160406-1637 EDT
nizak:
You have a coil, mechanical binding, or low voltage problem.
With the contactor in its correct machanical position do the following tests:
1. With no power to anything on the contactor (for your safety) try to move the solenoid plunger. This should move freely except for spring force. If it does not move freely, then you have to solve the binding problem or replace the contactor.
2. Use a 7.5 A Variac adjustable voltage supply connected to the contactor solenoid coil directly. Include a series switch if there is no switch on the Variac.
Use a meter to monitor coil voltage.
Set the Variac voltage to 120 V. Apply this 120 V to the contactor coil. Does the contactor pull in? If it does not pull-in and just buzzes, then does the solenoid plunger move at all? If it moves but does not seat look for binding or chips that may prevent seating.
Since you indicated that the plunger could be nudged to seat I don't believe it is chips, but could be binding.
Apply 140 V, but not for long. Does this make it seat without assistance? If so it would seem to imply a "weak" coil.
Measurements on an old A-B 709 #2 starter, 55 years old.
DC coil resistance 40.2 ohms. All starters of the same model probably will be closer than +/-5% of this value. A second identical model I have reads 40.1 ohms.
Pull-in voltage for no buzz (I describe this as a solid pull-in) 72 V.
Drop-out ( not the actual drop-out, but that point where buzzing starts as voltage is lowered) 50 V.
Other measurements at 120 V with a Kill-A-Watt EZ:
Current 0.26 A, meaning an AC impedance of 462 ohms.
Power input 8 W.
VA input 31.6 VA.
If you have shorted turns, then DC resistance will be lower than a good coil.
A sticking plunger with 120 V applied will have a much higher current for an AC coil. If the contactor is built with a DC solenoid and built in full-wave diode rectified then current won't change much with plunger position. This is a design that can reduce coil burnout problems, but does not create the large pull-in force that a shaded pole AC solenoid can produce.
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