STEM Problem Detail

Problem Statement

Prompt

Give me solution for question number 2.23.

Solution Rendering...

The transfer function for a second order system is given as
$$ \frac{C(s)}{R(s)} = \frac{\omega_n^2}{s^2 + 2\xi\omega_n s + \omega_n^2} $$ where, $$C(s)$$ = output, $$R(s)$$ = Input.

For a unit-step in put $$u(t)$$, $$R(s) = \frac{1}{s}$$. Therefore,
$$ C(s) = \frac{\omega_n^2}{s(s^2 + 2\xi\omega_n s + \omega_n^2)} $$

**Case I : Under -Damped system ($$\xi < 1$$)** : Roots of $$s^2 + 2\xi\omega_n s + \omega_n^2 = 0$$ will be complex.
$$ C(s) = \frac{A}{s} + \frac{Bs + C}{s^2 + 2\xi\omega_n s + \omega_n^2} $$
$$ \omega_n^2 = A(s^2 + 2\xi\omega_n s + \omega_n^2) + (Bs + C)s $$
$$ A + B = 0, \quad 2\xi\omega_n A + C = 0, \quad A = 1 $$
$$ B = -1, \quad C = -2\xi\omega_n $$
$$ C(s) = \frac{1}{s} - \frac{s + 2\xi\omega_n}{s^2 + 2\xi\omega_n s + \omega_n^2} = \frac{1}{s} - \frac{s + \xi\omega_n}{(s + \xi\omega_n)^2 + (\omega_n\sqrt{1-\xi^2})^2} - \frac{\xi\omega_n}{(s + \xi\omega_n)^2 + (\omega_n\sqrt{1-\xi^2})^2} $$
$$ C(s) = \frac{1}{s} - \frac{s + \xi\omega_n}{(s + \xi\omega_n)^2 + (\omega_n\sqrt{1-\xi^2})^2} - \frac{\xi}{(s + \xi\omega_n)^2 + (\omega_n\sqrt{1-\xi^2})^2} \omega_n\sqrt{1-\xi^2} $$

Taking Inverse Laplace Transform,
$$ c(t) = \left( 1 - e^{-\xi\omega_n t} \cos(\omega_n\sqrt{1-\xi^2} t) - \frac{\xi}{\sqrt{1-\xi^2}} e^{-\xi\omega_n t} \sin(\omega_n\sqrt{1-\xi^2} t) \right) u(t) $$
$$ c(t) = \left( 1 - e^{-\xi\omega_n t} \left[ \cos(\omega_n\sqrt{1-\xi^2} t) + \frac{\xi}{\sqrt{1-\xi^2}} \sin(\omega_n\sqrt{1-\xi^2} t) \right] \right) u(t) $$

Where,
$$ \phi = \cos^{-1}\xi $$

**Case II : Critically Damped System ($$\xi = 1$$)**
$$ C(s) = \frac{\omega_n^2}{s(s + \omega_n)^2} = \frac{A}{s} + \frac{B}{(s + \omega_n)} + \frac{C}{(s + \omega_n)^2} $$
$$ A = \lim_{s \to 0} s C(s) = \lim_{s \to 0} \frac{\omega_n^2}{(s + \omega_n)^2} = 1 $$
$$ B = \lim_{s \to -\omega_n} \frac{d}{ds} [s^2 C(s)] = \lim_{s \to -\omega_n} \frac{d}{ds} \left[ \frac{\omega_n^2}{s} \right] = \lim_{s \to -\omega_n} \frac{-\omega_n^2}{s^2} = -1 $$
$$ C = \lim_{s \to -\omega_n} (s + \omega_n)^2 C(s) = \lim_{s \to -\omega_n} \frac{\omega_n^2}{s} = -1 $$
$$ C(s) = \frac{1}{s} - \frac{1}{(s + \omega_n)} - \frac{\omega_n}{(s + \omega_n)^2} $$

Taking Inverse Laplace transform,
$$ c(t) = (1 - e^{-\omega_n t} - \omega_n t e^{-\omega_n t}) u(t) $$

**Case-III Over-Damped system ($$\xi > 1$$)** : Roots of $$s^2 + 2\xi\omega_n s + \omega_n^2 = 0$$ will be real.
$$ s = -\xi\omega_n \pm \omega_n\sqrt{\xi^2 - 1} $$
$$ C(s) = \frac{\omega_n^2}{s(s + \xi\omega_n - \omega_n\sqrt{\xi^2 - 1})(s + \xi\omega_n + \omega_n\sqrt{\xi^2 - 1})} = \frac{A}{s} + \frac{B}{(s + \xi\omega_n - \omega_n\sqrt{\xi^2 - 1})} + \frac{C}{(s + \xi\omega_n + \omega_n\sqrt{\xi^2 - 1})} $$
$$ A = \lim_{s \to 0} s C(s) = \frac{\omega_n^2}{(\xi\omega_n - \omega_n\sqrt{\xi^2 - 1})(\xi\omega_n + \omega_n\sqrt{\xi^2 - 1})} = \frac{\omega_n^2}{\omega_n^2(\xi^2 - (\xi^2 - 1))} = 1 $$
$$ B = \lim_{s \to -\xi\omega_n + \omega_n\sqrt{\xi^2 - 1}} (s + \xi\omega_n - \omega_n\sqrt{\xi^2 - 1}) C(s) = \frac{\omega_n^2}{(- \xi\omega_n + \omega_n\sqrt{\xi^2 - 1})(- \xi\omega_n + \omega_n\sqrt{\xi^2 - 1} + \xi\omega_n + \omega_n\sqrt{\xi^2 - 1})} = \frac{\omega_n^2}{(- \xi\omega_n + \omega_n\sqrt{\xi^2 - 1})(2\omega_n\sqrt{\xi^2 - 1})} $$
$$ B = \frac{\omega_n}{2\sqrt{\xi^2 - 1}(\sqrt{\xi^2 - 1} - \xi)} = \frac{\omega_n}{2(\xi^2 - 1) - 2\xi\omega_n\sqrt{\xi^2 - 1}} $$
$$ C = \lim_{s \to -\xi\omega_n - \omega_n\sqrt{\xi^2 - 1}} (s + \xi\omega_n + \omega_n\sqrt{\xi^2 - 1}) C(s) = \frac{\omega_n^2}{(- \xi\omega_n - \omega_n\sqrt{\xi^2 - 1})(- \xi\omega_n - \omega_n\sqrt{\xi^2 - 1} + \xi\omega_n - \omega_n\sqrt{\xi^2 - 1})} = \frac{\omega_n^2}{(- \xi\omega_n - \omega_n\sqrt{\xi^2 - 1})(-2\omega_n\sqrt{\xi^2 - 1})} $$
$$ C = \frac{\omega_n}{2\sqrt{\xi^2 - 1}(\xi + \sqrt{\xi^2 - 1})} = \frac{\omega_n}{2(\xi^2 - 1) + 2\xi\omega_n\sqrt{\xi^2 - 1}} $$
$$ C(s) = \frac{1}{s} + \frac{\omega_n}{2\sqrt{\xi^2 - 1}} \left[ \frac{1}{(s + \xi\omega_n - \omega_n\sqrt{\xi^2 - 1})} - \frac{1}{(s + \xi\omega_n + \omega_n\sqrt{\xi^2 - 1})} \right] $$

Taking Inverse Laplace transform,
$$ c(t) = \left[ 1 + \frac{1}{2\sqrt{\xi^2 - 1}} \left( e^{-(\xi\omega_n - \omega_n\sqrt{\xi^2 - 1})t} - e^{-(\xi\omega_n + \omega_n\sqrt{\xi^2 - 1})t} \right) \right] u(t) $$

**Case -IV : Undamped System ($$\xi = 0$$)**
$$ \frac{C(s)}{R(s)} = \frac{\omega_n^2}{s^2 + \omega_n^2} $$
$$ C(s) = \frac{\omega_n^2}{s(s^2 + \omega_n^2)}; \quad R(s) = \frac{1}{s} $$
$$ C(s) = \frac{A}{s} + \frac{Bs + C}{s^2 + \omega_n^2} $$
$$ \omega_n^2 = A(s^2 + \omega_n^2) + (Bs + C)s $$
$$ A + B = 0, \quad C = 0, \quad A = 1, \quad B = -1 $$
$$ C(s) = \frac{1}{s} - \frac{s}{s^2 + \omega_n^2} $$

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Problem #45