Steel ring air gap total reluctance

Total reluctance = total mmf ÷ total flux. The mmf is known, and the air-gap mmf (60 % of it) lets us find the flux from the air-gap reluctance.

Given: mean radius = 12 cm, so mean length l = 2π × 0.12 = 0.754 m; cross-section radius 1 cm, so A = π(0.01)² = 3.142×10⁻⁴ m²; gap lg = 2 mm; N = 550; I = 3 A.

Step 1: total mmf F = N I = 550 × 3 = 1650 AT

Step 2: air-gap mmf and reluctance

Fg = 0.6 × 1650 = 990 AT
Sg = lg/(μ0 A)
   = 2×10⁻³/(4π×10⁻⁷ × 3.142×10⁻⁴)
   = 5.066×10⁶ AT/Wb

Step 3: flux Φ = Fg/Sg = 990 / 5.066×10⁶ = 1.954×10⁻⁴ Wb

Step 4: total reluctance S = F/Φ = 1650 / 1.954×10⁻⁴ = 8.44×10⁶ AT/Wb

Check: since the gap takes 60 % of the mmf, S = Sg/0.6 = 5.066×10⁶/0.6 = 8.44×10⁶ AT/Wb. The steel part is 3.38×10⁶ AT/Wb (40 %).

Answer: total reluctance ≈ 8.44×10⁶ AT/Wb (A/Wb)