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Is there a simple way to estimate the amount of flux that is caught by a toroid coil (filled by some core material, like powder iron for example), which surrounds a straight wire with a current flowing through?

Without a magnetic material (air coil), one would simply have to use the known magnetic field of a straight wire and integrate over the cross section of the toroid. But I may be out of business for too long, but I see no simple way of accounting for the flux concentration effect of the core material. If the core would surround all the wire, it would, of course catch almost all the flux around the wire. But what if (like in reality) the core has only a height H which is substantially shorter than the length of the wire L?

Could the straight wire possibly be considered a single primary winding of a transformer, or do the "stray fields" farther away from the core play an important role?

oliver
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  • Search for theory of current transformers. You are just now planning to get or make one. –  Aug 26 '22 at 16:09
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    Yes quite, it is a transformer. The shared flux gives mutual inductance, and the wire/winding give self inductance. From these, you can calculate coupling factor and etc. Since flux linkage improves with permeability, you generally use a very high µ_r for the core (10k+ is available). – Tim Williams Aug 26 '22 at 16:11
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    Great idea, unfortunately already a well explored one. Look up "clamp current meter", works surprisingly well in practice too. – Ralph Aug 26 '22 at 17:36
  • Yes, it's exactly a single turn primary on a transformer and most of the flux will typically be confined to a toroidal core made of some plausible magnetic material like wound permalloy tape or ferrite. – Spehro Pefhany Aug 26 '22 at 19:17
  • @SpehroPefhany: one thing about that causes me difficulty in understanding - suppose the length of the wires is L=100 m and the height of the core (centered in the middle of the wire) is H=0.01 m, wouldn't that mean that the fraction of the magnetic flux which is 49.995 m away from the core is also confined to the core? Surely the spins in the core material form nothing but a dipole or something, don't they? If yes, they can't compensate the magnetic field of a straight wire... – oliver Aug 26 '22 at 20:27
  • Okay, the flux in the core from a tight single turn with twisted leads coming off will be almost the same as an infinitely large single turn passing through the core. The inductance of the "turn" will obviously be higher in the second case. – Spehro Pefhany Aug 26 '22 at 20:31
  • @Ralph: I wouldn't call it an idea because I knew about current clamps. The "idea" was more about using this inside a distribution cabinet for each circuit. I know there is at least one solution for that, but it's pricey and of the phone-home-to-china type. So, I wanted to do it myself, and hence, know about the theory of current clamps. But, alas, a quick search on google yielded only advertisements for the first trillion of results... – oliver Aug 26 '22 at 20:34
  • @SpehroPefhany: but the (mutual) inductance between primary (wire) and secondary (toroid) determines the voltage we measure in the secondary. So, do I understand you correctly that it makes a difference (in inductance) whether a clamp meter measures a straight wire or a section through a more or less tight loop? – oliver Aug 26 '22 at 20:51
  • The secondary doesn't see the difference (for a given primary current). The difference is leakage inductance in the transformer so it adds to the primary impedance. It's like a separate inductor in series with the primary (no significant coupling). – Spehro Pefhany Aug 26 '22 at 20:54
  • @SpehroPefhany: I'm afraid I don't know about leakage inductance in particular, and only a little about transformer theroy. I was hoping this could be understood solely by the properties of the primary wire and the core geometry/the concentrated flux, whereas the secondary doesn't play an active role (no current flows in it, so once the flux is known, the voltage can be simply calculated from the law of induction). What I think would possibly help me a lot, is a depiction of the field of the arrangement wire+toroidal core. – oliver Aug 26 '22 at 21:05

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