# A Novel Design of Permanent Magnets for the Air Gap Magnetic Field of Hollow-Cup Motor
## 2 Mathematical Model of AMF
### AMF Analysis Based on ESCM  
The AMF of the HCM is generated by PMs. In this paper, the ESCM is used instead of PMs; thus, the magnetic field generated in the air gap of the motor by a current-carrying coil equivalent to PMs is analysed [25; 26]. This paper focuses on the analysis of the AMF generated by the parallel magnetised PMs; the end-effects are ignored, and the inner and outer rotor cores are treated as ideal permeable bodies. On the basis of the boundary conditions of the AMF distribution of the HCM, the partial differential equations of the magnetic field distribution of the coil are solved using [PERSON]'s equation set. The AMF generated by the whole coil can be obtained by applying a superposition of the radial magnetic potential generated on each side of the coil:  
\[B(r,\theta)=\frac{\mu_{0}i}{\pi r}\sum_{m=1}^{\infty}\frac{R_{m}^{m}}{b^{m}} \left(\frac{R_{i}^{2m}+b^{2m}}{R_{0}^{2m}-R_{i}^{2m}}\right)\left(\frac{r^{m}} {R_{i}^{m}}+\frac{R_{i}^{m}}{r^{m}}\right)\sin(m\alpha)\cos(m\theta), \tag{1}\]  
where \(\mu_{0}\) is the air magnetic permeability, \(\alpha\) is the angle of the equivalent current coil with respect to the centre line of the PM, \(\eta\) is half the angle of the pair of coils, \(R_{o}\) is the ORC inner diameter, \(R_{i}\) is the IRC outer diameter, \((r,\theta)\) denotes the coordinates of any position within the air gap, \(b\) is the inner diameter of the PMs, and \(h_{m}\) is the PM thickness.  
The correspondence between the PMs and their equivalent currents is shown in Figure 2. When the PMs are magnetised in parallel and there are 2p PMs arranged alternately along opposite magnetisation directions, the superposition of magnetic fields in the air gap by different PMs is  
\[Z=\sum_{l=1}^{2p}\left(-1\right)^{l-1}\cos m\left[\theta-(l-1)\frac{\pi}{p} \right]. \tag{2}\]  
The remaining coils can be calculated in this way and then superimposed to obtain 2p pairs of the coil sets, producing the following magnetic flux density expression:  
\[B(r,\theta)=\frac{\mu_{0}i}{\pi r}\sum_{m=1}^{\infty}\frac{R_{o}^{m}}{b^{m}} \left(\frac{R_{i}^{2m}+b^{2m}}{R_{o}^{2m}-R_{i}^{2m}}\right)\left(\frac{r^{m} }{R_{i}^{m}}+\frac{R_{i}^{m}}{r^{m}}\right)\sin\left(m\alpha\right)Z. \tag{3}\]  
In the ESCM, the equivalent current along the radial side of the PM \(AB\), \(CD\) is equal in magnitude and opposite in direction, and the equivalent currents along the \(AD\) and \(BC\) circumferences sides of the PMs are equal in magnitude and opposite in direction.  
Since the AMF is generated by the superposition of several sets of coil currents, the magnetic field distribution must be obtained by integrating the current values on each  
Figure 2: Parallel magnetising tile PMs and the equivalent current coil model.  
The magnetic field strength at a point \((r,\theta)\) in the air gap can be obtained by summing the PM thickness integral along the AB-CD edge and the PM tensor angle integral along the BC and AD edges:  
\[B(r,\theta)=\int_{R_{\alpha}-h_{m}}^{R_{\alpha}}dB_{AB-CD}+\int_{-\eta}^{\eta} dB_{BC}+\int_{-\eta}^{\eta}dB_{AD}. \tag{7}\]  
The AMF is obtained by solving the above equation along the circumferential direction of the centre line of the air gap in the HCM, and the AMF can be changed by optimising the eccentricity value of the PMs.