Page 45 - Servo Motors and Industrial Control Theory -

2.7 Bode Diagram 37

Fig. 2.16 Linear and tor-
sional spring, which represent
a mechanical proportional
system

K
T θ

X,F

The gain margin and phase margin as defined in Nyquist plot can be obtained from
Bode diagram. This is for a stable system. If the amplitude ratio curve intersects the
0 db line below the 0 db line when the phase angle is – 180°, the closed system is
unstable. The Bode diagram shown in Fig. 2.12 gives the gain margin as 12 db. And
the phase margin is obtained as the difference between – 180° line and the actual
phase angle when the amplitude ratio is 0 db. This is 90° for the system shown in
Fig. 2.13.
It should be noted that the plot of frequency response of open loop transfer by
hand is very tedious. It is easy always to write a simple computer program to plot
Nyquist or Bode diagram. There is also commercial software to plot the above-
mentioned diagrams.
The Nyquist and Bode diagrams are the frequency response of the open loop
transfer function. At each frequency, the open loop response is a complex number.
Equation (2.7) can be used to calculate the frequency response of the closed loop
system. There are graphical methods of obtaining the closed loop frequency re-
sponse from the open loop frequency response. By the advent of computers, these
methods are not recommended.
The above analysis forms the basis of analysis of control systems. In the fol-
lowing examples, the analysis of some simple and complicated systems will be
covered. This includes some mechanical and electrical systems that find application
in servo motor systems.

Example 2 simple spring.
A simple linear and rotational spring is shown in Fig. 2.16. The relationship be-
tween the force F and displacement x assuming a linear relation can be written as
F = Kx
T = Kθ (2.21)

This relation, assuming the force F as input and displacement x as the output is
shown in block diagram form in Fig. 2.17. The output displacement x responds
instantly when a force F is applied. The term 1/ K is a zero order transfer function.
Example 3 Addition or subtraction of two signals.
Figure 2.18 shows a mechanical device that can be used to add or to subtract two
signals. The system consists of a mechanical bar with three possible movements.