Question

In a Second-order system of under-damped case, the decay ratio and overshoot are related as:

• A. Overshoot = (decay ratio)\(^2\)
• B. Overshoot = \( \sqrt{\text{Decay ratio}} \)
• C. Decay ratio = Overshoot
• D. Decay ratio = (Overshoot)\(^3\)

Hint:

Remember the exponential decay of oscillations in an under-damped system. Each successive peak is smaller than the previous one by a consistent factor related to the damping ratio. The decay ratio quantifies this reduction between peaks, and the overshoot is the first peak's magnitude.

Answer 1

The Correct Option is B

Step 1: Understand the characteristics of an under-damped second-order system. 
An under-damped second-order system exhibits oscillatory behavior before settling to its final steady-state value when subjected to a step input. The response is characterized by parameters such as damping ratio (\( \zeta \)), natural frequency (\( \omega_n \)), overshoot, settling time, and decay ratio. 
Step 2: Define overshoot and its relation to the damping ratio. 
Overshoot (OS) is the maximum peak value of the response curve measured from the final steady-state value. For a second-order system, the percentage overshoot (%OS) for a unit step input is given by: \[ \%OS = e^{-\frac{\pi \zeta}{\sqrt{1 - \zeta^2}}} \times 100\% \] The overshoot (as a fraction) is therefore: \[ OS = e^{-\frac{\pi \zeta}{\sqrt{1 - \zeta^2}}} \] 
Step 3: Define decay ratio and its relation to the damping ratio. 
The decay ratio (DR) is the ratio of the amplitudes of two successive peaks in the oscillatory response. The amplitude of the oscillations decays exponentially with a time constant related to the damping ratio. The peaks occur at times \( t_p = \frac{n\pi}{\omega_d} \), where \( \omega_d = \omega_n \sqrt{1 - \zeta^2} \) is the damped natural frequency and \( n = 1, 3, 5, ... \) for the peaks above the final value, and \( n = 2, 4, 6, ... \) for the valleys below the final value. The ratio of the amplitudes of two successive peaks (e.g., the first overshoot and the second undershoot, or the second undershoot and the third overshoot) separated by one period \( T = \frac{2\pi}{\omega_d} \) is \( e^{-\zeta \omega_n T} = e^{-\zeta \omega_n \frac{2\pi}{\omega_n \sqrt{1 - \zeta^2}}} = e^{-\frac{2\pi \zeta}{\sqrt{1 - \zeta^2}}} \). The decay ratio is typically defined as the ratio of the height of the second peak to the height of the first peak (both measured from the steady-state value). The first overshoot has a magnitude proportional to \( e^{-\frac{\pi \zeta}{\sqrt{1 - \zeta^2}}} \), and the next peak (undershoot) has a magnitude proportional to \( e^{-\frac{3\pi \zeta}{\sqrt{1 - \zeta^2}}} \). The peak after that (second overshoot) has a magnitude proportional to \( e^{-\frac{5\pi \zeta}{\sqrt{1 - \zeta^2}}} \). The decay ratio (ratio of successive peak heights) is then: \[ DR = \frac{e^{-\frac{3\pi \zeta}{\sqrt{1 - \zeta^2}}}}{e^{-\frac{\pi \zeta}{\sqrt{1 - \zeta^2}}}} = e^{-\frac{2\pi \zeta}{\sqrt{1 - \zeta^2}}} = \left( e^{-\frac{\pi \zeta}{\sqrt{1 - \zeta^2}}} \right)^2 \] So, \( DR = (OS)^2 \). 
Step 4: Relate overshoot and decay ratio. 
From the relationship derived above, \( DR = (OS)^2 \), we can also express the overshoot in terms of the decay ratio: \[ OS = \sqrt{DR} \] 
Step 5: Evaluate the given options. 
Option 1: Overshoot = (decay ratio)\(^2\) - This is incorrect; the decay ratio is the square of the overshoot.
Option 2: Overshoot = \( \sqrt{\text{Decay ratio}} \) - This is consistent with our derivation.
Option 3: Decay ratio = Overshoot - This is incorrect unless the overshoot is 0 or 1, which is not generally true for under-damped systems.
Option 4: Decay ratio = (Overshoot)\(^3\) - This is incorrect based on our derivation.
Step 6: Select the correct answer. 
The correct relationship between overshoot and decay ratio for an under-damped second-order system is Overshoot = \( \sqrt{\text{Decay ratio}} \).