To basic parameters are critical here: Induction and resistance value.
Induction, L, in Henry
Basically i simulated a buck converter circuit with the LTspice schematic tool which is also a simulator for electronic circuits, in order to find out a coarse min and max for the inductor value. buck in picture runs at 20kHz switching frequency. Apart from that, it was trial and error. I remember playing with low-voltage side of mains transformers for a short time, but a trafo has an iron core that is lossy and heats itself up from within fast and melts soon. So, I needed a core material with low loss: ferrite. I ended up using a speaker coil normally used for filtering audio current to a speaker I think. Could easily be, there's a better solution for this out there.
Resistance, R, in ohm
due to resistance a coil-component dissipates energy that heats itself up from within when a current flows through it. In a power converter current is high, so resistance must be as low as possible to keep the coil's core temperature below a certain limit: say 100C. Too much current and coil will melt itself from within, blowing converter up. Coil must be inside force ventilated casing and ucontroller must monitor temp inside casing and reduce current when temp goes too high. In my prototype 1kW buck converter in the picture, coil is glued to casing, over time i figured out coil could dissipate a max of 12Watts (i started lower and increased it slowly over time). With P = I*V, V = I*R and P = I^2*R we can calculate current that gives 12Watts dissipation: Imax = sqrt(P/R) Imax = sqrt(12/0,17) = 8,3 Amps. Up till now i didn't blow up a single coil (-yet-) (Before coil blows, fets blow first, started with one, but needs 3 in parallel for 1kW throughput, still coil is not at it max here and can handle more, could be, it can handle 1500 or even 2000W)
Eventually turned out i had to use a ferrite coil with windings that have the lowest resistance, in order words, that have the lowest number of windings, or lowest induction value, L, that can still provide sufficient induction. This boils, in turn, down to the minimum switching frequency you can use. The lower L gets, the higher your switching frequency must get. Losses are lower in the coil, but higher in the mosfets. A balance must be found there. I'm using 20kHz in the buck in the picture, what already works for a couple or years now.