Consider the standard second-order system of the form $\frac{{\omega }_n^2}{s^2+2\zeta {\omega }_{n}s+{\omega }_n^2}$ with the poles p and p^∗ having negative real parts. The pole locations are also shown in the figure. Now consider two such second-order systems as defined below:

System 1 : ${\omega }_n=3\mathrm{rad}/\sec$ and $\theta ={60}^{\circ }$

System 2 : ${\omega }_n=1\mathrm{rad}/\sec$ and $\theta ={70}^{\circ }$

The correct option is: Settling time of system 2 is more than that of system 1.

1. Analysis of the Pole Locations in the s-Plane:
From the provided image, we can observe the following elements in the s-plane diagram:
- The horizontal real axis is labeled as $\sigma$.
- The vertical imaginary axis is labeled as $j\omega$.
- A pole is plotted at location $p$ in the second quadrant, with its complex conjugate pole at $p^{\ast }$ in the third quadrant. Both lie on a dashed circular arc representing a constant natural frequency ${\omega }_n$.
- The angle $\theta$ is measured between the negative real axis and the radial line from the origin to the pole $p$.

For a standard second-order system, the poles are represented as:

$p,p^{\ast }=-\zeta {\omega }_n\pm j{\omega }_n\sqrt{1-{\zeta }^2}$

Using basic trigonometry on the right-angled triangle formed by the origin, the pole $p$, and the real axis, we find the relation between the damping ratio $\zeta$ and the angle $\theta$:

$\cos \left(\theta \right)=\frac{\zeta {\omega }_n}{{\omega }_n}=\zeta$

Thus, the damping ratio is given by $\zeta =\cos \left(\theta \right)$.

2. Settling Time Formula:
The settling time $T_s$ for a second-order system is determined by the real part of its poles, which is $\zeta {\omega }_n$. Using the 2% settling time criterion, we have:

$T_s=\frac{4}{\zeta {\omega }_n}$

This shows that settling time is inversely proportional to the product $\zeta {\omega }_n$.

3. Calculations for Each System:
Let us calculate the value of $\zeta {\omega }_n$ for both systems:

System 1:
- Given: ${\omega }_n=3\text{rad/sec}$ and $\theta ={60}^{\circ }$
- Damping ratio: ${\zeta }_1=\cos \left({60}^{\circ }\right)=0.5$
- Real part value:

${\zeta }_1{\omega }_{n1}=0.5\times 3=1.5\text{rad/sec}$

System 2:
- Given: ${\omega }_n=1\text{rad/sec}$ and $\theta ={70}^{\circ }$
- Damping ratio: ${\zeta }_2=\cos \left({70}^{\circ }\right)\approx 0.342$
- Real part value:

${\zeta }_2{\omega }_{n2}=0.342\times 1=0.342\text{rad/sec}$

4. Comparison and Conclusion:
Comparing the real pole coordinates of both systems:

${\zeta }_1{\omega }_{n1}=1.5>{\zeta }_2{\omega }_{n2}=0.342$

Since System 2 has a smaller real part ($\zeta {\omega }_n$) than System 1, its poles are closer to the imaginary axis. Consequently, its response decays more slowly, yielding a larger settling time:

$T_{s2}>T_{s1}$

Thus, the settling time of system 2 is more than that of system 1.