ECE GATE 2019 Question No 52

Question No. 52 ECE | GATE 2019

Consider a unity feedback system, as in the figure shown, with an integral compensator $ \frac ks $ and open-loop transfer function

$ G(s)=\frac1{s^2+3s+2} $

where K > 0. The positive value of K for which there are exactly two poles of the unity feedback system on the $ j\omega $ axis is equal to ______ (rounded off to two decimal places).

Answer : 5.99 to 6.01

GATE 2019 ECE

Solution of Question No 52 of GATE 2019 ECE Paper

$ \mathrm G(\mathrm s)=\frac1{\mathrm s^2+3\mathrm s+2} $

$ \mathrm C\cdot\mathrm E=1+\frac{\mathrm k}{\mathrm s}\mathrm G(\mathrm s)\;\mathrm H(\mathrm s) $

Syatem is UFB so, H(s) = 1

$ \begin{array}{l}\therefore\mathrm C\cdot\mathrm E=1+\frac{\mathrm k}{\mathrm s}\cdot\frac1{\mathrm s^2+3\mathrm s+2}=0\\\Rightarrow\mathrm s(\mathrm s^2+3\mathrm s+2)+\mathrm k=0\\\Rightarrow\mathrm s^33\mathrm s^2+2\mathrm s+\mathrm k=0\end{array} $

R-Array
s³	1	2	By applying the sufficient condition. For stability k>0 and k<6
s²	3	K
s¹	$\frac{6-\mathrm k}3$	0
s⁰	k

But system will have 2-poles on $j_ω_axis$ if auxiliary equation will be formed. Auxiliary equation will be formed if odd order row becomes zero.

if k = 6, s^-1 row becomes zero.

So, k = 6

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GATE Papers >> ECE >> 2019 >> Question No 52

Answer : 5.99 to 6.01

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