The Critical Damped System

A critically damped system is a type of dynamic system where the damping is set at the critical value, resulting in the fastest possible return to equilibrium without oscillation. It is commonly used in control systems and other applications where overshoot and oscillations are undesirable.

Behavior of a Critically Damped System

When a system is critically damped, it returns to its equilibrium position without any oscillation. This means that the response of the system is smooth and without any overshoot. The time it takes for the system to return to equilibrium is minimized, making it ideal for applications where quick settling time is important.

Effect of Increasing the Gain

When the gain of a critically damped system is increased, it will no longer behave as a critically damped system. Instead, it will transition into an underdamped system. This is because the increased gain causes the system to become more responsive and leads to overshoot and oscillations.

Underdamped System

An underdamped system is a dynamic system that exhibits oscillations and overshoot in its response. This occurs when the damping is less than the critical value, resulting in the system taking longer to return to equilibrium and oscillating around the equilibrium position.

Reasoning

Increasing the gain of a critically damped system causes the system to become more responsive. This increased responsiveness leads to a faster response, which in turn results in overshoot and oscillations. Therefore, the system can no longer be considered critically damped and instead behaves as an underdamped system.

Conclusion

In summary, when the gain of a critical damped system is increased, it will transition into an underdamped system. This change in behavior is due to the increased responsiveness of the system, leading to overshoot and oscillations in the system's response.