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Consider a second order all-pole transfer function model, if the desired settling time(5%) is 0.60 sec and the desired damping ratio 0.707, where should the poles be located in s-plane?
  • a)
    -5+j4√2
  • b)
    -5+j5
  • c)
    -4+j5√2
  • d)
    -4+j7
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Consider a second order all-pole transfer function model, if the desir...
Answer: b
Explanation: G = 1/√2
Gw = 5
s = -5+j5.
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Most Upvoted Answer
Consider a second order all-pole transfer function model, if the desir...
To determine the locations of the poles in the s-plane, we can use the settling time and damping ratio formulas for a second-order system.

The settling time formula for a second-order system is given by:

Ts = 4 / (ζωn)

where Ts is the settling time, ζ is the damping ratio, and ωn is the natural frequency.

Given that Ts = 0.60 sec and ζ = 0.707, we can rearrange the formula to solve for ωn:

ωn = 4 / (ζTs)

ωn = 4 / (0.707 * 0.60)

ωn ≈ 9.13 rad/s

The natural frequency (ωn) is related to the complex conjugate poles of the transfer function as follows:

ωn = sqrt(Re(s1)^2 + Im(s1)^2)

Given that the poles are complex conjugate pairs, we can express the poles as s1 = -ζωn ± jωn * sqrt(1 - ζ^2).

By substituting the given values of ζ and ωn, we can determine the locations of the poles:

s1 = -0.707 * 9.13 ± j * 9.13 * sqrt(1 - 0.707^2)

s1 ≈ -6.46 ± j4

Therefore, the poles should be located at approximately -6.46 + j4 and -6.46 - j4 in the s-plane.
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