Project ForgeInstrumentation & Process Control
Controller Modes
GATE 2013
Hard
A unit gain 2nd order underdamped process has a period of oscillation 1 second and decay ratio 0.25. The transfer function of the process is
- 1/(0.024s²+0.067s+1)
- B1/(0.067s²+0.024s+1)
- C1/(0.021s²+0.1176s+1)
- D1/(0.1176s²+0.021s+1)
Explanation
Decay ratio=exp(−2πζ/√(1−ζ²))=0.25 ⟹ ζ≈0.2155. Period T=2π/(ωn√(1−ζ²))=1s ⟹ ωn=2π/√(1−ζ²)≈6.434 rad/s, so τ=1/ωn≈0.1554s, τ²≈0.02416, and 2ζτ≈2×0.2155×0.1554≈0.067. Transfer function: 1/(τ²s²+2ζτs+1)=1/(0.024s²+0.067s+1).
Source: GATE CH 2013 (2013)
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