Question

Determine the transfer function of the system ***with an added inductor***. Then determine the state space representation of the system shown with an added inductor.

us Question Determine the Transfer Function of the systern in Problem #2, but now added into the circuit (figure below). (4 points) a) ith an inductor b) Determine the State-Space representation for the system with R. b6m

Answer 1

Transfer function from the circuit v_a = e + ia Rq + La dia/dt rightarrow (1) e = k_e theta_m rightarrow (2) d theta_n/dt = w_n rightarrow (3) therefore e = k_e w_m (4) rightarrow (4) substitute eqn (4) in (1) v_a(t) = k_e w_m(t) + i_a R_a + L_a dia/dt rightarrow (5) Torque of the motor T = k_t I_a rightarrow (6) Torque is directly proportional to armature current and mechanical Torque T = Im d^2 theta m/dt^2 + b d theta m/dt + k_m theta m rightarrow (r) therefore from eqn (6) & (3) k_t I_a(t) = I_m d w_m(t)/dt + b w_m(t) + k_m theta_m therefore Jm dw_m (t)/dt = k_t I_a(t) - k_m theta_m (t) - b w_n(t) \r\n d w_m(t)/dt = k_t/J_m I_a(t) - k_m/J_m theta m(t) - b/Jm wm(t) rightarrow (7) Apply Laplace Transform for equation (5) & (7) v_a(t) = k_e d theta m/dt + I_a(t) R_a + la dia/dt (t)