6. Frequency Response of Pure Capacitive Process

For a pure capacitive process \[G(s) = \frac{K_p}{s}\] Put \(s=j\omega\) \[G(j\omega) = \frac{K_p}{j\omega} = \frac{K_p}{j\omega}\frac{j\omega}{j\omega}=0-j\frac{K_p}{\omega}\]

The amplitude ratio is \[\text{AR} = |G(j\omega)| = \sqrt{0^2+\left(\frac{-K_p}{\omega}\right)^2} = \frac{K_p}{\omega}\]

The phase shift is \[\phi = \tan^{-1}\left(\frac{-Kp/\omega}{0}\right) = \tan^{-1}(-\infty) = -90^\circ\]