How to design wireless charging coils so they are resonant?

I am continuing from this question: What can I use to increase efficiency of inductive wireless energy transfer instead of factory manufactured ferrite plates?

I am trying to make somewhat efficient wireless energy transfer for charging a 190mAh 3.7V battery. The plan is to use a rectifier, a voltage regulator and a charging IC MCP738312(datasheet) in the device and generate AC at a few hundreds of kHz from a 12V coming from a 1A regulated wall power adaptor in the charger device.

I am making my own coils because of the size I want them to be by etching multiple layers of copper foil:

A new question arose: how do I design the coils so that they are resonant? It would be great to get an answer that would be easy to understand for someone without education in electrical engineering, if that's possible, as much as it is possible.

Answer

how do I design the coils so that they are resonant? It would be great to get an answer that would be easy to understand for someone without education in electrical engineering, if that's possible, as much as it is possible.

What you have without resonance is something like this: -

Image source

With a capacitor that resonates the receive coil you get a circuit like this: -

And this magnifies the induced voltage in the receive coil (\$V_{IN}\$ below) because it acts as a very resonant low pass filter like this: -

_{\$R\$ represents the coil and transmit driver losses.}

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\$V_{OUT}\$ can be made to peak at magnitudes like this: -

Interactive filter tool

With the values chosen, the circuit resonates at about 411 kHz with an amplitude gain (\$G_P\$) of 77.5 or nearly 38 dB. But, bear in mind, that when loading the circuit and taking real power from it, the peaking may drop to a few dB. However, even 6 dB (a doubling of voltage) is a great benefit for the selection of one or two parallel capacitors to hit the sweet-spot.

Also remember to use high-speed diodes in the receiver power rectifier. Something like the ubiquitous 1N400x type diode is wholly unsuitable because of its reverse recovery time.

You can also do a similar trick for the transmit coil but now, what happens is that you "tickle" a parallel coil and capacitor with a low current and get a much higher current. Sure, it can take several milliseconds to build up but who cares about that?