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Inductance, series mutual flux aiding

Short answer

Inductance is a coil’s opposition to current change, measured as flux linkage per unit current. For two coupled coils in series with mutual flux aiding, the same current induces additive emfs; equivalent inductance L_eq = L1 + L2 + 2M.

What is inductance? Derive the expression for two inductances in series, with mutual flux aiding each other.

Inductance is the property of a coil by which it opposes any change of current through it. It is measured as flux linkage per unit current: L = Nφ / I, in henry (H). One henry is the inductance of a coil that induces 1 V when the current changes at 1 A/s.

Series connection, mutual flux aiding

Let two coils of self inductance L₁ and L₂ and mutual inductance M be connected in series so that the current flows in the same sense through both. Then the flux of each coil adds to the flux of the other (aiding).

   i →        L1              L2
 o------( coil 1 )------( coil 2 )------o
             \_______ M _______/
        (both fluxes in the same direction)

The same current i flows in both coils. The emf induced in each coil has two parts: self-induced and mutually induced.

  • Coil 1: e₁ = L₁ di/dt + M di/dt
  • Coil 2: e₂ = L₂ di/dt + M di/dt

Total induced emf (opposing the applied voltage):

e = e₁ + e₂ = (L₁ + L₂ + 2M) di/dt

If the combination is replaced by one equivalent inductance L_eq, then e = L_eq di/dt. Comparing,

L_eq = L₁ + L₂ + 2M

For opposing flux (the current enters the dotted ends in opposite senses), the sign of M changes: L_eq = L₁ + L₂ − 2M.

✓ Answer checked by Claude Opus 5.5 (high effort)

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