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Equivalent inductance of two coupled coils in parallel

Short answer

For coupled coils in parallel, L_eq = (L1L2 − M²)/(L1 + L2 ∓ 2M). Using L1=4 H, L2=6 H, M=3 H: aiding (minus) yields 3.75 H; opposing (plus) yields 0.9375 H.

Two coils of inductance of 4 and 6 Henry are connected in parallel. If their mutual inductance is 3 Henry, calculate the equivalent inductance of the combination if mutual inductance: i) Supports the self inductance, and ii) Oppose the self inductance

For two coupled coils in parallel, with mutual inductance M:

L_eq = (L₁L₂ − M²) / (L₁ + L₂ ∓ 2M)

The minus sign in the denominator is for aiding (mutual flux supports self flux). The plus sign is for opposing.

Given L₁ = 4 H, L₂ = 6 H, M = 3 H.

  • L₁L₂ − M² = 24 − 9 = 15
  • L₁ + L₂ = 10

i) Mutual inductance supports (aiding)

L_eq = 15 / (10 − 2×3) = 15 / 4 = 3.75 H

ii) Mutual inductance opposes

L_eq = 15 / (10 + 2×3) = 15 / 16 = 0.9375 H

Answer: 3.75 H (aiding) and 0.9375 H (opposing).

✓ Answer checked by Claude Opus 5.5 (high effort)

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