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EE2022 Electrical Energy Systems
Lecture 11: Transformer and Per Unit Analysis
17-02-2012
Panida Jirutitijaroen
Department of Electrical and Computer Engineering
2/9/2012
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Detailed Syllabus
20/01/2012
20/01/2012
27/01/2012
27/01/2012
30/01/2012
03/02/2012
03/02/2012
06/02/2012
10/02/2012
10/02/2012
13/02/2012
17/02/2012
17/02/2012
27/02/2012
02/03/2012
02/03/2012
2/9/2012
Three-phase circuit analysis: Introduction to three-phase circuit. Balanced three-phase systems.
Three-phase circuit analysis: Delta-Wye connection, Relationship between phase and line quantities
Three-phase circuit analysis: Per-phase analysis, Three-phase power calculation
Guest Lecture “Energy & Environment, Smart Grid & Challenges Ahead” by Prof. J Nanda (IIT Delhi,
IEEE Fellow)
Generator modeling: Simple generator concept
Generator modeling: Equivalent circuit of synchronous generators
Generator modeling: Operating consideration of synchronous generators, i.e. Excitation voltage
control, real power control, and loading capability
Generator modeling: Principle of asynchronous generators
Transmission line modeling: Overhead VS Underground cable
Transmission line modeling: Four basic parameters of transmission line
Transmission line modeling: Long transmission line model, Medium-length transmission line model,
Short transmission line model
Transmission line modeling: Operating consideration of transmission lines i.e. voltage regulation,
line loadability, efficiency
Transformer and per unit analysis: Principle of transformer, Single-phase transformer
Transformer and per unit analysis: Single-phase per unit analysis
Transformer and per unit analysis: Three-phase transformer, Three-phase per unit analysis
Review : if time permits.
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Learning outcomes
Outline
Reference
IN THIS LECTURE
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Learning Outcomes
• Formulate equivalent circuits of various
components in electrical energy systems.
– Equivalent circuit of transformer
• Explain basic operations of different
components in electrical energy systems.
– Short circuit/ open circuit test
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Outline
• Fundamental concept of transformer
• Single Phase Transformer
– Ideal Transformer
– Reflected load
– Maximum power transfer
– Practical Transformer
• Transformer operation
– Short circuit test
– Open circuit test
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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References
•
Glover, Sarma, and Overbye, “Power System Analysis and
Design”.
–
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Chapter 3
EE2022: Three-phase circuit by P. Jirutitijaroen
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Magnetic flux
Electromagnetic induction
Dot notation
Ampere’s Law
Faraday’s Law
FUNDAMENTAL CONCEPT OF
TRANSFORMER
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Magnetic Flux
• DC source  Constant
magnetic flux
• AC source  Varying
magnetic flux
DC
/AC
What will happen if we have
another coil to link the
varying magnetic flux?
Source:
http://www.lanl.gov/news/i
ndex.php/fuseaction/1663.
article/d/20085/id/13276
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Electromagnetic Induction
• Recall Faraday’s law:
• When we link Coil 2 to the magnetic flux generated by coil
1, if the flux is varying, there will be induced electromotive
force (EMF) at Coil 2. The voltage, V2, will be generated by
the magnetic force across wire.
Source
http://yourelectrichome.b
logspot.com/2011/07/intr
oduction-to-coupledcircuit.html
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Dot Notation
• The direction of induced EMF depends on the direction of magnetic
flux i.e. location that the coil links magnetic flux. Dot notation is
used to indicate the direction of current out of Coil 2 in the
equivalent circuit.
●
●
+
-
-
●
●
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
+
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Magnetic Core
• We can better link the magnetic flux by using magnetic
core.
• Magnetic flux “Ф” is now confined in the core and links
both windings.
Note that ‘N’ refers to
number of turns.
We are now interest to relate V1 and V2, and relate i1 and i2.
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Faraday’s Law
• Recall that:
• Let
• Then
  2 cost 
e  2 N sin t   2 N cost  90
• We can write the above equation in a Phasor form.
E  N  j 
• Since magnetic flux
  BA ,
– B = flux density (Weber/m²) , A = cross-sectional area (m²).
• We can write:
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E  N  j BA
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Faraday’s Law
• For ideal transformer, we assume that the flux linkage
at coil 1 and coil 2 is the same i.e. there is no flux
linkage loss.
• We can now find relationship between the voltage at
two sides of the transformer as follows.
V1  N1  j   N1  j BA
V2  N 2  j   N 2  j BA
V1 N1

V2 N 2
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Ampere’s Law
• “Current passing through a conductor creates magnetic field
around it ”
Hdl

I
enclosed

•
•
•
•
B = μH
B = Magnetic flux density (Weber/m² or Tesla)
H = Magnetic field intensity (A/m)
μ = Magnetic core permeability (H/m)
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Ampere’s Law Applied to Transformer
Flux
Flux
• “Magnetic flux along the path equals the net current
enclosed by the path”
Hlpath  I enclosed
Hlpath  i1N1  i2 N 2
Bl path
i₁N₁
-i₂N₂

 i1 N1  i2 N 2
Magnetic permeability
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Magnetic Core Permeability
• Magnetic core permeability represent the ‘resistance’ that the
magnetic core will allow the magnetomotive force to pass
through.
• For ideal transformer, the ideal value of the permeability is
infinity.
• We can now see the relationship of the current from both
sides of the transformers.
Bl path

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 i1 N1  i2 N 2
i1 N1  i2 N 2
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Ideal transformer
Reflected load
Impedance matching
Practical transformer
SINGLE PHASE TRANSFORMER
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Ideal Transformer
Primary side
• Assumptions:
1. No resistance in both
windings.
2. No leakage flux around
the core.
3. No core resistive loss.
4. Core permeability is
infinite.
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V1 N1

V2 N 2
Secondary side
i1 N1  i2 N 2
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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An Ideal Transformer Model
• We represent an equivalent circuit of an ideal transformer
as shown below.
a:1
• Define turn ratio as:
N1
a
N2
• From Faraday’s and Ampere’s Law:
 N1 
 N1 
V2  aV2
V1  
i2   i1  ai1
 N2 
 N2 
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Complex Power
• Complex power at primary side,
*
 I2 
*
S1  V I  aV2    V2 I 2  S2
a
*
1 1
• is the same as the complex power at secondary side.
• This means that ideal transformer has no real/reactive
power losses.
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Example 1
• From the circuit below, what is the current at the
secondary and primary side?
10:1
100
V
(rms)
V1
V1
 10  V2 
 10 V
V2
10
Z = 100 Ω
V2
i2 
 0.1 A
100
i2
i1   0.01 A
a
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Reflected Load
• We can reflect the load from one side of a transformer to the
other side of a transformer.
• This trick allows us to combine the two separate
primary/secondary circuits for easy(?) calculation.
V1 aV2
2 V2
Z1  
a
 a2Z2
i1 i2
i2
a
a:1
Interest
to find
reflected
load “Z₁”
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Z₂
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
Z₁
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Impedance Matching
• When a voltage source V with internal resistance Rs is connected to
a load R, the amount of power at the load depends on the value of
load resistance R.
• Maximum power transfer occurs when R = Rs.
• In the case that we need to connect the voltage source to a load
that does not satisfy the above condition, we can use transformer
to match impedance for maximum power transfer.
• To find an appropriate transformer, we let Rs = a²R and find a
transformer turn ratio.
Rs a²R
Rs
~
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R
When
Rs ≠ R
a:1
~
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
R
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Ideal VS Practical Transformer
Ideal transformer
Practical transformer
1. Zero resistance in the both
windings.
2. No leakage flux around the
core.
3. No core resistive loss.
4. Core permeability is infinite
1. Winding losses (copper
losses) represented as
resistance in both windings.
2. Leakage flux around the core.
3. Core resistive losses
(hysteresis loss + eddy
current loss)
4. Magnetic core permeability is
finite.
How can we represent this effect in the circuit?
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Finite Magnetic Core Permeability
• For practical transformer,
Bl path

 i1 N1  i2 N 2
• Recall that
E1  N1  j BA ,
 N2 
i1 
  i2
N1  N1 
Bl path
 N2 
E1
i2
i1 
 
j   N1 
• We call this ‘Magnetizing current’.
• The ratio between the voltage across the coil (E₁) and magnetizing
current can be written as jω(..). Thus we use an inductor to
represent the effect of finite magnetic core permeability in the
equivalent circuit.
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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A Practical Transformer Model
Copper Leakage
losses reactance
Iron losses
(core)
Reflected copper losses and leakage
reactance of secondary winding.
Magnetizing
susceptance
Note that in Chapter 3 [Glover, Sarma, and Overbye, “Power System Analysis and
Design”], the core losses are represented as ‘shunt admittance’, Y = G –jB where G
and B is positive. The imaginary part is negative to represent inductive property.
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A Simplified Model
• Z₁ and Z₂ are series impedances representing the resistive loss and
flux linkage loss in the two windings.
• Y is a shut admittance representing iron core loss and magnetizing
susceptance.
• Typically Y is very small i.e. resistance is very large. This means that
the currents flowing through Z₁ and a²Z₂ are almost the same. We
can simply combine Z₁ and a²Z₂ to “Zeq”, the equivalent series
impedance.
a²Z₂
Z₁
Y
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Zeq = Z₁+a²Z₂
a:1
simplified
a:1
Y
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Transformer Parameter Tests
Short circuit test
Open circuit test
• To find equivalent series
impedance.
• Short circuit the secondary
side.
• Apply rated current at the
primary side.
• Measure real power and
voltage at the primary side.
• To find equivalent shunt
admittance.
• Open circuit the primary side.
• Apply rated voltage at the
secondary side.
• Measure real power and
current at the secondary side.
Zeq
P₁, V₁
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~
Y
a:1
Zeq
Y
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
a:1
~
P₂,I₂
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Example 2: Short Circuit Test
• Consider a single-phase 20kVA, 480/120 V 60 Hz transformer.
During short circuit test, rated current is applied to the
primary side. The voltage of 35 V and real power of 300 W are
measured. Find equivalent series impedance of this
transformer.
S
20  103
I1,rated 
Zeq = Req + jXeq
P₁ = 300 W
V₁ = 35 V
~
Y
480:120
Req 
Z eq 

rated
V1,rated
P1
I1,rated
V1
I1,rated
2
480
 41.667 A

300
 0.1728 
2
41.667

35
 0.84 
41.667
X eq  0.842  0.17282  0.822 
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Example 3: Open Circuit Test
• Consider the same transformer as Example 2. During
open circuit test: rated voltage applied to secondary
side, then I₂ = 12 A and P₂ = 200 W. Find equivalent
shunt admittance Y of this transformer.
Zeq
Y = G-jB
V2,rated  120 V, V1,rated  480 V
480:120
~
P₂ = 200 W Geq  P1 2  2002  0.000868 S
480
V1,rated
I₂ = 12 A
Y 
I1
V1,rated
I2 a
12 4


 0.00625 S
V1,rated 41.667
B  0.00625 2  0.000868 2  0.00619 S
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Saturation
• In practical transformer model, we
assume constant core permeability
and linear relationship between B
and H follows.
• In fact, the B-H curve for
ferromagnetic materials used for
transformer core is nonlinear and
has multiple values.
• As H increases, the core become
saturated i.e. the magnetic flux
density B increase at a much lower
rate.
• This effect is NOT included in the
equivalent circuit.
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B-H curve is approximated by a
dashed line.
EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
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Summary
• A turn ratio (a:1) is the ratio between the number of
turns on the primary side of transformer and that
on the secondary side.
• A load on the secondary side of transformer can be
reflected to the primary side of the transformer.
• We can use transformer for impedance matching by
choosing the turn ratio that makes reflected load
equal to internal resistance of voltage source.
• A practical transformer contains series impedance
and shunt admittance.
– Series impedance represents winding losses and flux
leakage losses
– Shunt admittance represents iron core losses and
magnetizing susceptance.
Zeq = Z₁+a²Z₂
• Short circuit test is used to find seires impedance by
short circuit the secondary side.
• Open circuit test is used to find shunt admittance by
open circuit the primary side.
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EE2022: Transformer and Per Unit Analysis by P. Jirutitijaroen
a:1
Y
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Next Lecture
• Single-phase per unit analysis
– We use this analysis to eliminate transformer
model in the circuit.
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