Govt. Exams
Entrance Exams
Peak overshoot = exp(-ζπ/√(1-ζ²))·100% = exp(-0.5π/√0.75)·100% ≈ 16.3%
At the gain crossover frequency, magnitude is 0 dB. When this equals phase crossover frequency, the phase is -180°, giving phase margin = -180° - (-180°) = 0°
For ramp input, ess = 1/Kv where Kv = lim s→0 s·G(s) = 10/2 = 5. Therefore ess = 1/5 = 0.2
The number of sign changes in the first column of the Routh array equals the number of poles in the right half-plane. A sign change indicates at least one pole in RHP, making the system unstable.
Closed-loop gain = G(s)/(1+G(s)). At DC, this becomes 100/(1+100) = 100/101 ≈ 0.99, showing the effect of feedback on reducing overall gain.
The polynomial factors as (s+1)(s+2)(s+3) = 0, giving poles at -1, -2, and -3. All poles are in the left half-plane, making the system stable.
Lag compensator (zero at -0.1, pole at -0.01, with pole closer to origin) adds gain at low frequencies without significantly affecting phase at crossover, improving steady-state accuracy.
Gain margin is the factor by which the system gain at phase crossover frequency can be increased before the system becomes unstable (phase reaching -180° at unity gain).
The derivative term acts as damping in the system. Increasing Kd increases damping, which reduces overshoot and oscillations without affecting steady-state error significantly.
For minimum phase systems, magnitude and phase are related through the Kramers-Kronig relations. Knowledge of magnitude plot uniquely determines phase plot.
