Previous Year Question
A unity-feedback system has open-loop transfer function G(s)H(s) = 16/(s² + 4s). The steady-state response c(t) will exhibit a resonance peak at a frequency of
Correct answer: B
Explanation
The closed-loop denominator is s²+4s+16, so ωn=4 rad/s and ζ=0.5. Since ζ<1/√2, a resonance peak exists at ωr=ωn√(1−2ζ²)=2√2 rad/s.
Formula
ωr=ωn√(1−2ζ²)=4√(1−0.5)=2√2 rad/s
Step-by-step solution
Correct Answer
B — 2√2 rad/s
Concept and Reasoning
The closed-loop denominator is s²+4s+16, so ωn=4 rad/s and ζ=0.5. Since ζ<1/√2, a resonance peak exists at ωr=ωn√(1−2ζ²)=2√2 rad/s.
Calculation / Key Relation
ωr=ωn√(1−2ζ²)=4√(1−0.5)=2√2 rad/s
Exam Takeaway
Find ωn and ζ from the closed-loop denominator before using the resonance formula.
Common Mistake
Do not use ωn itself as the resonance frequency unless the damping is negligible.
Common mistake
Do not use ωn itself as the resonance frequency unless the damping is negligible.
Exam tip
Find ωn and ζ from the closed-loop denominator before using the resonance formula.
Quick method
Find ωn and ζ from the closed-loop denominator before using the resonance formula.