When operating at rated voltage, a $2\mathrm{.}5-$kW$,120-$V$,1725$ r$/$min separately excited dc motor achieves a no$-$load speed of $1708$ r$/$min at a field current of $0\mathrm{.}85$ A$.$ The motor has an armature resistance of $163$ mA and a shunt$-$field resistance of $114.$ For the purposes of this problem you may assume the rotational losses to be negligible.

This motor is to be used to control the speed of a load whose torque is constant at $13\mathrm{.}7$ N $$ m over the speed range of $14001750$ r$/$min$.$ The motor will be operated at a constant armature voltage of $120$ V$.$ The field winding will be supplied from the $120-$V dc armature supply via a pulse$-$width modulation system and the average field voltage and current, and hence the motor speed, will be varied by adjusting the duty cycle of the pulse$-$width modulation; a duty cycle variation between $0$ and $1\mathrm{.}0$ will result in a linear variation of the average field voltage over the range of $0$ to $120$ V$.$

a$.$ Calculate the field current required to achieve operation at $13\mathrm{.}7$ N $$ m

torque and $1750$ r$/$min$.$ Calculate the corresponding PWM duty cycle D$.$

b$.$ Calculate the field current required to achieve operation at $13\mathrm{.}7$ N $$ m

torque and $1400$ r$/$min$.$ Calculate the corresponding P