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-

locus of poles with varying ζ

  • 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].

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