CE560 DEPARTING TEB EXPERIENCES DISTRACTION FROM A STABILIZER TRIM PROBLEM; AND AN ALTITUDE OVERSHOOT RESULTS.

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CE560XL TOOK OFF FROM TEB ON THE TEB 5 DEP AT XA46. I WAS PIC AND PNF. I NOTICED THE FLT DIRECTOR WAS SET TO THE NAV MODE AND DIRECTING A TURN THAT WAS NOT APPROPRIATE FOR THE DEP AS WE CLBED THROUGH 300 FT MSL. I CHANGED THE MODE TO HEADING WHICH WAS MORE APPROPRIATE FOR THE DEP WHICH CALLED FOR 'RWY HEADING TO 1500 FT THEN A R TURN TO 280 DEGS.' WHILE I WAS ATTENDING TO THE FLT DIRECTOR; I MISSED A 400 FT CALL AS DICTATED BY THE C560XL SOP. THIS CALL IS A PROMPT TO SELECT THE FLAPS TO THE '0 DEG' POSITION AND TURN ON THE YAW DAMP. IN SO DOING THIS; YOU ALSO CAUSE THE HORIZONTAL STABILIZER TO MOVE FROM THE TKOF TO THE CRUISE POSITION. AS WE CONTINUED ON THE DEP AT APPROX XA47; WE WERE GIVEN A CLB TO 6000 FT WHICH WE COMPLIED WITH AND LEVELED OFF APPROPRIATELY. AS THE CO-PLT (PF) BEGAN TO ALLOW THE AIRPLANE TO ACCELERATE WE THEN NOTICED A 'MASTER CAUTION LIGHT' AND A 'SEGMENT' LIGHT WHICH BROUGHT THE STABILIZER'S WRONG POSITION TO OUR ATTENTION. WHEN OUR ATTENTION WAS DRAWN TO THE CAUTION LIGHTS; THE ALTITUDE OF THE AIRPLANE INADVERTENTLY CLBED APPROX 300 FT. ONCE WE NOTICED THE CLB; WE CORRECTED THE STABILIZER ERROR AND IMMEDIATELY DSNDED TO THE CORRECT ALTITUDE. THE DEV DID NOT CAUSE A CONFLICT WITH ANOTHER AIRPLANE AND WAS NOT NOTED BY THE NY ATC. THE CONTRIBUTING FACTORS TO THE DEVIATION WERE FATIGUE AND THE DESIGN OF THE AIRPLANE'S MOVABLE STABILIZER. THIS WAS OUR SEVENTH CONSECUTIVE DAY OF DUTY. ADDITIONALLY; WE HAD WORKED FROM 12 TO 14.5 HRS EACH OF THE PREVIOUS 5 DAYS AND ONLY HAD 10 TO 10.5 HRS OF REST BETWEEN DUTY PERIODS EACH OF THE PREVIOUS THREE NIGHTS. WE HAD ARRIVED AT THE ARPT 1.5 HOURS PRIOR TO THE FLT DUE TO PREFERENCE OF THE CUSTOMER TO HAVE US THERE EARLIER THAN THE USUAL 1 HR PRIOR TO THE FLT. THIS OBVIOUSLY CUT 30 MORE MINS INTO OUR ALREADY SHORT NIGHT'S WORTH OF REST. THE STABILIZER REMAINING IN THE 'TKOF POSITION' AND THE AIRPLANE BEING AT A HIGHER AIRSPEED THAN IT NORMALLY WOULD IMMEDIATELY AFTER TKOF; CAUSED THE AIRPLANE TO CLB MORE READILY THAN IT WOULD HAVE HAD THE STAB BEEN IN THE 'CRUISE' POSITION.