2nd Order System Response

This page summarizes step and frequency responses of second order system of the form:

• Stable = (a₀ > 0 and a₁ > 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).