A second order control system exhibits 100% overshoot. Its damping ratio is:

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A second order control system exhibits 100% overshoot. Its damping ratio is:
1. Less than 1
2. Equal to 1
3. Greater than 1
4. Equal to zero

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Correct Answer - Option 4 : Equal to zero

Concept:

The transfer function of the standard second-order system is:

$TF = \frac{{C\left( s \right)}}{{R\left( s \right)}} = \frac{{ω _n^2}}{{{s^2} + 2ζ {ω _n}s + ω _n^2}}$

Characteristic equation: ${s^2} + 2ζ {ω _n} + ω _n^2 = 0$

ζ is the damping ratio

ωn is the undamped natural frequency

${M_p} = {e^{\frac{{ - ζ \pi }}{{\sqrt {1 - {ζ ^2}} }}}}$   ----(1)

Calculation:

Given:

Mp = 100%

From the above equation,

${M_p} = {e^{\frac{{ - ζ \pi }}{{\sqrt {1 - {ζ ^2}} }}}}$

$ln\;1 = \frac{{ - \zeta \pi }}{{\sqrt {1 - {\zeta ^2}} }}$ ; (ln 1 = 0)

So, ζ = 0

Note:

Mp is the maximum peak overshoot of the closed-loop transfer function

$M_p \ \alpha \ \frac{1}{ζ}$