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How to use double star winding motor with VFD?
With this kind of motor ,the best way is to choose the winding connection that link with torque requested by application corresponding to speed link with winding ,and rest is to set motor nameplate in
drive which is corresponding to winding connection used.

For Example:
LS90L2 motor 2 speeds form Leroy Somer ,motor nameplate :
@low speed = 0.75kW ;50Hz; 1395 rpm ; 220V ; 4.85A => Tn=5.13Nm
@high speed = 0.44kW ;50Hz; 670 rpm ; 220V ; 4.7A   => Tn =6.27Nm
We can that we have 22% of difference on torque at low speed compare to torque at high speed.

So,if I set the low speed nameplate in the drive , and use low speed motor coupling:
NPR = 0.44kW
UNS=220V
FRS =50Hz
NCR = 4.85A
(+ AutoTun)

Available torque up to 50Hz (670rpm) will be 6.27Nm.If I increase frequency up to 104Hz to reach 1395rpm ,then available torque will be 0.31 Nm (due to voltage limitation above 50Hz).Main part of 2 speed motor was not design to operate with drive so 87Hz operation(supply delta winding with 400V) can damage winding ,will not provide enough torque up to 104Hz.In addition,2 speed motor was designed to operate on network at 50Hz ,and not with pwm of drive (dv/dt,etc...),so drive pwm could be on issue on motor insulation (coating) .

Next, if I set the  high speed nameplate in the drive, and use high speed motor coupling:
NPR = 0.75kW
UNS=220V
FRS =50Hz
NCR = 4.7A
(+ AutoTun)
s
Available torque up to 50Hz (1395rpm) will be 5.13Nm,so torque at 670rpm (24Hz) will be 5.13Nm and 6.27Nm which is 22% less. Available torque could be increase through boost parameter (BOO=Yes ;FAB=30Hz,etc...) , in addition CLI will permit transient over torque up to 60 s (typically 120% of CLI will permit 130% of Cn).Result obtain with these setting will strongly depends of motor capability. Whatever, you have to set ITH to NCR setting to permit thermal monitoring of motor by the drive (to trip in fault if thermal state is to high).
So, setting could permit transient over torque up to 6.27Nm ,but not a constant operation at this point due to motor current increasing (this is why ITH setting is strongly important).

Conclusion :
Drive can works with 2 speeds motor if we set nameplate which corresponding to coupling ,but we can't reach performance of both nameplate/coupling in same time for a continuous operation.

PS : Drive can have an impact on motor insulation ,if motor is not design to operate with drive ,but to check this point the only way is to see with motor manufacturer.

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