For a second-order system, what is the damping ratio if the poles are located at

For a second-order system, what is the damping ratio if the poles are located at -2 ± j2?

For a second-order system, what is the damping ratio if the poles are located at -2 ± j2?

Step 1: Identify the poles of the system, which are given as -2 ± j2.
Step 2: From the poles, determine the real part (σ) and the imaginary part (ω). Here, σ = -2 and ω = 2.
Step 3: Since we need the damping ratio (ζ), we use the formula ζ = -σ / √(σ² + ω²).
Step 4: Calculate σ², which is (-2)² = 4.
Step 5: Calculate ω², which is (2)² = 4.
Step 6: Add σ² and ω² together: 4 + 4 = 8.
Step 7: Take the square root of the sum: √8 = 2.828.
Step 8: Substitute σ and the square root value into the damping ratio formula: ζ = -(-2) / 2.828.
Step 9: Simplify the equation: ζ = 2 / 2.828.
Step 10: Calculate the final value: ζ ≈ 0.707.

Damping Ratio – The damping ratio (ζ) is a measure of how oscillations in a system decay after a disturbance, calculated using the real and imaginary parts of the poles.
Poles of a System – The poles of a second-order system are complex numbers that determine the system's stability and response characteristics.
Second-Order System Dynamics – Understanding the behavior of second-order systems, including how damping affects the transient response.