Page 44 - Servo Motors and Industrial Control Theory -

36 2 Feedback Control Theory Continued

Fig. 2.15 Bode diagram 5
for the open loop transfer
function in the form of –10
/ ye = 0.25 / (0.25s s 2 . The Amp(ω) –25
+ 0.1s + 1) –40
phase angle reaches – 270° 0.1 1 10
ω
0
-90
Ψ(ω) –180

–270
0.1 1 10
ω

seen at the intersection point there is an exact point, which has happened by chance.
Otherwise more points are needed to be calculated. The phase angle between a unity
circle and Nyquist plot occurs at phase angle of 90°.

2.7 Bode Diagram

Bode diagram is an alternative way of presenting the frequency response of open
loop transfer function. The amplitude ratio in decibels is plotted against frequency
in logarithmic scale. On the same diagram or on separate diagram underneath the
amplitude ratio curve, the phase angle is also plotted against the frequency in loga-
rithmic scale. A typical Bode diagram is shown in Fig. 2.15.
The phase angle reaches – 270° for a third order transfer function as frequency is in-
creased. The graph of Fig. 2.15 shows that the phase angle reaches – 180° and then the
phase angle is reduced. This is because the acos function only supports 180 and – 180°.
The phase angle above –180° minus the reduction from –180°. The absolute phase
angle is above –180°. To find the direction of phase angle it is better to study the real
and imaginary values as shown below the equations used to plot the Bode diagram.

i:=− 1
0.1
z( ) :ω
(0 i ω+ · ) · ( 0.25ω− 2 + 0.li · ω+ 1)
ω
ω=
x( ) : Re(z( ))
ω=
ω
y() : Im(z())
ω=
M( ) : ( ( ))x ω 2 + ( ( ))y ω 2
x( ) ω
− 180acos
ψω M( ) ω
( ):=
π
ω
ω=
Amp( ) : 20log(M( ))