A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.

By reinterpreting the system, one can derive the closed-loop transfer function, which characterizes second-order systems as those with two poles. This closed-loop transfer function can be rearranged to highlight essential parameters: undamped natural frequency and damping ratio.

The damping ratio (ζ) is a critical parameter defined as the ratio of actual damping to critical damping. It provides insight into the system's behavior and stability. With these parameters identified, the closed-loop transfer function is often rewritten in a standard form to represent the second-order system comprehensively:

Here, ω_n denotes the undamped natural frequency, and ζrepresents the damping ratio. The damping ratio categorizes the system's response:

Underdamped (0 < ζ < 1): The system exhibits oscillatory behavior with gradually diminishing amplitude, indicating oscillations around the equilibrium point before settling down.

Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating, often desired in servo systems for swift and smooth positioning.

Overdamped (ζ > 1): The system returns to equilibrium without oscillations but more slowly compared to a critically damped system, leading to a slower response time, which might be less desirable in applications requiring rapid adjustments.

Understanding these parameters and their influence on the system's response is crucial for designing and tuning servo systems to meet specific performance criteria. The analysis of second-order systems through their transfer functions and damping characteristics allows engineers to predict and optimize system behavior under various operating conditions.