In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If ζ = 0, the system becomes undamped, leading to perpetual oscillations without any attenuation. The system continues to oscillate indefinitely, never reaching a steady state.
In a critically damped scenario, where ζ = 1, the system's poles are identical. For a unit-step input, the output equation is derived, and through inverse Laplace transformation, the response is obtained. This response does not exhibit oscillations and returns to equilibrium as swiftly as possible without overshooting.
In an overdamped scenario, where ζ > 1, the system's two integral components are real, negative, and unequal. When a unit-step input is applied, the output equation is formulated. The inverse Laplace transformation of this equation results in a response characterized by two decaying exponential terms. One of these exponential decays is significantly faster than the other when the damping ratio is much greater than unity. Consequently, the faster-decaying term can often be neglected, simplifying the response to resemble that of a first-order system. This approximation allows for the derivation of an approximate transfer function, simplifying analysis and design.
In summary, understanding the behavior of second-order systems in response to a unit-step input under various damping conditions is crucial for system design and analysis. The underdamped response is characterized by damped oscillations, while the undamped response features continuous oscillations. Critically damped systems achieve equilibrium swiftly without oscillations, and overdamped systems exhibit slower, non-oscillatory responses. These insights are vital for tuning systems to achieve desired performance characteristics, ensuring stability and accuracy in practical applications.
