# Inductance of planar (spiral) coil near solid copper surface

I'm working on a electromagnetic levitation system, consisting of a flat spiral (planar) inductor positioned above a solid copper surface. The planar coil is parallel to the copper surface. (Consider both the coil and the surface to be 100% pure copper, and the surface infinite in area and depth.)

I know how to calculate the inductance of a spiral inductor in free space. In fact, there are a number of calculators online, like this one.

My question is regarding the change of inductance as the coil is brought near the copper surface. My thinking is to consider the copper surface as a "shorted secondary" in a coreless transformer. This would imply that the inductance of the primary is actually the effective leakage inductance based on the system geometry. I suppose that if the primary (the spiral coil), were brought infinitesimally close to the secondary (copper surface), the leakage inductance would be zero. As the coil is moved away from the copper surface, the leakage inductance would increase (up to the full inductance of the coil in free space, at infinite distance from the copper surface below). I think it is probably ok to consider the resistances as negligible (effectively zero).

I am looking for two things:

1. Guidance on my thinking, confirmation that I'm on the right track, or correction if not.
2. A formula that I can apply to calculate [the change in] the coil's inductance as it is positioned at different distances to the copper surface. (Typical distances would be on the order of 1 mm for a 25 mm diameter coil, FWIW.)

I realize that I could actually construct this system in some form and measure the inductance, and perhaps even empirically derive a formula. But I'd rather model it first if possible.

• Eddy current should be in the model. the lost in near field is high due to eddy current. – M KS Nov 22 '18 at 21:05
• Eddy current is the whole reason it levitates... the current in the “shorted secondary” is the eddy current, and the inductance of the shorted secondary is assumed to be negligibly low. Am I missing something? – Kevin H. Patterson Nov 22 '18 at 21:09
• "shorted secondary" in a coreless transformer" I think this is not true. I am not an expert in this field, But with my knowledge, The transformer helps guide the magnetic flux, but we haven't that, and flux leakage can't be model with a simple model. Zero resistance also is not good for this case in reality. The lost with external copper near WPT coil is high. – M KS Nov 22 '18 at 21:32
• Think of it this way. If the coil and surface were both superconductive (truly zero resistance), and the coil were paralleled with a lossless capacitor, an alternating current would persist within the system and the coil would hover at a constant distance with no ongoing energy input. – Kevin H. Patterson Nov 22 '18 at 22:24
• A coreless transformer can certainly have a shorted secondary. Also, it is not the case that coreless transformers are inherently lossy. Leakage inductance is not loss— it is simply inductance that does not contribute to transfer of energy from the primary to secondary; the energy is simply returned to the primary side. – Kevin H. Patterson Nov 22 '18 at 22:29