Are you talking about a star delta motor starter, or a star delta winding on a transformer?
In the case of a star delta motor starter (I usually say (and hear) delta star), the basic purpose of any reduced voltage motor starter is to control the inrush current during startup. The delta star motor starter changes the voltage by changing the configuration of the (secondary) windings of a transformer which is part of the starter--in one configuration the windings are configured delta, in the other wye (or star), delta gives a lower voltage.
So the motor is started in the delta configuration, and switched to the wye after the motor is up to (or almost up to) speed.
Sorry; that's backwards. The first step is star, followed by delta. It is the motor windings, rather than the transformer windings, that are switched.
(In theory, if there were only one motor on a transformer, it might be possible to switch the transformer instead. I've never heard of such a case, but who knows?)
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They don't seem to be all that popular in the US. I've seen a very few, and have built/operated some closed-transition auto-transformer starters that worked fine.
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In vino veritas; in cervisia carmen; in aqua E. coli.