The motor on my dryer does not always switch from the start winding to the run winding
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I do not have a wiring diagram for this unit so the following is just what is normal.
The motors usually do not switch from start to run windings.
What they do is switch out the start winding when the motor gets close to operating speed. At rest, both the start and run windings are in circuit.
The start winding is switched out because if left in circuit the motor will overheat and trip it’s internal thermal protect.
The switch Normally Closed contacts open to disconnect the run winding and it’s Normally Open contacts are usually wired in parallel with the start switch to keep it running after you release the start switch.
So if your symptom is that the unit runs fine as long as you hold the start switch closed then odds are the Normally Open side of the switch is toast.
If it will not keep running and you do not have heat, there is probably a problem with the mechanical device in the motor and it is not activating the switches.
With luck this may be that it is just dirty or clogged with lint.