Problem #$6$: In the PWM control system as shown in the diagram, the PWM signal is used to control the speed of a DC motor. The PWM duty cycle determines the average voltage supplied to the motor, thereby controlling its speed. Assume the following:

$[7$ The pulse width of the PWM signal is adjusted using a potentiometer connected to the analog input pin AN$0$ of the PIC$24$ microcontroller.

$[3$ The potentiometer produces an output voltage ranging from $0$ V to $3\mathrm{.}3$ V $.$

$[7.$ The ADC on the PIC$24$ microcontroller is configured to read this voltage with a $10-$bit resolution $($values range from $0$ to $1023).$

$0$ The motor's speed is linearly proportional to the PWM duty cycle.

$.$ At $0\%$ duty cycle, the motor is stationary, and at $100\%$ duty cycle, the motor rotates at its maximum speed of $3000$ RPM$.$

A$)$ Calculate the motor's speed $($in RPM$)$ when the ADC value is $(10$ points$)$:

I$)0$

II$)256$

III$)1023$

B$)$ What percentage of the maximum speed will the motor reach for each ADC value? $(10$ points$)$