Arduino Nano or any board by battery power and solar charging

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Of course. Charging the batter from solar is totally outside the world of arduino. Powering the arduino from a battery is straight foward as long as you keep it in the right range (6-12V)

You can use a boost converter to turn < 5V into 5V at less amps.

Solar powered microcontrolling a is fickle endeavor. Lithium-polymer single cells output 3 to 4.2 volts, and tend to hover around 3.6-3.7 volts for most of their time. Some people don't use them because of a reputation for spontaneously, and without any reason, blowing up spectacularly. I've never seen it except on YouTube, and they had to severely torture the battery. The combination of size, form and a convenient voltage makes LiPolys my first choice for solar projects (perhaps until I get my hands on a LiFePO4 battery?)

I use a few common sense rules: 1) charge the battery in stages: constant current at 1C or less for the first 90% of the charge, then constant voltage for the last 10%, and 2) draw 2C or less during normal use. You can spike somewhat higher, but keep the running current as low as possible. 3) Keep 'em cool. They will last longer, and be less sad.

For that use a LiPoly charger chips like the Microchip 73831 or TI's BQ2970 on a breakout board. The breakouts are nice cause they typically have a JST battery connector, and a barrel jack and/or usb connector for power in. Then put an appropriate buck or boost converter on the battery voltage out, and voila, solar power.

The thing to watch out for with this circuit is over-volting the charger chip, and with solar, that's easier to do than you might think. I use larger 5 volt solar cells rather than one with a higher voltage.

Design your circuit so it is either off, or in deep sleep for as much time as you can. Your WiFi will probably be the biggest current draw. I've made several projects with this setup, including a scrolling 14 segment LED clock and announcement sign that has run for more than 4 years now, with zero maintenance.

Last week I discovered the 5252F chip. It's really cool and uses a NiMH cell for daytime storage. This 4 pin chip does all the work - it attaches to a low voltage (2-3 volts) solar cell on one pin, to a NiMH battery and an inductor on the second pin. The value of the inductor, 80-400uH, controls the current to the load on the third pin. A common ground for the entire circuit takes the forth pin. While the solar cell is illuminated, all of the current charges the battery. When the solar cell goes dark, a constant current runs from the battery to the load.

It's perfect for solar lawn ornaments, but might it work for your soil meter?

The soil meter only needs to be read so often. Perhaps a cap bank could store extra charge to run one.

ChrisTenone:
It's perfect for solar lawn ornaments, but might it work for your soil meter?

I don't think so, unfortunately. The OP will need a constant voltage supply.

@OP it might be possible to power your sensor for at least 2~3 months on 3xAA NiMH or 1x18650 Li-Ion battery, which may make solar recharging unnecessary.

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tvr4:
I am wondering the same thing. The outdoor solar lights I have do just this, although they do not need to run a microcontroller.

The LED solar light I have charges during the day and automatically turns the LED light on at night. It also uses a 3.7V 18650 rechargable lithium battery.

I took the back off the solar LED light and there is only one little board in there. Unfortunately I did not see an identification of any kind that would assist in finding the board.

Has it got a 4 pin chip on it?

Solar phone chargers with battery banks at Amazon

Keep the AVR in sleep mode most of the time. Small problem is Arduino clock drift, how small depends on what oscillator the Arduino is using. A crystal is very good, a resonator (what Uno has) can be off over 3 minutes a day and the 8MHz internal oscillator is worst but can be calibrated some.

In a week, how far off may not matter at least as long as Arduino time gets corrected before it does matter.
You can get an RTC and be sure but those require constant power.

Another thing your can do to save power is run a lower speed and voltage. At 8MHz and AVR can run on 3V.

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tvr4:
No it has a bunch of chips on there. I can't find a way to attach a pic to this message so here is one I took
Dropbox - File Deleted - Simplify your life

I've not seen a board like that in any of the lawn ornaments I've dissected. Looks like it's all resistors, transistors, and one big honking diode!