Roots Analysis in the s-Plane

To determine the number of roots of a polynomial in the left half of the s-plane, we typically use the Routh-Hurwitz stability criterion. This method helps us understand the stability of a system represented by its characteristic equation.

Polynomial Equation for Root Analysis

The given polynomial equation is:

\( s^3 + 5s^2 + 7s + 3 = 0 \)

This is a third-order polynomial, which means it has a total of three roots. Our goal is to find how many of these roots lie in the left half of the s-plane.

Routh-Hurwitz Criterion Application

The Routh-Hurwitz criterion involves constructing a Routh array from the coefficients of the polynomial. The first column of this array tells us about the location of the roots.

The general form of a third-order polynomial is \( a_3s^3 + a_2s^2 + a_1s + a_0 = 0 \).

Comparing this with our given equation \( s^3 + 5s^2 + 7s + 3 = 0 \), we have the coefficients:

• \( a_3 = 1 \)
• \( a_2 = 5 \)
• \( a_1 = 7 \)
• \( a_0 = 3 \)

Now, let's construct the Routh array:

Calculate the elements for the \( s^1 \) row:

\( b_1 = \frac{(a_2 \times a_1) - (a_3 \times a_0)}{a_2} = \frac{(5 \times 7) - (1 \times 3)}{5} = \frac{35 - 3}{5} = \frac{32}{5} \)

Calculate the elements for the \( s^0 \) row:

\( c_1 = \frac{(b_1 \times a_0) - (a_2 \times 0)}{b_1} = \frac{(\frac{32}{5} \times 3) - (5 \times 0)}{\frac{32}{5}} = \frac{\frac{96}{5}}{\frac{32}{5}} = 3 \)

So, the complete Routh array is:

Interpreting the Routh Array for Roots

Now, we examine the signs of the elements in the first column of the Routh array. The first column elements are:

• \( s^3 \): 1 (positive)
• \( s^2 \): 5 (positive)
• \( s^1 \): \( \frac{32}{5} \) (positive)
• \( s^0 \): 3 (positive)

All the elements in the first column are positive. There are no sign changes in the first column.

According to the Routh-Hurwitz criterion:

• The number of sign changes in the first column of the Routh array indicates the number of roots of the polynomial that lie in the right half of the s-plane.
• Since there are zero sign changes in our Routh array, there are zero roots in the right half of the s-plane.

Furthermore, because there are no zero rows or zero elements in the first column that would indicate roots on the imaginary axis (jω-axis), we can conclude there are no roots on the jω-axis.

The total number of roots for a third-order polynomial is 3. Since there are no roots in the right half of the s-plane and no roots on the imaginary axis, all three roots must lie in the left half of the s-plane.

Conclusion on Root Location

Based on the Routh-Hurwitz analysis, the polynomial \( s^3 + 5s^2 + 7s + 3 = 0 \) has all its roots in the left half of the s-plane. This implies that a system described by this characteristic equation would be stable.

Therefore, the number of roots in the left half of the s-plane is three.