**3. Analytical Expression of Cogging Torque by VWM**

The most widely used method for calculating cogging torque is the virtual work method (VWM). The virtual work method is also called the co-energy method [24]. The basic principle is that neglecting the variation in magnets and iron of PMBLDC motor, the cogging torque of an ideal lossless PMBLDC motor can be expressed as the derivative of co-energy in the air gap [25]. According to [26], cogging torque is produced because of the interaction between the PMs and the armature and the slot. Because of this interaction in the absence of a current, energy variation occurs inside the motor.

$$E\_{\upsilon} = E\_{\upsilon, I} + E\_{\upsilon, airgap} + E\_{\upsilon, PM} \tag{1}$$

where *E<sup>v</sup>* is the total energy variation, *Ev*.*<sup>I</sup>* is the energy variation in iron, *Ev*.*airgap* is the energy variation in airgap, and *Ev*.*PM* is the energy variation in PM.

When compared with the energy variation in the airgap and PM, only a minor variation occurs in iron. Therefore

$$E\_v \cong E\_{v, airgap} + E\_{v, PM} = \frac{1}{2\mu\_0} \int \int \int B^2 dv \tag{2}$$

Hence, cogging torque can be stated as

$$T\_{\rm cog} = -\frac{\partial E\_v}{\partial \alpha} \tag{3}$$

where, *µ*0, *B* and ∝ are the permeability of air, the magnetic flux density (magnetic induction) and the angle of rotation of the rotor, respectively. The distribution of magnetic induction certainly stated as

$$\mathbf{B}\left(\theta,\mathfrak{a}\right) = B\_{rs}(\theta)\frac{l\_m}{l\_m + \lg(\theta,\mathfrak{a})}\tag{4}$$

where *Brs*(*θ*) is the residual flux density along the periphery of the airgap, *l<sup>m</sup>* is the length of the permanent magnet, and *l<sup>g</sup>* is the effective length of airgap distribution. Equation (2) can be redrafted as

$$E\_{\upsilon} = \frac{1}{2\mu\_0} \int \int \int B\_{rs}^2(\theta) \left[ \frac{l\_m}{l\_m + \lg(\theta, \alpha)} \right]^2 dv \tag{5}$$

To obtain the magnetostatic energy within the motor, Fourier expansion of *Brs* 2 (*θ*) and h *lm lm*+*lg*(*θ*,*α*) i2 can be performed.

$$B\_{\rm rs}^{\,\,2}(\theta) = B\_{\rm rs0} + \sum\_{n=1}^{\infty} B\_{\rm rsn} \cos n\theta + B\_{\rm rsbn} \sin n\,\,\theta \tag{6}$$

$$\left[\frac{l\_m}{l\_m + \lg(\theta, \mathfrak{a})}\right]^2 = \mathcal{G}\_0 + \sum\_{n=1}^{\infty} \mathcal{G}\_n \cos n \text{ s } (\theta + \mathfrak{a}) \tag{7}$$

The analytical statement of cogging torque for asymmetrical magnets can be expressed as

$$T\_{\rm cog} = \frac{\pi z L s}{4 \mu\_0} \left( R\_r^2 - R\_s^2 \right) \sum\_{n=1}^{\infty} B\_{\rm rsanz} \sin n \text{ s.a.} + B\_{\rm rsbnz} \cos n \text{ s.a.} \tag{8}$$

where *Ls* is the length of the stack, *s* is the slot number, *Rr* is the rotor outer radius and *Rs* is the stator inner radius. The Fourier coefficients *Brsanz* and *Brsbnz* can be expressed as

$$B\_{rsanz} = \frac{2B\_{rs}^2}{ns\pi} \sin\frac{ns\pi\alpha\_p}{2p} \sum\_{k=1}^{2p} \cos ns \left[\frac{\pi}{p}(k-1) + \theta\_s\right] \tag{9}$$

$$B\_{rsbmz} = \frac{2B\_{rs}^2}{ns\pi} \sin\frac{ns\pi\alpha\_p}{2p} \sum\_{k=1}^{2p} \sin n\, s \left[\frac{\pi}{p}(k-1) + \theta\_s\right] \tag{10}$$

where, *θ<sup>s</sup>* is the shifting angle.

When symmetrical structure is adopted for the magnets (*θ<sup>s</sup>* = 0◦ ), *Brsbnz* is zero and cogging torque can be expressed as

$$T\_{\rm cog} = \frac{\pi z L s}{4 \mu\_0} (R\_r^2 - R\_s^2) \sum\_{n=1}^{\infty} B\_{\rm rsonz} \sin n \text{ s.a.} \tag{11}$$

Table 2 shows the analytical results of cogging torque with shifting angles ranging from 1◦ to 8◦ . Figure 4 shows a graphical representation of cogging torque with shifting angles ranging from 1◦ to 8◦ .

**Table 2.** Analytical results for cogging torque with shifting angles from 1◦ to 8◦ .

**Figure 4.** Analytical results of cogging torque with shifting angles from 1◦ to 8◦ .
