# Simulation of Electromagnetic Generator as Biomechanical Energy Harvester
## 3 Results and Discussion
Equation (6), shows the proportion of cogging torque:  
\[\mathrm{T}_{\mathrm{cog}}(\alpha_{\mathrm{r}})\ \propto-\sum_{n=1}^{\infty}[ \mathrm{G}_{a_{n\mathrm{NL}}}\ \mathrm{sin}(n\mathrm{N}_{L}\frac{\alpha_{\mathrm{p}}\mathrm{\tau}}{N_{ \mathrm{p}}})\ \mathrm{sin}\ (n\mathrm{N}_{L}\alpha_{\mathrm{r}})] \tag{6}\]  
Equation (7), shows the direction proportion of cogging torque while the rotor position \(\alpha_{\mathrm{r}}=2\mathrm{\pi}/\mathrm{N}_{\mathrm{L}}\):  
\[\mathrm{T}_{\mathrm{cog}}\propto\sum_{n=1}^{\infty}\mathrm{T}_{n}=-\sum_{n=1} ^{\infty}[\mathrm{G}_{a_{n\mathrm{NL}}}\ \mathrm{sin}(n\mathrm{N}_{L}\frac{\alpha_{\mathrm{p}}\mathrm{\pi}}{N_{ \mathrm{p}}})\ \mathrm{sin}\ \left(n\frac{\mathrm{\pi}}{2}\right)] \tag{7}\]  
The Fourier coefficient of the air gap relative permeance \(\mathrm{G}(\theta)\) is shown by Equation (8):  
\[\mathrm{G}_{a_{n\mathrm{NL}}}=\frac{2\mathrm{N}_{\mathrm{s}}}{\mathrm{\pi}} \left[\int_{\frac{\mathrm{\eta}}{2}}^{\frac{\mathrm{\eta}}{N_{\mathrm{s}}}} \mathrm{cos}(n\mathrm{N}_{L}\theta)\mathrm{d}\theta\ +\int_{0}^{\frac{\mathrm{ \eta}}{2}}\frac{\left(\frac{\mathrm{h}_{\mathrm{m}}}{\mathrm{\mu}_{\mathrm{r}} }+\mathrm{g}\mathrm{C}_{\varnothing}\right)^{2}\mathrm{cos}(n\mathrm{N}_{ \mathrm{L}}\theta)}{\left[\frac{\mathrm{h}_{\mathrm{m}}}{\mathrm{\mu}_{\mathrm{ r}}}+\left(\mathrm{g}-\mathrm{R}_{2}+\mathrm{R}_{2}\cos\left(\frac{\mathrm{b}_{ \mathrm{0}}}{2}-\theta\right)+\frac{\mathrm{\pi R}_{2}}{2}\left(\frac{\mathrm{ b}_{\mathrm{0}}}{2}-\theta\right)\right)\mathrm{C}_{\varnothing}\right]^{2}}\ \mathrm{d}\theta\right] \tag{8}\]  
where \(\mathrm{\alpha}_{\mathrm{r}}\) is the rotational angle of rotor, \(\mathrm{N}_{\mathrm{P}}\) is the number of permanent magnet pole, \(\mathrm{N}_{\mathrm{S}}\) is the number of slots, \(\mathrm{N}_{\mathrm{L}}\) is the least common multiple between \(\mathrm{N}_{\mathrm{P}}\) and \(\mathrm{N}_{\mathrm{S}}\), \(\mathrm{\alpha}_{\mathrm{p}}\) is the effective ratio of pole arc to pole pitch, \(\mathrm{b}_{\mathrm{0}}\) is the slot opening, \(\mathrm{h}_{\mathrm{m}}\) is the thickness of magnet, \(\mathrm{\mu}_{\mathrm{r}}\) is the magnet relative recoil permeability, \(\mathrm{C}_{\varnothing}\) is the flux concentration factor, \(\mathrm{g}\) is the mechanical air gap, \(\mathrm{R}_{2}\) is the outer radius of air gap and \(\mathrm{Ga}_{n}\mathrm{N}_{\mathrm{L}}\) is the Fourier coefficient  
Figure 11: Cogging torque. of air gap relative permeance. Equations (6)-(8) clearly shows the reduction in the slot opening to slot pitch ratio will reduce the cogging torque. The slot opening angle and slot pitch angle region in the generator geometry are shown in Figure 12.  
The initial generator design has a slot opening to slot pitch ratio of 1 and the geometry of the stator is shown in Figure 12a. To reduce the cogging torque, the \(\beta/\delta\) ratio has been varied by fixing the slot pitch angle to the initial value. Figure 13 shows the cogging torque by \(\beta/\delta\) ratio. As expected, the lower \(\beta/\delta\) ratio decreases the cogging torque. The lowest cogging torque, 0.14 Nm, was achieved at a \(\beta/\delta\) ratio of 0.2 but observed a ripple on the cogging torque peak. Hence the \(\beta/\delta\) ratio of 0.4 with cogging torque 0.21 Nm is chosen for this study. This will be 46% lower than the initial cogging torque. The proposed design has been simulated by varying the parameters as stated in Figure 11.0 The overall simulation outcome has been plotted to illustrate the power generated under two load conditions at different RPM. Figure 14 shows the output power generated at 360 RPM. Figure 11.1 Slot opening, \(\beta\) and slot pitch, \(\delta\). (**a**) Slot opening for an open slot design. (**b**) Slot opening for a semi-closed slot design. Figure 11.2 Cogging torque by \(\beta/\delta\) ratio. At 800 RPM generator was able to generate a peak power of 18.1 W under a 10-ohm load condition and 24.2 W under a 100-ohm load condition. The generated output power was further computed to obtain the root mean square value of the power.