Charging my Macbook with power inverter

The problem lies in one little word: Modified sine wave. Which typically has a rather tenuous connection with an actual sine wave. Rather, its voltage is a periodic step function with rather few steps (though I can't say for sure about the specific model you mention). You need an inverter that provides the latter. They exist, but tend to be more expensive. I use one made by Mascot, and it charges my macbooks just fine. My previous inverter, with its modified sine wave, did not.