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An under damped second order system having a transfer function of the form
has a frequency response plot shown in fig.
Q. The system gain K is
From the fig. P6.5.12, T(j0) = 1
An under damped second order system having a transfer function of the form
has a frequency response plot shown in fig.
Q. The damping factor ξ is approximately
The peak value of T(jω) occurs when the denominator of function T(jω)^{2} is minimum i.e. when
Consider the Bode plot of a ufb system shown in fig.
Q. The steady state error corresponding to a rampinput is
The Bode plot is as shown in fig.
Consider the Bode plot of a ufb system shown in fig.
The damping ratio is
From Fig.:
The Nyquist plot of a openloop transfer function G(jω)H(jω) of a system encloses the (1, j0) point. The gain margin of the system is
If Nyquist plot encloses the point (1, j0), the system is unstable and gain margin is negative.
Consider a ufb system
The angle of asymptote, which the Nyquist plotapproaches as ω → 0 is
Hence, the asymptote of the Nyquist plot tends to an angle of 90° as ω→ 0.
If the gain margin of a certain feedback system isgiven as 20 dB, the Nyquist plot will cross the negativereal axis at the point
Since system is stable, it will cross at s = 0.1
The transfer function of an openloop system is
The Nyquist plot will be of the form
Hence (B) is correct option.
Consider a ufb system whose openloop transfer function is
The Nyquist plot for this system is
Due to s there will be a infinite semicircle. Hence (C) is correct option.
The open loop transfer function of a system is
The Nyquist plot for this system is
∠GH(jw) = 270°+ 2 tan1ω
For ω = 0, GH(jω) = ∞∠ 270°
For ω = 1 , ∠GH(jω) = 180°
For ω = ∞, GH(jω) = 0 ∠  90°
As ω increases from 0 to ∞, phase goes 270° to 90°.
Due to s^{3} term there will be 3 infinite semicircle.
For the certain unity feedback system
The Nyquist plot is
∠GH(j(ω) = 90 ° tan^{1} ω  tan^{1} 2ω  tan^{ 1} 3ω,
For ω = 0, GH(jω) = ∞∠  90°,
For ω = ∞, GH(jω) = 0∠ 360°,
Hence (A) is correct option.
The Nyquist plot of a system is shown in fig. The openloop transfer function is
The no. of poles of closed loop system in RHP are
The openloop poles in RHP are P = 0. Nyquist path enclosed 2 times the point (1 + j0). Taking clockwise encirclements as negative N = 2.
N = P  Z, 2 = 0 Z , Z = 2 which implies that two poles of closedloop system are on RHP.
The openloop transfer function of a feedback control system is
Q. The Nyquist plot for this system is
If the damping of the system becomes equal to zero, which condition of the resonant frequency is likely to occur?
A unity feedback system has the open loop transfer function G(s)=1/((s−1)(s+2)(s+3))
The Nyquist plot of GG encircle the origin
The openloop transfer function of a feedback system is
Q. The Nyquist plot of this system is
The openloop transfer function of a feedback system is
The system is stable for K
RHP poles of openloop system P = 1, Z = P  N .
For closed loop system to be stable, Z = 0, 0 =1  N ⇒N = 1
There must be one anticlockwise rotation of point (1+ j0). It is possible when K > 1.
A unity feedback system has openloop transfer function
Q. The Nyquist plot for the system is
The intersection with the real axis can be calculated as {GH(jω)} = 0, The condition gives ω(2ω^{2} 1) = 0
With the above information the plot in option (C) is correct.
A unity feedback system has openloop transfer function
Q. The phase crossover and gain crossover frequenciesare
The Nyquist plot crosses the negative real axis
Hence phase crossover frequency is
The frequency at which magnitude unity is
A unity feedback system has openloop transfer function
The gain margin and phase margin are
∠GH(jω) = 90° tan^{1} ω tan^{1} 2ω ,
At unit gain ω_{1} = 0.57 rad/sec,
Phase at this frequency is ∠GH(jω_{1}) = 90° tan^{1} 0.57 tan^{1} 2(0.57) = 168.42°
Phase margin = 168.420+180° = 11.6°
Note that system is stable. So gain margin and phase margin are positive value. Hence only possible option is (D).
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