Write borkhausen criterion of oscillations.

The Routh-Hurwitz Criterion for Oscillations

The Routh-Hurwitz criterion is a mathematical tool used to determine the stability of a linear time-invariant (LTI) system. It provides a necessary and sufficient condition for all the roots of the characteristic equation of the system to have negative real parts, ensuring that the system is stable.

The Routh-Hurwitz criterion is based on the Routh array, which is a triangular array constructed from the coefficients of the characteristic equation. The elements of the Routh array are calculated using a specific set of rules, and the stability of the system is determined by analyzing the signs of the elements in the first column of the array.

Statement of the Routh-Hurwitz Criterion

The Routh-Hurwitz criterion states that an LTI system is stable if and only if all the elements in the first column of the Routh array have the same sign. If any element in the first column changes sign, the system is unstable.

Procedure for Applying the Routh-Hurwitz Criterion

1. Write down the characteristic equation of the system in descending powers of the variable (s).

2. Construct the Routh array from the coefficients of the characteristic equation.

3. Evaluate the elements of the Routh array using the following rules:

• The elements in the first row are the coefficients of the characteristic equation, arranged in descending powers of (s).

• The elements in the subsequent rows are calculated using the following formula:

> $$a_{i,j} = \frac{a_{i-1,j}a_{i-2,j+1} - a_{i-1,j+1}a_{i-2,j}}{a_{i-1,j}}$$

where (a_{i,j}) is the element in the (i^{th}) row and (j^{th}) column of the Routh array.

1. Determine the stability of the system by analyzing the signs of the elements in the first column of the Routh array:

• If all the elements in the first column have the same sign, the system is stable.

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

Example

Consider the following characteristic equation:

$$s^3 + 2s^2 + 3s + 4 = 0$$

The Routh array for this characteristic equation is:

$$\begin{array}{ccc} s^3 & 2 & 4 \\ s^2 & 3 & 0 \\ s^1 & 6 & 4 \\ s^0 & 2 & \end{array}$$

Since all the elements in the first column of the Routh array have the same sign (positive), the system is stable.

Applications of the Routh-Hurwitz Criterion

The Routh-Hurwitz criterion is widely used in control theory and related fields to analyze the stability of LTI systems. Some of its applications include:

• Determining the stability of feedback control systems

• Designing control systems with desired stability properties

• Analyzing the stability of electrical networks

• Studying the behavior of mechanical systems

• Investigating the stability of chemical processes

Conclusion

The Routh-Hurwitz criterion is a powerful tool for analyzing the stability of LTI systems. It provides a simple and effective way to determine whether a system is stable or unstable, and it has numerous applications in various fields of engineering and science.