Q41GATE 2004MCQ

The open loop transfer function of a unity feedback control system is given as $G\left ( s \right )=\frac{as+1}{s^{2}}$ The value of 'a' to give a phase margin of $45^{\circ}$ is equal to

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📖 Explanation

$G(s)=\frac{as+1}{s^2}$ $\omega _g$ = gain crossover frequency at which open gain is loop Phase margin $=180+\angle G(j\omega )$ $45^{\circ}=180^{\circ}+\angle G(j\omega_g )$ $\angle G(j\omega_g =-130^{\circ}$ $\angle G(j\omega_g )=\frac{ja \omega _g +1}{(j\omega _g)^2}$ $\;\;=-2 \times 90^{\circ}+tan^{-1}a\omega _g$ $\angle G(j\omega_g )=-180^{\circ}+tan^{-1}a\omega _g=-135^{\circ}$ $tan^{-1}a\omega _g=45^{\circ}$ $a\omega _g=1$ $\omega _g=\frac{1}{a}$ $Gain=|G(j\omega_g )|=1$ $\Rightarrow \frac{\sqrt{1+a^2\omega ^2_g}}{\omega ^2_g}=1$ $\Rightarrow \frac{\sqrt{1+a^2 \times \frac{1}{a^2}}}{1/a^2}=1$ $\frac{1}{a^2}=\sqrt{2}$ $a=\frac{1}{2^{1/4}}=0.841$

Q43GATE 2001MCQ

The polar plot of a type-1, 3-pole, open-loop system is shown in figure. The closed loop system is

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📖 Explanation

Nyquist plot will be $\therefore$ Number of encirclement of (-1,j0)=-2 and number of right sided poles in open loop system=0 $\;\;\;N=P-Z$ $\therefore \;\; -2=0-2$ $\Rightarrow \;\;Z=2$ $\therefore$ Closed loop system is unstable with two poles on the right half of s-plane.

Q44GATE 2001MCQ

The asymptotic approximation of the log-magnitude versus frequency plot of a minimum phase system with real poles and one zero is shown in figure. Its transfer functions is

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📖 Explanation

Type: 2 Poles : 0, 0, 2, 25 Zeroes : 5 $Gain=m \log \omega +20 \log K$ $54=-40 \log (0.1)+20 \log K$ $\Rightarrow \;\; K=5$ $\therefore \;\; T.F.=\frac{5(1+0.2s)}{s^2(1+0.5s)(1+0.04s)}$ $\;\;\;=\frac{50(s+5)}{s^2(s+2)(s+25)}$