### With Varying Damping Factor ζ Pole Distribution of a Second order Linear Time Invariant system when subjected to an Unit Step Input

From the time response analysis of a second order system the characteristic equation of a system can be written as-
$s^{2} + 2\zeta\omega_{n}s + \omega_{n}^2 = 0$
where $\omega_n$ is undamped natural frequency

For this second degree equation we can have 2 values of $s$ , those are-
$s_{1} = - \zeta\omega_{n} + j\omega_{n}\sqrt{1-\zeta^2}$
and $s_{2} = - \zeta\omega_{n} - j\omega_{n}\sqrt{1-\zeta^2}$

Now, $\mid{s_1} \mid = \mid{s_2} \mid = \sqrt {{(\zeta \omega_n)^2} + (\omega_n \sqrt{1-\zeta^2})^2} = \pm\omega_n$

It is evident that for any value of $\zeta$ , the values of the poles ; i.e the value of $s$ remains constant ; i.e $\omega_n$ .
Take a look at the locus of the poles with change in the value of $\zeta$ below-

• Here  $\theta = cos^{-1}\zeta$
• $\omega_d$ is the damped natural frequency.
• It’s evident from the figure that if $\zeta =0$ ; i.e for a limitedly stable system,the mod value of the pole will be $\omega_n$ and there will be two poles at $\pm\omega_n$ .
• With the increase in value of $\zeta$ pole will be shifted along with the perimeter of the half-circle with radius $\omega_n$ .The locus of the pole ; i.e the shown red half circle will be on the left side of the $j \omega$ axis in the value of $s_1$ $s_2$ the real part is always negative ;i.e $- \zeta\omega_n$ .
•  When in case of a critically damped system damping factor $\zeta =1$ ,both of the poles are at $- \omega_n$.
•  In the range $0< \zeta <1$ the poles traverse along the perimeter of the half circle(remember the radius is $\omega_n$ ),one pole starting from $+j \omega_n$ and the other one from $-j \omega_n$, towards the left side of the $j \omega$ axis, creating that half circular locus and they meet at $- \omega_n$ when $\zeta$ becomes $1$ .
• Now when $\zeta$ continues to increase beyond $1$ ; i.e for a over damped system, there stays no imaginary part in the expressions of the poles, they break away from the point $- \omega_n$ and start travelling in opposite direction along the real ( $\sigma$ ) axis.

One can say that for $\zeta = 0$ , the poles are on the imaginary axis. And for $\zeta = 0$ the response of a second order system to a unit step input keep oscillating like sine wave
Say the response is $c(t)$ .
Then the response for $\zeta = 0$ becomes-
$c(t) = 1- cos {\omega_n}{t}$ [This will be proved in some other post].

So for basic understanding it can be concluded that for a second order Linear Time Invariant (LTI) system, damping factor $\zeta$ is the decider for the stability of that system.

[Details on LTI system stability will be discussed in some other post].