Coupled Mode Theory for Wireless Power Transfer

Hello, many papers used coupled mode theory for wireless power transfer design.
It looks interesting because we can know the energy exchange in the system.
But mostly, they don't explain in details about this theory. Is there any body know well about this theory? Can you give me some good documents.
Thank you.

See it -> https://www.youtube.com/watch?v=r1UT4NuygmQ

The question has been gelling in my mind, as to which process is more feasible, or more efficient, or more prevalent:

(a) magnetic coupling, or

(b) electromagnetic/photon transmission?

When we recall that an old-fashioned crystal radio receiver can operate without batteries, by drawing its power from a broadcast antenna miles away, then a new realm of possibilities comes to mind.

I would not be surprised if researchers keep close-mouthed about the details. Efficient wireless power transfer is yet future, and there's a fortune to be made by whoever succeeds at it.

Existing wireless charging (e.g. Qi for phone charging) uses inductive coupling.

As long as coil dimensions and distances are small against wavelength, the coupling reduces to pure AC magnetic fields problem and can be analyzed respectively.

I'm not sure what "coupled mode theory" is about, but related to wireless power transfer, it sounds like a pretentious title for a simple problem.

Thank you for your replies.
My wireless power transfer system uses magnetic coupling to transfer power. Normally, frequency is about 10 ~ 100 kHz for electric vehicles, and about 6.78 MHz for mobile devices like laptops.

To analyze this system, it is possible to use Equivalent circuit, reflected load theory or coupled mode theory. People often said in papers that Coupled mode theory is familiar to physicists who want to know about the nature of energy exchange. Therefore, I want to explore this method.

But it looks complicated because it uses differential equations. Plz see a picture below.

Do you have any idea? Thank you

Thank you every one. I almost understood about this coupled mode theory.
It is necessary to review circuit theory, complex number and differential equations carefully.

You have a sinusoidal source and are looking for the stationary solution of the differential equations. You can get it much easier by analyzing the coupled resonators as AC network.

Your schematic has two coils close together. If we ask which electrical principle is operating here, then it looks as though it might be the transformer principle. However it is very different from transformers that we are used to working with. Those have tightly wrapped windings (for efficient coupling), and often have a ferrite or metal core.

This system has loose coupling, and no common core. It is inefficient. Nevertheless it has been used successfully in recent years, to charge batteries in hearing aids and electric toothbrushes, without wires. Since it is so convenient, that makes it a practical system, whether efficient or not.

So the theory behind it might be as simple as a transformer, but there is more about it that we need to think about. There might be a different name we give it, and a different range of parameters which we apply, and a different way we can optimize it, etc.

Yes, two coupled inductors make a transformer. Coupling factor of a usual transformer is high, e.g. > 0.99, here it's respectively low, e.g. 0.1 to 0.5, or even less. But the math is essentially the same.

The difference between an electric transformer and a Wireless power transfer system is the resonance.
Transformers utilize non-resonant inductive coupling while the other uses resonant inductive coupling.
Therefore, in circuit of a WPT system, it consists of Inductor and Capacitor for resonance. Coupling coefficient for this case may be very low, eg. 0.01, at far distance (2 meters)