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GATE 2016 EC (EC2) – Question 56

Control Systems · Routh-Hurwitz and Nyquist Stability Criteria · 2 marks · Numerical answer

The feedback system has $G(s)=1/[(s+1)(s+2)(s+3)]$. Find the positive gain k for which loop gain margin is 0 dB and phase margin is 0°.

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Correct answer: 59.5 to 60.5

Explanation

The unity-feedback characteristic polynomial is $s^3+6s^2+11s+6+k$. At the stability boundary, Routh's condition is $6\cdot11=6+k$, so k=60.

The same result follows by setting s=jω: the imaginary part vanishes at $\omega^2=11$, and the real part requires k=60. At that frequency the loop is −1, giving both zero margins. Thus **k=60**.

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