Transfer Function

Standard Second Order System

Definition 1: Standard second order system

The standard second order system has a Transfer Function given by $H(s)=\frac{w^2}{s^2+2\xi \omega s+{\omega }^2}$

where $\omega >0$ and $\xi \ge 0$.

The quadratic formula tells us that the poles of $H(s)$ are $s_{\pm }=-\xi \omega \pm \omega \sqrt{{\xi }^2-1}$

There are hence three different forms of the solution based on where the poles lie.

Case 1: Two real distinct pole.

Case 4: Two complex conjugate poles with a zero real component: If ξ2 − 1 = 0, or simply if ξ = 0, then we have complex conjugate poles at s± = ±ωj These roots have zero real parts and hence the system may not be well behaved is there is resonance!