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Master-Level Question 1

Question:
A second-order control system has the transfer function:

G(s) = 25 / (s² + 6s + 25)

Determine the damping ratio (ζ), natural frequency (ωn), and percentage overshoot.

Solution by Our Expert:

Comparing with the standard second-order form:

G(s) = ωn² / (s² + 2ζωn s + ωn²)

We obtain:

ωn² = 25

ωn = 5 rad/s

2ζωn = 6

2ζ(5) = 6

ζ = 0.6

Percentage overshoot:

PO = e^((-ζπ)/√(1-ζ²)) × 100

PO = e^((-0.6π)/√(1-0.36)) × 100

PO ≈ 9.48%

Answer:
Natural Frequency = 5 rad/s
Damping Ratio = 0.6
Percentage Overshoot ≈ 9.48%

Master-Level Question 2

Question:
A three-phase induction motor operates at a supply frequency of 50 Hz with 4 poles. If the rotor speed is 1440 rpm, determine the synchronous speed and slip.

Solution by Our Expert:

Synchronous speed:

Ns = (120 × f) / P

Ns = (120 × 50) / 4

Ns = 1500 rpm

Slip:

s = (Ns - Nr) / Ns

s = (1500 - 1440) / 1500

s = 0.04

Slip percentage = 4%

Answer:
Synchronous Speed = 1500 rpm
Slip = 4%

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