Modelling large air gap transformer

Question

I am trying to model a wireless energy transmitter as a transformer equivalent circuit with a large air gap. Because of the large air gap I’m assuming that for a realistic model fringing fluxes would have to be taken into account, I can’t find an easy to understand equivalent circuit that allows me to model the effect of a large air gap (or other mediums e.g concrete+air) by some formula for the coupling coefficient (I think?) that allows for a large air gap because most seem to assume the air gap is small.

Any help would be greatly appreciated.

Answer 1

A large air gap will give you a large leakage inductance, which appears as series inductance in your model.

Normally a transformer designer tries to minimize leakage inductance since it negatively affects regulation, but sometimes it's necessary (as in your case) or even an advantage (as in the case of discharge tube transformers).

Answer 2

You can use the model of a transformer as suggested in Spehro's answer but it's a little complex. This is because at a large distance between sending and receiving coils, the leakage inductance gets bigger BUT the magnetizing inductance gets smaller (shown as kL\$_P\$ on Spehro's diagram). It is shown this way correctly in Spehro's diagram (also note the "1-k" term) but, when ever I simulate these scenarios I use coupled inductors of fixed value with a variable coupling coefficient to account for air gap.

This means that L1 (driving inductor) is a simple inductor and small series resistor and ditto for L2 (receiving inductor). Why do it my way? Almost without doubt, if you are to achieve maximum power throughput on large gaps you'll need to parallel tune L1 with a capacitor and the same applies with L2 - this means you can easily set the tuning capacitors and their respective inductors to maximize power throughput at a large distance then forget about those values - at short distances (coupling approaching 100%) they will naturally de-tune each other but that is what you'd expect for this type of circuit: -

Note how when coupled at a factor of 0.5 there are two distinct peaks in the spectrum. As coupling falls the peaks gradually come together eventually becoming one.

Regarding the difference between concrete and air and the difference it may make - it won't - only ferromagnetic materials or conducting materials will alter the relationship - ferromagnetic materials will enhance the coupling and conducting materials (due to eddy currents being induced in them) will lower the coupling factor.