04 - Second Order Systems

Overdamped System:

\(\zeta > 1\): \[Y(t) = K_p\left[1-e^{-\zeta t/\tau}\left(\cosh\left\{\sqrt{\zeta^2 - 1}\,\frac{t}{\tau}\right\}+\frac{\zeta}{\sqrt{\zeta^2-1}}\sinh\left\{\sqrt{\zeta^2 - 1}\,\frac{t}{\tau}\right\} \right)\right]\] Instead of the above big formula, we shall make use of the following: \[Y(t) = K_p\left(1+\frac{\tau_1e^{-t/\tau_1}}{\tau_2-\tau_1}-\frac{\tau_2e^{-t/\tau_2}}{\tau_2-\tau_1} \right)\] from writing the transfer function as: \[G(s)=\frac{K_p}{(\tau_1s+1)(\tau_2s+1)}\]