01․ The transfer function of a plant is T(s) = 5/(s+5)(s² + 5 + 1). The second order approximation of T(s) using dominant pole concept is
Using dominant pole concept, T(s) = 5/(s+5)(s² + 5 + 1) T(s) = 5/5(s/5+1)(s² + 5 + 1) T(s) = 1/(s² + 5 + 1)
02․ The phase margin of a system having the loop transfer function G(s)H(s) = 2√3/s(s+1) is
The characteristic equation for given system is s² + s + 3.46 = 0 Phase margin ≈ 100*ζ Damping ratio = 0.27 Phase margin = 27° ≈ 30°
03․ A system has poles at 0.01 Hz, 1 Hz and 80 Hz; zeros at 5 Hz, 100 Hz and 200 Hz. The approximate phase of the system response at 20 Hz is
Using dominant pole technique, all the pole and zeros are insignificant except a pole at 0.01 Hz. Therefore, given system consists of only a pole at 0.01 Hz. Therefore, the approximate phase of the system is -90°.
04․ The Laplace transform of a transportation lag of 5 seconds is
Dead time in real system is also known as transportation lag. Transfer function of transportation lag = e-Ts Therefore, transportation lag of 5 seconds can be written as e-5s.
05․ A property of phase lead compensation is that the
Lead compensator characteristics:
- 1. For sinusoidal input, the phase of controller output leads by tan-1 (ωTd).
- 2. In terms of filtering property it acts as high pass filter.
- 3. It reduces rise time.
- 4. It increases bandwidth.
- 5. It increases stability of the system.
- 6. It reduces peak over shoot.
06․ Which of the following is the correct expression for the transfer function of an electrical RC phase lag compensating network?
Phase lag compensator acts as a low pass filter. Therefore transfer function for lag compensator = 1/(1+RCS).
07․ The value of 'a' in to give phase margin = 45o will be
Characteristic equation is Natural frequency ωn = 1 rad/s Phase margin = 100*ζ Damping ratio ζ = 45/100 = 0.45 2ζωn = a Therefore, a = 0.9
08․ A unity feedback control system has forward loop transfer function as e-Ts/s(s+1). Its phase value will be zero at frequency ω1. Which of the following equations should be satisfied by ω1?
Given transfer function = e-Ts/s(s+1) G(jω) = e-Tjω/jω(jω+1) ∠G(jω) = -ωT - π/2 - tan-1ω At ω = ω1;∠G(jω) = 0 -ω1T - π/2 - tan-1ω1 = 0 - tan-1ω1 = ω1T + π/2 - ω1 = -cot(Tω1) ω1 = cot(Tω1)
09․ In polar plots if the critical point '-1+j0' is enclosed then the system is --------
A point is said to be enclosed by a contour if it lies to the right side of the direction of the contour. A point is said to be encircled if the contour is a closed path. In polar plots if the critical point '-1+j0' is enclosed then the system is said to be unstable and if it not enclosed then the system is said to be stable.
10․ Resonant peak of a marginally stable system is
Resonant peak is the maximum value of magnitude occurring at resonant frequency ωr. Resonant peak µr = 1/(2ζ√(1-ζ²)) For marginally stable system, damping ratio ζ = 0 Therefore, resonant peak = Infinity