## 2 Design and Analysis Methods
### Pm Dimension and Coil Mmf  
Maximum flux linkage \(\psi_{pm}\), airgap flux density \(B_{mg}\), and magnet thickness \(h_{m}\) can be expressed as follows [32].  
\[\psi_{pm}=B_{mg}b_{pm}L_{e}n_{c}N_{ct} \tag{11}\]  
\[B_{mg}\cong\frac{B_{r}}{1+k_{fr}}.\frac{h_{m}}{h_{m}+g}\Rightarrow\frac{h_{m}} {h_{m}+g}\cong\frac{B_{mg}}{B_{r}}\left(1+k_{fr}\right) \tag{12}\]  
where \(k_{fr}\) is the fringing factor in the range of 0.1\(\ldots\)0.2, \(b_{pm}\) is the pole arc or magnet width, \(n_{c}\) is the coil number per phase, and \(N_{ct}\) is the turn number per coil. \(B_{r}\) = 1.25 \(T\) and \(H_{c}\) = 959 kA/m for NeFeB N40 UH PMs. Electromagnetic torque and coil mmf can be expressed as Equations (13) and (14).  
\[T_{e}=3p\cdot\frac{\Psi_{pm}}{\sqrt{2}}\cdot l_{ph}=3p\cdot\frac{B_{mg}b_{pm}L _{e}n_{c}N_{ct}}{\sqrt{2}}\cdot l_{ph} \tag{13}\]  
\[N_{ct}I_{ph}(rms)=\frac{T_{e}\sqrt{2}}{3 pB_{mg}b_{pm}L_{e}n_{c}} \tag{14}\]To increase the winding factor and torque and reduce cogging torque, a 36-stator slot and 32-rotor pole configuration are used. So, the coil number per phase \(n_{c}\) is 12 for the 3-phase machine. The machine functions similarly to a DC machine if the voltage waveforms are applied to the rotor position by a PWM and with phase commutation every 60\({}^{\circ}\) (electrical). The 120\({}^{\circ}\) broad, smooth current blocks cause the machine to operate in two phases in a steady state, ignoring phase commutation. The sum of two-phase EMFs in series and emf per phase can be expressed as Equations (15) and (16) [32].  
\[E_{LL}=V_{dc}-2R_{s}i_{a}-2L_{s}\bigg{(}\frac{di_{a}}{dt}\bigg{)}_{=0} \tag{15}\]  
\[e_{1}=\frac{V_{dc}-2R_{s}i_{a}}{2} \tag{16}\]  
where \(Ia\) is the flat-topped value of the squarewave current equal to the inverter input current, \(V_{dc}\) is the DC input voltage that supplies the inverter, \(R_{s}\) is the stator phase resistance, and \(Ls\) is the synchronous inductance. When a PM brushless motor is powered by an inverter, and the DC bus voltage is known, the fundamental harmonic of the six-step, three-phase inverter's output AC line-to-line voltage is [28]  
\[V_{1L}=\frac{\sqrt{6}}{\pi}V_{dc}\approx 0.78V_{dc} \tag{17}\]  
The ratios of the first harmonic amplitudes to the maximum values of the normal components of the armature reaction magnetic flux densities in the d-axis and q-axis, respectively, are known as the form factors of the armature reaction (\(k_{fd}=B_{ad1}/B_{ad}\) and \(k_{fd}=B_{ad1}/B_{ad}\)). Additionally, the armature reaction form factors for the surface PM rotor are \(k_{fd}=k_{fd}=1\)[28]. For a square-wave excitation, the EMF induced in a phase turn and turn number per phase, respectively  
\[e_{1}=\pi\sqrt{2}fN_{1}k_{w}\phi_{g}=\pi\sqrt{2}fN_{1}k_{w}\alpha_{i}B_{mg} \tau_{p}L_{e} \tag{18}\]  
\[N_{1}=\frac{e_{1}}{\pi\sqrt{2}fk_{w}\alpha_{i}B_{mg}\tau_{p}L_{e}}\cong\frac{ \epsilon V_{1L}/\sqrt{3}}{\pi\sqrt{2}fk_{w}\alpha_{i}B_{mg}\tau_{p}L_{e}} \tag{19}\]  
The phase current can be calculated from Equations (14) and (19). So, the copper area \(S_{Cu}\) and copper diameter \(d_{cu}\) can be expressed as follows:  
\[S_{cu}=\frac{I_{ph}}{j_{cont}},\qquad d_{cu}=2\sqrt{\frac{S_{cu}}{\pi}} \tag{20}\]  
where current density \(j_{con}=4\ldots 6.5\) A/mm\({}^{2}\).