ALT OVERSHOT BY THE SINGLE PLT OF AN AC90 PART 135 CHARTER EXPERIENCING AN ELEVATOR TRIM PROB ON CLBOUT FROM ZZZ; US.

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I DEPARTED RWY 34 AT ZZZ; WITH CEILINGS APPROX 1500 FT AGL AND A CLRNC LIMIT OF 5000 FT. APPROX 2 MINS AFTER DEPARTING; ZSE TOLD ME THEY COULD TURN ME N EARLIER IF I COULD ACCEPT ZZZ VOR; THEN PORTLAND. I TOLD SEATTLE THAT WOULD WORK; AND STARTED REFOLDING THE ENRTE CHART TO SEE WHAT RADIAL WAS OFF OF THE ZZZ VOR. AT THE SAME TIME; WHILE I WAS ABOUT HALFWAY THROUGH THE CLBING 200 DEG L TURN BACK TO THE VOR; I NOTICED THE DME QUIT WORKING. I THEN TURNED ON THE AUTOPLT TO REDUCE SOME OF MY WORKLOAD. I SWITCHED THE DME TO THE #2 NAV AND STILL NOTHING; SO I STARTED WONDERING ABOUT THE INTEGRITY OF THE VOR SIGNAL. THE AUTOPLT WAS ACTING A LITTLE ERRATIC (SLIGHT OSCILLATIONS IN THE PITCHING AXIS); BUT WITH THE MODERATE TURB WE WERE IN; I DIDN'T FEEL IT WOULD BE NOTICEABLE TO THE PATIENT; NURSE OR PARAMEDIC WHO WERE LOCATED IN THE CTR OF THE ACFT; CLOSER TO THE PIVOTING POINT. BY STABILIZING THE YOKE; I FELT I WAS DAMPENING THE OSCILLATIONS; AND DECIDED TO LEAVE THE AUTOPLT ON FOR A FEW MORE MINS. WITH MY R HAND; I STARTED CHKING THE DETENTS ON THE NAV RADIOS TO MAKE SURE THEY WERE NOT INBTWN FREQS. AT THIS POINT; I WAS COMING UP ON 4800 FT; SO I DECIDED TO USE THE ELECTRIC ELEVATOR TRIM ON THE YOKE WITH MY L THUMB TO LEVEL THE AIRPLANE; EVEN THOUGH THAT WOULD TURN THE AUTOPLT OFF. WHEN I STARTED TRIMMING THE NOSE DOWN; THE AIRPLANE ABRUPTLY PITCHED UP -- AT LEAST 20 DEGS; AND I IMMEDIATELY TURNED THE AUTOPLT OFF WITH THE DISCONNECT ON THE YOKE WITH MY L THUMB; AND ALSO AT THE AUTOPLT CTL HEAD ON THE LOWER CTR CONSOLE WITH MY R HAND. THEN WITH BOTH HANDS; I PUSHED THE NOSE DOWN TO GET BACK TO 5000 FT. AS I WAS DOING ALL OF THIS WHILE IN IMC CONDITIONS; I NOTIFIED SEATTLE THAT I WAS THROUGH 5300 FT OR SOMETHING LIKE THAT; BUT I'M NOT EXACTLY SURE; AS I WAS TRYING TO GET THE AIRPLANE BACK UNDER CTL AND UNDERSTAND WHAT WAS HAPPENING. AFTER WHAT I THINK WAS ONLY 1 OSCILLATION; I GOT THE AIRPLANE BACK LEVEL AND MANUALLY TRIMMED OUT; AND CONTINUED THE FLT TO PORTLAND; WHERE I MEL'ED THE AUTOPLT AND DME. THE PATIENT ON THIS PART 135 AIR AMBULANCE FL WAS NEXT TO ME GETTING SICK DURING ALL OF THIS; (SHE THREW-UP TWICE ON THIS FLT); BUT I HAVE LEARNED TO TUNE THESE LITTLE DISTRS OUT. I FEEL THE CAUSE OF THIS DEV WAS THE AUTOPLT TRIMMING THE AIRPLANE TO A VERY HIGH NOSE UP ATTITUDE; ALONG WITH THE FAILURE OF THE DME AND A CHANGE IN MY CLRNC ALL OCCURRING AT THE SAME TIME; WHILE IN IMC CONDITIONS WITH MODERATE TURB.