01․ Find the relation of damped natural frequency (ω

_{d})Damped natural frequency (ω

_{d}) = (ω_{n}) √(1-ζ²)02․ A second order control system has F(jω) = 100/(100 - ω² + 10√2 jω). Find the resonant peak?

Resonant peak is the maximum value of magnitude occurring at resonant frequency ω

_{r}. Resonant peak µ_{r}_{}= 1/(2ζ√(1-ζ²)) Given transfer function F(jω) = 100/(100 - ω² + 10√2 jω) Compare to standard second order equation, Damping ratio ξ = 1/√2 Resonant peak = 103․ For a second order under damped system subjected to unit step input the time response has first peak value to be 4 times the second over shoot. Determine the damping ratio ξ ?

First peak value = 1 + M

_{p1}Second peak overshoot = M_{p2}Where, Mp = Peak over shoot = e^{-nξπ/√(1-ζ²)}M_{p1}= e^{-ξπ/√(1-ζ²)}M_{p2}= e^{-3ξπ/√(1-ζ²)}From give data, 1 + M_{p1}= M_{p2}By solving above equation, Damping ratio ξ = 0.08704․ Find the bandwidth of the system, when rise time of the system is give as 1msec?

Bandwidth is the range of frequencies over which the magnitude has a value of 1/√2. Band width indicates the speed of response of the system. Wider bandwidth indicates that, faster response.
Bandwidth = 0.35/Rise time = 0.35/(1*10

^{-3}) = 350 Hz05․ If the bandwidth of the system is very large, the system response is

Bandwidth is the range of frequencies over which the magnitude has a value of 1/√2. Band width indicates the speed of response of the system. Wider bandwidth indicates that, faster response.
Bandwidth = 0.35/Rise time

06․ The magnitude of a transfer function G(jω)H(jω) at gain cross over frequency is

Stability from frequency response plots
1 + G(s)H(s) = 0
G(s)H(s) = -1 + j0
Put s = jω
G(jω)H(jω) = -1 + j0
|G(jω)H(jω)| at gain cross over frequency = 1
∠G(jω)H(jω) at phase cross over frequency = -180°

07․ Find the conditions for the system to be stable?

Stable system: Gain margin and phase margins are positive and Gain cross over frequency is less than phase cross over frequency.
Unstable system: Gain margin and phase margins are negative and Gain cross over frequency is greater than phase cross over frequency.
Marginally stable system: Gain margin and phase margins are zero and Gain cross over frequency is equal to phase cross over frequency.

08․ A second order under damped system has damping ratio of 0.3. Find the phase margin of the system?

Phase margin is the allowable phase lag.
∠G(jω)H(jω) at gain cross over frequency = φ
Phase margin = 180° + φ
And also
Phase margin ≈ 100*ξ
Therefore, for given system Phase margin = 30°

09․ If the magnitude of G(jω)H(jω) at phase crossover frequency is 0.5. Find the gain margin of the given system?

Gain margin is the allowable gain.
|G(jω)H(jω)| at phase crossover frequency = X
Gain margin = 1/X
Gain margin in db = 20 log(1/X)
For given system gain margin = 1/0.5 = 2

10․ A second order system has peak over shoot = 50% and period of oscillations 0.2 seconds. Find the resonant frequency?

Peak over shoot = 0.5
Peak over shoot M

_{p}= e^{-nξπ/√(1-ζ²)}= 0.5 Damping ratio ξ = 0.215 Time = 0.2 f_{d}= 1/0.2 = 5 Hz Damped frequency ω_{d}= 2πf_{d}= 31.41 rad/s Damped frequency ω_{d}= (ω_{n}) √(1-ζ²) Natural frequency (ω_{n}) = 32.16 rad/s Resonant frequency (ω_{r}) = (ω_{n}) √(1-2ζ²) = 30.63 rad/s<<<1819202122>>>