This form of the control system is called the standard form of second-order system.
The closed-loop transfer function \(C(s)/R(s)\) of the second-order system,
\(\displaystyle G(s) = \frac{\omega_{n}^2}{s\left(s + 2\zeta\omega_{n}\right)}\)
Since the system is an unity feedback system,
\(\displaystyle \mathrm{CLTF} = \frac{C(s)}{R(s)} = \frac{G(s)}{1 + G(s)} = \frac{\mathrm{num}\left[G(s)\right]}{\mathrm{den}\left[G(s)\right] + \mathrm{num}\left[G(s)\right]}\)
\(\displaystyle \mathrm{CLTF} = \frac{\omega_{n}^2}{s\left(s + 2\zeta\omega_{n}\right) + \omega_{n}^2}\)
Then, the general second-order transfer function is
\(\boxed{\displaystyle \mathrm{CLTF} = \frac{C(s)}{R(s)} = \frac{\omega_{n}^2}{s^2 + 2\zeta\omega_{n}s + \omega_{n}^2}}\)
where \(\zeta\) is the damping ratio; and \(\omega_{n}\) is the undamped natural frequency, expressed in radians per second (rad/s).
For the damping attenutation \((\alpha = \sigma_{d})\), expressed in nepers per second (Np/s):
\(\boxed{\displaystyle \alpha = \sigma_{d} = \zeta\omega_{n}}\)
For the damped natural frequency \((\omega_{d})\), expressed in radians per second (rad/s):
\(\boxed{\displaystyle \omega_{d} = \omega_{n}\sqrt{1 - \zeta^2}}\)
Pole Location
Solving for the poles of the general second-order transfer function,
The characteristic equation is \((\displaystyle s^2 + 2\zeta\omega_{n}s + \omega_{n}^2 = 0)\)
\(\displaystyle s^2 + 2\zeta\omega_{n}s + (\zeta\omega_{n})^2 - (\zeta\omega_{n})^2 + \omega_{n}^2 = 0\)
\(\displaystyle \left(s + \zeta\omega_{n}\right)^2 = (\zeta\omega_{n})^2 - \omega_{n}^2\)
\(\displaystyle s = -\zeta\omega_{n} \pm \sqrt{(\zeta\omega_{n})^2 - \omega_{n}^2}\)
Then, the poles are located at
\(\boxed{\displaystyle s = -\zeta\omega_{n} \pm \omega_{n}\sqrt{1 - \zeta^2} = -\sigma_{d} \pm j\,\omega_{d}}\)
which can be graphically represented on the \(s\)-plane plot or pole-zero map.
The natural frequency \((\omega_{n})\) can also be found from the poles by using the pythagorean theorem,
\(\displaystyle \omega_{n}^2 = \sigma_{d}^2 + \omega_{d}^2\)
\(\boxed{\displaystyle \omega_{n} = \sqrt{\sigma_{d}^2 + \omega_{d}^2}}\)
For the value of the angle \(\theta\),
By trigonometry,
\(\displaystyle \cos(\theta) = \frac{\sigma_{d}}{\omega_{n}} = \frac{\zeta\omega_{n}}{\omega_{n}}\)
\(\boxed{\displaystyle \theta = \cos^{-1}(\zeta)}\)
or
\(\displaystyle \cot(\theta) = \frac{\sigma_{d}}{\omega_{d}} = \frac{\zeta\omega_{n}}{\omega_{n}\sqrt{1 - \zeta^2}}\)
\(\boxed{\displaystyle \theta = \cot^{-1}\left(\frac{\zeta}{\sqrt{1 - \zeta^2}}\right)}\)
