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-
where is undamped natural frequency
For this second degree equation we can have 2 values of , those are-
It is evident that for any value of , the values of the poles ; i.e the value of remains constant ; i.e .
Take a look at the locus of the poles with change in the value of below-
- is the damped natural frequency.
- It’s evident from the figure that if ; i.e for a limitedly stable system,the mod value of the pole will be and there will be two poles at .
- With the increase in value of pole will be shifted along with the perimeter of the half-circle with radius .The locus of the pole ; i.e the shown red half circle will be on the left side of the axis in the value of the real part is always negative ;i.e .
- When in case of a critically damped system damping factor ,both of the poles are at .
- In the range the poles traverse along the perimeter of the half circle(remember the radius is ),one pole starting from and the other one from , towards the left side of the axis, creating that half circular locus and they meet at when becomes .
- Now when continues to increase beyond ; i.e for a over damped system, there stays no imaginary part in the expressions of the poles, they break away from the point and start travelling in opposite direction along the real ( ) axis.
One can say that for , the poles are on the imaginary axis. And for the response of a second order system to a unit step input keep oscillating like sine wave
Say the response is .
Then the response for becomes-
[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 is the decider for the stability of that system.
[Details on LTI system stability will be discussed in some other post].