Consider a three$-$phase permanent$-$magnet synchronous motor with the following mo$-$ del parameters: stator resistance Rs $=1\mathrm{.}5$; stator inductance Ls $=20$ mH; and permanent$-$magnet flux linkage f $=0\mathrm{.}49$ Vs$.$

$($a$)$Using the waveforms given below, determine the number of pole pairs np and jus$-$ tify your answer. $[$Tip: It might be easiest to study the interval t $=0\mathrm{.}3...0\mathrm{.}4$ s$.]$

$($b$)$For the mechanical angular speed M $=2$ $16$ rad$/$s$,$ calculate the correspon$-$ ding supply frequency $($Hz$),$ electrical angular speed $($rad$/$s$),$ and rotation speed $($r$/$min$).$

$($c$)$Using the waveforms, approximate the total inertia J and justify your answer. The friction can be omitted. $[$Tip: In the interval t $=0\mathrm{.}1..\mathrm{.}0\mathrm{.}15$ s$,$ you can assume the electromagnetic torque to be constant.$]$

$($d$)$ Calculate the stator voltage us in rotor coordinates at t $=0\mathrm{.}35$ s$.$