Inductors do seem mysterious, partially because more than in any other kind of part, the imperfections and side effects show through. You can make a pretty perfect resistor if you're careful. You can make a modestly perfect cap with some effort. Mostly you can ignore the extraneous effects on both Rs and Cs in audio work. You can't with inductors.
An inductor is any conductor which forms a magnetic field - which is all of them. A circular magnetic field encircles every conductor. If you hold a wire in your right hand with your thumb lined up with the wire pointing in the direction that current flows, your fingers point in the direction in which the M-field circles the wire. M-field is proportional to current.
If you form the wire into a loop, then the M-field in the middle of the loop is all in the same direction, and reinforces. The M-field on the outside is all spread out and less intense. If you make many loops with a relatively common center area, the M-fields all add in the middle, and you can make an area of intense M-field.
As you build up M-field, you force energy into the M-field. It's stored there. When you stop forcing energy in, the M-field relaxes and the energy comes back out. Since it went in as a current, it comes back out as a current. This is "inductive kickback". The energy stored in the M-field is going to come back out - Mother Nature told it to act that way - and it does not much care what voltage it takes. So it will go to ANY voltage to make the current come back out. It happens that if you put a voltage across an inductor, that starts the current across the inductor increasing. The inductor impedes the change by the relation V = L* di/dt. In fact, this is one definition of inductance: L = V* dt/di, or inductance is the voltage divided by the rate of change of current.
You can measure inductances this way with a pulse inductance test. You put a pulse of voltage across an inductor and watch the current on a scope. The current starts at 0 and rises in a linear ramp. You can then compute di/dt directly from the scope trace, divide that into the voltage, and you have the inductance value.
When you put AC currents into an inductor, you're continously changing the M-field, and that takes work. So the inductor impedes the flow of AC current. This impedance is Xl = 2*pi*F*L big. The inductor has no impedance to DC, only its resistance. It takes a while to load up to a certain current, but when that current stops changing, the inductor M-field is constant, and it does not impede the DC flow, except by its resistance.
We use ferromagnetic cores in inductors because these materials concentrate M-field and make the volume of space they occupy much more M-field friendly. For the same turns of wire, filling the loops with iron may increase the inductance of the same wire coil by a factor of several thousand. Obviously, we ...like... iron cored inductors. They're much smaller and cheaper than the huge coils of wire we'd otherwise have to use.
But Mother Nature told us that there is no such thing as a free lunch (that's the Second Law of Thermodynamics, loosely translated). The iron materials which increase inductance bring a whole set of quirks and odd imperfections of their own. They saturate (there's a limit to the field strength they can take) and they distort (hysteresis losses in the material have this effect) as well as heating up (due to eddy current losses).
So what do we make of this mess?
(1) Don't use inductors unless you have to. This is the solution adopted by the industry. Lots of brainpower has gone into how to not use an inductor.
(2) If you have to, you need to know that the signal you will use in the inductor does not step over the bounds of what the inductor can do. That is, it must not saturate the inductor or burn it up by overheating the resistance or the eddy current losses in the core.
(3) For us amateurs, look for inductors which have been made cheaply for something else that we can pervert to the use.
For lots of us, that's a small tranformer, which is where we started this thread.
Paul, you're mostly correct. If I have a wound inductor core with two windings, I can measure the inductance on each winding and get inductances I'll call L1 and L2. These are proportional to the number of turns squared, that is L1 = Kc*N1*N1, where Kc is the core constant that makes the numbers come out right. L2 = Kc*N2*N2.
If we hook N1 and N2 in series aiding, then the total turns is N1+N2, so Ltotal = Kc*(N1+N2)*(N1+N2), or Kc*(N1^2+N2^2+2N1*N2) So you actually get a bit more than L1+L2. If you hook N1 and N2 in series opposing, then the resulting Lt is Kc*(N1-N2)*(N1-N2) = Kc*(N1^2+N2^2 - 2*N1*N2). So the inductances don't strictly add and subtract. But they do get larger and smaller in the directions you propose. You get two different inductances in the direction you expect, but it's not a direct add and subtract of the inductances, as a result of inductance being proportional to the square of the turns.
For measuring inductors, the easiest for me has always been a pulse inductance test, but I have oscilloscopes. It may be easier for you to measure if you put a known frequency into a series resistor/inductor and measure the voltages across the resistor and inductor with an AC meter. The voltages are proportional to the impedance of each, so you can compute the impedance of the inductor by the voltage divider rule and from that L = Xl/2*pi*F.
