A step response shows how a system reacts when its input suddenly changes from one constant value to another. Engineers use it to predict whether a design will respond quickly, smoothly, or with unwanted oscillations. Second-order step responses are especially important because many mechanical, electrical, thermal, and control systems behave like mass spring damper models.
The shape of the response reveals stability, speed, overshoot, and settling behavior.
Key Facts
- Standard second-order transfer function: G(s) = omega_n^2 / (s^2 + 2 zeta omega_n s + omega_n^2)
- Damping ratio zeta controls response type: 0 < zeta < 1 underdamped, zeta = 1 critically damped, zeta > 1 overdamped
- Damped natural frequency: omega_d = omega_n sqrt(1 - zeta^2) for 0 < zeta < 1
- Percent overshoot for an underdamped system: PO = 100 e^(-zeta pi / sqrt(1 - zeta^2))
- Approximate 2 percent settling time: T_s = 4 / (zeta omega_n) for typical underdamped second-order systems
- Peak time for an underdamped response: T_p = pi / omega_d
Vocabulary
- Step response
- The output of a system after its input suddenly changes to a new constant value.
- Damping ratio
- A dimensionless number zeta that measures how strongly oscillations are reduced in a second-order system.
- Underdamped
- A response with 0 < zeta < 1 that oscillates before settling to the final value.
- Critically damped
- A response with zeta = 1 that returns to the final value as fast as possible without oscillating.
- Overshoot
- The amount by which the response exceeds its final steady-state value, usually expressed as a percentage.
Common Mistakes to Avoid
- Confusing natural frequency with damped natural frequency is wrong because omega_n describes the undamped system while omega_d is the actual oscillation frequency when damping is present.
- Assuming the fastest response always has the most overshoot is wrong because critical damping can be fast without overshoot, while low damping usually creates oscillations.
- Using the settling time formula for every damping case is wrong because T_s = 4 / (zeta omega_n) is an approximation mainly used for typical underdamped second-order responses.
- Treating overdamped and critically damped responses as identical is wrong because overdamped systems do not overshoot but usually reach the final value more slowly than critically damped systems.
Practice Questions
- 1 A second-order system has zeta = 0.5 and omega_n = 10 rad/s. Find the damped natural frequency omega_d and the approximate 2 percent settling time T_s.
- 2 For an underdamped system with zeta = 0.6 and omega_n = 8 rad/s, calculate the percent overshoot using PO = 100 e^(-zeta pi / sqrt(1 - zeta^2)).
- 3 A position control system must reach its target quickly without passing beyond it. Which damping condition is usually preferred, underdamped, critically damped, or overdamped, and why?