Question

D.C. motor is shown below, where the inductance L and the resistance R model the armature circuit. The voltage V_brepresents the back-emf which is proportional to dθ/dt via K_f. The torque T generated by the motor is proportional to the i via a constant K_t. In this application, let the constants K_t = K_f. The inertia J represents the combined inertia of the motor and load. The viscous friction acting on the output shaft is b. Attached to the shaft is a N1:N2 gear train that decreases the output velocity. There is no load on the motor. (Note: DO NOT neglect the electrical characteristics of the motor with relationship to the mechanical characteristics.)

For the transfer function Gp(s) , find the following parameters (if applicable):

1. ξ
2. ω_n
3. Settling Time T_S
4. %OS
5. Rise Time Tr
6. Peak Time Tp
7. Is the Underdamped, Overdamped, critically damped? Why?
8. Is there steady state error for this system shown?  

T. NI:N2 out e Vi imput + final be Rotor where K K

Answer 1

Torque T = K_t I - (1) Back emf V_theta = K_f d theta/dt - (2) Apply KCL to the armature circuit V_i = IR + L dI/dt + V_b - (3) Let V_i = V_a Load Torque J d^2 theta/dt^2 + b d theta/dt = T_m = K_t I - (4) Take Laplace transform of equation (1), (3), (4) V_b (s) = K_f S theta (s) - (5) (SL + R) I = V_a (s) - V_b (s) - (6) (s^2 J + sb) theta (s) = T_m (s) = K_t I_a (s) - (7) from equation (6) T (s) = V_a (s) - V_b (s)/SL + R = V_a (s) - K_f s theta (s)/SL + R - (8) from equation (7) I (s) = (JS^2 + Sb)/K_t theta (s) - (9) form equation (8) & (9) V_a (s) - K_f s theta (s)/SL + R = (Js^2 + sb)/K_t theta (s) Vertical-bar V_o (s) - K_f s theta (s)/SL + R Vertical-bar K_t = (Js^2 + sB) theta (s) \r\n V_a (s) = Vertical-bar (SL + R) (JS^2 + Sb)/K_t + K_f s Vertical-bar theta (s) theta (s)/V_a (s) = K_t/[(SL + R) (SJ + b) + K_t K_f] S