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,

(III.125)

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

(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,