Single-phase induction motors are not self-starting
without an auxiliary stator winding driven by an out of phase
current of near 90°. Once started the auxiliary winding is
optional.
The auxiliary winding of a permanent split capacitor
motor has a capacitor in series with it during starting and
running.
A capacitor-start induction motor only has a capacitor
in series with the auxiliary winding during starting.
A capacitor-run motor typically has a large
non-polarized electrolytic capacitor in series with the auxiliary
winding for starting, then a smaller non-electrolytic capacitor
during running.
The auxiliary winding of a resistance split-phase
motor develops a phase difference versus the main winding
during starting by virtue of the difference in resistance.