i used .the left connector´s. It was more luck that it works because i don´t know why there are on the right side another connectors .
I couldn´t say which one is the positive and which is nagative because i didn´t test it. I´ve connect it like on my pictures and it work´s.
I´ve used the left connector´s because they are the same then on S3 and Note 2. I think the right connector´s are for NFC because the original Samsung Cover is "extreme" fat
With my solution there is a little camber in the Battery Case and nothing else. That means that you can use any nice thin Protection Cover for the S4 and it charges through the Cover.