I just now saw your vector question, Mr. C; I don't know how to answer it!
Been a long time since I've studied and applied vectors. But the 90 degrees is the inductive phase shift from the motor windings. So, assuming the DC coil resistance to be negligible, then the vector would indeed be 45 degrees, correct? So the square root of 2 appears as a consequence of 1/sin45. Or am I totally off base on that?
I was a little hesitant about putting 120v AC Put the load in series with the AC voltage in series with a 4w to 100w incand. bulb to limit current. No drama with this method.
I put it on the concrete floor in the basement plugged in and let it ring. for several minutes to see if it would get hot. It barely got warm so I am not worried about a fire hazard especially when it runs for only 10-11 seconds at a time with a 40 second rest for moderate rain fall.
I'd say you have shown Due Diligence and exercised your Duty of Care.
The diagram was for two out-of-phase 'former windings supplying a load.