Transformers
Equivalent Series Impedance:
Transformer bank of three single-phase transformers
Z 0 = Z1 = Z 2 = Z l
Three-phase transformer with a three-leg core
Z1 = Z 2 = Z l
Z0 > Zl
Wye-Delta Wound Transformers
Wiring connection will always cause a phase shift
Positive Sequence rotates by a +30 degrees from HV to LV side
Negative Sequence rotates by a -30 degrees from HV to LV side
Zero Sequence does not rotate
© March 2004
Power Systems I
1
Zero-Sequence in a Transformer
Dependent on the winding connection on each side
Dependent on the ground connection
wye or delta
ungrounded or grounded
Types of connections
wye-grounded to wye-grounded
wye-grounded to wye
wye-grounded to delta
wye to wye
wye to delta
delta to delta
© March 2004
Power Systems I
2
Zero-Sequence in a Transformer
wye-grounded to wye-grounded
wye-grounded to wye
wye-grounded to delta
© March 2004
Power Systems I
3
Zero-Sequence in a Transformer
wye to wye
wye to delta
delta to delta
© March 2004
Power Systems I
4
Generators
Synchronous Reactance
Dependent on the time period of the study
Steady-state
Transient
Sub-transient
Generator is modeled as a wye-connected machine for the
positive- and negative-sequence impedances
The zero-sequence is dependent on the winding connections of
the machine
© March 2004
wye-grounded
wye-grounded through an impedance
wye
delta
Power Systems I
5
Solidly Grounded Generators
ZS
E
E
E
ZS
ZS
© March 2004
Power Systems I
6
Solidly Grounded Generators
A E 012 = E abc
 Ea 


E abc =  Eb 
 Ec 
0
E 012 =  Ea 
 0 
X 0 = X1 = X 2 = X d
© March 2004
jXd
Zero Seq
Pos Seq
E
Neg Seq
Power Systems I
jXd
jXd
VG0
VG1
VG2
7
Impedance Grounded Generators
ZS
E
Zn
E
E
ZS
ZS
© March 2004
Power Systems I
8
Impedance Grounded Generators
0
E 012 =  Ea 
 0 
 jX d + 3Z n
0
Z 012 = 

0
Zero Seq
0
Xd
0
0 
0 
X d 
Pos Seq
Neg Seq
© March 2004
jXd+3Zn
Power Systems I
E
jXd
jXd
VG0
VG1
VG2
9
Wye Connected Generators
ZS
E
E
E
ZS
ZS
© March 2004
Power Systems I
10
Wye Connected Generators
0
E 012 =  Ea 
 0 
0 0
Z 012 = 0 X d
0 0
Zero Seq
0 
0 
X d 
Pos Seq
Neg Seq
© March 2004
jXd
Power Systems I
E
jXd
jXd
VG0
VG1
VG2
11
Delta Connected Generators
ZS
E
E
ZS
E
ZS
© March 2004
Power Systems I
12
Delta Connected Generators
0
E 012 =  Ea 
 0 
0 0
Z 012 = 0 X d
0 0
Zero Seq
0 
0 
X d 
Pos Seq
Neg Seq
© March 2004
jXd
Power Systems I
E
VG0
jXd
jXd
VG1
VG2
13