EE 740
Professor Ali Keyhani
Lecture #3
Ideal Transformers
The ideal transformer:
I1
+
- V1
IT
I2
V2
N1
I1
+
-
V1
N2
V1/N1 = V2/N2
N1I1 = N2I2
S1 = V1I1*
S2 = V2I2*
S1 = S2
a = V1/V2 = N1/N2
V1 = aV2
I1
I2
V1
V2
N1
Z2 = V2/I2
I1 = I2/a
Z2
N2
Z2’ = V1/I1 = (aV2) / (I2/a) = a2Z2
Real transformer:
I1
R1 jX1
Ic
V1
Rc
R2
I2/a
Im
V1
V2
jXm
N1
N2
jX2 I2
Load
I2
V2
I1
I2
+
+
V1
V2
-
-
I1
I2
V1
V2
N1 IT N2
a=
V1
V2
I1
=
N1
N2
R1
jX1
Ic
V1
Rc
I2/a
I2
R2
Im
jX2
Load
jXm
N1
N2
R2’ = a2R2
X2’ = a2X2
I1
R1
Ic
V1
R2’ jX2’
jX1
Rc
I2
Im
Load
jXm
N1
N2
I1
+
- V1
IT
I2
V2
N1
I1
+
-
I2
V1
N2
V2
a = V1 = N1
V2
N2
R1jX1
I1
Ic
V1
R2
I2/a
Im
Rc
V1
jX2 I2
Load
V2
jXm
N1
N2
R2 & X2 are referred to
winding 1
I1
R1
Ic
V1
R2’ jX2’
jX1
Rc
I2
Im
Load
jXm
N1
N2
I1
Req
jXeq
I2
V2
V1
Load
N1
N2
Req = R1 + R2’
Xeq = X1 + X2’
I1
Neglecting excitation
current
jXeq
I2
V1
V2
Load
N1
N2
Determination of Req and Xeq from short-circuit test.
Isc
Vsc
Req
jXeq
Vsc
Ideal Transformer
Real Transformer
Vsc is normally equal to five percent of the rated voltage of winding #1, or:
Vsc = .05 x V1rated
Measure Isc, Vsc, Psc
Psc = /Isc/2 Rs1
Rs1 = Psc / (/Isc/2)
Zsc1 = /Vsc1/Isc1/ = /Req + jXeq/ = /Rs1 + jXs1/
Xs1 =  Zsc2 - Rs2
Isc Req jXeq
Isc
Vsc
Vsc
Vsc = .05 x V2rated
Measure Isc, Psc, Vsc
The same procedure as before however, the short-circuit impedance is referred to
winding #2.
The per-unit system:
pu = Actual Quantity / Base Value of Quantity
Name plate rating: An electric element has a nameplate rating which specifies the safe
operating voltage, current and power for the element.
Example: A light bulb is rated at 120V and 50Watts.
Base Values: The base values are normally selected equal to base values.
Pb = Qb = Sb = S1(rated)
Vb = VL-N(rated)
Zb = Vb/Ib = Vb / (Sb/Vb) = Vb2/Sb
Yb = 1/Zb
Ppu = PA/Pb
Qpu = QA/Qb
Spu = SA/Sb
Vpu = VA/Vb
Ipu = IA/Ib
Zpu = ZA/Zb
The nameplate rating for transformers, generators and machines are given in percent
(which is the same as pu, i.e. percent% = pu) using the device power rating as Sb and
voltage rating as Vb.
Example: A 3 generator is rated 300 MVA, 20 kV and ten percent reactance. Compute
the impedance of the machine in ohms.
Sb = 300 MVA/3 = 100 x 106 MVA
Vb = 20 kV/ 3 = 11.56 x 103 V
Zb = Vb2/Sb = (11.56 x 103)2 / 100 x 106 = 1.34 
Zgen = j.1 x (1.34) = j.134 
I1
Rs
jXs
I2
V1(rated)
LV
V2(rated)
N1
N2
S(rated)
Rs’
I1
jXs’
V1(rated)
HV
N1
a=
N2
=
V1(rated)
V2(rated)
N1
N2
S(rated)
V2(rated)
P.U. Equivalent Circuit of a Transformer.
Select:
Vb1 = V1(rated)
Sb = S(rated)
Select:
Vb2/Vb1 = V2(rated) / V1(rated) or Vb2 = (V2(rated) / V1(rated))Vb1
Zb1 = Vb1/Ib1 = Vb12/Sb Sb = Vb1Ib1
Zb2 = Vb2/Ib2 = Vb22/Sb
Ib2 = Sb/ Vb2
Zb1 / Zb2 = Vb12/ Vb22 = Vb12/( V2(rated) / V1(rated))2 Vb12 = 1/(1/a)2 = a2
Zb1 = a2Zb2
Zpu1 = Rs+jXs/Zb1 = Rs+jXs/ a2Zb2 = (Rs/ a2) +(jXs/ a2)/Zb2 = R’s+jX’s/ Zb2
= (Rs/ a2 + jXs/ a2)1/ Zb2
or Zpu1 = Zpu2
Rs
jXs
V1(rated)
V2(rated)
N1
N2
Vb1
Vb1 = V1(rated) Sb = S(rated 1 phase)
Vb2 = V2(rated)
Zb1 = (Vb1)2/Sb Zb2 = (Vb2)2/Sb
I1 pu
V2 pu
Sb
Vb1
Vb2
V2(rated)
N1
Vb1 = V1(rated)
N2
Zpu
Vb1
Zpu = Rs+jXs/Zb1
Rs/a2 + jXs/a2
Vb2
I2 pu
V1 pu
Zpu
V1(rated)
Zpu
Vb2
1:1
Sb
Vb1
Zpu
I1 pu
V1 pu
Vb2
I2 pu
Sb
V2 pu
Sb = S1(rated) Vb2 = V2(rated)
Zb1 = (Vb1)2/Sb Zb2 = (Vb2)2/Sb
Zb2 = Zb1/a2
Zpu2 = Rs/a2 + jXs/a2 =1/a2(Rs+jXs)(1/Zb2) =
1/a2(Rs+jXs)(a2/Zb1) = Zpu
Transformation between two bases:
ZL
Selection 1:
Sb1 = SA Vb1 = VA
Then:
Zb1 = (Vb1)2 / Sb1 Zpu1 = ZL/ Zb1
Selection 2:
Sb2 = SB Vb2 = VB
Then:
Zb2 = (Vb2)2 / Sb2 Zpu2 = ZL/ Zb2
Zpu2 / Zpu1 = ZL / Zb2 x Zb1 / ZL = Zb1 / Zb2
= (Vb1)2 / Sb1 x Sb2 / (Vb2)2
Zpu2 = Zpu1 (Vb1 / Vb2)2 (Sb2 / Sb1)
“1” = old
“2” = new
Zpu (new) = Zpu (old) (Vb (old) / Vb (new))2 (Sb (new) / Sb (old))