Question

Find the location of the poles of second-order systems with the following specifications: [Section: 4.6] (a) %OS=15;Ts​=0.5 second (b) %OS=8;Tp​=10 seconds (c) Ts​=1 second ;Tp​=1.1 seconds

Answer 1

1. We can start by calculating the damping ratio (ζ) using the formula
`ζ = (-ln(%OS/100))/(sqrt((pi^2)+(ln(%OS/100))^2)).`
2. Once we have the damping ratio, we can find the natural frequency (ωn) using the formula ωn
`= (4)/(ζ*Ts).`
Remember, these formulas assume a second-order system in the form of
`H(s) = K/(s^2 + 2ζωn*s + ωn^2),`where K is the system gain.

(b) For a second-order system with %OS of 8 and Tp (peak time) of 10 seconds:
1. Calculate the damping ratio (ζ) using the formula ζ = (-ln(%OS/100))/(sqrt((pi^2)+(ln(%OS/100))^2)).
3. Find the complex conjugate poles using the formula s1,2 = -ζωn ± jωn*sqrt(1-ζ^2).

(c) For a second-order system with Ts of 1 second and Tp of 1.1 seconds:
1. Calculate the damping ratio (ζ) using the formula
`ζ = (1)/(sqrt(2))*((Tp/Ts)-1).`
3. Find the complex conjugate poles using the formula
`s1,2 = -ζωn ± jωn*sqrt(1-ζ^2).`
Remember, these formulas assume a second-order system in the form of H(s) = K/(s^2 + 2ζωn*s + ωn^2), where K is the system gain.