Final answer:
The rotor copper loss of the induction motor can be calculated using the formula Pᵣ = 3 * Iᵣ² * Rᵣ. Given that the rotor copper loss is 6 kW and the rotor frequency fᵣ is 3 Hz, we can find the current in the rotor using the formula Iᵣ = √(Pᵣ / (3 * Rᵣ)) ≈ 1.84 A. The power absorbed by the air gap (Pₐ) can be calculated using the formula Pₐ = Pᵣ - Pₛ, where Pₛ is the mechanical power developed.
Step-by-step explanation:
The rotor copper loss in a three-phase induction motor can be calculated using the formula P₁ = 3 × I₁² × R₁, where I₁ is the current in the rotor and R₁ is the resistance of the rotor. Given that the rotor copper loss is 6 kW and the rotor frequency f₁ is 3 Hz, we can find the current in the rotor using the formula P₁ = 3 × (I₁)² × R₁. Rearranging the equation, we can solve for I₁:
I₁ = √(P₁ / (3 × R₁)) = √(6 kW / (3 × (2π × 3 Hz))) ≈ 1.84 A
Next, we can find the power absorbed by the air gap (P₂) using the formula P₂ = P₁ - Ps, where Ps is the mechanical power developed. In this case, Ps is given by the formula Ps = 2π f₁ |s| T. Substituting the given values, we have:
Ps = 2π × 3 Hz × |s| T = 6π |s| T
Since the motor is a four-pole motor, it completes two revolutions per cycle (f₁). Therefore, the time for one revolution (T) is given by T = 1 / (2 f₁) = 1 / (2 × 3 Hz) = 1 / 6 s. Substituting this value into the expression for Ps, we have:
Ps = 6π |s| (1 / 6 s) = π |s| W
Thus, the air gap power absorbed is P₂ = P₁ - Ps = 6 kW - π |s| W.
Lastly, we can find the mechanical power developed (P₃) using the formula P₃ = 2π f₁ |s| W. Substituting the given values, we have:
P₃ = 2π × 3 Hz × |s| W = 6π |s| W.
Therefore, the air gap power absorbed P₂ is equal to the rotor copper loss P₁ - P₃:
P₂ = 6 kW - 6π |s| W = 6000 W - 6π |s| W = 6000 W - 18.85 |s| W.