INSTRUCTOR WITH INST STUDENT HAS ROUGH ENG WHILE EXECUTING MISSED APCH. INSTRUCTOR TAKES CTLS AND RETURNS TO LAND ON INTERSECTING RWY.

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MY STUDENT AND I DEPARTED PTK ENRTE TO FNT FOR SOME PRACTICE APCHS. WE WERE VECTORED IN FOR THE ILS 27 APCH INTO FNT. THE APCH WENT WELL; NO INDICATIONS OF ANY PROBLEMS WITH THE ENG. (PREFLT CHKED OUT WELL; NO PROBLEMS FOUND). AFTER THE APCH; I; THE INSTRUCTOR; CALLED FOR THE STUDENT TO EXECUTE THE MISSED APCH. HE ADVANCED THE PWR AND BEGAN THE CLB. THE MISSED APCH WAS EXECUTED AT THE DH. THE ENG ACCEPTED THE PWR WITH NO HESITATION. AT APPROX 1800 FT MSL; THE ENG SUDDENLY LOST ALL BUT IDLE PWR. I TOOK THE CTLS FROM THE STUDENT. I INSTRUCTED HIM TO REMOVE HIS HOOD AND READ ME THE EMER CHKLIST AS I BEGAN AN IMMEDIATE L TURN BACK TO THE ARPT. THE FIRST RWY THAT LINED UP WAS RWY 09. WE WERE TOO HIGH AND PICKING UP AIRSPD ON THE DSCNT. I PUT THE ACFT INTO A HARD SLIP TO THE L. I BEGAN TO EXTEND FLAPS WHEN I SAW WE WERE IN THE FLAP RANGE. THE RWY IS 7199 FT LONG. I USED APPROX 6500 FT. WE HAD A STRONG QUARTERING TAILWIND. IT WAS APPROX 20 KTS IN INTENSITY. I STOPPED SHORT OF THE END OF THE RWY AND THE LNDG WAS TERMINATED IN A NORMAL FASHION ONCE THE SPD WAS BLED OFF. NO DAMAGE WAS SUSTAINED IN THE ENTIRE EPISODE. BOTH STUDENT AND I WERE NOT HURT. WE MANAGED TO KEEP IDLE PWR IN THE ENG AND TAXI BACK UNDER OUR OWN PWR. ANY ADVANCE OF THE THROTTLE FROM IDLE WOULD CAUSE THE ENG TO RUN VERY ROUGH AND SOUND LIKE IT WAS ABOUT TO DIE OUT. DAYS LATER; THE MECH FOUND THE CARB TO BE CAUSE OF THE PWR LOSS. I AM FILING THIS RPT BECAUSE I UNDERSTAND THAT THE FAA IS GOING TO FILE AN INCIDENT RPT PERTAINING TO THIS OCCURRENCE. I DO NOT RECALL IF I DECLARED AN EMER DURING THE INCIDENT. AFTER CAREFUL REVIEW OF THE INCIDENT; I AM HARD-PRESSED TO FIND A BETTER APCH THAT WOULD HAVE HANDLED THE PWR LOSS IN A BETTER WAY. I CAN ONLY SAY THAT MAYBE I SHOULD HAVE ATTEMPTED TO CIRCLE AROUND TO RWY 18 AND MADE AN UPWIND LNDG. THE WIND WAS FROM THE SW.