2nd Order System Response
This page summarizes step and frequency responses of second order system of
the form:
 Stable = (a_{0} > 0 and a_{1} > 0).

Settling time rule from 1st order system (i.e. response is within 2% of
steady state value at 4τ) does not hold (see example 4).
Step Response
Input: step A
Output:
Frequency Response
Input: Asin(ωt)
Output (steady state amplitude):
Output (steady state amplitude trend):
Change Response Speed
σ times faster:
 Gain (K): unchanged
 New natural frequency (undamped frequency): σω_{n}
 Damping ratio (ζ): unchanged
Examples
 Example 1: Speed up response 5 times.
 Example 2: Design critically damped amplifier for servo.
 Example 3: Design to a specific zeta.
 Example 4: Critically damped and undamped system (and settling time).