Here is a simple example to show you how off-grid system is calculated, it will help you understand solar off-grid system better.
Appliance: DC 12V40W.
Working Time: 8hs/day.
Average Peak Sun Hour: 5hs/day
Days of Self-autonomy: 3 days.
Rainy days interval: 7 days.
Controller current and voltage.
Appliance Current=Appliance Power / Appliance Voltage = 40 / 12 ≈3.3A
So the panel current=Appliance current x Appliance working time / Average Sunshine Duration = (3.3 x 8) / 5 = 5.28A
So the panel power=Panel current x Panel voltage = 5.28 x 17.2 = 91W
Normally, the voltage of solar panels for 12V battery is 17.2V.
As rainy days interval is 7 days, and days of autonomy are 3 days, we need to enlarge the panel size to provide extra current to battery for rainy days:
Panel power = 91 x (10 / 7) = 130W
Considering the power loss and other factors, we give a safety factor 1.2 for this:
So the final panel power = 130 x 1.2 = 156 ≈ 160W
Appliance Current=Appliance Power / Appliance Voltage = 40 / 12 ≈ 3.3A
Daily Current Capacity = Appliance Current x Working time = 3.3A x 8h = 26.4AH
Battery Capacity = Daily capacity x (1+ Days of autonomy) = 26.4 x (1 + 3) = 105.6 AH
For consideration of battery service life, at most 50% of its capacity can be discharged.
So the battery size = 105.6 AH x 2 = 211 AH
So we choose two pcs of 120 AH12V battery and connect in series.
Controller voltage = Appliance Voltage = 12V
Controller current = Max [appliance current, panel current] = Max[40/12,160/17.2] = 9.3A
Safety Factor = 1.2
So the current = 9.3 x 1.2 = 11.16A
So a 12V15A controller is needed.