Figure P11.35a is the circuit diagram of a speed-control

Chapter , Problem 11.35

(choose chapter or problem)

Figure P11.35a is the circuit diagram of a speed-control system in which the dc motor voltage \(v_a\) is supplied by a generator driven by an engine. This system has been used on locomotives whose diesel engine operates most efficiently at one speed. The efficiency of the electric motor is not so sensitive to speed and thus can be used to drive the locomotive at various speeds. The motor voltage \(v_a\) is varied by changing the generator input voltage \(v_f\). The voltage \(v_a\) is related to the generator field current \(i_f\) by \(v_a=K_f i_f\).

Figure P11.35b is a diagram of a feedback system for controlling the speed by measuring it with a tachometer and varying the voltage \(v_f\). Use the following values in SI units.

\(\begin{aligned}& L_f=0.2 \quad R_f=2 \quad K_t=1 \\& L_a=0.2 \quad R_a=1 \quad K_b=K_T=0.5 \\& K_f=50 \quad I=10 \quad c=20 \\&\end{aligned}\)

a. Develop a state variable model of the plant that includes the generator, the motor, and the load. Include the load torque \(T_L\) as a disturbance.

b. Develop a proportional controller assuming all the state variables can be measured. Analyze its steady-state error for a step command input and for a step disturbance.