Question

Consider the following equation \(2s^4 + s^3 + 3s^2 + 5s + 10 = 0\); How many roots does this equation have in the right half of s-plane?

• A. One
• B. Two
• C. Three
• D. Four

Hint:

In Routh array calculations, if any coefficient in the first column is negative while others are positive, the system is guaranteed to be unstable.
Each sign change represents one root in the right half of the s-plane, so always double-check the arithmetic of the cross-multiplications to avoid counting errors.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks us to find the number of roots of the given characteristic equation in the right half of the s-plane (RHP).
Roots in the right half of the s-plane cause the system to be unstable because they correspond to positive exponential terms in the time domain.
We can determine the number of RHP roots without explicitly solving the fourth-order polynomial by utilizing the Routh-Hurwitz stability criterion.

Step 2: Key Formula or Approach:
The Routh-Hurwitz stability criterion involves constructing a Routh array from the coefficients of the polynomial.
For a polynomial of the form: \[ a_0 s^n + a_1 s^{n-1} + a_2 s^{n-2} + \dots + a_n = 0 \] We construct rows starting from $s^n$ down to $s^0$.
The number of sign changes in the first column of the Routh array is exactly equal to the number of roots of the equation with positive real parts (lying in the right half of the s-plane).

Step 3: Detailed Explanation:
The given characteristic equation is: \[ 2s^4 + s^3 + 3s^2 + 5s + 10 = 0 \] The coefficients are: $a_0 = 2$, $a_1 = 1$, $a_2 = 3$, $a_3 = 5$, $a_4 = 10$.
Let us construct the Routh array:
The first two rows are filled using the coefficients directly:
Row $s^4$: \(2 \quad 3 \quad 10\)
Row $s^3$: \(1 \quad 5 \quad 0\)
Now, we calculate the coefficients for the subsequent rows:
For row $s^2$, the first element is: \[ b_1 = \frac{(1 \times 3) - (2 \times 5)}{1} = \frac{3 - 10}{1} = -7 \] The second element is: \[ b_2 = \frac{(1 \times 10) - (2 \times 0)}{1} = 10 \] Thus, row $s^2$ is: \(-7 \quad 10\)
For row $s^1$, the first element is: \[ c_1 = \frac{(-7 \times 5) - (1 \times 10)}{-7} = \frac{-35 - 10}{-7} = \frac{-45}{-7} = \frac{45}{7} \approx 6.43 \] Thus, row $s^1$ is: \(\frac{45}{7} \quad 0\)
For row $s^0$, the first element is: \[ d_1 = \frac{(\frac{45}{7} \times 10) - (-7 \times 0)}{\frac{45}{7}} = 10 \] Thus, row $s^0$ is: \(10\)
Let us examine the first column of the completed Routh array:
Row $s^4$: \(+2\) (positive)
Row $s^3$: \(+1\) (positive)
Row $s^2$: \(-7\) (negative)
Row $s^1$: \(+6.43\) (positive)
Row $s^0$: \(+10\) (positive)
Now we count the number of sign changes in this first column:
1. From $+1$ (row $s^3$) to $-7$ (row $s^2$): first sign change (positive to negative).
2. From $-7$ (row $s^2$) to $+6.43$ (row $s^1$): second sign change (negative to positive).
There are no other sign changes since $+6.43$ to $+10$ is positive to positive.
The total number of sign changes is 2.
Therefore, there are exactly 2 roots of the characteristic equation in the right half of the s-plane.

Step 4: Final Answer
Thus, the equation has two roots in the right half of the s-plane, which corresponds to option (B).