- B is flux/area, and being a spacial quantity, is /turn.

- H is amps/length, and being a spacial quantity, is *turn (i.e., amp-turns).

yep, so just to be crystal clear, your saying as N increases, B decreases while H increases? So more turns would help if a transformer were saturating?

Yes and no. In a

*given core and geometry*, holding all else constant, B and H are always proportional (B = mu_r * mu_0 * H), so if B reduces, H reduces as well.

An in-circuit explanation of that would be, more turns means more flux handling (before saturation) and more inductance, so the current drops, so B and H both drop.

If you're controlling something else at the same time, like the gap (to keep the same desired inductance), then H must rise (because you're asking the same amps, but more turns = more amp-turns = more H). B and H are then related effectively by a different mu_r.

B is generally better to use, because B is equal throughout a loop, but H depends on the material it's measured in (H is discontinuous at the surface of a permeable material).

To figure this, you can imagine the core, with magnetic path length l_e, being shrunk by a factor of mu_r (of the core material), to represent an equivalent air gap. Consider the purpose, why magnetic cores are used for in the first place: free space isn't very inductive, so to get higher inductivity (and thus, higher inductance from a given pile of wire), we want to concentrate the magnetic field into a smaller (thinner, wider) volume than the space naturally around the coil.

So, if you take l_e / mu_r, you get the air gap equivalent of the core. Add to this the actual air gap you've used (which is probably a lot larger, anyway) to get l_eff, the equivalent air gap of the assembly.

A_L = mu_0 * A_e / l_eff

So, the equivalent air gap of almost all ferrites is vanishingly small: a very average EE core set might have l_e around 100mm, but with mu_r = 2000 or so, it amounts to an equivalent 0.05mm. This is handy to know, since any accidental gap on the order of l_eff will have a dramatic impact on the apparent permeability of the core -- if the core halves meet with more than, say, 20um RMS surface finish, or tilt or general out-of-touching-ness, mu_eff will be much lower than you were expecting.

And, it follows, anywhere you require an extremely high mu (e.g., common mode chokes), you need very well fitting faces -- often, such cores aren't made from cut pieces, but completely solid pieces instead (toroids, figure-8 shapes, etc.).

I agree with this for most cases, but for this specific case I'm aiming for a specific magnetising inductance, which is why I'm adding an air gap in an attempt to allow more turns and a lower B. That's one of the uses of an air gap right?

Yes, precisely the use.

Note that energy density goes as B^2 / (2*mu), so the energy storage in ferrite is abysmal (almost nil, because mu > 1000 for most), and most filter chokes are wound on gapped, or distributed gap (e.g., powdered iron) materials. An air core choke has the highest energy density, because it won't saturate and because mu = mu_0, as low as it can go. But it isn't very practical, because the copper resistance gets in the way, and it gets really bulky. Practical inductors are a matter of optimization for the engineer, rather than theoretical formulations derived by the physicist.

Though this is worth keeping in mind if you ever have very peaky situations. Ferrite beads are completely and utterly futile for ESD and EFT, but air core coils can stand a chance (assuming they're big enough to do anything at all, which is kind of unlikely for casual use?). Snubbers ranging from switching circuits (especially using SCRs) up to photoflash and more can benefit, because although the peak power sucks, the pulsed operation means the average dissipation isn't a thermal problem. (A photoflash application might have peak currents in the 100s of A, even for a compact DSLR application. Jjust a few microhenry is enough to have an impact on the microsecond range transients, and a permeable cored inductor would be required to handle each and every ampere of that transient without saturating to do the job -- there's no way you could get the core small enough to matter at that current level!).

I think this might have been where I was going wrong, I used the pk-pk of the current, I didn't divide by 2. But if L is the magnetising inductance, doesn't that increase rapidly with an increase of N? Wouldn't that make B increase with N?

No, B decreases with N. If current is constant, then for rising L, you must be driving it with much more voltage or much lower frequency! Remember, Phi = L * I, and B = Phi / (N * A_e).

edit: I just tested this, the difference between the RMS of the total cycle and average of half a cycle is only a couple of volts in this case.

Yeah, a waveform like that, I'd approximate with a worst-case square wave of peak amplitude

*times worst case input variation* and full duty cycle. You'll probably end up with 10-30% overhead, which is fine; all the more to help prevent it from saturating at high temperature (where Bsat is lower).

Tim