Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

Instrumentation and Process Control

Electrical Engineering (EE) : Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

The document Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Instrumentation and Process Control.
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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. 

Response of a First-Order System to a Sinusoidal Input 
Consider a simple first order process,

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev                          125

Let the sinusoidal input u(t) = A sin ωt perturb the system. Then the output of the process will be
Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

Computing the constants C1,C2and C3 and taking inverse Laplace Transform of the above equation we obtain,

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev                  127

After sufficiently long time  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev , the first term disappears as  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev  Hence, using the identity eq. (91) we obtain,

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev         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  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

•  Output wave lags behind the input wave with a phase difference of  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

Fig. III.13 shows the Input and Output wave profile for a frequency response analysis.

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

 Complex Plane and Frequency Response Analysis

Consider a complex number W = a + jb
Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

The modulus (or absolute value or magnitude) of W is  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev  and the argument (or phase angle) is  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev . Let us put  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev  in the transfer function of the first order process as

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev              129
As  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev  is now a complex number, the modulus and argument can be calculated as,
Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev                       130

The last two relationships indicate the amplitude ratio and phase lag for the ultimate response of the first order process. Hence the observations in the last subsection can also be stated in the light of the above results as follows:

•  Sinusoidal output wave has the same frequency as that of input sinusoid

•  Amplitude Ratio between the output wave and input wave is  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

•  Output wave lags behind the input wave with a phase difference of  Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev

Example of frequency response of a second order system
The process is
Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev                       131

Put and Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev calculate 
Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev                 132

then

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev           133

and

Frequency Response Analysis Electrical Engineering (EE) Notes | EduRev             134

 

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