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GATE 2017 EC – Question 58

Control Systems · Routh-Hurwitz and Nyquist Stability Criteria · 2 marks · Multiple choice

Which one of the following options correctly describes the locations of the roots of the equation $s^4 + s^2 + 1 = 0$ on the complex plane?

  1. Four left half plane (LHP) roots
  2. One right half plane (RHP) root, one LHP root and two roots on the imaginary axis
  3. Two RHP roots and two LHP roots
  4. All four roots are on the imaginary axis

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Correct answer: (C) Two RHP roots and two LHP roots

Explanation

Treat the equation as a quadratic in $s^2$, $s^2 = \frac{-1 \pm j\sqrt{3}}{2} = e^{\pm j2\pi/3}$. Taking square roots gives $s = \pm e^{\pm j\pi/3}$, which are $e^{j\pi/3}$, $e^{-j\pi/3}$, $-e^{j\pi/3}$ and $-e^{-j\pi/3}$. Two of them have positive real part and two have negative real part, with none on the imaginary axis.