The characteristic equation of a feedback system is s³ + Ks² + 5s + 10 = 0. For a stable system, the value of K should not be less than

Concept:

The characteristic equation for a given open-loop transfer function G(s) is

1 + G(s) H(s) = 0

To find the closed system stability by using RH criteria we require a  characteristic equation. Whereas in remaining all stability techniques we require open-loop transfer function.

The nth order general form of CE is

a0 sn + a1 sn-1 + a2sn-2 + __________an-1 s1 + an

RH table shown below

Necessary condition: All the coefficients of the characteristic equation should be positive and real.

Sufficient Conditions for stability:

1. All the coefficients in the first column should have the same sign and no coefficient should be zero.

2. If any sign changes in the first column, the system is unstable.

And the number of sign changes = Number of poles in right of s-plane.

Calculation:

Characteristic equation: s3 + Ks2 + 5s + 10 = 0

By applying Routh tabulation method,

\(\begin{array}{*{20}{c}} {{s^3}}\\ {{s^2}}\\ {{s^1}}\\ {{s^0}} \end{array}\left| {\begin{array}{*{20}{c}} 1&5\\ K&{10}\\ {\frac{{5K - 10}}{K}}&0\\ {10}&{} \end{array}} \right.\)

The system to become stable, the sign changes in the first column of the Routh table must be zero.

5K – 10 > 0 and K > 0