8 Questions
What is the damping ratio of the second order system in the given example?
0.5
What is the un-damped natural frequency of the second order system in the given example?
2 rad/sec
What is the general transfer function of a second order system?
C(s) = R(s) / (s + ζωn s + ωn2)
What are the two poles of the second order system in the given example?
-ωnζ + ωn√(ζ2 - 1) and -ωnζ - ωn√(ζ2 - 1)
What does the damping ratio (ζ) represent in a second order system?
The degree of resistance to change in the system output
What is the characteristic of an overdamped second order system?
Two real distinct poles
What is the physical significance of ωn in a second order system?
The frequency of oscillation of the system without damping
What is the condition for a second order system to be overdamped?
ζ > 1
Study Notes
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)
Understand the characteristics of a general second-order system through its transfer function, including undamped natural frequency and damping ratio.
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