Inductance from flux linkages

Inductance L of a coil is the flux linkage produced per unit current. Consider a coil of N turns carrying current i. Let φ be the flux linking each turn.

Derivation

1. The flux produced is proportional to the current (for a linear magnetic path), so the flux linkage is proportional to i:

   Nφ = L i ⟹ L = Nφ / i

2. When the current changes, the flux linkage changes. By Faraday's law the induced emf is

   e = −d(Nφ)/dt = −N dφ/dt

3. Since Nφ = L i and L is constant,

   e = −L di/dt

   The negative sign shows (Lenz's law) that the induced emf opposes the change of current.

4. Taking magnitudes, the inductance in terms of flux linkage and current change is

L = N dφ / di = e / (di/dt)

For a finite change: L = N Δφ / Δi = Δ(flux linkage) / Δi.

So 1 henry means 1 weber-turn of flux linkage per ampere, or 1 V induced when current changes at 1 A/s.