III.5 Frequency Response Analysis
When a linear system is subjected to sinusoidal input perturbation, its ultimate response after a long time also becomes a sinusoidal wave, however with different amplitude and a phase shift. This characteristic constitutes the basis of frequency response analysis. One needs to study how the amplitude and phase shift change with the frequency of the input perturbation.
III.5.1 Response of a First-Order System to a Sinusoidal Input
Consider a simple first order process,
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(III.125)
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Let the sinusoidal input u(t) = A sin ωt perturb the system. Then the output of the process will be
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(III.126)
|
Computing the constants
and
and taking inverse Laplace Transform of the above equation we obtain,


(III.127)
|
After sufficiently long time
, the first term disappears as
Hence, using the identity eq. (III.91) we obtain,


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(III.128)
|
Hence we observe that
• Sinusoidal output wave has the same frequency as that of input sinusoid
• Amplitude Ratio between the output wave and input wave is 

• Output wave lags behind the input wave with a phase difference of 

Fig. III.13 shows the Input and Output wave profile for a frequency response analysis.
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Fig. III.13: Input and Output wave profile for a frequency response analysis
III.5.2 Complex Plane and Frequency Response Analysis
Consider a complex number
![]()
The modulus (or absolute value or magnitude) of W is
![]() ![]() ![]()
As
![]()
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We encounter the following two terminal situations:
• As
and hence
. That means at very low frequency, the profile of AR approaches a constant value with slope zero. This is low frequency asymptote .


•As
and hence
.That means at very high frequency, the profile of approaches a value which is inverse of the square of the frequency. This is high frequency asymptote .


• The two asymptotes meet at corner frequency 

• The profile of amplitude ratio will have three different shapes that depends upon the value of the damping coefficient. It is understood that


• For overdamped process
,
where
is a non-negative quantity. Hence,
or
is always less than one.





• For critically damped process
,
. Hence
is always less than 1



• For underdamped process
,
. Hence there will be some values of frequency where
is greater than 1



The profile of phase shift can be analyzed in the similar manner:
• As
then 


• As
then 


• As
then 


Note that the phase shift leaps by a full
as the order of the process increases by one.

III.6.3 Bode Diagram of pure time delay
It is worth to study the frequency response characteristics of pure time delay
that has
and
. The following figure shows the Bode Diagram of pure time delay.



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Note that
as
.


III.6.4 Bode Diagram for processes in series Let
and
are two processes in series. Let signals
and
be the input and output of the process
and
signals
and
be the input and output of the process
.(see the following figure).









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III.7 Nyquist Plot
A Nyquist plot is an alternative way of representing the frequency response characteristics. It uses Cartesian coordinates in two dimensions whose ordinate represents the imaginary axis and abscissa represents the real axis. A specific value of frequency defines a point on the table with ![]() ![]() ![]()
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