Two Inductors in Series

Inductors in series add, exactly as resistors do, and for the same structural reason: the same current flows through both, each opposes a change in that current with its own $L\,dI/dt$, and the voltages add around the loop. Divide out the common $dI/dt$ and $L_t = L_1 + L_2$. A 100 mH and a 220 mH choke in series make 320 mH, and nothing about the arithmetic is subtle.

What is subtle is the assumption. The addition holds only if the two coils are magnetically independent — if neither sees the other's flux. Where they do share flux the total becomes $L_1 + L_2 \pm 2M$, larger when the fields aid and smaller when they oppose, and with two identical coils at $k = 1$ the aiding case gives $4L$ rather than $2L$. Two toroids at opposite ends of a board are independent enough; two windings on one core are not, and neither are two solenoids end to end on the same axis.

The practical uses run both ways. Series connection is how you reach an inductance value nobody stocks, and how a filter designer trades one large part for two smaller ones with better current ratings. It is also how a common-mode choke works, in reverse: two windings deliberately coupled on one core so that common-mode current sees $L + M$ and differential-mode current sees $L - M$, which is very nearly nothing. That component only functions because series inductors do not simply add when they are coupled.

Beyond coupling, three things the addition does not capture. Current rating does not add — the string can only carry what its weakest member will take before saturating, and saturation in one part collapses the total inductance without warning. Resistance does add, so a series string has more copper loss and a lower Q than one part of the same total inductance. And self-resonant frequency falls: each inductor carries its own winding capacitance, and the combination resonates lower than either alone, so a series pair chosen to make a high-frequency choke may stop behaving as an inductor well below the frequency you needed it at.