ELEKTROTEHNIŠKI VESTNIK 92(3): 115-122, 2025 
IZVIRNI ZNANSTVENI ČLANEK 
 
A Comprehensive Study of Transformer Winding Losses Using 
the Finite Element Method 
Peter Kosmatin
1
, David Nedeljković
2
 
1
Kolektor Etra, Šlandrova ulica 10, 1231 Ljubljana – Črnuče 
2
 Univerza v Ljubljani, Fakulteta za elektrotehniko, Tržaška 25, 1000 Ljubljana, Slovenija 
E-pošta: davidn@fe.uni-lj.si 
 
Abstract. The paper presents a study of the transformer winding losses caused by eddy and circulating currents 
using the finite element method. The losses are evaluated using different transformer simulation models and by 
including the phase interaction effect. The increase in the winding losses is also evaluated when the transformer is 
exposed to geomagnetically induced currents. A new calculation method using a 2D axisymmetric geometry is 
proposed. In comparison with a 3D simulation model based on detailed manufacturing data for 14 power 
transformers, the proposed procedure demonstrates an acceptable level of accuracy, achieved within a short 
computational time. Consequently, this method can be used as a valuable design tool for optimising the 
transformer geometry. 
 
Keywords: Eddy currents, circulating currents, geomagnetically induced currents, finite element analysis, power 
transformers 
 
 
Študija izgub v navitjih transformatorja z uporabo metode 
končnih elementov 
V prispevku je predstavljena študija izgub v navitjih 
transformatorja, ki jih povzročajo vrtinčni in krožni tokovi, z 
uporabo metode končnih elementov. Izgube so ovrednotene z 
uporabo različnih simulacijskih modelov transformatorjev in z 
vključitvijo učinka fazne interakcije. Povečanje izgub v 
navitju se je ocenilo tudi za primere, ko je transformator 
izpostavljen geomagnetno induciranim tokovom. Posledično 
je predlagana nova metoda izračuna z uporabo 2D osno- 
simetrične geometrije. Za ovrednotenje metode je bila 
opravljena primerjava njenih rezultatov z rezultati 3D 
simulacijskega modela, zgrajenega na podlagi podrobnih 
proizvodnih podatkov za 14 projektiranih in izdelanih 
energetskih transformatorjev. Primerjava kaže, da predlagana 
metoda zagotavlja sprejemljivo natančnost s kratkim 
računskim časom, zato se lahko uporablja kot učinkovito 
načrtovalsko orodje za optimizacijo geometrije 
transformatorja. 
 
Ključne besede: Vrtinčni tokovi, geomagnetno inducirani 
tokovi, krožni tokovi, metoda končnih elementov, energetski 
transformatorji 
 
1 INTRODUCTION 
Due to the longevity of power transformers, their high 
operating efficiency is of paramount importance. As 
transformer power and voltage ratings increase, the 
accurate estimation of losses during the design process 
becomes increasingly crucial. To ensure 
competitiveness on the global market, transformer 
manufacturing companies must optimise their designs 
within technical and technological limitations [1, 2]. 
This involves optimising not only the quantity of 
installed materials but also the mathematical models 
employed to predict the parameters of the constructed 
transformer. The key parameter guaranteed by the 
manufacturer after signing a construction contract with 
a customer is the value of the short-circuit losses during 
a normal transformer operation. 
High-power transformer design incorporates an 
iterative geometry optimisation procedure, using a 
previously manufactured transformer with similar 
characteristics as a starting point. Although some design 
optimisation algorithms have been found among the 
published papers [3-7], the product uniqueness forces 
power transformer designers to rely on their experience 
and the established design methods within the company. 
With each modification of the transformer geometry, 
numerous calculations must be performed to verify the 
transformer compliance with the adopted technical 
specifications and the corresponding standards. Besides 
their accuracy, the calculation methods must be time-
efficient to optimise the design process. The paper 
addresses one of the calculation procedures and presents 
a basis for evaluating additional load losses on a given 
transformer geometry under various operating 
conditions. 
The load losses are the most dominant component 
Received: 20 March 2025 
Accepted: 21 April 2025 
 
Copyright: © 2025 by the authors 
Creative Commons Attribution 4.0 
International License 
 

116 KOSMATIN, NEDELJKOVIĆ 
with the values of approximately 80% of the dissipated 
high-power transformer electromagnetic losses. Their 
predominant portion is generated within the transformer 
windings, where losses can be categorised into two 
specific types: direct current (DC) ohmic losses and 
stray losses, the latter arising from leakage fields. The 
stray losses can be further divided into eddy current 
losses and circulating current losses present where the 
winding is constructed with parallel conductors to 
reduce the eddy current losses. Any additional losses, 
such as the skin-effect losses, are negligible in the total 
winding losses of a high-power transformer [8]. The 
first step in assessing the stray winding losses is to 
determine the leakage field distribution within the 
transformer windings. Typically,  an analytical approach 
is used that achieves a satisfactory balance between the 
accuracy and computational efficiency [9, 10]. 
Nowadays, when computers with a high computational 
power are available to transformer designers, the 
numerical finite elements method (FEM) for leakage 
field calculations can be included within the design 
procedure. The applicability of FEM in the transformer 
design is well documented in [11-14], however, a 
practical geometry meshing is required to achieve an 
acceptable computational speed, especially when the 
geometry is modelled in a 3D space. Bearing this in 
mind, this study focuses on finding a suitable winding 
stray losses calculation procedure using an FEM-based 
computer software (Ansys Maxwell [15], FEMM [16]). 
Such a procedure can then be used in the transformer 
design optimisation process. A comparison is made 
between various 2D and 3D transformer modelling 
possibilities, taking into account their accuracy and 
computational speed. The evaluation is made on a 
sample of 14 transformers of a rated power from 10 
MVA to 500 MVA. 
The winding losses are also calculated at the presence 
of geomagnetically induced currents (GIC) [17-19] 
which occur when the geomagnetic disturbance affects 
the distribution of the electric potential on the Earth 
surface, resulting in a flow of induced currents in low-
resistance paths that include transmission lines coupled 
to neutral-grounded transformer windings. For a 
transformer operating at 50 Hz, GIC with a low 
frequency (a few mHz) can therefore be treated as a DC 
source. The presence of GIC flowing through the 
transformer windings can lead to a high saturation of the 
core, resulting in the distortion of the stray magnetic 
field inside the transformer [20-23]. Consequently, the 
winding stray losses increase and the hot-spot 
temperature rises as well [24]. The simulated values of 
the injected GIC into the transformer model are based 
on the measurements [25], where the maximum GIC 
value of 67 A per phase is measured in the Finnish 
power system. 
 
2 STRAY LOSSES OF THE TRANSFORMER 
WINDINGS 
A time-varying leakage field is formed when the 
transformer windings are not perfectly coupled and 
some of the magnetic flux generated by the load 
currents is distributed within the windings and outside 
the core. While some leakage field is usually needed to 
limit the fault currents, it causes stray losses in the 
exposed conductive materials (windings, tank, clamps, 
etc.). 
 For the leakage field calculation, analytical methods 
have been developed [9]. They are commonly used in 
the initial transformer design. However, for an accurate 
determination of the magnetic field distribution, a 3D 
mathematical model is developed [26] and enhanced 
with FEM-based computer programs. 
 Using Ansys Maxwell as a field calculation tool, 
designers have several choices of how to construct the 
transformer model, either a 2D or a 3D model, with the 
former offering an axisymmetric or xy-plane option. 
With FEMM, a freeware program, only 2D modelling is 
supported. The options will be compared later, below in 
terms of their accuracy and computational speed. 
 
2.1 Eddy Current Losses 
When the distribution of the magnetic field for a given 
transformer geometry is known, the winding eddy 
current losses can be evaluated analytically. The 
classical approach can be used, where the conductor is 
described as an infinitely long strand (Figure 1). A 
strand with a rectangular cross-section is affected by 
external magnetic field H Y [9, 10]. By applying the 
Maxwell's equations, the magnetic field distribution in 
the strand obtained by the diffusion equation is: 
 
2
2
,

=  

yy
HH
xt
 (1) 
where µ and σ are the permeability and conductivity of 
the strand material, respectively. Due to the load current 
and GIC, the time and spatial dependence of external 
magnetic field H Y has the following form: 
 ( ) ( ) ( )
0
ˆ
,.

=+
jt
Y GIC
H x t H x e H x (2) 
Where 
0
ˆ
H is the peak value of the sinusoidal magnetic 
field waveform due to the load current, and H GIC is the 
magnetic field DC value due to GIC. For solving (1), 
the magnetic field boundary condition on either side of 
the conductor is: 
 
0
ˆ
,.
2


 = +


jt
y GIC
b
H t H e H (3) 
Due to the small dimensions of the conductor, peak 
sinusoidal magnetic field 
0
ˆ
H at the conductor centre can 
be used, or it can be calculated as a mean value between 
the readings on either side of the conductor. Therefore, 
magnetic field distribution H y within a conductor is: 
 

A COMPREHENSIVE STUDY OF TRANSFORMER WINDING LOSSES USING THE FINITE ELEMENT METHOD 117 
 
Figure 1. Rectangular strand in an external magnetic field with 
a y-component [9]. 
 
 ( )
( )
0
cosh
ˆ
,.
cosh
2

  
=+

  


jt
y GIC
jx
H x t H H e
b
j
 (4) 
The generated eddy current losses are then calculated 
using the ohmic loss component of the Poynting 
theorem: 
 
2
2
11
ˆ
.
2
=  = =

   eddy
V V V
P dv dv J dv J E J (5) 
Based on the presented electromagnetic problem in 
Figure 1 with current density J having only the z-
component, the eddy current losses per unit length (l) in 
the z-direction are expressed as: 
 ( )
( ) ( )
( ) ( )
2
2
2
2
2
2
0 2
2
2
2
0
2
ˆ
sinh
2 cosh
2
ˆ
sinh sin
.
cosh cos
b
eddy y
b
b
jt
b
PH
c
dx
lx
c j H e
j x dx
b
j
qb qb c q H
qb qb
+
−
+ 
−

==

 
=   =

  


 − 
=

+




 (6) 
Where q is the reciprocal value of the skin depth: 
 ,
2
  
= q (7) 
while c and b are the strand dimensions. Besides the 
skin depth, the argument in the trigonometric and 
hyperbolic functions is the strand dimension 
perpendicular to the direction of external magnetic field 
H Y. The eddy current losses are commonly expressed 
per a unit volume and written separately for the axial 
0,
ˆ
z
H and radial 
0,
ˆ
r
H component of the magnetic field 
(polar coordinate system): 
 
( ) ( )
( ) ( )
( ) ( )
( ) ( )
2
, 0,
2
, 0,
ˆ
sinh sin
;
cosh cos
ˆ
sinh sin
.
cosh cos
eddy z z
eddy r r
P q H qc qc
V c qc qc
P q H qb qb
V b qb qb
  −
=

  +


  −
=

  +


 (8) 
Even though the external magnetic field has an 
additional DC component due to GIC (2), the eddy 
current losses depend only on the time-varying 
magnetic field component. To eliminate the DC bias, it 
is necessary to obtain a half of the peak-to-peak 
magnetic field affecting the conductor. Therefore, the 
equation (8) can be used. The total winding eddy current 
losses can then be calculated as a sum of all (N) 
individual conductor losses: 
 ( ) ( )
,,
1
N
eddy r eddy z
eddy
i
PP
P i i
VV
=

=+



. (9) 
For the windings with a low number of turns that carry a 
high-load current, there is also a considerable azimuthal 
component of eddy current losses [27]. Based on the 
given reference, the azimuthal eddy current losses can 
be neglected for the transformers of the disk type or the 
multiple-layer type coils. The 14 transformers, for 
which the additional winding losses are evaluated, are 
constructed with these coil types. Consequently, the 
azimuthal eddy current losses are neglected. 
 Another aspect to consider when modelling the 
transformer and calculating the magnetic field is the 
interaction between the phases. Leakage magnetic field 
H i imposed on the conductor is the vector sum of the 
fields formed by the adjacent phases: 
 
2
j
j 3
00
2
22
jj
22
33
2 0
2 cosh j
3
2
2 cos 1
3
ˆˆ
ˆ
1 1 ,
ˆ
t
t
i
i
H H e a H e
H
a e e a a a
H
 






 
=


 
=  =


=  +  




= −  + + = − +





 (10) 
where a is the magnetic leakage field modification 
parameter due to the phase interaction. Its value 
depends on the specific position of the conductor and 
varies around its circumference. As the conductor is 
affected by the cosine function of the magnetic leakage 
field, the peak radial field value would be expected at a 
0° phase angle along the entire length of the conductor. 
However, due to the interaction of the adjacent phase, 
the phase angle of the peak magnetic leakage field 
varies around the circumference of the conductor, as 
shown in the example, given in Figure 2. Therefore, to 
increase the accuracy of the eddy current losses 
calculation, the winding must be segmented along its 
circumference, and the equation (8) can be used 
correctly. 
 
2.2 Circulating Current Losses 
The value of the ohmic and eddy current losses is highly 
dependent on the dimensions of the conductors that 
carry the high current. In order to reduce the ohmic 
losses, it is therefore necessary to divide the conductor 
into parallel strands, while maintaining the overall 
cross-section of the conductor. As a result, the 
circulating currents can occur if there is no proper 
transposition between the parallel conductors. 
z
y
x
c
+b/
H
Y
H
Y
2
-b/2

118 KOSMATIN, NEDELJKOVIĆ 
 
Figure 2. Radial component of the magnetic leakage field at a 
given phase angle around a particular conductor circumference 
(example). 
 
 An ideal transposition ensures that the parallel 
conductors share the same amount of the magnetic 
leakage field and therefore there is no current 
circulation. Analytical [28, 29] or numerical [9, 30] 
methods can be used to evaluate the circulating current 
losses with a particular transposition scheme. In high-
power transformers, the circulating losses are usually 
focused to the low-voltage winding, and even there only 
if a continuously transposed cable is not used. 
 In this paper, the currents through the n p parallel 
conductors with a transposition scheme shown in 
Figure 3 are numerically calculated. The winding 
scheme shown with n p-1 transpositions is relatively easy 
to construct and it the windings are assumed to be 
placed centrally on the limb. 
 A transformer with parallel conductors is modelled in 
a 2D space according to the used transposition (Figure 
4). The short-circuit test is analyzed with a rated current 
excitation on the high-voltage winding, and the 
circulating current losses are calculated. 
 
3 RESULTS AND DISCUSSION 
The simulations are carried out on a set of 14 
transformers, with their data provided in Table 1. The 
values are provided by the power transformer 
manufacturer along with a detailed geometry and 
material specification. 
 
3.1 Eddy Current Losses of the Transformer 
Windings 
To calculate the winding eddy current losses, a high-
power five-limb transformer (Transformer no. 12 in 
Table 1) is modelled in a 3D space (Figure 5). Besides 
the core and the winding elements, the model includes 
most of the installed conductive components, such as tie 
plates, clamps and tank shunts, which further affect the 
distribution of the stray field inside the transformer. 
Despite using the surface impedance boundary 
conditions [11], the model requires a dense finite 
element mesh for an accurate magnetic field calculation.  
Table 1: Rated values of the analysed transformers 
Tr. 
no. 
Rated power 
[MVA] 
Rated voltage 
HV/LV 
[kV/kV] 
Ohmic losses 
[kW] 
1 10 115/38.5 37.7 
2 16 46/23 69.2 
3
a
 25 138/22 54.6 
4
a
 27.5 115/10.5 59.4 
5
a
 30 48/23 78.2 
6
a
 40 132/22 83.0 
7
a
 50 135/23 96.3 
8 63 140/46 174.7 
9 90 132/33 182.9 
10 100 132/10.5 252.7 
11 125 330/115 302.2 
12
b
 300 400/115 458.5 
13
b
 350 400/121 480.5 
14
b
 500 415/145 758.5 
In the presence of a voltage regulation winding, a transformer is 
simulated at the rated tap position. 
a
 Transformer with the coils wound with rectangular parallel 
conductors 
b
 Five-limb transformer 
 
 
 
Figure 3. Transposition scheme for the np parallel conductors. 
 
 
 
Figure 4. In-plane 2D transformer model for circulating 
currents evaluation. 
 
While the transformer length, height and depth may be 
several metres, at least one dimension of the constructed 
elements is in the range of a few tens of millimetres. 
Meshing and analysing such a model is a demanding 
task, even for a high-end specialised personal computer. 
Since modelling with long magnetic field computation 
time is impractical for the transformer design 
optimisation, a simplified 3D model including a core, 
windings and tank without magnetic shunts is built. 
0 0.5 1 1.5 2 2.5 3
0
50
100
150
200
conductor length [m]
H
0,r
2
 [A
2
/m
2
]
 
 
t = 0° t = 20° t = 90°
...
...
...
...
...
...
...
...
...
...
...
...
...
1 2
2
3
4
n
p
 -1
n
p
n
p
n
p
 -1 n
p
 -2
3
1
2
3
1
2 1
3 4
4 5
5 6
n
p 1
1 2
n
p
n
p
 -1
n
p
 -2 n
p
 -3 n
p
 -4
n
p
 -1 n
p
 -2 n
p
 -3
n
p
tank wall
fine
regulation
winding
coarse
regulation
winding
high-
voltage
winding
low-
voltage
winding
tertiary
winding
core

A COMPREHENSIVE STUDY OF TRANSFORMER WINDING LOSSES USING THE FINITE ELEMENT METHOD 119 
 
 
Figure 5. Top half of the detailed 3D transformer model. 
 
 Due to the symmetry of the transformer, it is 
sufficient to model only its ¼, thus reducing the mesh 
size. In order to evaluate the reduced accuracy resulting 
from the model simplifications, both a detailed and a 
simplified model of the transformer no. 12 (Table 1) are 
built. 
 Because of the small difference between the 
calculated total eddy current losses of the two models 
(Table 2), the simplified version can be used. Due to the 
large ohmic loss component (458.5 kW in Table 1), 
such a decision is even more justified when the total 
winding losses are evaluated. 
 With the Ansys Maxwell adaptive meshing, the 
number of the finite elements for the model analysis to 
converge to the targeted energy error is roughly five 
times higher for the detailed transformer model (1.8·10
6
 
finite elements). Although the number of the finite 
elements is greatly reduced within the simplified model, 
the relatively long computational time is still 
unacceptable for the use in the design optimisation 
process. Consequently, a single-phase hybrid 2D 
axisymmetric model is proposed to further reduce the 
computational time while maintaining an acceptable 
accuracy required by the manufacturing company. 
 The model enables a magnetic field analysis for 
multiple cross-sections to simulate the variety of 
transformer geometries around the winding 
circumference. The windings are therefore divided into 
four sections (Figure 6). For the three-limb 
transformers, the magnetic field of two different 
sections is analysed (sections a and b in Figure 6). The 
total eddy current losses are then calculated by 
considering the section proportion to the circumference 
of the conductor. Such an approach takes into account 
the yoke and tank effect on the magnetic field 
distribution. For a five-limb transformer design, the 
third cross-section (section c in Figure 6) is analysed to 
determine the effect of the side limb on the stray 
magnetic field. 
 The comparison of the results for the detailed 3D 
model (Table 2; total losses 35.5 kWh) and the hybrid 
2D model (Table 3; total losses 36.7 kWh shows, that 
the difference in the calculated total eddy current losses 
is 3.4%. 
 
Figure 6. Three-limb (above) and five-limb (below) 
transformer sections for the 2D eddy current losses calculation 
– a) tank effect; b) yoke effect; c) side-limb effect. 
 
Table 2: Winding eddy current losses for transformer no. 12 
A – Detailed 3D transformer model 
B – Simplified 3D transformer model 
Winding 
Radial eddy 
current 
losses [kW] 
Axial eddy 
current 
losses [kW] 
Winding 
eddy current 
losses [kW] 
A B A B A B 
Tertiary 1.4 1.4 0.1 0.0 1.5 1.4 
Low-voltage 2.7 2.7 11.5 12.2 14.2 14.9 
High-voltage 1.7 1.5 15.9 15.5 17.6 17.0 
Coarse 0.5 0.4 1.5 1.3 2.0 1.7 
Fine 0.0 0.0 0.2 0.2 0.2 0.2 
Total losses 6.3 6.0 29.2 29.2 35.5 35.2 
 
Table 3: Winding eddy current losses for transformer no. 12 
Hybrid 2D transformer model 
Winding 
Radial eddy 
current losses 
[kW] 
Axial eddy 
current losses 
[kW] 
Winding eddy 
current losses 
[kW] 
hybrid classic hybrid classic hybrid classic 
Tertiary 1.5 1.2 0.0 0.0 1.5 1.2 
Low-voltage 2.9 2.3 11.4 11.9 14.3 14.2 
High-voltage 1.4 1.4 16.4 17.3 17.8 18.7 
Coarse 0.4 0.3 2.5 2.8 2.9 3.1 
Fine 0.0 0.0 0.2 0.2 0.2 0.2 
Total losses 6.2 5.2 30.5 32.2 36.7 37.4 
 
 This is acceptable for a practical estimation of the 
total transformer losses. The advantage of the described 
hybrid 2D model is that it provides a valid result with a 
reduced computational time, depending on the mesh 
size and its number of the finite elements. While 
maintaining the adaptive meshing settings, the number 
of finite elements for the hybrid solution is significantly 
lower than for the simplified 3D model. By using the 
2D model of the transformer no. 12, the solution is 
analysed with 1.2·10
5
 finite elements. Traditionally, 
only one cross-section of the transformer is analysed. 
Based on the distribution of the stray magnetic field, the 
eddy current losses are integrated over the volume of all 
the windings. The results of this approach are presented 
a
a
b c
a
a
b b
a
a
c b
a
a
b a
a
a
b b
a
a
a b

120 KOSMATIN, NEDELJKOVIĆ 
in Table 3 in the columns labelled “classic”. Section b 
(Figure 6) is used for the analysis. Due to the yoke 
effect, the axial eddy current losses are predictably 
higher than those of the hybrid model. Alternatively, the 
radial eddy current losses are slightly lower due to the 
predominance of the axial component of the stray 
magnetic flux path at the winding ends. Overall, for the 
given transformer geometry (transformer no. 12), the 
calculated total eddy current losses when analysing only 
one cross-section (37.4 kWh) are by 5.4% higher than 
the losses calculated using the detailed 3D model 
(35.5 kWh). 
 Table 4 shows the winding eddy current losses for 
the entire set of 14 transformers. The results of the 2D 
hybrid model are not significantly different from the 
results of the 3D model. Therefore, the hybrid 2D model 
is used to evaluate the increase in the windings eddy 
current losses when the transformer is exposed to GIC. 
Table 4 shows the calculated eddy current losses at the 
occurrence of 33.5 A/phase and 67 A/phase GIC. Under 
GIC conditions, when the transformer core is 
completely saturated, the path of the stray flux in the 
transformer is even more determined by the structural 
elements. By using the proposed hybrid 2D model, these 
elements can easily be incorporated with just a slight 
increase in the mesh size. 
 
3.2 Circulating Current Losses of the Transformer 
Windings 
The circulating current losses in normal and GIC 
conditions (Table 5) are calculated only for the 
transformers (Table 1) that carry coils wound with 
rectangular parallel conductors. The rest of the modelled 
transformers are wound with the continuously 
transposed cable, which inherently eliminates the 
circulating current effect. Due to the requirements of a 
highly detailed winding model (Figure 4), which 
consequently leads to a significant increase in the 
number of the finite elements, the 2D xy-plane geometry 
with a transient solution is used. This geometry is not 
ideal for studying the winding losses, as it does not 
represent all of the stray magnetic field conditions 
around the winding circumference. These results are 
therefore intended as a guide to predicting short-circuit 
losses for a particular transformer design. 
 Figure 7 presents the load current distribution 
through the parallel conductors of the low-voltage 
winding of the transformer no. 3 (Table 1), using the 
transposition scheme from Figure 3. The currents 
through the parallel conductors are obtained by 
simulating a short-circuit test. The transformer model is 
energized at the rated current on the high-voltage side 
with the rated tap position. 
 
Table 4: Winding eddy current losses increase due to GIC 
Tr. 
no. 
Total 3D eddy 
current losses 
[kW] 
Total 2D eddy 
current losses 
[kW] 
Total 2D eddy 
current losses 
[kW] 
GIC = 33.5 A 
Total 2D eddy 
current losses 
[kW] 
GIC = 67 A 
1 2.6 2.7 3.9 6.5 
2 3.3 3.4 4.5 6.1 
3 3.3 3.3 6.5 11.5 
4 4.6 4.5 7.3 12.2 
5 8.0 8.1 9.1 12.8 
6 6.1 5.9 10.2 15.1 
7 9.6 9.8 14.5 21.6 
8 11.5 11.2 12.8 16.5 
9 16.6 16.2 17.8 20.0 
10 20.8 18.9 20.9 22.9 
11 16.5 16.3 19.7 25.4 
12 35.5 36.7 39.7 44.8 
13 35.7 33.5 39.7 45.0 
14 47.1 47.3 48.3 51.8 
 
Table 5: Winding circulating current losses 
XY-plane 2D transformer model 
Tr. 
no. 
Circulating 
current losses 
[kW] 
Circulating 
current losses 
[kW] 
GIC = 33.5 A 
Circulating 
current losses 
[kW]  
GIC = 67 A 
3 1.0 1.5 2.0 
4 0.8 1.0 1.2 
5 1.0 1.1 1.2 
6 1.0 1.2 1.5 
7 1.0 1.2 1.4 
 
 The used transposition scheme is valid for 
symmetrical magnetic field conditions with regard to 
the centre of the winding. If there is a large difference 
between the distances of the winding to the top and to 
the bottom yoke, the stray flux paths at the ends of the 
winding will be different and the evenly spaced 
transposition of the parallel conductors will need to be 
revised. This difference is unavoidable due to the 
transformer manufacturing processes and is between 
10% and 20% for the low power range transformers 
with rectangular conductors. In order to evaluate the 
transposition scheme under these conditions, the 
winding positions of the transformer no. 3 are modified 
to take into account the 20% difference between the 
upper and lower winding distances from the yoke. The 
results of the circulating current losses for this 
modification are shown in Table 6. Comparison of the 
simulation results with those for the centrally positioned 
windings (Table 5) shows that the differences are 
negligible and the chosen transposition scheme is 
appropriate. 
 

A COMPREHENSIVE STUDY OF TRANSFORMER WINDING LOSSES USING THE FINITE ELEMENT METHOD 121 
 
Figure 7. Distribution of the load current through the 
low-voltage winding parallel conductors (transformer no. 3). 
 
Table 6: Winding circulating current losses 
The XY-plane 2D transformer model with the 
windings offset from the centre 
Tr. 
no. 
Circulating 
current losses 
[kW] 
Circulating 
current losses 
[kW] 
GIC = 33.5 A 
Circulating 
current losses 
[kW]  
GIC = 67 A 
3 1.0 1.5 2.1 
 
4 CONCLUSION 
A hybrid 2D transformer model is presented to evaluate 
the winding eddy current losses. Compared to a detailed 
3D model used as a reference, the proposed modelling 
solution significantly reduces the mesh size (by a factor 
of 10) with an acceptable decrease in the accuracy of the 
calculated magnetic field distribution in the power 
transformer. A transformer designer can therefore use 
the proposed calculation procedure on a regular basis.  
 A hybrid 2D model is also used to evaluate the 
winding losses in the presence of GIC. It is shown that 
GIC contribute to a substantial increase in the winding 
losses. Consequently, transformer designers should 
adjust the transformer geometry and eventually use 
some additional construction elements to make the 
transformer more robust to the GIC phenomenon. As to 
the circulating current losses, there is a further increase 
in the additional winding losses for the transformers 
wound with parallel rectangular conductors, especially 
when subjected to GIC. 
 In the future, additional laboratory measurements 
beyond standard tests will be conducted to verify the 
simulation results. The proposed eddy current loss 
calculation procedure will thus be fully implemented, 
providing an efficient tool in the design of power 
transformers. 
 
ACKNOWLEDGMENTS 
The research was supported by the Slovenian Research 
and Innovation Agency through the research programme 
“Electric Power Converters and Controlled Drives” 
(grant no. P2-0258). 
 
REFERENCES 
[1] C. A. Charalambous, A. Milidonis, A. Lazari, and A. I. 
Nikolaidis, “Loss Evaluation and Total Ownership Cost of 
Power Transformers-Part I: A Comprehensive Method,” IEEE 
Trans. Power Deliv., vol. 28, no. 3, pp. 1872–1880, Jul. 2013. 
[2] P. S. Georgilakis, “Recursive Genetic Algorithm-finite Element 
Method Technique for the Solution of Transformer 
Manufacturing Cost Minimisation Problem,” IET Electr. Power 
Appl., vol. 3, no. 6, pp. 514–519, Nov. 2009. 
[3] E. I. Amoiralis, M. A. Tsili, D. G. Paparigas, and A. G. Kladas, 
“Global Transformer Design Optimization Using Deterministic 
and Nondeterministic Algorithms,” IEEE Trans. Ind. Appl., vol. 
50, no. 1, pp. 383–394, Jan. 2014. 
[4] M. A. Arjona, C. Hernandez, and M. Cisneros-Gonzalez, 
“Hybrid Optimum Design of a Distribution Transformer Based 
on 2-D FE and a Manufacturer Design Methodology,” IEEE 
Trans. Magn., vol. 46, no. 8, pp. 2864–2867, Aug. 2010. 
[5] L. D. S. Coelho, V. C. Mariani, F. A. Guerra, M. V. F. da Luz, 
and J. V. Leite, “Multiobjective Optimization of Transformer 
Design Using a Chaotic Evolutionary Approach,” IEEE Trans. 
Magn., vol. 50, no. 2, pp. 669–672, Feb. 2014. 
[6] M. A. Tsili, E. I. Amoiralis, A. G. Kladas, and A. T. Souflaris, 
“Optimal Design of Multi-winding Transformer Using 
Combined FEM, Taguchi and Stochastic-deterministic 
Approach,” IET Electr. Power Appl., vol. 6, no. 7, pp. 437–454, 
Aug. 2012. 
[7] E. I. Amoiralis, M. A. Tsili, and A. G. Kladas, “Transformer 
Design and Optimization: A Literature Survey,” IEEE Trans. 
Power Deliv., vol. 24, no. 4, pp. 1999–2024, Oct. 2009. 
[8] M. Vitelli, “Numerical Evaluation of 2-D Proximity Effect 
Conductor Losses,” IEEE Trans. Power Deliv., vol. 19, no. 3, 
pp. 1291–1298, Jul. 2004. 
[9] S. V. Kulkarni, Transformer engineering: design, technology, 
and diagnostics, 2nd ed. Boca Raton: CRC Press, 2013. 
[10] R. M. Del Vecchio, Ed., Transformer design principles: with 
applications to core-form power transformers, 2nd ed. Boca 
Raton, FL: CRC Press, 2010. 
[11] L. Kralj and D. Miljavec, “Stray Losses in Power Transformer 
Tank Walls and Construction Parts,” in 2010 XIX International 
Conference on Electrical Machines (ICEM), 2010, pp. 1–4. 
[12] S. V. Kulkarni, J. C. Olivares, R. Escarela-Perez, V. K. 
Lakhiani, and J. Turowski, “Evaluation of Eddy Current Losses 
in the Cover Plates of Distribution Transformers,” Sci. Meas. 
Technol. IEE Proc. -, vol. 151, no. 5, pp. 313–318, Sep. 2004. 
[13] G. B. Kumbhar, S. M. Mahajan, and W. L. Collett, “Reduction 
of Loss and Local Overheating in the Tank of a Current 
Transformer,” IEEE Trans. Power Deliv., vol. 25, no. 4, pp. 
2519–2525, Oct. 2010. 
[14] R. Escarela-Perez, S. V. Kulkarni, J. Alvarez-Ramirez, and K. 
Kaushik, “Analytical Description of the Load-Loss Asymmetry 
Phenomenon in Three-Phase Three-Limb Transformers,” IEEE 
Trans. Power Deliv., vol. 24, no. 2, pp. 695–702, Apr. 2009. 
[15] ANSYS Maxwell. ANSYS, Inc. 
[16] D. Meeker, FEMM: Finite Element Method Magnetics. 
[17] K. Zheng, D. Boteler, R. J. Pirjola, L. Liu, R. Becker, L. Marti, 
S. Boutilier, and S. Guillon, “Effects of System Characteristics 
on Geomagnetically Induced Currents,” IEEE Trans. Power 
Deliv., vol. 29, no. 2, pp. 890–898, Apr. 2014. 
[18] R. Horton, D. H. Boteler, T. J. Overbye, R. Pirjola, and R. C. 
Dugan, “A Test Case for the Calculation of Geomagnetically 
Induced Currents,” IEEE Trans. Power Deliv., vol. 27, no. 4, 
pp. 2368–2373, Oct. 2012. 
[19] E. Bernabeu, “Modeling Geomagnetically Induced Currents in 
Dominion Virginia Power Using Extreme 100-Year Geoelectric 
Field Scenarios-Part 1,” IEEE Trans. Power Deliv., vol. 28, no. 
1, pp. 516–523, Jan. 2013. 
1 4 7 10 13 16 19 22 25
0
10
20
30
40
consecutive parallel wire
I
rms
 [A]
  ideal distribution

122 KOSMATIN, NEDELJKOVIĆ 
[20] L. Marti, A. Rezaei-Zare, and A. Narang, “Simulation of 
Transformer Hotspot Heating due to Geomagnetically Induced 
Currents,” IEEE Trans. Power Deliv., vol. 28, no. 1, pp. 320–
327, Jan. 2013. 
[21] L. Marti, J. Berge, and R. K. Varma, “Determination of 
Geomagnetically Induced Current Flow in a Transformer From 
Reactive Power Absorption,” IEEE Trans. Power Deliv., vol. 
28, no. 3, pp. 1280–1288, Jul. 2013. 
[22] P. R. Price, “Geomagnetically Induced Current Effects on 
Transformers,” IEEE Trans. Power Deliv., vol. 17, no. 4, pp. 
1002–1008, Oct. 2002. 
[23] A. Rezaei-Zare, “Behavior of Single-Phase Transformers Under 
Geomagnetically Induced Current Conditions,” IEEE Trans. 
Power Deliv., vol. 29, no. 2, pp. 916–925, Apr. 2014. 
[24] R. Girgis and K. Vedante, “Methodology for Evaluating the 
Impact of GIC and GIC Capability of Power Transformer 
Designs,” in 2013 IEEE Power and Energy Society General 
Meeting (PES), 2013, pp. 1–5. 
[25] R. J. Pirjola, A. T. Viljanen, and A. A. Pulkkinen, “Research of 
Geomagnetically Induced Currents (GIC) in Finland,” in 2007 
7th International Symposium on Electromagnetic Compatibility 
and Electromagnetic Ecology, 2007, pp. 269–272. 
[26] R. S. Girgis, D. J. Scott, D. A. Yannucci, and J. B. Templeton, 
“Calculation of Winding Losses in Shell-Form Transformers 
for Improved Accuracy and Reliability Part-1: Calculation 
Procedure and Program Description,” IEEE Trans. Power 
Deliv., vol. 2, no. 2, pp. 398–403, Apr. 1987. 
[27] Ž. Janic, Z. Valkovic, and Ž. Štih, “Helical Winding’s Magnetic 
Field in Power Transformers,” Electr. Eng., vol. 91, no. 3, pp. 
161–166, Nov. 2009. 
[28] B. Stojčić and D. Miljavec, “Current Distribution in the Low-
voltage Winding of the Furnace Transformer,” Int. J. Electr. 
Power Energy Syst., vol. 43, no. 1, pp. 1251–1258, Dec. 2012. 
[29] H. J. Kaul, “Stray-Current Losses in Stranded Windings of 
Transformers,” Power Appar. Syst. Part III Trans. Am. Inst. 
Electr. Eng., vol. 76, no. 3, pp. 137–146, Apr. 1957. 
[30] S. Isaka, T. Tokuinasu, and K. Kondo, “Finite Element 
Analysis of Eddy Currents in Transformer Parallel 
Conductors,” IEEE Trans. Power Appar. Syst., vol. PAS-104, 
no. 10, pp. 2730–2737, Oct. 1985. 
Peter Kosmatin received his B.Sc. and Ph.D. degrees from 
the University of Ljubljana, Faculty of Electrical Engineering, 
Slovenia, in 2006 and 2012, respectively. He is currently 
employed at Kolektor Etra, a power transformer manufacturer 
located in Ljubljana, Slovenia, where he works as a power 
transformer designer. 
 
David Nedeljković received his B.Sc., M.Sc., and Ph.D. 
degrees from the University of Ljubljana, Faculty of Electrical 
Engineering, Slovenia, in 1991, 1996, and 1998, respectively. 
In 1993, he joined the same faculty where he is currently 
employed as an Associate Professor and Vice Dean for 
Quality and Sustainable Development. His research interests 
include active power filters, power converters, electric 
vehicles, and control of electrical drives.