SR20
Flight Crew Flight Instructor
Aircraft Equipment Problem Critical; Deviation / Discrepancy - Procedural Published Material / Policy
Flight Crew Landed As Precaution; Flight Crew Returned To Departure Airport; General Maintenance Action
Aircraft
After being cleared to takeoff Runway XXR we begin to execute a short field takeoff with the student flying the plane.  All indications were in the green during takeoff roll and rotation.  Upon reaching about 400 MSL CHT #3 showed a red indication with a max of 500 degrees and was accompanied by vibrations and a burning smell which dissuaded me from believing it was a faulty sensor.  I took controls at this point.  I had just switched frequencies to ZZZ departure and I immediately notified them that I would like to return to ZZZ for landing.  Once on frequency with Tower I told them i wished to land full stop but I did not declare an emergency.  By this time i made the decision to attempt to climb to ZZZ1 as I did not want an engine failure at 400 AGL.  I was able to climb to ZZZ1 and fly a mostly normal pattern and land on Runway XXR.  While on final I went power idle and the CHT went to a reading of about 100 degrees which was now significantly lower than the other CHT temperatures.  We taxied back and wrote up the plane to be assessed by Maintenance.
SR20 flight instructor reported an abnormally high cylinder head temperature and burning smell during initial climbout. An uneventful return to departure airport was executed.