Answer:
The required current in the 600-turn exciting coil to establish a flux of 100 μWb in the air gap is approximately 7.96 Amperes.
Explanation:
To find the required current, we can use the magnetic circuit equation:
Φ = B × A × l
where Φ is the magnetic flux, B is the magnetic field, A is the cross-sectional area, and l is the length.
Given information:
l1 = 10 cm
l2 = l3 = 18 cm
Cross-sectional area of l1 path (A1) = 6.25 cm²
Cross-sectional area of l2 and l3 paths (A2) = 3 cm²
Length of the air gap (l_gap) = 2 mm = 0.2 cm
Relative permeability of the core material (μ_r) = 800
Number of turns in the coil (N) = 600
Flux in the air gap (Φ) = 100 μWb = 100 × 10^-6 Wb
First, we need to calculate the magnetic field in the air gap using the formula:
B_gap = Φ / (A_gap × l_gap)
where A_gap is the cross-sectional area of the air gap.
A_gap = A1 + 2 × A2 = 6.25 cm² + 2 × 3 cm² = 12.25 cm²
B_gap = (100 × 10^-6 Wb) / (12.25 cm² × 0.2 cm) = 0.325 T
Next, we can calculate the required current in the coil using Ampere's Law:
B_core × A_core = B_gap × A_gap
where B_core is the magnetic field in the core and A_core is the cross-sectional area of the core.
Since there is no leakage and fringing, the magnetic field in the core is constant and equal to the magnetic field in the air gap.
B_core = B_gap = 0.325 T
A_core = A1 = 6.25 cm²
Now we can calculate the required current:
B_core × A_core × l1 + B_core × A_core × (l2 + l3) = μ₀ × μ_r × (N / l_gap) × I
μ₀ = 4π × 10^-7 T·m/A
I = (B_core × A_core × (l1 + l2 + l3)) / (μ₀ × μ_r × (N / l_gap))
Substituting the given values:
I = (0.325 T × 6.25 cm² × (10 cm + 18 cm + 18 cm)) / (4π × 10^-7 T·m/A × 800 × (600 / 0.2 cm))
Simplifying the expression:
I ≈ 7.96 A
Therefore, the required current in the 600-turn exciting coil to establish a flux of 100 μWb in the air gap is approximately 7.96 Amperes.