Question

You have a motor such that if you give it 12 Volt, it will eventually reach a steady state speed of 200 rad/s. If it starts from rest, then it takes 1.2 second for it to reach 63.2% of the steady state speed. You attach a sensor to the motor which measures its angle very accurately. The measured angle is used by the controller whose aim is to provide voltage to the motor and make the angle of the motor same as the desired angle.
Required:
a. What is the transfer function of the motor, from voltage to speed?
b. Draw the block diagram of the system with plant, controller and the feedback path.
c. What is the system type (i.e. the type of GK)?
d. If the desired speed is a constant (say 100 rad/s). What should be range of gain K of the proportional controller, so that steady state error in the speed is less than 10%? (i.e. what number K will ensure that, (desired speed - actual steady state speed)/desired speed < 0.1 ?)
e. What happens the above % error in the steady state speed as the gain K is increased?

Answer 1

Answer:

a) $\frac{Ws}{Es} = \frac{200}{1+1.2s}$

b) attached below

c) type zero system

d) k > $\frac{g}{200}$

e) The gain K increases above % error as the  steady state speed increases

Explanation:

Given data:

Motor voltage  = 12 v

steady state speed = 200 rad/s

time taken to reach 63.2% = 1.2 seconds

a) The transfer function of the motor from voltage to speed

let ; $\frac{K1}{1+St}$ be the transfer function of a motor

when i/p = 12v then steady state speed ( k1 ) = 200 rad/s , St ( time constant ) = 1.2 sec

hence the transfer function of the motor from voltage to speed

= $\frac{Ws}{Es} = \frac{200}{1+1.2s}$

b) draw the block diagram of the system with plant controller and the feedback path

attached below is the remaining part of the detailed solution

c) The system is a type-zero system because the pole at the origin is zero

d) ) k > $\frac{g}{200}$