\_\mathrm{L}}\\ i_{0\_\mathrm{L}}\end{array}\right]+\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{l}{L}_{d\_\mathrm{L}}{i}_{d\_\mathrm{L}}\\ L_{q\_\mathrm{L}}{i}_{q\_\mathrm{L}}\\ L_{0\_\mathrm{L}}{i}_{0\_\mathrm{L}}+{\psi }_{0\_\mathrm{f}}\end{array}\right]+{\omega }_{\mathrm{e}}\left[\begin{array}{ccc}& -1& \\ 1& & \\ & & \end{array}\right]\left[\begin{array}{l}{L}_{d\_\mathrm{L}}{i}_{d\_\mathrm{L}}+{\psi }_{d\_\mathrm{f}}\cos ({\theta }_L-{\theta }_f)\\ L_{q\_\mathrm{L}}{i}_{q\_\mathrm{L}}+{\psi }_{d\_\mathrm{f}}\sin ({\theta }_L-{\theta }_f)\\ L_{0\_\mathrm{L}}{i}_{0\_\mathrm{L}}+{\psi }_{0\_\mathrm{f}}\end{array}\right]\end{array}$$

(11)

Therefore, the voltage equation can be expressed as:

$$\{\begin{array}{l}{u}_{d\_\mathrm{L}}=R{i}_{d\_\mathrm{L}}+L_{d\_\mathrm{L}}\frac{d{i}_{d\_\mathrm{L}}}{dt}-{\omega }_{\mathrm{e}}{\psi }_{q\_\mathrm{L}}\\ u_{q\_\mathrm{L}}=R{i}_{q\_\mathrm{L}}+L_{q\_\mathrm{L}}\frac{d{i}_{q\_\mathrm{L}}}{dt}+{\omega }_{\mathrm{e}}{\psi }_{d\_\mathrm{L}}\end{array}$$

(12)

where ω_e is the electrical angular velocity of the motor (rad/s), and u_{d_L}, u_{q_L} are the d_L-axis and q_L-axis voltage in the dq_L coordinate system, respectively.

The electromagnetic torque can be expressed as:

$$\begin{array}{l}{T}_{\mathrm{e}}=\frac{3}{2}p(i_{q\_\mathrm{L}}{\psi }_{d\_\mathrm{L}}-i_{d\_\mathrm{L}}{\psi }_{q\_\mathrm{L}})\\ =\frac{3}{2}p({\psi }_{d\_\mathrm{f}}(i_{q\_\mathrm{L}}\cos {\theta }_{\mathrm{s}}-i_{d\_\mathrm{L}}\sin {\theta }_{\mathrm{s}})+i_{d\_\mathrm{L}}{i}_{q\_\mathrm{L}}(L_{d\_\mathrm{L}}-L_{q\_\mathrm{L}}))\end{array}$$

(13)

3.2. Design of the Motor Control System

To reduce copper loss and enable the motor to output rated torque, it is necessary to identify the MTPA curve of the motor, which is finding the minimum value of i_{d_L}² + i_{q_L}² under a certain T_e. To solve this problem, the Lagrange’s extreme value theorem was employed and the Lagrange multiplier λ was taken to establish the auxiliary function F, which is:

$$F=i_{d\_\mathrm{L}}^2+i_{q\_\mathrm{L}}^2+\lambda \left\{(3/2)p\left[{\psi }_{d\_\mathrm{f}}(i_{q\_\mathrm{L}}\cos {\theta }_{\mathrm{s}}-i_{d\_\mathrm{L}}\sin {\theta }_{\mathrm{s}})+i_{d\_\mathrm{L}}{i}_{q\_\mathrm{L}}(L_{d\_\mathrm{L}}-L_{q\_\mathrm{L}})\right]-T_{\mathrm{e}}\right\}$$

(14)

According to the Lagrange’s extreme value theorem, the required relationship between i_{q_L} and i_{d_L} is the extreme value point of the function F, which can be expressed as:

$$\frac{\partial F}{\partial i_{d\_\mathrm{L}}}=0,\frac{\partial F}{\partial i_{q\_\mathrm{L}}}=0,\frac{\partial F}{\partial \lambda }=0$$

(15)

By substituting the data and discarding the root, the solution was obtained as:

$$i_{d\_\mathrm{L}}=\left(1.2822-\sqrt{1.644+3.66(0.915i_{q\_\mathrm{L}}^2+0.376i_{q\_\mathrm{L}})}\right)/1.83$$

(16)

The current trajectory at this time is shown in Figure 11a, where the brown line is the current trajectory with the condition of MTPA, and the other lines are the torque contour lines.

To further obtain the relationship between torque and current, (13) and (16) need to be combined. However, this will result in a higher-order equation, and the solution of it will cause a large computational burden in the control system. To solve this problem, the relationship between T_e, i_{d_L}, and i_{q_L} is obtained by the numerical method, and the results are shown in Figure 11b.

In summary, the control system of the optimal model is established as shown in Figure 12, where θ_e is the angle between the +A-axis and the d_L-axis. The parameters of the proposed control system are shown in

Table 4. Parameters of the control system.

Item	Unit	Value
Phase resistance	Ω	2.04
Number of pole pairs	-	2
No-load flux linkage	Wb	0.2259
Moment of inertia	kg·m2	0.001274
Direct axis inductance	H	0.0845
Quadrature inductance	H	0.237
Motor sampling time	μs	25
Inverter sampling time	μs	25
Current loop sampling time	μs	50
Speed loop sampling time	μs	500
Quadrature current loop Kp	-	1066.5
Quadrature current loop Ki	-	9180
Direct axis current loop Kp	-	380.25
Direct axis current loop Ki	-	9180
Speed loop Kp	-	0.0072
Speed loop Ki	-	0.16

. As a comparison, the conventional MTPA control system is established, and the system parameters are consistent with those shown in Table 4. It is worth mentioning that the conventional scheme locates the d-axis at the position of ψ_{d_f}, and the dq-axis given current meets (17) [20,21].

$$i_d^{*}=\frac{{\psi }_{d\_f}}{2(L_{d\_L}^{}-L_{q\_L}^{})}-\sqrt{\frac{{\psi }_{d\_f}^2}{4{(L_{q\_L}^{}-L_{d\_L}^{})}^2}+i_q^{{*}^2}}$$

(17)

where $i_d^{*}$, $i_q^{*}$ are the dq-axis given current of the conventional MTPA control system.

3.3. Simulation Results

The control systems of proposed system and conventional MTPA system were started at a given speed of 1000 rpm, loaded half of the rated torque at 0.4 s, and loaded with the rated torque at 0.8 s. The response of the proposed system is shown in Figure 13. The response of the conventional system is shown in Figure 14.

As shown in Figure 13a, the proposed system only takes about 0.05 s to reach the rated speed, which fully reflects the advantages of the rapid acceleration due to its high torque density. During the loading process, the system only takes about 0.15 s to recover to a steady state, so the dynamic performance is excellent. In the steady state, the given speed can be tracked without error. As shown in Figure 13c and Figure 14c, when the rated load is loaded at rated speed, the peak current values of the conventional and proposed control system are 5.39 A and 4.38 A, respectively, which indicates that the proposed system has a higher TPA. It is worth mentioning that during the no-load state, the conventional control system still has current output of about 1.61 A peak value, which is due to the fact that the current output is needed to cancel the effects caused by the flux offset. What’s more, the current peak value in the rated state of the proposed system is 4.38 A, so the error is only 0.38 A compared with the FEA. This is because the influence of harmonic magnetic fields is ignored in the modeling process, which also generates torque. However, this slight error can be canceled by an integral part of the PI controller in the real system. Therefore, the simulation results fully prove the accuracy of the modeling and the effectiveness of the control system.

4. Conclusions

In this paper, an optimal design and a control system for the SIPM were proposed using asymmetric barriers to improve torque density by approximating the maximum value of the magnetic torque and the reluctance torque. Based on the FEA results, compared with the basic model, it was found that the torque density, power and efficiency of the proposed model increased by 14.6%, 3.5% and 0.16%, respectively, and the torque ripple decreased by 1.7%. After optimization, the torque density of the optimal model has increased by 16.4% compared with the basic model. As shown in simulation results, compared to the conventional control system, the proposed control system saves 18.7% current when outputting rated torque. And the proposed control system took only 0.05 s to reach the rated speed. The current error was only 0.38 A compared with the FEA, which can be canceled by PI controller in the real system.