What is the result of the characteristic equation for critically damped systems?

• A. Two distinct complex roots
• B. Two identical real roots
• C. Two distinct real roots
• D. No real roots

Answer: (B)

In the context of control systems and differential equations, critically damped systems are characterized by their response to disturbances. The characteristic equation of a second-order system can be generally represented as:

[ s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 ]

Here, ( \zeta ) (zeta) represents the damping ratio and ( \omega_n ) is the natural frequency. For a critically damped system, the damping ratio ( \zeta ) is equal to 1. When substituting ( \zeta = 1 ) into the equation, it simplifies to:

[ s^2 + 2\omega_n s + \omega_n^2 = 0 ]

This quadratic equation can be solved using the quadratic formula:

[ s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

In the case of critical damping, the discriminant ( b^2 - 4ac ) (which corresponds to ( (2\omega_n)^2 - 4(1)(\omega_n^2) )) equals zero, resulting in two identical (or repeated) real roots: