A great way to understand inductors

Nice analogy!

The reason for using an active xo in the bass is to avoid the series inductor between the amp and the woofer driver. Otherwise with a passive xo and series inductor, the amplifier must 'spin up' and 'spin down' this inductor while attempting to control the woofer on the other side...
 
I don't like the mass inertia analogy. The use of the words "spinning up" and "down" seems to suggest a slow response to an applied voltage, but the current through an inductor is only lagging the voltage by 90° (in case of an ideal inductor). It follows the voltage perfectly, only 90° "later".
A capacitor does the opposite, the current leads the voltage by 90°.

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What the reactance of an inductor does mean is that it increases if frequency increases, increasingly lowering the amplitude of a signal passing through it. This could affect the damping factor of the amp at different frequencies. In an active XO, the filters are placed before the amp so the driver only "sees" the wire and speaker impedances.

The "problem" Steve mentioned is just the result of how energy is stored in an inductor. But it's also quite useful, think of boosting a voltage to light a fluorescent tube or create sparks in those old fangled petrol/gas internal combustion engines.
 
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I prefer, rather than analogies such as water through a pipe, springs/dashpots, etc it is best to describe and understand the actual reason. Not to get too deep into physics, granted - we don't necessarily benefit from talking about orthogonal fields and magnetic dipoles.

Inrush current in a transformer is simply the result of driving too much flux in the core. Worst case is when the transformer (or at least one of the windings in a three-phase design) is energized at voltage zero crossing. Residual magnetism is an additional factor, but we can leave it out for now, let's not get too technical.

Blue is steady state flux. When energized at zero voltage (green), the blue flux needs to rise from -1pu to +1pu. But flux unfortunately did not start out at -1pu : it started out at 0pu in the core (red). The red flux is therefore requested to rise from 0pu to +2pu. However, the core is not sized to handle +2pu flux. The magnetics saturate, which causes the inductance of the winding to fall off very quickly, with the resulting decreased impedance. Low impedance with fixed voltage gives high current (black). You can see the current does not rise significantly until the core saturates (red flat tops). A high inductance winding draws very little current; a low inductance winding (due to saturated iron) draws huge current.

Over many cycles, the AC excitation from the system tends to pull the core out of saturation as the iron runs along the B-H curve, and the current starts to get more symmetrical about the x-axis. But those first few cycles can run extremely high with DC offset as the winding looks more like a small-ohm resistor rather than a high-L inductor.
 

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So not some magic flywheel? But not some spring either?
The spring can be used as an analogy for a capacitor, not an inductor.

Personally, I see the flywheen as something that takes a lot of doing getting going and when it's going, it takes a lot of doing to get it stopped.
But I guess, if you consider the 90° lag of current "a lot of doing", it would be valid.

If you suddenly stop the current flowing through an inductor, the magnetically stored energy still needs to go somewhere, and as a result the voltage shoots up. This may be dissipated in (parasitic) capacitances or actually lead to an arc if the voltage gets high enough. The parallel resistor that Steve mentions is another way to dissipate this energy. You'll often also see diodes across coils that can do the same thing.
 
The basic physics is that when a wire passes perpendicularly through a magnetic field (whether the wire is moving or the magnetic field is moving), it generates a voltage, or back electromotive force (EMF). This back EMF opposes the current flow or if there is no driving current, then the effect acts as a generator. I find thinking in these real terms helps my understanding of inductors and transformers.