$1-$ Model a single$-$phase full wave controlled bridge Inverter for speed control of a DC motor using on MATLAB$/$Simulink$.[5$ marks$]$

$2-$ Present simulation results that demonstrate the relationship between input voltage and motor speed. Include appropriate graphs and circuit diagrams. $[5$ marks$]$

triggering angle

$=40$

$V_{RMS}=40$V

$f=50$ Hz

R$=5$ ohm

$L=100mH$

$3-$ Integrate a PID controller into the system with the Single$-$Phase Full Bridge Inverter and analyze its effect on the motor's speed and system stability. $[5$ marks$]$

Provide simulation results illustrating the system's response with the PID controller. Discuss improvements in speed regulation and response time. Include system diagrams and waveforms.

Hint: Consider DC motor load as RLE load to find relation between vout and firing angle

$4-$ Investigate the power losses in the Single$-$Phase Full Bridge Inverter across a range of motor speeds, from half$-$rated speed to rated speed. Analyze how these losses affect the system's overall efficiency. $[5$ marks$]$

The process block diagram is illustrated in Figure $3.$

Requirements:

Software: MATLAB$/$Simulink

Components: DC motor and necessary blocks $($provided on Moodle$)$

Documentation: Clearly present all steps, including circuit diagrams, graphs, equations, and analyses. Demonstrate a thorough understanding of the processes and calculations involved.