The figure below on the left shows an armature controlled de servomotor driving a load through a gear train, which is commonly used in a closed-loop control system. The schematic diagram below on the right represents the armature circuit rotating simply due to the voltage ea(t) applied and the fixed magnetic field B by a permanent magnet. The armature voltage as an electrical parameter ea(t) is considered to be the input to the system. The resistance and inductance of the armature circuit are Ra and La. respectively. vý(t) is the back emf and directly proportional to the rotational speed of the armature as v,(t) = Kpwm(t), where K, is a constant of proportionality called the back emf constant. The torque developed by the motor is proportional to the armature current, Tm(t) = Kglq(t), where Kę is the constant of proportionality and called the motor torque constant. When motor drives a load, the equivalent inertia and viscous damping at the amature are Jm and Dm. respectively. These entities include the corresponding armature and load parameters. (a) Obtain the transfer function of G,(s) = "mO and G2(s) = in terms of electrical and mechanical parameters like K,. Kp, Jm. Dm Ra La- (b) Obtain G, (s) and G2(s) with the assumption of La = 0, which is usual for de machines since La « Ra- (c) A de motor develops 60 Nm of torque at a speed of 500 rad/s when 12 volts are applied. It stalls out at this voltage with 120 Nm of torque. If the inertia and damping of the amature are 7 kg-m² and 3 Nm- s/rad, respectively, find the transfer functions, G,(s) and G2(s), of this motor with the assumption of La = 0, if it drives a load with 108 kg-m² inertia and 9 Nm-s/rad damping through a gear train as shown below on the left. Ea(s) Ea(s) wL(s) Ea(s) (d) Find also the transfer function from the input voltage to the output speed, i.e., G3(s) (e) Plot the step responses of G2(s) and G3(s) for t= 0 to 20 sec. Remember that the input voltage is 12 V. (f) Determine the number of rotations within the first 18 secs from the 0, vs. t plot. (g) What is the steady-state value of the speed in rad/sec and also in rpm? Confirm it from the w̟ vs. t plot. ed) Motor N1 = 12 R. N2 = 25 N3 = 25 Armature v) Rotor circuit N4 = 72 Load ell