A $\backslash (25\backslash$mathrm$\{$~kW$\},250\backslash$mathrm$\{$~V$\},1,400\backslash$mathrm$\{$rpm$\}\backslash )$ DC shunt motor has an armature resistance $\backslash (\backslash$mathrm$\{$R$\}\_\{\backslash$mathrm$\{$a$\}\}=0\mathrm{.}15\backslash$Omega $\backslash ).$ The magnetisation characteristic at $\backslash (1,400\backslash$mathrm$\{$rpm$\}\backslash )$ is shown in Figure $2.$ At no$-$load conditions, the motor runs at $\backslash (1,400\backslash$mathrm$\{$rpm$\}\backslash )$ and the armature current is measured as $12$ A $.$

Figure $2$ DC motor magnetisation curve at $1,400$ rpm

$($a$)$ Explain how a commutator works in a DC machine. Use a diagram of a typical commutator structure to illustrate your answer.

$[5$ marks$]$

$($b$)$ Find the value of the resistance of the shunt field for the given motor conditions.

$[4$ marks$]$

$($c$)$ Find the speed and the electromagnetic torque when rated current flows in the armature of the given motor.

$[5$ marks$]$

$($d$)$ Given that mechanical losses are $1\mathrm{.}25$ kW $,$ calculate the output mechanical power and the motor efficiency for the conditions of part $($c$).$

$[6$ marks$]$