A $3-$phase BLDC motor operates at a constant speed of $1800$ rpm with a torque constant Kt$=0\mathrm{.}2$Nm$/$AKt$=0\mathrm{.}2$Nm$/$A$.$ The motor is powered by a $400$ V DC supply, and the back EMF per phase is given by:

e$($t$)=250$sin$(100$t$)$V$.$e$($t$)=250$sin$(100$t$)$V$.$

The motor's phase resistance is R$=0\mathrm{.}3$R$=0\mathrm{.}3,$ and the load torque fluctuates by $10\%10\%.$ Assuming Pout$=$PinPout$=$Pin$,$ determine:

a$.$ The instantaneous power output of the motor.

b$.$ The variation in motor speed due to load torque fluctuation, assuming the motor has a moment of inertia J$=0\mathrm{.}01$kg$$m$2$J$=0\mathrm{.}01$kg$$m$2.$