Question

Answer Question B-3-13 above with the instructions listed

B-3-13. Consider the system shown in Figure 3-42. An armature-controlled dc servomotor drives a load consisting of the moment of inertia Ji. The torque developed by the motor is T.The moment of inertia of the motor rotor is J. The angular displacements of the motor rotor and the load er circuit Eo(s)/E(s) of the element are 0 and 0, respectively. The gear ratio is n 0/0m. Obtain the transfer function O(s)/E(s) L R ww T JL Figure 3-42 Armature-controlled dc servomotor system Problem 1 (20 points) For problem B-3-13, assuming n=1, answer the following 1. Derive the differential equations of the system. 2. Build the block diagram of the system showing all variables relations e (s) Simplify the block diagram to get the transfer function Ei(s) 3. 4. What is the transfer function if the output is the speed instead of angular displacement, i.e w (s) ? Ei(s) 5. If the inductance of the motor is very small such that it can be ignored, What is the new transfer function Ei(s)

Answer 1

Consider the following dc servomotor system.

Figure 1

In a armature controlled DC servomotor, the field current is held constant, and the armature current is controlled through the armature voltage .

In this case, the motor torque increases linearly with the armature current.

That is,

…… (1)

Where,

is a constant that depends on the chosen motor.

Write the voltage relationship for the armature side of the motor.

…… (2)

Where,

is “back EMF” induced by the rotation of the armature windings in a magnetic field.

The back emf is directly proportional to angular velocity.

Substitute in equation (2).

Apply Laplace transform to the equation .

…… (3)

Draw the free body diagram of the inertial load.

Figure 2

Applying ’s law for the rotational motion of the motor gives

But

Therefore,

…… (4)

Substitute for in equation (4).

…… (5)

We know that,

Therefore,

…… (6)

Substitute for in equation (5).

…… (7)

Write the expression for load torque.

…… (8)

Substitute for and for in equation (7).

…… (9)

Apply Laplace transform to equation (9).

…… (10)

Substitute for in equation (3).

Therefore, the transfer function of the system is

.