How is the size of my battery backup system determined?

Each piece of equipment or appliance in the home (draws) a specific amount of power expressed in Watts. Watts = the applied voltage expressed as E, times the current expressed as I, passing through the device (W= E x I or Watts= Volts x Amps for this example).  While in operation this power is basically supplied by the systems batteries and the solar panel array which is sized to recharge or replace the energy drawn from the batteries during the normal periods of sunlight.  Power is related to the current capacity of the batteries (expressed in Ampere Hours or AH) and determines the length of time the energy can be drawn from the battery at a rate to allow normal operation of the loads attached.  The amount of energy consumed during the daily period must be put back into the batteries, or the batteries will eventually discharge below a usable level causing the system to fail.  A 100 watt lamp consumes power at a rate of 100 watts per hour for the period of time it is on. If the lamp is on for 1 hour it consumes 100 watt hours. In order to make the power available later to turn on the lamp we must recharge that 100 watt hours back into the batteries plus about 12% (112 watts) for applicable system losses.  The lamp consumes 100 watts at 120 volts so while the lamp is on the current required is relatively small (100W/120V = 0.83 amps).  The power that comes from a battery is at a lower voltage, in this test case at 24 volts. Therefore, the current flowing from the battery for the period of time the lamp was on would be 4.67 amps (112W/ 24V = 4.67 amps). In Virginia we have 5  hours (annual  average) of peak solar energy production to replace the power consumed by the lamp back into the battery.  Therefore to size the solar panels required to charge the batteries at a rate to reflect the available hours of solar production is 112 watts / 4.8 hours = 23 watts/hour. So one 24 volt solar panel (typical panel voltage) rated at 28 watts can be used to keep the battery charged. Of course that battery capacity must be capable of supplying the power needed based on the total time that the devices are being used, and at the power supply rate to match the energy that is pulled out of the battery. With larger loads, like a window AC unit, more current is needed at a higher rate of power supply. This then requires a larger battery storage system. The inverter (converts the battery power from DC to standard AC power) also needs to be sized to meet the load demand, not only the constant load, such as a light, but also the surge load caused when starting motors such as those used in an air conditioner fan and refrigerator compressors. It takes about 3 to 7 times as much energy to start the motors as it takes to operate them. For instance a 5000 BTU WINDOW AC unit requires 1,465 watts to operate but it may require 4,395 watts for a short duration to start. If the battery systems demand capacity is not built large enough to start the motors the motor will not start.