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Well 20 watt mounted with a 40-50 degree angle should get 4 hours of usable sun per day.
That’s 80 watts on clear days. Do you get many of those where you are?
I have yet to get a real watt power draw number on a Talking House unit. I’d guess that it ends up in 2-5 watt power draw. The ATU is passive with no power required.
To run the Talking House based on this 24 hours a day you would need:
2 watts x 24 hours = 48 watt hours
5 watts x 24 hours = 120 watt hours
The 2 watt low end is able to be ran from the 20 watt panel with excess of 32 watt hours thrown into battery, minus inherent losses with solar and batteries. Which in real world is revised downward like this:
20 watt solar panel = 15 watts
15 watts x 4 hours = 60 watt hours
60 watt hours is real tight on the 2 watt low end end. Only sticking a maximum of 12 watt hours above use into the batteries, which isn’t going to make batteries happy.
On the high end your 20 watt panel produces about 50% of what you need at 5 watt sustained power draw.
So the choices here are really to first get a real wattage consumption figure for the Talking House unit you have on order. This can be done with a Kill-a-Watt unit which is cheap and plug and play (we all should have one or similar in our testing gear).
Beyond that, once you confirm power number, you can choose to run less hours like daylight only (which is very common in AM band). That should proportionally reduce your consumption.
2 watts x 12 hours = 24 watt hours
5 watts x 12 hours = 60 watt hours
With the 2 watt draw, running 12 hours a day, it would be possible to run this transmitter for 2 days without total discharge (of course depends on your battery which should be total draw for a day multiplied by say 5 — 24 watt hours @ 12V = 2AH of battery 60 watt hours @ 12V = 5AH of battery… So 10AH or 25AH of battery.