Hmm, the datasheet gives a different figure, mu_e = 198. And that's for one gapped and one ungapped core, giving a total 0.5mm air gap. Note that two gapped cores together will make 1mm gap!

You can gap much more, typically you want mu_e in the 20-60 range for best performance. Should be able to get more amps out of those bad boys, though you won't be able to fit too many more turns on there with the insulated wire. It's fine to add a gap with spacer material, cardboard, plastic, fiberglass, whatever. Note that when you gap the whole core (center limb and legs), the air gap around the path is double the mechanical offset!

The relations for a core, or generally a known geometry to which these parameters apply, are:

integral V dt = N Phi = N B Ae

H = N I / l_e

B = mu H = mu_0 mu_r H

L in henry == V s / A = phi / I

L = N B Ae / (H l_e / N) = N^2 A_e B / (H l_e) = N^2 A_e mu H / (H l_e) = mu N^2 A_e / l_e

The mu A_e / l_e part of course is A_L.

When the path is inhomogeneous, we can define a mu_eff that includes l_e and l_g; this is what the calculator and datasheet above do.

A better way to think of it may be that flux density, and therefore flux through a given winding (and therefore, say for a square pulse, the voltage and duration), remains constant; by varying gap, you're varying what current is required to achieve that flux density. Therefore as gap goes up, energy storage goes up. (L drops proportionally, but energy goes as current squared, so energy is proportional to air gap length.)

There's some fudge to this, particularly at large gaps, where the fringing field becomes more significant (effectively, A_e becomes larger in the gap); or at low mu, where the leakage is high. (For example, bunched turns on a low-mu toroid have higher inductivity than evenly spaced turns do.)

Transformer design is a bit easier than inductor design, because you only need to know flux, not magnetization; magnetization is intentionally very low (i.e., minimal idle current, maximum winding impedance). To fully design inductors, you need to know how much ampacity is required, so the resistivity of copper is a factor, as well as the winding area and fill factor. Sometimes the area double-product (A_e A_w) is listed in catalogs for this reason. That, or you go back and forth a few times, adjusting gap, turns, wire size, core size or stack, etc. The effort to solve it in closed form is kind of not worthwhile when you only have so many cores to choose from and you can just run the numbers on all of them.

Tim