How can I figure out why my Apple TV drops connection when using AirPlay
There are multiple tools available on your Mac.


open Console and look around the time stamp of the event (lost connection) to see what is it reporting.
Use Ping to test the connection speed and stability


Apple TV: Go to the Settings menu on Apple TV, select General > Network -note the IP address (you will need it later).


Start Network Utility 


Ping your Apple TV IP, you might want to run a continuous ping to see if it changes significantly over time.



Read more "how to" here:

Testing TCP port connectivity between computers and devices