Question

The first two rows of Routh's tabulation of a third order equation are as follows:

S${}^3$ 2 2
S${}^2$ 4 4

This means there are :

• A. two roots at s = $\pm$ j and one root in right half s-plane
• B. two roots at s = $\pm$ j2 and one root in left half s-plane
• C. two roots at s = $\pm$ j2 and one root in right half s-plane
• D. two roots at s = $\pm$ j and one root in left half s-plane

Answer 1

The correct option is D two roots at s = $\pm$ j and one root in left half s-plane

Routh-Array



The third row vanishes.

An auxiliary equation is formed using elements of $2^{nd}$ row.

Auxiliary equation
A(s) = 4$s^2+4=0\Rightarrow s=\pm j.$

The derivative of this auxiliary equation is taken w.r.t. s and the coefficients of the differentiated equation are taken as the elements of $3^{rd}$ row.

$\frac{dA\left(s\right)}{ds}=8s$

Routh-Array



There is no root in RHS of s-plane.

Two roots of s = $\pm$ j, so one root is in LHS of s-plane.