Page 53 - Servo Motors and Industrial Control Theory -

2.7 Bode Diagram 45

Fig. 2.28 A servo position 1 θ
control system with position K o
and derivative feedback s s(0.5s + 1)

1 + K s
d

In fact, the process can be repeated to plot the root locus if required. It can be seen
that all roots move towards infinity. The real root moves on the negative real axis
towards infinity and the two complex roots, which always appear in conjugate move
towards infinity on the right hand side of the s-plane. In fact, there are always the
same amount of loci as the order of the characteristic equation. If there are zeros in
the open loop transfer function, some loci tend to move towards these zeros; other-
wise they move to infinity.
Example 8 Servo with velocity feedback.
As was shown in the previous example, position control systems with two integra-
tors inherently are unstable. To introduce damping and to stabilize the system, a ve-
locity feedback is added. The derivative term always introduces noise in the system.
In practice, a small DC motor known as tacho generator is attached to the motor,
which produces a voltage proportional to velocity of the motor. This is shown in
Fig. 2.28.
The closed loop transfer function can be obtained as

k
θ s (0.5s + ) s
2
o = (2.36)
θ i 1+ (1 k sk+ d )
2
s (0.5s + ) s
The right hand side of the denominator is known as the open loop transfer func-
tion. It shows that with the addition of the velocity feedback, a zero appears that
attract the unstable roots. Therefore, the system becomes more stable. Simplifying
Eq, (2.36) gives
θ k (2.37)
o =
θ i 0.5s + s + k ks + k
2
3
d
With the characteristic equation of

05. s + s + kksk+= 00. (2.38)
2
3
d