Well.. what's a core? Powdered iron, well, that's made of... iron. Which is conductive. So you can guess it will carry eddy currents -- not much, that's why it's powdered -- but always some, it doesn't simply go away. You can effectively model this as an inductor in series with a capacitor; this model goes in parallel with the magnetizing inductance (which, mind, is itself only a model). The capacitor represents the cutoff frequency due to skin effect, and the resistor represents the primary reflected resistance of the core material itself.
Skin effect actually goes as 1/sqrt(f) or so, whereas a capacitor's impedance goes as 1/f. It's not a very good model, especially over a wide range of frequencies. Eliminating the capacitor entirely (so the resistance is equivalent parallel resistance, EPR) goes as f^0, which is literally just as good -- the error factor is merely inverted. So, most times you might as well model the losses as EPR. In either case, mind that it's an approximation for a given frequency range.
Laminated iron cores fit this model very well. In a prior project, I had chosen a toroidal transformer, 10kVA capacity, and suitable for square waves in the mid audio range. (It was a neat project, worked just as it should; and yes, it was loud...) When we got the parts in, I did a small signal test:
- The inductance was relatively low, as expected: a quirk of iron is it's "sticky", i.e., it doesn't start being a good magnet (mu ~ 20k) until you put some field through it. The initial permeability might be under 1000.
- At low frequencies, applying a square wave voltage results in a triangle wave current. So it looks like an inductor. The current varies inversely with frequency (I = V / (2*pi*F*L), of course), from very low frequencies up to maybe 500Hz.
- If you look closely at the triangle waveform, it's got a bit of a step when it flips direction, not just a simple direction reversal. At low frequencies, it's not very noticeable, but at medium frequencies, because the triangular component shrinks so much, it becomes very obvious, and at high frequencies, it even dominates. An R || L model exhibits exactly this behavior: the resistor delivers a square wave current, while the inductor delivers a triangular current waveform. The sum is measured externally.
- At high frequencies (2kHz or so), something different happens. The triangle wave component is now so small that the current waveform looks like a square wave with only a slight tilt on it. But now the current actually rises slightly with increasing frequency -- maybe 10% per octave. Indeed, the parallel resistance is dropping in this range. I think an explanation for this effect is, as the skin depth becomes more shallow, the magnetic field is squeezed out of the core, and any time the volume occupied by a field drops, so does the inductance; thus the impedance drops. True, according to the scope, we're not talking inductance anymore, but in the presence of phase shift, you end up with loss, or something.
Now, all these effects are characteristic of laminated steel only, but the general form remains applicable even to ferrites. Which, by the way, are still conductive; try probing a bare MnZn core (most common type) some time. (The other main material is NiZn ferrite, which is much less conductive, and also has a brown streak ('streak' in the mineralogical sense). It finds use in the MHz+ range.) Hysteresis rather than eddy current loss is dominant in this material, but losses and phase shifts still work out the same. (By the way, ferrite cores over a few inches thickness -- for only the biggest applications of course -- also exhibit skin effect, so it's useless to make single cores bigger than this.)
Ferrites exhibit cutoff frequencies, though it's usually assumed characteristic of the material rather than the geometry. The direct tradeoff is permeability, so a 10k mu material might drop off around 50kHz, while an 800 mu material is usable to nearly a MHz.
Tim
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