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I recently made a coupled inductor on an EE core and wanted to calculate the coupling coefficient at different airgaps. The windings were each on their own outer leg and the airgap was inserted in all three legs and the coupling coefficient was calculated by following equation:

$$ k = \frac{M}{\sqrt{L_1\cdot L_2}}$$

Where \$k\$ is the coupling coefficient, \$M\$ is the mutual inductance and \$L\$ is the inductance of each winding.

I measured the required above parameters for different airgaps (0mm to 2mm with 0.2mm increment) and noticed that the coupling coefficient was constant at around 0.33 for all airgaps, why is this? Before doing the measurement I expected to see better coupling coefficient at lower airgaps and lower coupling coefficient at higher airgaps.

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    \$\begingroup\$ There will be very little fringing from such small gaps : virtually all the flux entering the gap at one pole will exit via the other, therefore the coupling is roughly constant. The reluctance, OTOH, is strongly dependent on the gap length. Once the gap approaches the pole dimensions, I would expect to see coupling start to reduce. \$\endgroup\$ Dec 20 '21 at 23:08
  • \$\begingroup\$ Why are you posting this when it’s so similar to your original question that you appeared to understand my answer: electronics.stackexchange.com/questions/596093/…. So what has changed between the previous question and answer and now? Do you understand the effect of the central limb of the EE core? \$\endgroup\$
    – Andy aka
    Dec 20 '21 at 23:34
  • \$\begingroup\$ Me previous question was the confusion about low coupling coefficient at no airgap versus a higher coupling with an airgap, which made sense due to the magnetic flux short in the center leg. But now I see a constant coupling coefficient no matter how big the airgap is. So that is purely due to the center leg of the EE core? And does that mean that in this configuration, the most coupling we can get is 0.33? It sure seems so. \$\endgroup\$
    – James W.
    Dec 20 '21 at 23:50
  • \$\begingroup\$ Please tabulate a couple of results from this experiment so that I can consider the values you measured. Please also explain how you made the measurements. There is a much simpler method of measuring k directly without the need for the formula of course. \$\endgroup\$
    – Andy aka
    Dec 21 '21 at 7:42
  • \$\begingroup\$ A picture/drawing would also help showing the cores and the coils wound on them. The devil will be in the detail. \$\endgroup\$
    – Andy aka
    Dec 21 '21 at 8:28
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With that 2-E setup, coupling is set by how the primary flux splits into the secondary and the 'magnetically shorting' central leg.

Adding a leg-area air-gap into all three 'leg-area' legs doesn't really change the reluctance ratio of the central and the secondary leg, so the flux split ratio and hence the coupling coefficient would be expected to remain roughly constant.

It's not until the air-gaps become large enough such that the leg reluctance becomes an appreciable fraction of the air reluctance around the primary that you'd expect the coupling coefficient to drop significantly.

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