## Star delta

####
Kennelly's
Star-Delta Transformation

A
star network of three impedances Z_{AN}, Z_{BN} and Z_{CN}
connected together at common node N can be transformed into a delta network
of three impedances Z_{AB}, Z_{BC} and Z_{CA} by the
following equations:

Z_{AB} = Z_{AN} + Z_{BN} + (Z_{AN}Z_{BN}
/ Z_{CN}) = (Z_{AN}Z_{BN} + Z_{BN}Z_{CN}
+ Z_{CN}Z_{AN}) / Z_{CN}

Z_{BC} = Z_{BN} + Z_{CN} + (Z_{BN}Z_{CN}
/ Z_{AN}) = (Z_{AN}Z_{BN} + Z_{BN}Z_{CN}
+ Z_{CN}Z_{AN}) / Z_{AN}

Z_{CA} = Z_{CN} + Z_{AN} + (Z_{CN}Z_{AN}
/ Z_{BN}) = (Z_{AN}Z_{BN} + Z_{BN}Z_{CN}
+ Z_{CN}Z_{AN}) / Z_{BN}

Similarly,
using admittances:

Y_{AB} = Y_{AN}Y_{BN} / (Y_{AN} + Y_{BN}
+ Y_{CN})

Y_{BC} = Y_{BN}Y_{CN} / (Y_{AN} + Y_{BN}
+ Y_{CN})

Y_{CA} = Y_{CN}Y_{AN} / (Y_{AN} + Y_{BN}
+ Y_{CN})

In general
terms:

Z_{delta} = (sum of Z_{star} pair products) / (opposite Z_{star})

Y_{delta} = (adjacent Y_{star} pair product) / (sum of Y_{star})

####
Kennelly's
Delta-Star Transformation

A
delta network of three impedances Z_{AB}, Z_{BC} and Z_{CA}
can be transformed into a star network of three impedances Z_{AN}, Z_{BN}
and Z_{CN} connected together at common node N by the following
equations:

Z_{AN} = Z_{CA}Z_{AB} / (Z_{AB} + Z_{BC}
+ Z_{CA})

Z_{BN} = Z_{AB}Z_{BC} / (Z_{AB} + Z_{BC}
+ Z_{CA})

Z_{CN} = Z_{BC}Z_{CA} / (Z_{AB} + Z_{BC}
+ Z_{CA})

**Similarly,
using admittances:
Y _{AN} = Y_{CA} + Y_{AB} + (Y_{CA}Y_{AB}
/ Y_{BC}) = (Y_{AB}Y_{BC} + Y_{BC}Y_{CA}
+ Y_{CA}Y_{AB}) / Y_{BC}
Y_{BN} = Y_{AB} + Y_{BC} + (Y_{AB}Y_{BC}
/ Y_{CA}) = (Y_{AB}Y_{BC} + Y_{BC}Y_{CA}
+ Y_{CA}Y_{AB}) / Y_{CA}
Y_{CN} = Y_{BC} + Y_{CA} + (Y_{BC}Y_{CA}
/ Y_{AB}) = (Y_{AB}Y_{BC} + Y_{BC}Y_{CA}
+ Y_{CA}Y_{AB}) / Y_{AB} **

In general terms:

Z_{star} = (adjacent Z_{delta} pair product) / (sum of Z_{delta})

Y_{star} = (sum of Y_{delta} pair products) / (opposite Y_{delta})