Solution

For two ideal inductors with self‑inductances$L_1$and $L_2$ and mutual inductance coefficient $M = L_{12}$, connected in series

the total magnetic flux may be reinforced or partially cancelled depending on the relative orientation of the windings. The total inductance of the circuit is:

$\boxed{L_{\text{total}} = L_1 + L_2 \pm 2L_{12}}$

The positive sign is taken when the currents in both coils produce magnetic fields that add (series‑aiding connection, fluxes in the same direction); the negative sign corresponds to fields that oppose each other (series‑opposing connection, fluxes in opposite directions).