Task $1$

$(25$ pts$)$

Consider a DC series motor as illustrated in FIG. Q$1.$ The field winding is shunted by a branch consisting of a divertor and a switch $($sw$)$ connected in series. $\backslash ($ R$\_\{\backslash$text $\{$e $\}\}\backslash )$ represents the resistance of the armature, whilst $\backslash ($ R$\_\{$r$\}\backslash )$ is the resistance of the fleld winding. The torque of the motor varies as the square of the motor's speed. The voltage drops due to the resistances of the windings and the brush are assumed to be negligible when compared to the back EMF generated by the armature. If the switch $($sw$)$ is open, the motor runs at $1000$ rpm and its supply current is $11$ A $.$

FIG. Q$1.$ Serial motor schematic $($Note: $\backslash ($ R $\backslash ),$ and $\backslash ($ R $\backslash ),$ are respectively the armature resistance and the field resistance. The resistance of the divertor is changeable. $\backslash ($ E$\_\{\backslash$text $\{\}\}\backslash )$ is the back EMF of the armature.$)$

$($a$)$ Calculate the new speed and the new armature current after the switch $($sw$)$ is closed, assuming that the resistance of the divertor is twice as much as the field resistance.

$[12$ pts$]$

$($b$)$ Find the resistance of the divertor that doubles the motor's speed if the switch $($sw$)$ is closed. Please express your answer as a function of the field esistance.