Page 18 - Servo Motors and Industrial Control Theory

10 1 Feedback Control Theory

1.7 Frequency Response

For harmonic input in the form of x( t) = A sin ωt, the output for linear system will
be in the form of y( t) = B sin( ωt − Ψ). The phase angle Ψ, and amplitude ratio
M = B A/ can be obtained from the transfer function. If s is replaced by iω, then
the amplitude ratio and phase angle can be obtained from the resulting complex
number in the following manner
( y iω ) 1 (1.35)
( xiω ) = iτω + 1

Multiplying the numerator and denominator by the conjugate of the denominator
yields
( y iω ) = 1 − τω i (1.36)
( xiω ) τω + 1 τω + 1
2
2
2
2
Then the amplitude ratio and phase angle may be obtained as

M = realpart 2 + imaginarypart 2 (1.37)

 imaginarypart
ψ =tan − 1   realpart   (1.38)

hence,
M = 1 (1.39)
+
1 τω 2
2
ψ = tan (τω− − 1 ) (1.40)

It is a common practice to plot the amplitude ratio in decibel and phase angle in
degrees against frequency changing from zero to infinity. The frequency response is
usually plotted in db and log of frequency. These diagrams for the first order trans-
fer function are shown in Fig. 1.6.
The important points are at ω =1/τ where the amplitude ratio is M = 0.707,
M = − 3 db (in decibels; i.e. 20 log(m)) and Φ: = − 45°. This frequency is known as
the break frequency.
It can be seen from the frequency response curve and Eq. (1.32) that at low fre-
quency, that is, at frequency much lower than ω 1/τ , the amplitude ratio is ap-
proximately 0 db, and at frequency much higher than ω 1/τ , the amplitude ratio
becomes a line with slope of 20 db/decade. These two lines are the asymptotes of the