The number of roots in the left half of the s-plane for a system having characteristic equations : s³ + 5s² + 7s + 3 = 0 is : 

Explanation:

In control system analysis, determining the stability and behavior of a system involves understanding the location of the roots (or poles) of its characteristic equation in the complex plane. The characteristic equation is typically derived from the system's transfer function or differential equations describing the system dynamics.

Given the characteristic equation:

s³ + 5s² + 7s + 3 = 0

We need to determine the number of roots (poles) that lie in the left half of the s-plane. This is crucial because the location of these roots directly affects the stability of the system. Roots in the left half-plane (with negative real parts) indicate a stable system, whereas roots in the right half-plane (with positive real parts) indicate instability.

Routh-Hurwitz Criterion:

One common method to determine the number of roots in the left half-plane is the Routh-Hurwitz criterion. This criterion uses the coefficients of the characteristic polynomial to form the Routh array, which can then be analyzed to determine the number of roots with positive real parts (i.e., in the right half-plane).

Formation of the Routh Array:

For the given characteristic equation:

s³ + 5s² + 7s + 3 = 0

The Routh array is formed as follows:

The elements in the first column are calculated as follows:

Thus, the Routh array is:

Analysis of the Routh Array:

To determine the number of roots in the left half-plane, we need to count the number of sign changes in the first column of the Routh array. The first column is:

1, 5, 6.4, 3

There are no sign changes in this sequence, indicating that all the roots have negative real parts (i.e., they are all in the left half-plane). Therefore, there are zero roots in the right half-plane, and all roots are in the left half-plane.

Conclusion:

The number of roots in the left half of the s-plane for the given characteristic equation s³ + 5s² + 7s + 3 = 0 is:

Option 4: Three

This is the correct option. All roots are in the left half-plane, indicating the system is stable