Page 40 - Servo Motors and Industrial Control Theory

32 2 Feedback Control Theory Continued

Fig. 2.10 A feedback control
system with first order open x + 1 y
loop transfer function τs + 1
–

The closed loop transfer function becomes

()
y = Gs (2.12)
x 1 Gs+ () ()H s

To obtain the Nyquist Plot only the frequency response of the open loop transfer
function ( G( s)H( s)) is considered. To obtain the frequency response s is replaced
by iω. The result of this calculation will be a complex number with real part and
imaginary part. The amplitude and phase angle are obtained as

2
Amplitude ratio = real part + imaginary part 2 (2.13)

imaginary part
tan( )ϕ = (2.14)
real part

It is not necessary to go in details of proof of Nyquist stability criteria. In Eq. (2.10),
it can be seen that G( s)H( s) might become – 1 at certain frequency. Therefore, the
Nyquist stability criteria for open loop stable system are that the frequency response
of the open loop transfer function G( iω)H( iω) must not encircle the – 1-point. It
should be stressed that in Nyquist plot the frequency response of open loop transfer
function is plotted in polar coordinate.
Example 1 System with first order open loop transfer function.
A system with first order open loop transfer function and unity feedback is
shown in Fig. 2.10.
1
() =
Giω
i τ ω− 1

It was shown in previous chapter that the amplitude ratio and phase angle are given
as
1
M = (2.15)
2
+
1 τω 2