Inductance and inductors in parallel

Inductance

Inductance (L) is the property of a coil by which it opposes any change of current through it. It is the flux linkage per ampere:

L = NΦ/I (henry)

From Faraday's law, the induced EMF is e = −L di/dt, so L is the EMF induced per unit rate of change of current. 1 henry produces 1 volt when the current changes at 1 A/s.

Inductors in parallel (no mutual coupling)

               i1 →
           +---(L1)---+
   i →     |   i2 →   |
 o---------+---(L2)---+---------o
           |   i3 →   |
           +---(L3)---+

All inductors have the same voltage v across them, and the supply current divides: i = i₁ + i₂ + i₃.

For each inductor, v = L di/dt, so

di₁/dt = v/L₁, di₂/dt = v/L₂, di₃/dt = v/L₃

Differentiate the current equation:

di/dt = di₁/dt + di₂/dt + di₃/dt = v (1/L₁ + 1/L₂ + 1/L₃)

For the equivalent inductor, di/dt = v/L_eq, so

1/L_eq = 1/L₁ + 1/L₂ + 1/L₃

For two inductors: L_eq = L₁L₂/(L₁ + L₂).

Notes

• L_eq is smaller than the smallest inductance.
• The current divides inversely to the inductances: i₁ = i × L₂/(L₁ + L₂).
• Example: 6 H and 3 H in parallel give 2 H.
• If the coils are magnetically coupled with mutual inductance M, L_eq = (L₁L₂ − M²)/(L₁ + L₂ ∓ 2M) (minus for aiding flux, plus for opposing).