Many of the Fixya experts must have seen this problem of Live chats disconnecting. A livechat disconnect almost always ends with a bad rating. However there are some tips that can help you reduce the disconnects and the bad ratings associated with them.
1)DO NOT press the the F5 key when you are in a live session. This will disconnect you from the session and displays the message "Sorry this chat has been answered by another expert"
If a chat disconnects go to your profile immediately and open the transcript.

This will open the live chat transcript

I have noticed that if you post on the transcript your post will appear in the chat window of the customer(Although there is some lag).

Explain the customer that the session had been diconnected accidentally and request them to post the live chat request again and inform them that "They wont be charged again for that". With some luck you may be able to pick the reposted chat request.This has almost always worked for except on a recent chat with a user called nipa123.
Other reasons for disconnects
2)You router went offline: If this happen immediately reset your router. Once its back online you can continue typing in the chat utility
3)You closed the session by mistake: If this happens Follow tip 1
The success of avoiding a bad rating due to a live chat disconnect depends on how quickly you respond after the disconnect. If you don't post back quickly thinking that you may get that 6$ anyway all you will get is an IA rating and nothing else.
Hope this helps the new experts on the site.
Thanks for reading,
Sai.