9. AR and ф Various Systems for Sinusoidal Input

System Transfer function Amplitude ratio (AR) Phase shift (\(\phi\))
First order \(\frac{K_p}{\tau_p s+1}\) \(\frac{K_p}{\sqrt{\tau_p^2\omega^2+1}}\) \(\tan^{-1}(-\omega \tau_p)\)
Pure capacitive \(\frac{K_p}{s}\) \(\frac{K_p}{\omega}\) \(-90^o\)
Second order \(\frac{K_p}{\tau^2s^2+2\zeta\tau s+1}\) \(\frac{K_p}{\sqrt{(1-\tau^2\omega^2)^2+(2\zeta\tau\omega)^2}}\) \(\tan^{-1}\left(-\frac{2\zeta\tau\omega}{1-\tau^2\omega^2}\right)\)
Dead time \(e^{-\tau_d s}\) 1 \(-\tau_d \omega\)
Systems in series \(G_1(s)G_2(s)\cdots G_N(s)\) (AR)\(_1\)(AR)\(_2\cdots\)(AR)\(_N\) \(\phi_1+\phi_2+\cdots+\phi_N\)
System Transfer function Amplitude ratio (AR) Phase shift (\(\phi\))
Proportional controller \(K_c\) \(K_c\) 0
PI controller \(K_c\left(1+\frac{1}{\tau_I s}\right)\) \(K_c\sqrt{1+\frac{1}{(\omega\tau_I)^2}}\) \(\tan^{-1}\left(\frac{-1}{\omega\tau_I}\right)\)
PD controller \(K_c(1+\tau_Ds)\) \(K_c\sqrt{1+\tau_D^2\omega^2}\) \(\tan^{-1}(\tau_D\omega)\)
PID controller \(K_c\left(1+\frac{1}{\tau_Is}+\tau_Ds\right)\) \(K_c\sqrt{\left(\tau_D\omega-\frac{1}{\tau_I\omega}\right)^2+1}\) \( \tan^{-1}\left(\tau_D\omega-\frac{1}{\tau_I\omega}\right)\)