The motor design variables were: Voltage of the DC bus, *U*_{DC}; Shaft Power, Pc; angular speed, *ω*; Motor Efficiency, *η*_{c}; mean radius at magnets center position, r; magnet poles number, p; flux intensity of the magnets, *B*_{mag}; magnet thickness tm, Stator filling factor, FF; magnet pole length, L; magnet pole width, W and the airgap thickness of 2mm. So, in the first design iteration, all these variables values were assigned. In the case of *U*_{DC}, it had to be a multiple of 3.7V because the SEM competition rules mandate the use of a lithium-ion battery. In the first design iteration, the value was set to 7.4V but the final design point uses 15V and the actual vehicle’s battery uses 22.2V to reach a top speed sufficiently above the design speed. shows the values that were used for Pc, 15W, for *ω*, 29.4rad/s and the maximum radius of the motor, 340mm/2=0.17m.

Table 1 Motor requirements

The main constrain for the motor development was the motor cost and the main cost share was found to be for the magnets. So, these were chosen for their low cost; their geometric suitability to the in-wheel motor, and peak magnetic flux density rating. The adopted magnets were N52 flux rated NdFeB magnets with L=0.030m; W=0.012 and tm=0.012 m costing 60€/kg. So, the mean radius of the magnets was set at 0.15m (0.17-0.03/2=0.155 subtracted of a 5mm radial margin). The magnetic flux density of these N52 specified magnets reaches a *B*_{mag}=1.48T. A stator filling factor of FF = 0.3 was assumed. The major unknowns were the number of magnet poles and the efficiency that the motor could be designed for. So, during the design iterations, p was varied in the range of 32 to 40 and *η*_{c} from 0.96 to 0.99.

To model the motor, the first step is to obtain the available AC tension, *U*_{ABrms} from the U (1).

$$\begin{array}{r}{U}_{A{B}_{rms}}=\frac{\sqrt{3}}{2\sqrt{2}}U\end{array}$$(1)The motor electrical power is obtained from the shaft power, using the design point motor efficiency (2).

$$\begin{array}{r}{P}_{e}=\frac{{P}_{c}}{{\eta}_{c}}\end{array}$$(2)From the motor electrical power, the motor current is obtained (3).

$$\begin{array}{r}{I}_{rms}=\frac{{P}_{e}}{U}\end{array}$$(3)The motor phase current is also obtained from the motor electrical power, (4).

$$\begin{array}{r}{I}_{ABCrms}=\frac{1}{\sqrt{3}}\frac{{P}_{ce}}{{U}_{A{B}_{rms}}}\end{array}$$(4)Knowing motor phase current, the motor total Joule power loss is calculated with Equation (5) assuming half the losses will occur due to internal flow drag (windage loss) and eddy-currents in the copper. So, the windage power loss is equal to the Joule power loss.

$$\begin{array}{r}{3RI}^{2}=\frac{({P}_{ce}-{P}_{c})}{2}\end{array}$$(5)The phase winding resistance is found from the total Joule power loss by Equation (6).

$$\begin{array}{r}R=\frac{{RI}^{2}}{{3I}_{ABCrms}}\end{array}$$(6)The motor resistance voltage drop, which is the required electromotive force to reach the motor phase current is calculated from Equation 7).

$$\begin{array}{r}{U}_{emf}=3R{I}_{AB{C}_{rms}}\end{array}$$(7)The required motor effective back electromotive force, *U*_{bemfrms} , is obtained from Equation (8).

$$\begin{array}{r}{U}_{bem{f}_{rms}}=U-{U}_{emf}\end{array}$$(8)The motor constant *K*_{t} is now obtained from Equation (9).

$$\begin{array}{r}{K}_{t}=\frac{{U}_{bem{f}_{rms}}}{\omega}\end{array}$$(9)The angle corresponding to half electric revolution is calculated from the motor pole pair number *p* from Equation (10).

$$\begin{array}{r}\alpha =\frac{2\pi}{p}\end{array}$$(10)The duration of the coil motion from one magnet to the next is calculated from Equation (11).

$$\begin{array}{r}{t}_{mag}=\frac{\alpha}{\omega}\end{array}$$(11)The motor stator mean perimeter is given by Equation (12).

$$\begin{array}{r}{p}_{er}=2\pi r\end{array}$$(12)The stator outer perimeter is given by Equation (13).

$$\begin{array}{r}{p}_{e{r}_{out}}=2\pi r\left[r+\frac{L}{2}+0.005\right]\end{array}$$(13)The stator inner perimeter is given by Equation (14).

$$\begin{array}{r}{p}_{e{r}_{inn}}=2\pi r\left[r-\frac{L}{2}-0.005\right]\end{array}$$(14)The distance between opposing magnets faces, for the air gap in each face of the stator plus the stator itself with thickness, *t*_{s}, (see Figure 2) is determined by Equation (15).

$$\begin{array}{r}g={t}_{s}+4\end{array}$$(15)Regarding the mean flux density at the stator, it was considered to be 90% of the peak magnet flux density and proportional to the total thickness of the magnets surrounding the stator coil, 2t (see Figure 2) is determined by Equation (16).

$$\begin{array}{r}{B}_{coil}=0.9{B}_{mag}\frac{2t}{(g+2t)}\end{array}$$(16)The wave winding coil area subjected to the magnetic flux is calculated according to Equation (17).

$$\begin{array}{r}A=LWp\end{array}$$(17)The magnetic flux is calculated from Equation (18).

$$\begin{array}{r}\varphi =AB\end{array}$$(18)The pole width available at the mean stator perimeter is *p*_{er}/2. The desired magnet width was checked as *p*_{er}/4. It should be close to *W* such that the gap between successive rotor magnets was close to the value of *W*. It could never approach *2W* or the magnets could not be fitted in the rotor.

The flux gradient in the stator coil is considered as $\frac{2\varphi}{{t}_{mag}}$A maximum single coil voltage drop was calculated by Equation (19).

$$\begin{array}{r}RI=1.25R{I}_{ABCrms}\end{array}$$(19)So, the required maximum phase voltage is given by Equation (20).

$$\begin{array}{r}{U}_{ABC}=\frac{{U}_{{AB}_{rms}}}{\sqrt{3}}-RI\end{array}$$(20)The required turns per coil is obtained from Equation (21). The actual design point had to correspond to a finite *N* number.

$$\begin{array}{r}N=\frac{{U}_{ABC}}{\frac{2\varphi}{{t}_{mag}}}\end{array}$$(21)The length of coil wire per turn per coil is obtained from Equation (22).

$$\begin{array}{r}lp=2p\left(L+0.01\right)+0.6{p}_{e{r}_{out}}+0.6{p}_{e{r}_{inn}}\end{array}$$(22)The total wire length per coil is obtained from Equation (23).

$$\begin{array}{r}{l}_{N}=N({l}_{p}+1)\end{array}$$(23)The minimum copper coil wire section area was calculated from Equation (24) considering a copper wire resistivity at 40*∘*C of 2.06×10^{−8} m.

$$\begin{array}{r}{A}_{N}=2.06128\times {10}^{-8}\frac{{l}_{N}}{R}\end{array}$$(24)The corresponding diameter was calculated from Equation (25).

$$\begin{array}{r}{d}_{wire}=2\sqrt{\frac{{A}_{N}}{\pi}}\end{array}$$(25)The minimum copper volume per phase was calculated from Equation (26).

$$\begin{array}{r}{V}_{wire}={A}_{N}{l}_{N}\end{array}$$(26)The motor stator copper mass for the required copper volume per phase was calculated by Equation (27) considering a copper density of 8930kg/m^{3}.

$$\begin{array}{r}{m}_{wire}=8930{V}_{wire}\end{array}$$(27)The stator disk area perpendicular to the axial axis was calculated from Equation (28).

$$\begin{array}{r}{A}_{S}=2\pi \left[{\left(r+\frac{L}{2}+0,005\right)}^{2}-{\left(r-\frac{L}{2}+0,005\right)}^{2}\right]\end{array}$$(28)The required stator thickness was calculated considering the filling factor from Equation (29).

$$\begin{array}{r}{t}_{s}=\frac{3{V}_{wire}}{{A}_{S}{F}_{F}}\end{array}$$(29)The required motor magnets mass was calculated, considering the filling factor from Equation (30) and considering a NdFeB magnet density of 7500kg/m^{3}.

$$\begin{array}{r}{m}_{mag}=2p(7500WL{t}_{m})\end{array}$$(30)In order to decide what pole count, p, would be used in the motor, a parametric study was performed to see how much mass the motor would have due to the desired design efficiency. So, calculations where performed to determine the copper mass in the stator windings due to the adoption of different pole count in function of the design point efficiency of the motor. The study was performed in a range of 32 to 40 poles. The magnet mass for 32 poles was estimated at 2.1 kg and for 40 poles a value of 2.6 kg was expected. The results are presented in Figure 3. The results for the total mass of rotor magnets plus stator copper are presented in Figure 4. The adopted design was considered a good compromise between magnet mass, the corresponding cost and the value of efficiency that corresponded. A higher magnet poles number would increase even further the total magnet mass and, thus, motor cost.

Figure 3 Motor stator copper mass due to pole count in function of design point efficiency

Figure 4 Motor mass versus design point efficiency

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