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At no load, I connected a 380V motor to start in the star connection for 5 seconds and thereafter change to the delta connection. I understand that torque will change since the current will change as well. In fact, the torque should increase as the connection changes from star(wye) to delta... But what about speed?

When I look at it, it seems as though the motor actually rotates faster when the starter switches to delta. However, I've been told that speed remains the same.

I do not have access to a tachometer to test it myself, so it's hard to tell. Does speed increase as well or does it remain the same?

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Speed will increase but not dramatically. A 3ph induction motor never runs synchronous with the AC power frequency; there is always the phenomena known as "slip" and slip is needed to induce a voltage into the rotor that in turn circulates a current and generates a rotor magnetic field.

The amount of slip is dependant on mechanical loading of the rotor; more loading means more slip and a reduction in rotor speed and, even on no mechancial load, there is friction and windage losses just due to the rotor spinning.

If you then re-connect from star to delta formation you are providing more torque and that can overcome a part of these "no load" mechanical losses and hence the speed increases a little bit.

The picture below shows the effect of three different voltages and the resultant change in speed (given a particular torque-speed load): -

enter image description here

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