How do I find the Arch-mage at the end of the quest "Under Sarhaal"?

Solution 1:

Check to see if he isn't sitting on a chair in his quarters. I just spent 15 minutes looking for him but could hear him coughing.

Solution 2:

As I recall, the arch-mage is either in his study, in the Hall of Elements, or wandering the courtyard.

If you're on PC, you can try loading a previous save where you can get to him, open the console, click on Savos Aren, then type prid [id], where [id] is the ID listed in the top-center of the console window (without the brackets, should look like prid 15d31 or something), and hit enter. Now, load your broken game and do one of two things. First, open the console, hit up (should load the previous command entered, prid [id] in this case), then enter to select Savos without clicking him. Type moveto player, and hit enter. Exit the console and he should be standing in front of you. The second option, if you'd rather be legit about it, is to type player.moveto [id]. It should teleport you to him, so you can see where he's at. If he's somewhere accessible, reload and run to him.

You can also try leaving the college and going somewhere else, like Whiterun, then sleeping for 3 or 4 days, then going back to the college to see if he got reset. Doesn't require the console and works on non-PC versions, but may not be effective. Also, make sure you don't over-write your only autosave with him accessible, or you won't get his reference ID (which changes every time you start a new character, as far as I'm aware, so you can't just ask me what mine says or start a new game to get it).