Calculation of Short Circuit Reactance and Electromagnetic Forces in Three
Phase Transformer by Finite Element Method
1
S. Jamali2 , M. Ardebili1 , K. Abbaszadeh1
Assistant professor the electrical engineering, Department of K.N. Toosi University, Tehran, Iran
2
M.S Student of the electrical engineering, Department of K.N. Toosi University, Tehran Iran
abbaszadeh@eetd.kntu.ac.ir
current density of them and permeability of transformer
core. Then this model is divided to triangular elements.
By using magneto static analysis of the finite element
method, the magnetic vector potential of three nodes of
each triangular element is calculated and therefore the
flux distribution over the model is obtained.
Then, the flux density of each element is evaluated.
Because of this reason that magnetic vector potential of
each element is considered as a linear function of x and y,
the flux density of each element becomes a constant
value.
After this step, leakage reactance of transformer can
be calculated by using several post processing methods
such as magnetic energy method.
A model of total three phase transformer is also
considered and since the magnetic vector potential of
each node is evaluated, the self inductances and mutual
inductance of coils is calculated by a post processing
method and then leakage inductance of each winding is
calculated.
The accuracy of these post processing methods is
verified by comparison with experimental result.
For calculation of radial and axial electromagnetic
forces upon the transformer coils, two components of
leakage flux density in x and y directions are calculated.
By analysis of leakage flux distribution over the
transformer window, it can be realized that the
asymmetrical LV and HV windings can result in
asymmetrical forces.
Abstract: A new and simple procedure for
determination of leakage reactance and analysis of
electromagnetic forces that acting upon the transformer
coils is presented in this paper. Before manufacturing
and within the design process, it is required that we
model the transformer and analyze the transformer
condition using this model.
The analysis of
electromagnetic forces is essential for mechanical
considerations. This study is accomplished by using
two dimensional planar models utilizing the finite
element method. Just by modeling the transformer
window and using FEM, the magnetic vector potential
is calculated over each node. Then by using three post
processing procedures, the leakage reactance of
transformer coils is calculated and the results are
verified by comparison with experimental result. The
radial and axial electromagnetic forces are calculated
over the transformer coils and the effect of
asymmetrical of winding is analyzed.
I. INTRODUCTION
Electrical machines are usually represented, in
electrical engineering, by means of their equivalent
circuits. Accurate calculation of short circuit reactance
of transformer is essential for modeling of transformers.
However, the calculation of leakage reactance of
transformer coils is performed in many papers by using
different analytical and numerical methods [1] - [6], but
most of the analytical methods are not accurate,
especially when the axial length of HV and LV windings
are not equal, and in several of numerical methods the
whole or a half of a three phase transformer is modeled.
In this paper we determine leakage reactance of
transformer by using FEM and just by modeling the half
of transformer window for a three phase core type
transformer and therefore the required time for our
calculation is decreased.
Furthermore the forces that acting upon the transformer
coils have to be evaluated before manufacturing and then
required mechanical support considerations must be
performed during manufacturing process. For these
reasons, evaluation of magnetic field distribution of
transformer is essential for calculation of electromagnetic
forces and number of useful electric parameters such as
leakage reactance. For our purpose, just the modeling of
transformer window is adequate and therefore an
appropriate model of transformer window is defined
considering the construction and position of coils and the
II. MODELING AND FORMULATION
A. Model definition
The transformer that we considered for our studying
is a 30MVA, (63/20) KV, YnD three phase core type
power transformer that its HV winding has 480turns and
its LV winding has 264turns.
A two dimensional model of the window of this
transformer is defined considering construction and
dimensions of coils of this transformer, as shown in
Fig.1.
The boundary condition of this model is homogeneous
numan boundary condition over the external rectangle
and means that the flux lines come to limbs and yokes
vertically. This is reasonable due to this fact that the
relative permeability of core and yoke is very larger than
that of air and copper.
1725
υ : The reluctivity i.e. the inverse of magnetic
permeability ( µ ).
And,
A: is magnetic vector potential.
For 2D models in x-y plane, the non-zero component
of A is the z component of magnetic vector potential
which is function of x and y only.
Therefore the equation (4) takes the following scalar
form:
∂
∂A
∂
∂A
(υ
) + (υ
) = Js
(5)
∂x ∂x
∂y ∂y
In [3] it is shown that the ‘core effect’ has negligible
effect on the leakage reactance calculation and therefore
modeling of transformer window is adequate for our
purpose.
1.2
numan
1
core
numan
0.8
numan
boundury
condition
yoke
0.6
Solving equation (5), magnetic vector potential can
be obtained and solving equation (3), magnetic flux
density can be calculated.
By using magneto static analysis of PDE TOOLBOX of
MATLAB software, the flux distribution for our defined
model is obtained, as shown in fig.2.
0.4
0.2
primary winding
secondary winding
numan
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Fig 1: transformer model
B. Discretization of the defined model
By using PDE TOOLBOX of MATLAB software,
the model is discretized by triangular elements.
In the following section, we determine the partial
differential equation that is required for leakage reactance
calculation and electromagnetic force analysis.
1.2
1
0.8
0.6
C. Formulation
In this section, the partial differential equation that
governs calculation of leakage reactance problem is
determined.
Amperes law states that:
∇× Η = J
0.4
0.2
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Fig.2. Flux distribution over the defined model for the transformer
model
(1)
Where:
H: magnetic field intensity
J: total current density
We will assume that H is only due to source currents
i.e. no permanent magnets are present.
1) Linear magneto static analysis: Current density J in
equation (1) is due to current sources i.e. current
densities of the transformer’s primary and secondary
winding.
We have following relation between magnetic field
intensity and magnetic flux density:
The magnetic vector potential over the defined model
is given as a three dimensional figure .3.
0.025
0.02
0.015
0.01
0.005
B = µ .H ⇒ H = υ .B
(2)
0
1.5
0.4
1
And the relation between magnetic flux density and
magnetic vector potential is:
B = ∇× A
(3)
0.3
0.2
0.5
0.1
0
0
Fig.3. Magnetic vector potential distribution
Hence:
∇ × (υ .∇ × A) = J
The distribution of magnetic flux density over the
defined model is given in the figure.4.
(4)
Where:
1726
Color: -ux Height: -ux
1.2
0.12
0.12
1
0.15
0.1
0.1
0.8
0.1
0.08
0.08
0.6
0.05
0.06
0.06
0.4
0
0.04
0.04
0.2
-0.05
1.5
0.02
0.4
1
0.2
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.1
0
0.3
0.02
0.3
0
0
Fig.4. Magnetic flux density distribution (Tesla)
Fig.7. Axial magnetic flux density distribution (Tesla)
Figs.5, 6and 7 are the three dimensional pictures of
absolute, radial and axial flux densities over the defined
model, respectively.
With a glance to these figures, it is clear that the
value of radial leakage flux density is much smaller than
that of axial leakage flux density.
D. Reactance calculation
The leakage reactance of a transformer arises from
this fact that all the flux produced by one winding does
not link the other winding. The magnitude of this leakage
flux is the function of the geometry and construction of
the transformer.
In this stage of analysis, we able to calculate leakage
reactance of transformer, by using two post processing
methods.
1) Magnetic energy storage method: Due to this fact that
we considered the magnetic vector potential of each
triangular element as a linear function of x and y,
therefore the radial and axial component of magnetic flux
density and therefore absolute value of B of each
triangular element become fix values, as shown in
following equations.
Color: abs(B) Height: abs(B)
0.12
0.2
0.1
0.15
0.08
0.1
0.06
0.05
0.04
0
1.5
0.4
1
0.3
0.02
A = C1 + C 2 x + C 3 y
0.2
0.5
0.1
0
Bx =
Fig.5. Absolute magnetic flux density distribution (Tesla)
0.04
0.05
0.02
0
-0.01
-0.02
0.4
1
0.3
0.2
0.1
(7)
(8)
(9)
Where, C1, C2 and C3 are fix coefficients. Bx and By
are radial and axial component of magnetic flux
densities.
After the calculation of absolute value of B for each
element, the magnetic energy that stored in window
space can be calculated by using two following formulas.
1
W = ∫∫ B.H .dxdy
(10)
2
1
W = ∫∫ J . A.dxdy
(11)
2
From these two formulas, the former is more accurate
than latter. In the former equation, the integration is
performed on the whole window space whereas in the
0.01
0
-0.05
1.5
∂A
∂x
B = B x2 + B y2
0.03
0.5
∂A
∂y
By = −
Color: uy Height: uy
0
(6)
0
-0.03
-0.04
0
Fig.6. Radial magnetic flux density distribution (Tesla)
1727
The self inductance of the primary winding, for state that
the primary winding is excited by magnetizing current
and the secondary winding is open circuit, as shown in
Fig.8, can be calculated using the following equation:
Aavg1 − Aavg 2
L p = N 2p
(16)
I
Where:
Np: is the turn number of primary winding and I is
defined as following:
(17)
I = ∫∫ J .ds
latter equation, integration must be performed only on
the winding area.
Once the magnetic energy is calculated, leakage
reactance of transformer for each phase referred to
primary can be calculated using following formula.
(4 × π × f × W × t )
Xl =
(12)
i 2p1 + i s'21
Where:
Xl: is leakage reactance of transformer referred to
primary side (per phase).
f: is the supply frequency.
W: is the magnetic energy that calculated using (10) and
(11) equations.
t: is the depth of our defined model.
In other words, I is the ampere-turn of primary
winding.
Aavg1 and Aavg 2 are the average values of magnetic
i p1 : is the instantaneous current of one phase of primary
vector potential of two sheets of primary winding of one
phase, respectively.
The mutual inductance of primary winding and
secondary winding can be calculated by the following
equation:
Aavg 3 − Aavg 4
M ps = N p N s
(18)
I
Where:
Ns is number of turns of secondary winding.
Aavg 3 and Aavg 4 are the average values of magnetic
'
winding and i s1 is the instantaneous current of the same
phase of secondary winding (referred to primary
winding).
2) Linkage and mutual flux method: The leakage
reactance of transformer can be calculated by using
linkage and mutual flux calculations.
In this method, the whole three phase transformer is
modeled and A=0 is set as boundary condition on the
rectangle that enclose the three phase transformer model.
Firstly the primary winding of three phase transformer is
excited and the secondary winding is open circuited.
Then the secondary winding is excited and the primary
winding is open circuited.
In each state, the linkage flux of exited winding and
the mutual flux of excited winding with other open
circuit winding can be calculated using following
procedure.
The linkage flux of each winding can be calculated
using the following formula:
(13)
ψ = ∫∫ B.ds = ∫ A.dl
Each phase of each winding modeled as two same
sheets; therefore if we apply linear integration on each
winding for one phase, considering this fact that the
nonzero component of magnetic vector potential is z
component, we can write the linear integration as the
following form:
(14)
ψ = A1 − A2
Where: A1 and A2 are the magnetic vector potential
value of two sheets of each winding of one phase,
respectively.
Each phase of each winding is considered as two
sheets with uniform current densities; therefore the
average value of magnetic vector potential for each sheet
can be calculated as following equations:
∫∫ A.ds
Aavg (i ) =
(15)
S
For each sheet, the above integration must be
performed on the sheet area where S is the area of sheet.
vector potential of two sheets of secondary winding of
the same phase, respectively.
I in this case is the product of secondary turn number
and Since the Lp and Mps are calculated, the leakage
inductance the primary current.
of primary winding can be calculated using the
following relation:
l p = L p − M ps
(19)
2.5
2
1.5
1
0.5
0
-0.5
0
0.5
1
1.5
2
2.5
3
Fig.8. Core flux distribution for the case that primary winding is excited
and secondary winding is open circuit
The leakage inductance of the secondary winding can
be calculated in a same procedure.
1728
The self inductance of the secondary winding, for
state that the secondary winding is excited and the
primary winding is open circuit, can be calculated using
the following equation:
Aavg 3 − Aavg 4
Ls = N s2
(20)
I
The mutual inductance of secondary winding and
primary winding can be calculated by the following
equation:
Aavg1 − Aavg 2
M sp = N s N p
(21)
I
Since the Ls and Msp are obtained, the leakage
inductance of secondary winding can be calculated using
the following relation:
(22)
l s = L s − M sp
to withstand such forces, but this is usually not validated
by tests, and accurate calculations are essential. Utilities
are becoming increasingly concerned with this, and now
often require documentation from manufacturers that
their transformers are designed to withstand short
circuits, based on computer programs that they both are
familiar with and have confidence in.
Since the current density of the coils is known,
Electromagnetic forces upon the transformer coils can be
calculated, if the radial and axial leakage flux density be
known.
We have these quantities from (7) and (8) equations.
Radial forces are dominant and due to interaction of
current and axial component of leakage flux density, as
shown in the following equation:
r r
(24)
F = ∫∫ ( J × B ).l.dx.dy
x
The equivalent leakage reactance of transformer
referred to primary can be calculated using following
equation:
Np 2
X l = 2πf (l p + (
) ls )
(23)
Ns
y
Where:
J: is current density of the coils.
l: is the depth of model.
The above integration must be performed on the
winding area. Radial forces usually produce tensile stress
in the outer winding and compressive stress in the inner
winding. Compressive stress can cause buckling, and the
winding must be properly supported by axial spacer bars.
The distribution of radial forces that acting upon the
transformer coils is shown in fig (10).
3) Comparison of results: The results obtained from these
methods are compared with the experimental result, as
shown in the table I.
The experimental leakage reactance of the under
study transformer in percent is 13%.
With a glance to this table, we can deduce that our
modeling and post processing methods are very accurate.
5
x 10
1.5
5
x 10
Table1: Experiment and Simulation Results of Leakage Reactance
Leakage reactance in
method
per unit
Percentage error
Experimental
method
0.13
0%
Energy storage
method using
0.1227761
0.7224%
equation (10)
Energy storage
method using
0.1227455
0.7254%
equation (11)
Linkage
And
0.1215231
0.8477%
Mutual flux method
2
1
1
0.5
0
0
-1
-0.5
-2
1.5
-1
0.4
1
0.3
0.2
0.5
-1.5
0.1
0
0
Fig.9. Distribution of radial forces that acting upon
the transformer coils
Axial forces due to interaction of current and radial
leakage flux density are usually compressive, as shown
in the following equation:
r r
(25)
F y = ∫∫ ( J × B x ).l.dx.dy
As shown in the table the results obtained from our
modeling is a little smaller than experimental result. This
little difference is reasonable due to this fact that the lead
of winding, that coming out from the transformer, is not
modeled in our modeling and thus the little of leakage
flux is not considered in our modeling. In high current
transformer the leakage flux of high current leads
become much larger and therefore for increasing the
accuracy, the high current leads must be considered in
the modeling.
E. Force Analysis
Forces at short circuit are calculated as flux density
time current time length. The windings must be designed
The above integration must be performed on the
winding area. These forces tend to bend the conductors in
axial direction, and their sum total act on the coilclamping ring and other clamping structures. The
distribution of axial forces upon the transformer coils is
shown in the figure.10.
The distribution of axial forces upon the LV winding
of transformer is not symmetrical due to especial shape
1729
energy storage method that presented in other papers is
also presented in this paper for result comparisons.
Taking into account the fact that the values obtained
using this modeling coincide with the experimental
result, the described method establishes an industrial
application to the modeling and design of three phase
transformer.
Electromagnetic forces that acting upon the
transformer coils is also analyzed in this paper. Radial
and axial electromagnetic forces on the coils is calculated
and shown as the diagrams.
The modeling that shown in this paper, allowing us to
know the transformer behavior before manufacturing
them and, thus reducing the design time and cost.
of flux distribution and flux diverting in the middle of
LV winding toward to core limb in the case that the axial
length of HV and LV winding is not equal. Note to Fig.2.
4
x 10
4
4
x 10
6
3
4
2
2
1
0
0
-2
-1
-4
-2
-6
1.5
-3
0.4
1
0.3
-4
REFRENCES
0.2
0.5
0.1
0
0
-5
[1]. J.Wang, A.F.Witulski, J.L.Vollin, T.K.Phelps, G.I.Gardwell,
Fig.10. Distribution of radial forces that acting upon
the transformer coils
”Derivation, Calculation and Measurement of Parameters for
a Multi–Winding Transformer Electrical Model”, IEEE
Transaction, pp. 220-226, 1999.
Where ampere-turns are perfectly balanced so that the
leakage flux pattern is symmetrical, then the leakage field
is axial over the major part of the coil height. But since
the flux lines dispersing in the radial direction in the
vicinity of the winding ends, the axial flux density tends
to decrease, and the resultant flux density at the ends can
be resolved into the radial component causing axial
forces. Due to the core effect, these axial forces are
unequally distributed between the outer and inner
winding.
The axial forces at the top and bottom are in opposite
direction. In case the ampere-turns are perfectly balanced
and the leakage flux pattern is symmetrical, the resultant
forces on the winding would be zero. Any axial
displacement between the magnetic centers of HV and
LV windings will result in a net axial force, tending to
increase the displacement even further. However, in
practice, a complete balance of all element of winding
can not be achieved entirely for a number of results like
provision to tapping, dimensional accuracy and stability
of windings, etc.
In case of that symmetrical winding arrangement in
axial direction having uniform current distribution, there
is no resulting force against the yoke, and the winding
tend to compressed in the axial direction only.
Different yoke distances and tapping in the main
windings and uneven current distribution in the axial
direction can cause the force integral to reach a final
value greater than zero.
[2]. J.E. Hayek, “Short-Circuit Reactance of Multi-Secondaries
Concentric Winding Transformers”, IEEE Transaction, pp.
462-465, 2001.
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Magot,
X.
Margueron,
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Keradec,”PEEC-Like
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Smajic,
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Madzarevic,
S,
Berberevic,”Numerical
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[10]. Martin J. Heathcote,”J&P Transformer”, Reed Educational
III. CONCLUSION
and Professional publishing Ltd, 1998.
A new model and a new post processing algorithm
for calculation of short circuit reactance, within the
design process, is presented in this paper. Magnetic
1730