Question

The transfer function of a system is \(TF = \frac{{{s^3} + 2{s^2} + 3s + 1}}{{{s^3} + {s^2} + 2s + 1}}\) How may roots are lying on the right half side of S-Plane for the numerator and denominator for the transfer function?
1. 0, 0
2. 1, 0
3. 0, 1
4. None of the above

Answer 1

Correct Answer - Option 1 : 0, 0

Concept:

According to the Routh tabulation method,

The system is said to be stable if there are no sign changes in the first column of Routh array

The number of poles lie on the right half of s plane = number of sign changes

Calculation:

Numerator equation: s³ + 2s² + 3s + 1

By applying Routh tabulation method,

\(\begin{array}{*{20}{c}} {{s^3}}\\ {{s^2}}\\ {{s^1}}\\ {{s^0}} \end{array}\left| {\begin{array}{*{20}{c}} 1&3\\ 2&1\\ {2.5}&0\\ {2.5}&{} \end{array}} \right.\)

As there are no sign changes, the number of roots lies on the right half of S-plane = 0

Denominator equation: s³ + s² + 2s + 1

By applying Routh tabulation method,

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

As there are no sign changes, the number of roots lies on the right half of S-plane = 0