Correct Answer - Option 1 : relative stability
Damping ratio:
The ratio of ti me constant of critical damping to that of actual damping is known as damping ratio(ζ).
Time constant for critical damping = 1 / ωn
Times constant for actual damping = 1 / ζωn
\(ζ = \frac{{\frac{1}{{{ω _n}}}}}{{\frac{1}{{ζ {ω _n}}}}}\)
ωn = Natural frequency
Maximum overshoot (MP):
- It is straight way difference between the magnitude of the highest peak of time response and magnitude of its steady state.
- Maximum overshoot is expressed in term of percentage of steady-state value of the response.
- As the first peak of response is normally maximum in magnitude, maximum overshoot is simply normalized difference between first peak and steady-state value of a response.
- The amount of maximum overshoot directly indicates the relative stability of the system.
\({M_P} = \frac{{c\left( {{t_P}} \right) \;-\; c\left( ∞ \right)}}{{c\left( ∞ \right)}}\)
c(tP) = response at rise time
c(∞) = response at steady state
Peak overshoot: It is the maximum error at the output.
\({{M}_{P}}={{e}^{-\left( \frac{ζ \pi }{\sqrt{1-{{ζ }^{2}}}} \right)}}\)
It is indicative of damping in the system. The peak overshoot is more for fewer values of the damping factor.
ζ ↓ → MP ↑ ⇒ which is not good for the system
so Peak overshoot and damping ratio are the measures of relative stability.