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Wireless power transfer, where to start?

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I'm not asking for you guys to do my homework for me. But I am wondering where to start. I have a project to transmit wireless power to a micro controller that can read a small sensor. Specifically it looks like It'll need about 2 mA at 5 volts. Storing up power for several seconds to run it for 10 ms to take samples is an option but that adds a whole other facet to the project (if anyone wants to share wisdom about feasibility or methods for that too, they are much appreciated). The distance I need to cover is variable, as good as I can get. If it's 2 cm then that's the limit, if 2 m is somehow possible, great. A directed antenna with a narrow focus is fine. The size of the antenna is variable but for practical purposes it probably should not be more than 0.75 m in diameter. The power available for the transmitter is pretty good, if I really want to go nuts, up to 11.1 V @ 50 A is available. My parts budget is up to maybe $500.

Anyway, I'm not asking to be handed a full design or anything. Where should I start my research? Do you build these things using air core transformer equations? Or is it more like a radio transmitter? What's a good source on how to do this? I have a basic understanding of electrical engineering, so even if you point me to a textbook or research paper to look at that would help. What equations do I need? What kind of circuits and antennas have been successful for this sort of thing? What precautions should I take? I know it's a very open ended question, but I haven't been able to find much to go on.
Many P.O.V. clocks use wireless power transfer and might be a good thing to start googleing.

Does the system need to work in any orientation / position, or can the power transmitter be directional?

For the 10mW you need, you could just use a well focussed light source and photovoltaic cell.
Possibly infra-red, if you don't want an obvious light beam? (I have no idea if photovoltaic cells work with IR though, offhand..)

Anything else, beyond the range of simple transformer-style coupling, would involve radio transmission and that may be a bit dodgy from the possible interference & legal aspects; many parts of the world only allow microwatt power emission levels for continuous carrier type systems without licencing.

You could probably get somewhere from 10 - 50cm with two moderately large tuned loop style coils running at around 100 KHz or so?
That's pretty similar to some types of RFID system, as the low frequency versions of those use coupled coils (in the reader and card/tag) to power up the electronics in the card or tag.

Researching those could be a good starting point?
I did some design work on 125 kHz card readers. The transmitter and receiver coils are tuned to 125 kHz. I got about 10 cm range from a 10 cm coil, but the limit was getting the data back, not getting power to the card, but if you're getting the data back some other way you don't need to worry.

I also tried to get one to read with a single turn antenna of about 0.7m diameter. It didn't work, but there was enough power to activate the card at about 0.5 m range, and the problem was getting the data back from the card. I used a test coil to prove that the card was being activated. The current in the antenna was quite large, 8 A, if I remember correctly. I used a transformer to step up the current, and the tuning capacitors were on the low current side of the transformer, so the transformer turns ratio came into the tuning calculations. I used a ferrite ring as the transformer core.

Texas Instruments made a long range RFID system which could read a card at 1 m or so, but it used a different return frequency for the data. I think that it was 125 kHz one way and 132 kHz the other.

Keyless cars have a battery in the fob, and the 125 kHz is to trigger the key and to send data to it. The signal back to the car is at 433 MHz or something in that band, and is powered by the battery in the fob.
Thanks for the input. I've done some work with card readers. And in this case it doesn't need to carry a signal back. Do you know what equations I'd need to figure out the efficiency for a given antenna size and frequency? Or for that matter if there is any guide or table on antenna designs? I see a lot of these use a circular antenna, for applications like powering a drone it often appears to be a very large circular antenna with many turns, but I haven't found a good source on how to calculate what the size or number of turns should be, or how much current to push through it.

My results using a function generator so far are in a similar efficiency range to what you're describing. I may just have to figure this out experimentally. I tried using air gap transformer equations but they predict a far higher efficiency than I am getting. This may be due to them being in an ideal environment, and in reality the environment I am in has metal around, the ring isn't perfect, and the alignment isn't perfect.
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