Question

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?

• A. Four left half plane (LHP) roots
• B. One right half plane (RHP) root , one LHP root and two roots on the imaginary axis
• C. Two RHP roots and two LHP roots
• D. All four roots are on the imaginary axis

Answer 1

The correct option is C Two RHP roots and two LHP roots

$q\left(s\right)=s^4+s^2+1=0$
$\begin{array}{cccc}{s}^4& 1& 1& 1\\ s^3& 4& 2& 0\\ s^2& 0.5& 1& 0\\ s^1& -6& 0& 0\\ s^0& 1& 0& 0\end{array}$

$\frac{dA\left(s\right)}{ds}=4s^3+2s^1$

There are two sign changes in the first column of the R_H table and the order of auxillary equation is 4. So, four poles are symmetric about origion.
$\therefore 2$ RHP roots and $2$ LHP roots.