DA42 with GFC700 may have yaw damper
Yes, I have flown a DA42 with GFC700 which had a yaw damper.
never saw a DA40 G1000 with a KAP150
neither did I, I think it is a KAP140. I used to fly a DA40 ten years ago with this AP, but it had no yaw damper.
FWIW we have a KFC150 in our Mooney with a yaw damper but it’s an Stec. It’s a completely independent unit.
it may be that the plane is out of trim in the yaw axis. There is a little trim tab attached to the rudder. If you have to input right rudder to center the ball, you need to bend the tab to the left. This causes a deflection of the rudder to the right, which is what you need.
I have the KAP150 on the Mooney with yaw damper, I have two modes on the yaw damper ‘ON’ or ‘AUTO’ (and OFF…). I usualy select ‘ON’ and then add some rudder input throught the yaw damper with a knob when/if required, but I never used any other system.
That one has a “Century IIB autopilot turn and bank” which is interesting in being a rather basic AP but with a yaw damper!