Second-Order System Transfer Function

Second-Order System Transfer Function

• A general second-order system is characterized by the transfer function: C(s) = ωn² / (s² + 2ζωn s + ωn²)

Un-damped Natural Frequency (ωn)

• ωn is the un-damped natural frequency of the second-order system
• ωn is the frequency of oscillation of the system without damping

Damping Ratio (ζ)

• ζ is the damping ratio of the second-order system
• ζ is a measure of the degree of resistance to change in the system output

Example: Determining ωn and ζ

• Given a second-order system transfer function: C(s) = 4 / (s² + 2s + 4)
• By comparing with the general 2nd order transfer function, we can determine: ωn² = 4 => ωn = 2 rad/sec, and 2ζωn = 2 => ζωn = 1 => ζ = 0.5

System Poles

• The two poles of the system are: -ωnζ ± ωn √(ζ² - 1)

System Categorization Based on ζ

• According to the value of ζ, a second-order system can be categorized into four types:
• Overdamped: when the system has two real distinct poles (ζ > 1)