Inductors in Series and Parallel
Series adds flux linkage to grow the inductance
Wire a second inductor L2 in series with L1 = 4 mH, in one line, and slide L2. The same current threads both coils, the flux linkages join, and the series inductance L_s = L1 + L2 grows straight away. The parallel combination L_p of the same two is always smaller than either. Set L2 so the series inductance reaches the target of 10 mH.
In series the flux linkages add
Wire inductors in series, in one line, and the same current I passes through both coils in turn. The flux linkage λ each coil makes is proportional to the current (λ = LI), so the two linkages add directly: λ = L1·I + L2·I = (L1 + L2)·I. When the current changes, the voltage by which both coils oppose it, V = L1·dI/dt + L2·dI/dt, also adds. So series inductances simply add: L_s = L1 + L2. The combination is larger than either one alone.
In parallel the reciprocals add
In parallel the same voltage V sits across both coils and the current splits in two. From V = L·dI/dt the rate of change in each branch is dI1/dt = V/L1 and dI2/dt = V/L2, and the total current is their sum, so dI/dt = V/L1 + V/L2 = V·(1/L1 + 1/L2). Matching this to V = L_p·dI/dt gives 1/L_p = 1/L1 + 1/L2, that is L_p = L1·L2/(L1+L2), so the parallel inductance is always smaller than the smaller of the two. The flux now has two paths to take.
It is on the same side as resistors
Resistors add in series and add reciprocals in parallel. Inductors do exactly the same: they add in series and add reciprocals in parallel. The one that is reversed is the capacitor — capacitors add in parallel and add reciprocals in series. When in doubt, picture the shared current: in series the same current goes through both, so the opposing voltages add, and that is why the inductance grows. It is the very same logic by which resistors add in series.
Back to the first screen
The larger L2, the longer the series bar grew straight away, reaching 10 mH when L2 was 6 mH (L1 = 4). The parallel bar of the same two stayed shorter than the smaller of them throughout. Series grows because the same current passes through both and the flux linkages add; parallel shrinks because the flux splits into two paths and reciprocals add. It mirrors the capacitor exactly in reverse, yet matches the resistor exactly.