11 - Frequency Response

For a linear system with transfer function \(G(s)=Y(s)/X(s)\), for input \(X(t) = A\sin(\omega t)\), output at steady state (i.e., ultimate response) is given by \[Y_{ss}(t) = A|G(j\omega)| \sin(\omega t+\phi)\]

The ultimate response as \(t\rightarrow\infty\) is sinusoidal with frequency \(\omega\).

The amplitude ratio is \[\text{AR} = \frac{A|G(j\omega)|}{A} = |G(j\omega)|\]

The output sinusoidal wave has been shifted by the angle \[\phi = \text{argument of }G(j\omega)\]