EE 369
POWER SYSTEM ANALYSIS
Lecture 9
Transformers, Per Unit Calculations
Tom Overbye and Ross Baldick
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Announcements
• For lectures 8 to 10 read Chapter 3
• Homework 7 is 5.8, 5.15, 5.17, 5.24, 5.27,
5.28, 5.29, 5.34, 5.37, 5.38, 5.43, 5.45; due
10/22.
• Homework 8 is 3.1, 3.3, 3.4, 3.7, 3.8, 3.9, 3.10,
3.12, 3.13, 3.14, 3.16, 3.18; due 10/29.
• Homework 9 is 3.20, 3.23, 3.25, 3.27, 3.28,
3.29, 3.35, 3.38, 3.39, 3.41, 3.44, 3.47; due
11/5.
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Per Unit Change of MVA Base
Parameters for equipment are often given using
power rating of equipment as the MVA base
To analyze a system all per unit data must be on
a common power base
NewBase
Z OriginalBase

Z

Z
pu
actual
pu
Hence ZOriginalBase

pu
ZOriginalBase

pu
2
Vbase
/
OriginalBase
S Base
NewBase
S Base
OriginalBase
S Base
2
Vbase
NewBase
S Base
NewBase
 Z pu
NewBase
 Z pu
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Per Unit Change of Base Example
•A 54 MVA transformer has a leakage
reactance of 3.69% (on its own MVA base).
•What is the reactance on a 100 MVA base?
100
X e  0.0369 
 0.0683 p.u.
54
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Transformer Reactance
Transformer reactance is often specified as a
percentage, say 10%. This is a per unit value
expressed as a percentage on the power base
of the transformer.
Example: A 350 MVA, 230/20 kV transformer
has leakage reactance of 10%. What is p.u.
value on 100 MVA base? What is value in
ohms (230 kV)?
100
X e  0.10 
350
 0.0286 p.u.
2
230
0.0286 
 15.1 
100
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Three Phase Transformers
•There are 4 different ways to connect 3f
transformers
D-D
Y-Y
Usually 3f transformers are constructed so all windings
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share a common core
3f Transformer Interconnections
D-Y
Y-D
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Y-Y Connection
Magnetic coupling with An/an, Bn/bn & Cn/cn
VAn
VAB
IA 1
 a,
 a,

Van
Vab
Ia a
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Y-Y Connection: 3f Detailed Model
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Y-Y Connection: Per Phase Model
Per phase analysis of Y-Y connections is exactly the
same as analysis of a single phase transformer.
Y-Y connections are common in transmission systems.
Key advantages are the ability to ground each side
and there is no phase shift is introduced.
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D-D Connection
Magnetic coupling with AB/ab, BC/bb & CA/ca
VAB
I AB 1 I A 1
 a,
 ,

Vab
I ab a I a a
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D-D Connection: 3f Detailed Model
To use the per phase equivalent we need to use
the delta-wye load transformation
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D-D Connection: Per Phase Model
Per phase analysis similar to Y-Y except impedances
are decreased by a factor of 3.
Key disadvantage is D-D connections can not be
grounded; not commonly used.
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D-Y Connection
Magnetic coupling with AB/an, BC/bn & CA/cn
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D-Y Connection V/I Relationships
V AB
V AB
a
 Van , also Vab  3 Van 30
Van
a
V An 30
V AB 30
Hence
Vab  3
and Van  3
a
a
For current we get
I AB 1
  I a  a I AB
Ia a
I A  3 I AB   30  I AB
1
I a  a
I A30
3
1

I A30
3
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D-Y Connection: Per Phase Model
Note: Connection introduces a 30 degree phase shift!
Common for transmission/distribution step-down since
there is a neutral on the low voltage side.
Even if a = 1 there is a sqrt(3) step-up ratio
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Y-D Connection: Per Phase Model
Exact opposite of the D-Y connection, now with a
phase shift of -30 degrees.
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