Ungrounded Wye Unbalanced Detection Discussion (Page 2 of 3)
The polyphase analysis for an unbalance three phase, three wire system, where all three phases are unequal will take some math to determine the extent of unbalance.
For this ungrounded WYE connected bank (See Figure 2).
Figure 2
We can look at polyphase circuit analysis to try to develop more userfriendly equations. However, before we attempt to simplify anything we need to do the math. The analysis can be shown in this case with , the following formulas were developed:
Appling Kirchhoff’s voltage law around each closed loop from Figure 1 we have the following
E_{BA}  V_{N’A’} + V_{N’B’} = 0
E_{BC}  V_{N’B’ }+ V_{N’C’} = 0
E_{AC}  V_{N’C’} + V_{N’A’} = 0
Substituting
V_{N’A’ }= I_{A’N’}Z_{1}, V_{N’B’} = I_{B’N’}Z_{2}, V_{N’C’ }= I_{C’N’}Z_{3},
Then We Have
E_{BA} = I_{A’N’}Z_{1}I_{B’N’}Z_{2
}E_{CB} = I_{B’N’}Z_{2}I_{C’N’}Z_{3
}E_{AC} = I_{C’N’}Z_{3}I_{A’N’}Z_{1
}
The Current Law (Kirchhoff’s) at Node "N" will BE
I_{A’N’ }+ I_{B’N’} + I_{C’N’ }= 0
I_{B’N’} = I_{A’N’}  I_{C’N’
}
Substituting for I_{B’N’ }We Get
Rewritten We Have
Using Determinants
=
I_{A’N’}=
Appling Kirchhoff’s Voltage Law
E_{BA}+E_{AC}+E_{CB} = 0 or, E_{BA}+E_{CB} = E_{AC}
Substituting E_{BA}+E_{CB} with E_{AC
}I_{A’N’ }=
Or
I_{A’N’ }=
I_{C’N’ }=
I_{B’N’} =
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