Analytical study on the cogging torque in surface-mounted PM machines;effect of armature notches and PM segments

Abstract

This paper presents the cogging torque calculation in radial-flux surface-mounted permanent-magnet machines based on energy approach, air-gap flux density and permeance functions. Magnet segmentation and teeth notching are two important techniques in reducing the cogging torque, so in this paper an analytical approach is investigated to model these effects on the produced cogging torque. This approach employs new methods to model the teeth notch and PM segments based on Fourier expansion and simplified flux distribution. The analysis results and comparisons verify the validity of the model that has advantages in sensitivity analysis and structure optimizations of surface-mounted permanent-magnet machines from the aspect of cogging torque. This model could be extended for other types of PM electrical machines.

Keywords

Analytical cogging torque notch PM machine segmented PM

1. Introduction

Permanent magnet (PM) machines are widely used in high-performance and high-torque applications. However, in slotted PM machines, there exists cogging torque which may cause noise and vibration and affect the control accuracy and machine performance. Cogging torque produces from the interaction between PMs and slotted iron structure in nearly all types of PM machines with variable air-gap. Analytical calculation of cogging torque has been performed by the energy methods [1–3], using flux density function and air-gap permeance function [4,5].

In the design techniques to reduce the cogging torque, some papers focused on PM poles parameters such as pole arc [6,7], magnet shifting and skewing [8] and so on. There are researches on armature parameters such as teeth pairing [9], teeth notching [5], teeth skewing, teeth chamfering [10–12] and non-uniformly distributed teeth [13]. In addition, optimization methods to reduce the cogging torque and enhance the back-emf have been investigated [14–21].

[22] proposes a dual notched design of radial-flux PM motors and the effect of different notch shapes on the cogging torque. Similar analysis is reported for flux-switching PM machine in [23] and [24]. The embedding of notches in the armature teeth is a design method, where the optimal position and shape of these notches is found through the optimization. It is well-known that there is a trade-off between the cogging torque reduction and magnetic flux decrement by the integration of notches.

The magnet segmentation is another way to minimize the cogging torque. The key idea of this method is to set the air-gap flux density distribution by segmenting the magnet pole. This method can be applied for surface-mounted PM and interior PM machines motors by adjusting the optimal width and the displacement between them as well as imposing notches into the segments [25,26].

The optimizations methods mentioned above require the finite element analysis (FEA) [27] or analytical prediction of cogging torque. Although the electromagnetic analysis can be achieved by the numerical methods, the analytical solution can facilitate the optimal design procedure. Analytical models have been proven to be effective tools in the analysis of electromagnetic machines by providing fast and accurate solutions and are very good compromise between simplicity and accuracy. In contrast, FEA requires considerable time and memory, especially for optimization, where structure modelling and its field solution should be performed repeatedly. But, for the analytical model, the field solution uses the magnetic field equations and so is fast. Consequently, it is possible to merge the magnetic field equations in a simple or an intelligent optimization algorithm that increases the convergence rate.

So, the main contribution of this paper is introducing an analytical approach to model the notches on armature teeth as well as rotor segmented PMs for calculating the cogging torque of radial-flux surface-mounted PM machine. This new model is based on simplified air-gap flux and permeance functions, is flexible to geometrical parameters of notches and segments, and provides a convenient tool for the analysis of cogging torque and the optimization of machine structure taking into account two significant techniques at the same time.

Although the energy method for cogging torque calculation and Fourier expansion model for air-gap flux density and permeance are known, but according to the best knowledge of the authors, these techniques have not applied to analysis of notches on armature teeth and rotor segmented PMs simultaneously in cogging torque calculation of radial-flux surface-mounted PM machine.

Sections 2 and 3 present the detailed explanation about the analytical torque prediction and the flux density and permeance calculations, based on these the analytical model has been introduced. Then, Section 4 presents the numerical field and torque calculations that are based on FEA with a commercial software. The objective is to compare results and verify the analytical model.

Fig. 1.

A typical shape of a surface-mounted PM machine, (a) structure, (b) planar model of air-gap flux and permeance.

2. Analytical prediction of cogging torque

A typical geometry of the radial-flux surface-mounted PM machine is shown in Fig. 1(a) while the following assumptions are considered in the study:

The permeability of core is infinite and the permeability of PM is the same as that of air.

The slots are simplified to be trapezoidal.

The magnetic field crosses the air gap perpendicularly and in a straight line.

End effect and flux leakage is considered to be negligible.

Cogging torque that prevents the motor to rotate smoothly, is caused by the magnetic co-energy variation in PM machines as rotor rotates. It can be defined as [13]: $\begin{eqnarray}\displaystyle T_{\text{cogg}}=-{\displaystyle \frac{\partial W}{\partial {\alpha}}} & & \displaystyle\end{eqnarray}$ (1) where W is the magnetic co-energy and 𝛼 is the rotor angular position. By assuming a negligible stored energy in the core: $\begin{eqnarray}\displaystyle W\approx W_{\text{gap}}\text{} & =\text{} & \displaystyle {\displaystyle \frac{1}{2{\mu}_{0}}}\int _{V}B^{2}({\theta})dV={\displaystyle \frac{1}{2{\mu}_{0}}}\int _{V}^{B_{}^{}}({\theta})G^{2}({\theta},{\alpha})dV\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle {\displaystyle \frac{1}{2{\mu}_{0}}}\int _{0}^{2{\pi}}B_{r}^{2}({\theta})G^{2}({\theta},{\alpha})L_{s}{\displaystyle \frac{{\pi}r_{2}^{2}-{\pi}r_{1}^{2}}{2{\pi}}}d{\theta}\end{eqnarray}$ (2) where θ is the angle along the circumference of the air-gap, B (θ) and B _r(θ) are the distribution of flux density in the air-gap and the residual flux density of PMs, respectively and G (θ, 𝛼) is the air-gap permeance function. Also, L _s is the axial stack length and r ₁ and r ₂ are the outer radius of the rotor and inner radius of the stator, respectively.

The Fourier expansion of G ²(θ, 𝛼) and $B_{r}^{2}({\theta})$ are: $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}B_{r}^{2}({\theta})=B_{r0}+\displaystyle \mathop{\sum }_{n=1}^{\infty }B_{rn}\cos (nN_{p}{\theta})\\ G^{2}({\theta},{\alpha})=G_{0}+\displaystyle \mathop{\sum }_{n=1}^{\infty }G_{m}\cos (mN_{s}({\theta}+{\alpha}))\end{array}\right. & & \displaystyle\end{eqnarray}$ (3) where N _p and N _s are number of poles and slots, respectively.

Using Eqs (1)–(3) and the trigonometric functions, the cogging torque can be expressed as: $\begin{eqnarray}\displaystyle T_{\text{cogg}}({\alpha})\text{} & =\text{} & \displaystyle -{\displaystyle \frac{\partial }{\partial {\alpha}}}\left[{\displaystyle \frac{L_{s}}{4{\mu}_{0}}}(r_{2}^{2}-r_{1}^{2})\int _{0}^{2{\pi}}B_{r}^{2}({\theta})G^{2}({\theta},{\alpha})d{\theta}\right]\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle {\displaystyle \frac{L_{s}}{4{\mu}_{0}}}(r_{2}^{2}-r_{1}^{2})\int _{0}^{2{\pi}}\left[\mathop{\sum }_{n=0}^{\infty }B_{rn}\cos (nN_{p}{\theta})\cdot \mathop{\sum }_{m=0}^{\infty }G_{m}mN_{s}\sin (mN_{s}{\alpha}+mN_{s}{\theta})\right]d{\theta}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle {\displaystyle \frac{L_{s}}{4{\mu}_{0}}}(r_{2}^{2}-r_{1}^{2})\int _{0}^{2{\pi}}\mathop{\sum }_{n=0}^{\infty }\mathop{\sum }_{m=0}^{\infty }{\displaystyle \frac{B_{rn}G_{m}mN_{s}}{2}}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad \times ∼[\sin (nN_{p}{\theta}+mN_{s}{\alpha}+mN_{s}{\theta})+\sin (mN_{s}{\alpha}+mN_{s}{\theta}-nN_{p}{\theta})]d{\theta}.\end{eqnarray}$ (4) Let $\frac{n}{m}=\frac{N_{s}}{N_{p}}$ , so Eq. (4) could be simplified as: $\begin{eqnarray}\displaystyle T_{\text{cogg}}({\alpha})={\displaystyle \frac{{\pi}L_{s}N_{s}}{4{\mu}_{0}}}(r_{2}^{2}-r_{1}^{2})\mathop{\sum }_{m=1}^{\infty }B_{rn}\cdot G_{m}\cdot m\cdot \sin (m\cdot N_{s}\cdot {\alpha})=\mathop{\sum }_{m=1}^{\infty }T_{\text{cogg},m}\sin (m\cdot N_{s}\cdot {\alpha}). & & \displaystyle\end{eqnarray}$ (5)

3. Design description

3.1. Residual flux density of PMs function

Figure 1(b) shows the B _r(θ) function for the uniformly distributed solid PMs of machine of Fig. 1(a). For this flux density distribution, the Fourier expansion coefficients of $B_{r}^{2}({\theta})$ are simply as follows: $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}B_{r0}={\alpha}_{p}B_{r}^{2}=B_{r}^{2}{\displaystyle \frac{N_{p}{\theta}_{b}}{2{\pi}}},\\ B_{rn}={\displaystyle \frac{2}{n{\pi}}}B_{r}^{2}\sin (n{\pi}{\alpha}_{p})={\displaystyle \frac{2}{n{\pi}}}B_{r}^{2}\sin \left(n{\displaystyle \frac{N_{p}}{2}}{\theta}_{b}\right);\quad n=1,2,\ldots \end{array}\right. & & \displaystyle\end{eqnarray}$ (6) where B _r is the PM residual flux density and 𝛼_p is the pole arc to pitch ratio.

Fig. 2.

Planar model of rotor and stator, (a) segmented PM rotor, (b) notched stator.

For the segmented PM, as shown in Fig. 2(a), the Fourier expansion coefficients can be expressed as follows: $\begin{eqnarray}\displaystyle B_{n}\text{} & =\text{} & \displaystyle {\displaystyle \frac{N_{r}}{{\pi}}}\mathop{\sum }_{i=1}^{\frac{y_{n}}{2}}\left[\int _{(i-1){\displaystyle \frac{{\theta}_{b}}{y_{n}}}+{\displaystyle \frac{{\theta}_{b}}{2y_{n}}}-{\displaystyle \frac{{\theta}_{s}}{2}}}^{(i-1){\displaystyle \frac{{\theta}_{b}}{y_{n}}}+{\displaystyle \frac{{\theta}_{b}}{2y_{n}}}+{\displaystyle \frac{{\theta}_{s}}{2}}}B_{r}^{2}\cos (nN_{r}{\theta})d{\theta}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +\left.\int _{\frac{2{\pi}}{N_{r}}-{\displaystyle \frac{{\theta}_{b}}{2}}+(i-1)\frac{{\theta}_{b}}{y_{n}}+\frac{{\theta}_{b}}{2y_{n}}-{\displaystyle \frac{{\theta}_{s}}{2}}}^{\frac{2{\pi}}{N_{r}}-\frac{{\theta}_{b}}{2}+(i-1){\displaystyle \frac{{\theta}_{b}}{y_{n}}}+{\displaystyle \frac{{\theta}_{b}}{2y_{n}}}+{\displaystyle \frac{{\theta}_{s}}{2}}}B_{r}^{2}\cos (nN_{r}{\theta})d{\theta}\right];\quad n=1,2,\ldots .\end{eqnarray}$ (7) Above expression is valid for even number of segments (y _n). The same expression for odd number of PM segments is: $\begin{eqnarray}\displaystyle B_{n}\text{} & =\text{} & \displaystyle {\displaystyle \frac{N_{r}}{{\pi}}}\mathop{\sum }_{i=1}^{\frac{y_{n}+1}{2}}\left[\int _{(i-1){\displaystyle \frac{{\theta}_{b}}{y_{n}}}}^{(i-1)\frac{{\theta}_{b}}{y_{n}}+{\displaystyle \frac{{\theta}_{s}}{2}}}B_{r}^{2}\cos (nN_{r}{\theta})d{\theta}+\int _{\frac{2{\pi}}{N_{r}}-{\displaystyle \frac{{\theta}_{b}}{2}}+{\displaystyle \frac{{\theta}_{b}}{2y_{n}}}-{\displaystyle \frac{{\theta}_{s}}{2}}}^{\frac{2{\pi}}{N_{r}}-{\displaystyle \frac{{\theta}_{b}}{2}}+{\displaystyle \frac{{\theta}_{b}}{y_{n}}}-{\displaystyle \frac{{\theta}_{s}}{2}}}B_{r}^{2}\cos (nN_{r}{\theta})d{\theta}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼{\displaystyle \frac{N_{r}}{{\pi}}}\mathop{\sum }_{i=1}^{\frac{y_{n}-1}{2}}\left[\int _{(i){\displaystyle \frac{{\theta}_{b}}{y_{n}}}-\frac{{\theta}_{s}}{2}}^{(i){\displaystyle \frac{{\theta}_{b}}{y_{n}}}}B_{r}^{2}\cos (nN_{r}{\theta})d{\theta}+\int _{\frac{2{\pi}}{N_{r}}-{\theta}_{b}+{\displaystyle \frac{{\theta}_{b}}{y_{n}}}}^{\frac{2{\pi}}{N_{r}}-{\theta}_{b}+{\displaystyle \frac{{\theta}_{b}}{y_{n}}}+{\displaystyle \frac{{\theta}_{s}}{2}}}B_{r}^{2}\cos (nN_{r}{\theta})d{\theta}\right];\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad \quad n=1,2,\ldots .\end{eqnarray}$ (8)

3.2. Air-gap permeance function

Air-gap permeance function can be effectively controlled by notches on tooth face. The model presented in this paper considers the notch depth by a simplified magnetic circuit; it is considered that the air–gap permeance function is independent of the radius.

Figure 1(c) shows the G (θ, 𝛼) for the un-notched stator machine of Fig. 1(a). For this permeance function, the Fourier expansion coefficients of G ²(θ, 𝛼) are as follows: $\begin{eqnarray}\displaystyle \left.\begin{array}{@{}l@{}}G_{0}={\alpha}_{s}g_{1}^{2}=g_{1}^{2}{\displaystyle \frac{N_{s}{\theta}_{a}}{2{\pi}}},\\ G_{m}={\displaystyle \frac{2}{m{\pi}}}g_{1}^{2}\sin (m{\pi}{\alpha}_{s})={\displaystyle \frac{2}{m{\pi}}}g_{1}^{2}\sin \left(m\frac{N_{s}}{2}{\theta}_{a}\right);\quad m=1,2,\ldots \end{array}\right. & & \displaystyle\end{eqnarray}$ (9) where g ₁ is equal to $\left(\frac{h_{m}}{h_{m}+h_{g}}\right)$ , h _m and h _g are the PM and air-gap length, respectively and 𝛼_s is the stator teeth arc to pitch ratio.

For notched stator, as shown in Fig. 2(b), the Fourier expansion coefficients, for even number of notches (z _n) on the stator teeth, can be expressed as follows: $\begin{eqnarray}\displaystyle G_{m}\text{} & =\text{} & \displaystyle {\displaystyle \frac{N_{s}}{{\pi}}}\mathop{\sum }_{i=1}^{\frac{z_{n}}{2}}\left\{\int _{(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}}^{(i-1)\frac{{\theta}_{a}}{z_{n}}+{\displaystyle \frac{{\theta}_{a}}{2z_{n}}}-{\displaystyle \frac{{\theta}_{n}}{2}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}+\int _{(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}+{\displaystyle \frac{{\theta}_{a}}{2z_{n}}}-{\displaystyle \frac{{\theta}_{n}}{2}}}^{(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}+{\displaystyle \frac{{\theta}_{a}}{2z_{n}}}+{\displaystyle \frac{{\theta}_{n}}{2}}}g_{2}^{2}\cos (mN_{s}{\theta})d{\theta}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼\int _{(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}+{\displaystyle \frac{{\theta}_{a}}{2z_{n}}}+\frac{{\theta}_{n}}{2}}^{i\frac{{\theta}_{a}}{z_{n}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}+\int _{\frac{2{\pi}}{N_{s}}-\frac{{\theta}_{a}}{2}+(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}}^{\frac{2{\pi}}{N_{s}}-{\displaystyle \frac{{\theta}_{a}}{2}}+(i-1)\frac{{\theta}_{a}}{z_{n}}+{\displaystyle \frac{{\theta}_{a}}{2z_{n}}}-{\displaystyle \frac{{\theta}_{n}}{2}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +\left.\int _{\frac{2{\pi}}{N_{s}}-\frac{{\theta}_{a}}{2}+(i-1)\frac{{\theta}_{a}}{z_{n}}+\frac{{\theta}_{a}}{2z_{n}}-\frac{{\theta}_{n}}{2}}^{\frac{2{\pi}}{N_{s}}-\frac{{\theta}_{a}}{2}+(i-1)\frac{{\theta}_{a}}{z_{n}}+\frac{{\theta}_{a}}{2z_{n}}+\frac{{\theta}_{n}}{2}}g_{2}^{2}\cos (mN_{s}{\theta})d{\theta}+\int _{\frac{2{\pi}}{N_{s}}-\frac{{\theta}_{a}}{2}+(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}+\frac{{\theta}_{a}}{2z_{n}}+{\displaystyle \frac{{\theta}_{n}}{2}}}^{\frac{2{\pi}}{N_{s}}-{\displaystyle \frac{{\theta}_{a}}{2}}+(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}+{\displaystyle \frac{{\theta}_{n}}{z_{n}}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}\right\};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \qquad m=1,2,\ldots\end{eqnarray}$ (10) where g ₂ is equal to h _m∕(h _m + h _g + h _n) and h _n is the notch depth.

There is a similar expression for odd number of notches as follows: $\begin{eqnarray}\displaystyle G_{m}\text{} & =\text{} & \displaystyle {\displaystyle \frac{N_{s}}{{\pi}}}\mathop{\sum }_{i=1}^{\frac{z_{n}+1}{2}}\left\{\int _{(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}}^{(i-1)\frac{{\theta}_{a}}{z_{n}}+{\displaystyle \frac{{\theta}_{n}}{2}}}g_{2}^{2}\cos (mN_{s}{\theta})d{\theta}+\int _{(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}+\frac{{\theta}_{n}}{2}}^{(i-1){\displaystyle \frac{{\theta}_{a}}{z_{n}}}+\frac{{\theta}_{a}}{2z_{n}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +\left.\int _{\frac{2{\pi}}{N_{s}}-\frac{{\theta}_{a}}{2}}^{\frac{2{\pi}}{N_{s}}-\frac{{\theta}_{a}}{2}+\frac{{\theta}_{a}}{2z_{n}}-{\displaystyle \frac{{\theta}_{n}}{2}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}+\int _{\frac{2{\pi}}{N_{s}}-\frac{{\theta}_{a}}{2}+{\displaystyle \frac{{\theta}_{a}}{2z_{n}}}-\frac{{\theta}_{n}}{2}}^{\frac{2{\pi}}{N_{s}}-{\displaystyle \frac{{\theta}_{a}}{2}}+\frac{{\theta}_{a}}{z_{n}}-\frac{{\theta}_{n}}{2}}g_{2}^{2}\cos (mN_{s}{\theta})d{\theta}\right\}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +∼{\displaystyle \frac{N_{s}}{{\pi}}}\mathop{\sum }_{i=1}^{\frac{z_{n}-1}{2}}\left\{\int _{(i){\displaystyle \frac{{\theta}_{a}}{z_{n}}}-\frac{{\theta}_{n}}{2}}^{(i){\displaystyle \frac{{\theta}_{a}}{z_{n}}}}g_{2}^{2}\cos (mN_{s}{\theta})d{\theta}+\int _{(i){\displaystyle \frac{{\theta}_{a}}{z_{n}}}-{\displaystyle \frac{{\theta}_{a}}{2z_{n}}}}^{(i){\displaystyle \frac{{\theta}_{a}}{z_{n}}}-{\displaystyle \frac{{\theta}_{n}}{2}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad +\left.\int _{\frac{2{\pi}}{N_{s}}-{\theta}_{a}+{\displaystyle \frac{{\theta}_{a}}{z_{n}}}}^{\frac{2{\pi}}{N_{s}}-{\theta}_{a}+{\displaystyle \frac{{\theta}_{a}}{z_{n}}}+\frac{{\theta}_{n}}{2}}g_{2}^{2}\cos (mN_{s}{\theta})d{\theta}+\int _{\frac{2{\pi}}{N_{s}}-{\theta}_{a}+{\displaystyle \frac{{\theta}_{a}}{z_{n}}}+{\displaystyle \frac{{\theta}_{n}}{2}}}^{\frac{2{\pi}}{N_{s}}-{\theta}_{a}+{\displaystyle \frac{3{\theta}_{a}}{2z_{n}}}}g_{1}^{2}\cos (mN_{s}{\theta})d{\theta}\right\};\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad m=1,2,\ldots .\end{eqnarray}$ (11)

4. Simulation results

4.1. Cogging torque and back-EMF

The energy method presented in Section 2 and magnetic field and permeance functions presented in Section 3 are used to calculate the cogging torque for a sample radial-flux surface-mounted PM machine. The machine parameters are presented in Table 1.

The cogging torque angle interval of the machine is: $\begin{eqnarray}\displaystyle t_{\text{cogg}}=\frac{2{\pi}}{lcm(N_{s},N_{r})}=5^{\circ } & & \displaystyle\end{eqnarray}$ (12) where lcm (. ) is the least common multiplier function.

Figures 3–5 show the analytical solution results for the produced cogging torque. The machine of Table 1 also has been simulated by FEA where the machine structure and numerical results are presented in Figs 6, 7. In FEA, the magnetic field distributions at different relative positions are obtained to calculate the cogging torque using Maxwell’s stress tensor. The simulation for different relative positions within an angle interval is enough.

Table 1

Surface-mounted PM machine parameters

Parameter	Value	Parameter	Value
N _s	9	θ_a	24°
N _r	8	θ_n	2.4°
𝛼_s	0.6	h _n	0.5 mm
𝛼_p	0.6	z _n	4
r _i	13 mm	θ_b	27°
r _o	14 mm	θ_s	4.72°
L _s	100 mm	h _m	3 mm
h _g	1 mm	y _n	4
H _c	900000 A/m	B _r	1.2 T

Fig. 3.

Analytical results, (a) cogging torque for notched teeth, (b) cogging torque for segmented PMs, (c) cogging torque considering both.

Fig. 4.

Analytical results, (a) permeance function considering notched teeth and segmented PMs, (b) air-gap flux density function considering notched teeth and segmented PMs.

Fig. 5.

Analytical results, cogging torque coefficients.

Fig. 6.

Numerical results, (a) machine structure, (b) cogging torque for notched teeth, (c) cogging torque for segmented PMs, (d) cogging torque considering both.

Fig. 7.

Numerical results, (a) flux lines, (b) magnetic flux density distribution, (c) back-EMF.

Fig. 8.

Cogging torque coefficients, (a) effect of number of teeth notches, (b) effect of number of PM segments.

The results are obtained for three conditions: the machine has teeth notches but the PMs are not segmented, the PMs are segmented but the machine teeth have not notches, teeth notches and PM segments are both considered.

Figure 7(c) shows the steady state back-EMF waveform of the machine; there are high frequency harmonics due to effect of teeth notches and PM segments.

Comparing analytical and numerical results of Figs 3 and 6 shows acceptable correlation in cogging torque. The differences are due to simplification of air-gap flux density and permeance, and the flux paths.

Both analytical and numerical methods are suitable to evaluate the operation and performances of electromagnetic devices and systems. The main advantages of numerical methods such as FEA are ability to analyze complex systems and flexibility in analysis, while one of the main disadvantages is high computational time and memory. Analytical methods on the magnetic field and electromagnetic systems analysis have ability to facilitate the optimal design process and control system design and operation.

Fig. 9.

Cogging torque coefficients, (a) effect of changing number of teeth notches and PM segments, (b) effect of teeth notches depth.

4.2. Harmonics analysis

The analytical expression for the cogging torque shows that the cogging torque depends on the machine design parameters such as number of poles, number of slots, tooth and PM width and so on. But, as in this research the focus is on the tooth notches and PM segments, we investigate the effects of these parameters on the harmonic content of the cogging torque.

Considering the machine of Table 1 as the basic design, the effect of three main parameters are analyzed.

Figure 8(a) shows the cogging torque versus the number of notches while the number of PM segments is held constant. The effect of number of PM segments for constant number of notches is shown in Fig. 8(b). It should be noted that in these analyses, the total (sum) width of tooth notches and PM segments are held constant.

Clearly, cogging torque, as well as its harmonic content, varies by the change of mentioned parameters. In Fig. 8(a), the 24th component dominates all other components for all numbers of notches. In Fig. 8(b) the 24th component for y _n = 8 and 8th component for y _n = 6 have the considerably higher values compared to other components.

For similar number of notches and PM segments, as shown in Fig. 9(a), numbers 8 and 6 result in significant increase in one of components, so are not proper choices.

The effect of notch depth is analyzed in Fig. 9(b) that shows the linear dependency of each component to the change of this parameter.

It is possible to combine the effect of these parameters to reduce a specific component.

So, it can be concluded that in addition to the design parameters such as number of poles, number of slots, tooth and PM width and so on, the teeth notch and PM segment parameters could be selected optimally to achieve a low cogging torque.

4.3. Effect of electric loading

Normally the cogging torque and back-EMF are analyzed at open-circuit conditions. But, the electric loading and magnetic saturation have important effects on them so that, the magnitude of the under-load cogging torque, is considerably larger and the under-load back-EMF waveform contains larger harmonics and so there are higher torque ripples [28].

Therefore, in load conditions, the cogging torque, back-EMF and torque ripples of PM machines is analyzed by the numerical methods such as FEA [29].

5. Conclusion

In this paper, an analytical approach for cogging torque calculation of radial-flux surface-mounted PM machines including the effect of teeth notch and PM segmentation has been presented. This method uses the simplified flux distribution and could take into account different geometrical parameters of notches and segments that is useful in sensitivity analysis and reduction of the cogging torque of the machine.

Comparing analytical and numerical results shows that the presented model is suitable to predict the cogging torque. Moreover, this model could be extended to other types of PM electrical machines.

The effects of teeth notches and PM segments parameters on the harmonic content of the cogging torque are also investigated. These analyses could be used for optimal design of the machine.

References

Young-Un

and Dae-kyong

, Analytical prediction of the cogging torque in external-rotor single-phase BLDC motors with tapered air-gap, International Journal of Applied Electromagnetics and Mechanics 59(2) (2019).

Tootoonchian

and Nasiri-Gheidari

, Cogging force mitigation techniques in a modular linear permanent magnet motor, IET Electr. Power Appl. 10(7) (2016).

Zhu

Hua

and Wu

, Cogging torque suppression in flux-reversal permanent magnet machines, IET Electr. Power Appl. 12(1) (2018).

Zhu

Z.Q.

and Howe

, Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors, IEEE Trans. Magn. 28(2) (1992).

Schlensok

Gracia

M.H.

and Hameyer

, Combined numerical and analytical method for geometry optimization of a PM motor, IEEE Trans. Magn. 42(4) (2006).

Yang

Wang

Zhang

, The optimization of pole arc coefficient to reduce cogging torque in surface-mounted permanent magnet motors, IEEE Trans. Magn. 42(4) (2006).

Kim

K.C.

Koo

D.H.

Hong

J.P.

, A study on the characteristics due to pole-arc to pole-pitch ratio and saliency to improve torque performance of IPMSM, IEEE Trans. Magn. 43(6) (2007).

Gulec

and Aydin

, Magnet asymmetry in reduction of cogging torque for integer slot axial flux permanent magnet motors, IET Electr. Power Appl. 8(5) (2014).

Wang

Jin

M.J.

Fei

W.Z.

, Cogging torque reduction in permanent magnet flux-switching machines by rotor teeth axial pairing, IET Electr. Power Appl. 4(7) (2010).

10.

Wang

Yang

and Fu

, Study of cogging torque in surface-mounted permanent magnet motors with energy method, Journal of Magnetism and Magnetic Materials 267(1) (2003).

11.

Hwang

S.M.

Eom

J.B.

Jung

Y.H.

, Various design techniques to reduce cogging torque by controlling energy variation in permanent magnet motors, IEEE Trans. Magn. 37(4) (2001).

12.

Xiaofeng

Wei

and Zhongze

, Cogging torque minimisation in FSPM machines by right-angle-based tooth chamfering technique, IET Electr. Power Appl. 12(5) (2018).

13.

Wang

Qiao

, Reducing cogging torque in surface-mounted permanent-magnet motors by nonuniformly distributed teeth method, IEEE Trans. Magn. 47(9) (2011).

14.

Asgari

Yazdanpanah

and Mirsalim

, A dual-stator machine with diametrically magnetized PM: analytical air-gap flux calculation, efficiency optimization and comparison with conventional dual-stator machines, Scientia Iranica , to be published.

15.

Roshandel

T.N.

and Shoulaie

, Pole-shape optimization of permanent-magnet linear synchronous motor for reduction of thrust ripple, Energy Conversion and Management 52(1) (2001).

16.

Suzuki

Okamoto

and Wakao

Sh.

, Average-torque-maximization and cogging-torque-minimization of permanent-magnet-assisted synchronous reluctance motor using topology optimization, International Journal of Applied Electromagnetics and Mechanics 60(S1) (2019).

17.

Scuiller

, Magnet shape optimization to reduce pulsating torque for a five-phase permanent-magnet low-speed machine, IEEE Trans Magn. 50(4) (2014).

18.

Xue

Zhou

, Analytical prediction and optimization of cogging torque in surface-mounted permanent magnet machines with modified particle swarm optimization, IEEE Trans. Ind. Electron. 64(12) (2017).

19.

Mohammadi

and Mirsalim

, Analytical design framework for torque and back-EMF optimization, and inductance calculation in double-rotor radial-flux air-cored permanent-magnet synchronous machines, IEEE Trans Magn. 50(1) (2014).

20.

Mohammadi

Vahidi

Mirsalim

and Lesani

, Simple nonlinear MEC based model for sensitivity analysis and genetic optimization of permanent magnet synchronous machines, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 24(1) (2015).

21.

Hua

Cheng

Zhu

Z.Q.

and Howe

, Analysis and optimization of back EMF waveform of a flux-switching permanent magnet motor, IEEE Trans Energy Conv. 23(3) (2008).

22.

H.C.

B.S.

and Lin

C.K.

, A dual notched design of radial-flux permanent magnet motors with low cogging torque and rare earth material, IEEE Trans Magn. 50(11) (2014).

23.

Wang

and Jung

S. Y.

, Reduction on cogging torque in flux-switching permanent magnet machine by teeth notching schemes, IEEE Trans Magn. 48(11) (2012).

24.

Hao

Lin

and Zhang

, Cogging torque reduction of axial-field flux-switching permanent magnet machine by rotor tooth notching, IEEE Trans Magn. 51(11) (2015).

25.

Abbaszadeh

Rezaee Alam

and Saied

S.A.

, Cogging torque optimization in surface-mounted permanent-magnet motors by using design of experiment, Energy Conversion and Management 52(10) (2011).

26.

Abdollahi

S.E.

and Vaez-Zadeh

, Reducing cogging torque in flux switching motors with segmented rotor, IEEE Trans Magn. 49(10) (2013).

27.

Xing

Zhaoxue

Jibin

and Tao

, An analytical model of unbalanced magnetic pull for PMSM used in electric vehicle: Numerical and experimental validation, International Journal of Applied Electromagnetics and Mechanics 54(4) (2017).

28.

Azar

Zhu

Z.Q.

and Ombach

, Influence of electric loading and magnetic saturation on cogging torque, back-EMF and torque ripple of PM machines, IEEE Trans Magn. 48(10) (2012).

29.

Yazdanpanah

and Mirsalim

, Hybrid electromagnetic brakes: Design and performance evaluation, IEEE Transn Energy Conv. 30(1) (2015).