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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,

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Let the sinusoidal input u(t) = A sin Ï‰t perturb the system. Then the output of the process will be

Computing the constants C_{1},C_{2}and C_{3} and taking inverse Laplace Transform of the above equation we obtain,

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

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Hence we observe that

â€¢ Sinusoidal output wave has the same frequency as that of input sinusoid

â€¢ Amplitude Ratio between the output wave and input wave is

â€¢ 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.

**Complex Plane and Frequency Response Analysis**

Consider a complex number W = a + jb

The modulus (or absolute value or magnitude) of W is and the argument (or phase angle) is . Let us put in the transfer function of the first order process as

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As is now a complex number, the modulus and argument can be calculated as,

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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:

â€¢ Sinusoidal output wave has the same frequency as that of input sinusoid

â€¢ Amplitude Ratio between the output wave and input wave is

â€¢ Output wave lags behind the input wave with a phase difference of

**Example of frequency response of a second order system**

The process is

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Put and calculate

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then

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and

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35 videos|37 docs|8 tests

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### Frequency Response Example

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