Analysis of temperature distribution in amorphous core dry - type cast - resin transformers via a finite element method

 is the non-enclosed dry-type transformer. 
Hence, the major cooling is a heat conduction between the winding and resin, where the 
resin and core mainly depend on a natural convection. The effects of the heat generation, 
due to eddy currents, in mechanical parts such as clamps and bolts were neglected [2], [3]. 
The structure of transformer components is complicated, so under the accepted assumption, 
the transformer can be considered as comprised of five major components: amorphous core, 
low voltage windings, high voltage windings, air duct and fictitious surface acts as a 
boundary of the air around transformer. 
56
entity transformer’s 1/4
coordinates are obtained by the heat diffusion
where: 
(W/m k), 
surfaces of the model. Winding losses, convection and radiation relations are used to get a 
complete graph of the temperatures at every location i
2.1. 
where 
transfer coefficient for convection from outer surface (W/m2K)
of surface (K) and 
transfer coefficient, we use the correlations used in 
where
coefficient for convection from outer surface [W/(m
air W/(m K),
the Prandtl number, 
viscosity of air (m
distance (m).
that 
P 
According to the symmetry of the structure
In order to solve the equation (1), many boundary conditions must be applied on the 
Natural convection
The rated heat transfer of convection is obtained by 
The outer su
The above equations are valid over the Grashoff’s range (10
β, ν
. H
T

 
, and 
. Hai, 
 is the temperature (K), 
ρ 

" 
"
is the density of material in (kg/m
is heat transfer rate per unit area at the outer surface (W/m
 is the
 Gr
k
, P
rface of high voltage acts as vertical plate so, to get the value of heat 
ℎ(
∗
air are dependent on unknown temperatures [5], [6].
. 
Figure 1. 
Tair
)
 average heat flux in outer surface (W/m
 is 
2/sec), 
V. Binh
 is the
=

Grashoff number for uniform heat flux, 
β
 is studied in this paper




 is the volumetric expansion of air (1/K), 
, “


 ambient temperature (K). 

g is the acceleration of gravity (m/sec
Analysis of temperature distribution 
Schematic view of a dry


, 	


̇ is the heat source (W/m

"
	
 +
̇
=



̇
ℎ
=




+
(


, i.e.
̇

3), and 
−


 [4]
̇
=

∗
,	
-
 shown in 
. The temperatures in 2D cylindrical
:
̇

c 
),
[2]
	
2
type transformer


is the s
n the model.
[5]
 which are given below:
∗
. K)],
3), 
, [6] 
=

K
k
pecific heat in (J/kg K).


2), 
 kair
Nu
ỹ thuật điều
f
 is the thermal conductivity 
, T
"

hz is the
 is
z 
5
 a finite element method
igure 1
s is the local temperature 

,	
 thermal conductivity of 
is Nusselt number, 
2) and 
 ≤ 
. 
ν is the kinematical 
Gr∗
, the model of
2
local heat transfer
Z
 ≤ 10
khi
), h 
 is the vertical 
ển & Điện tử
is
10) [9]. Note 
 the
	(3a
 heat 
-b
P
.”
	(1)
	(2)
-c)
r is 
Nghiên cứu khoa học công nghệ 
Tạp chí Nghiên cứu KH&CN quân sự, Số 65, 02 - 2020 57
2.2. Natural convection in air duct 
As presented, it is preferable to use the experimental equations which are used by 
Pierce[6], these are given below: 

" = ℎ( − ), ℎ() =


 (4a-b) 
with 
  = (1 + )

Φ

 =  (1 + ) 








−


 +


,	 for	Φ ≤ 60	 (5) 
Φ =
(/)∗
[(/)∗]/
, 	∗ =
"

 , (6) 
where qd ′′ is the local convection heat flux in inner or outer cylinder (W/m2), hz is local 
heat transfer coefficient for convection from inner or outer wall (W/m2 K), b is the width 
of air duct (m), C1 = 0.697 and C2=1, L is the total height of cylinder (m), q′′ = qi is the 
heat flux in inner cylinder, and q0 is the heat flux for outer cylinder (W/m2), and R = q0/qi 
is the inner and outer cylinder. 
For a rectangular duct, b was replaced with B given by the following equation: 
 = 2
	
	
= 2

()
 (7) 
2.3. Natural radiation on the outer surface 
The outer surface is a vertical cylindrical surface and is assumed as a vertical plate. 
Two types of heat transfer can occur in the outer surface, including radiation and natural 
convection, which are described as follows. 
For the radiation for the outer surface, the heat losses have a considerable effect on the 
total distribution of temperature in the model. The radiation heat transfer also occurs at out 
surface off winding and iron core. It can be calculated as given below [6]: 

" = (
 − 
), (8) 
where 
" is the heat transfer rate per unit area by radiation(W/m2), ε is the emissivity 
coefficient of surface, σ is the Stephan Boltzman’s coefficient (5.67*10-8 W/m2 K4), Fij is 
the view factor, T1 is temperature of first surface (K) and T2 is temperature of second 
surface(K) or temperature of air. 
3. THERMAL EQUIVALENT CIRCUIT 
The governing equations for describing heat transfer have similar formats with the 
ones for the electric circuits, also known as “thermal equivalent circuit”. For instance, the 
Fourier’s law in heat transfer and Ohm’s law in electric circuits is analogous. The thermal-
circuit can then be solved with analogous Kirchhoffs’circuit laws. The analogous 
quantities have been summarized into Table 1 
Table 1. Thermal – Electric Analogy. 
Electric Thermal 
Current I (A) Rated heat transfer q (W) 
Electrical potential difference E(V) Temperature T(℃) 
Electric resistance Re (Ω) Thermal resistance Rth (℃/W) 
58
3.1. 
illustrated in 
voltage (LV) and disk
the geometric symmetry, the structure is interpreted into a 2D network model topology 
shown in 
network topology, as labeled in the figure
temperature of each node can be obtained. According to the symmetry of the structure, we 
calculate the model of entity transformer’s 1/8, shown in 
Figure 2. 
transformer loss components, which are the no
thermal behavior within the electromagnetic devices is composed by conduction, 
convection (also known as heat advection) and radiation 
dimensional heat flow makes it convenient to set up a heat transfer network corresponding 
to the actual transformer structures. By consi
circuit will include the heat conductors, and heat current sources. In establishing such an 
equivalent heat circuit for dry
by regarding the transformer to b
assembly; air ducts.
nodes, representing the complete volume of a part of the transformer. It is also regarded 
that 
resistances to the flow of heats. Thus, the realistic distributed heat sources and heat 
conductance are represented by several concentrated heat sources, equivalent thermal 
conductors. Then, the equivalent heat circuit of a dry
presented
P 
Thermal circuit 
The core and coil structure schematic of a vacuum cast dry
The temperature rise of the transformer components are caused by all of the 
For the global heat flow description, individual parts are represented by corresponding 
the heat convection, conduction and radiation paths will be simulated by the heat 
. H. Hai, 
fig. 4. 
 in [2], [3].
fig. 2
Schematic view of a dry
, P
[4]
. 
 [2]
. Each part of the transformer structure has a corresponding node in the 
V. Binh
equivalent for dry
. Surrounding the amorphous core, the coil consists of layers of low 
 high voltage (HV) windings which are casted in epoxy. Because of 
, “Analysis of temperature distribution 
(a)
of entity transformer’s 1/8
-type transformer, the proposed thermal model is developed 
e comprised of three major components: core
-type transformer core and coil 
-
type power transformer
, so after the model has been solved, the
-load and loa
dering the heat transfer methods thermal 
-type power transformer is already
figure 1.
(b).
[5]. The electrical analogy to one
K
d losses. It is known that the 
ỹ thuật điều
 a finite element method
-
(b)
(a) 
type transformer are 
and Schematic 2D 
 khiển & Điện tử
 average 
; 
 coil 
.”
-
Nghiên cứu khoa học công nghệ 
Tạp chí Nghiên cứu KH&CN quân sự, Số 65, 02 - 2020 59
Figure 3. Thermal and fluid flow network model topology are a 2D model 
and the left side dash line is the symmetric axis. 
3.2. Determination of thermal model inputs and parameters 
The heat is generated in the core and the coil assemblies from two sources: the copper 
losses, which equation 12R and stray losses in the winding, and the core losses, which are 
the sum of hysteresis and eddy-current power. In this paper, a 2605SA1 type amorphous 
core dry-type transformer is analyzed calculation of iron core and winding thermal. In 
order to obtain the input quantities of the established thermal model, copper losses and 
core losses of a power transformer are investigated with varying load and temperature. 
Generally, the copper losses are proportional to the square of the load current: 
, = , 


 
235 + 
235 + 
.	(9) 
where is the purely resistive loss at current load, ,is the purely resistive loss at rated 
load,  is the operating load current, 	 is the rated load current,  is the current winding 
temperature, and  is the winding temperature while finding the rated losses. The values 
of thermal model inputs are shown in Table 2. 
Table 2. Thermal model inputs. 
 Core P0 (W) Low voltage PCu (W) High voltage PCu (W) 
Heat Source 4.84 31.3875 31.3875 
4. RESULTS AND DISCUSSIONS 
4.1. Finite element model 
In this paper, the transformer temperature rise is analyzed by the two-dimensional finite 
element software. The front and right side of the model are set for the symmetric 
boundary. The bottom face is ground, and the top, left and the back side are outlet. The 
relative pressure stress of fluid outlet is zero with the environment temperature of 35℃. 
60
graph, the maximum core temperature is about 73
maximum value at middle. The maximum value of high and low voltage winding 
temperature is 114
average
rise is about 5.3%. The error is within 10%, meet the engineering requirements.
P 
The temperature field distributions are presented in 
Through the calculation, the relative error of high and low voltage 
. H
Figure 4. 
Averange 
temperature
Figure 5. 
. Hai, 
temperature are shown in Table
, P. 
Thermal equivalent circuit of thermal model for amorphous core.
℃
Thermal equivalent circuit of thermal model for amorphous core.
V. Binh
. The calculation of the average temperature and the stimulation 
TEC Model 
, “
Table 3.
100.33
Analysis of temperature distribution 
The contrast between the simulation and the tec model.
℃ 
 3. 
Simulation 
105.63
℃. The winding temperature reaches the 
0C
figure 5. It can be seen from 
Kỹ thuật điều
Relative error/%
 a finite element method
5.3
winding temperature 
khiển & Điện tử
.”
the 
Nghiên c
Tạp chí Nghi
4.2. Flow field analysis
external gas enters the transformer from the lower side, where the gas inside the 
transformer heat up flows out from the top surface with forming a cycle process. The 
maximum 
clamp, the airflow around the clamp change direction causing the wind speed is very small 
in the upper clamp.
based on the analogy between thermal and electric circuits. The proposed thermal model 
can calculate temperatures of the main parts of dry
proposed model were compared with professional softwa
showed reasonable accuracy. This analysis t is possible to predict the value and location of 
hot spot of transformer. This will be very useful for transformer performance.
University of Science and Technology (HUST).
[1]
[2]
[3]
[4]
Figure 
Amorphous core dry
Acknowledgment: 
. “Transformer Applications of Amorphous Alloys in Power Distribution Systems Rating 
. Summary and Status of the State
. E. Rahimpour and D. Azizian, “
type transformers
10.1007/s00202
. L. W. Pierce, 
transformer windings” 
Georgia,” 
. W. Wu and J. A. Kern, 
transformer,”
ứu khoa học công nghệ 
6
wind speed is 0.39m/s in top surface. Due to the influence of transformer top 
ên c
 gives the axial section wind velocity vector graphics. It can be seen that the 
ứu KH&CN 
IEEE Trans. Power Deliv.
-
 Conf. Proc
Figure 6. 
This work is supported by the fund (
,” 
006
“An inv
-type cast
-0008
quân s
Electr. Eng.
Linden W. Pierce, Member General Electric Company Rome, 
-4.
estigation of the temperature distribution in cast
“Temperature rise predic
. - 
ự, Số
Graphics for wind velocity vector.
5
-
IEEE SOUTHEASTCON
. CONCLUSIONS
resin transformer therma
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Analysis of temperature distribution in cast
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ABSTRACT 
PHÂN TÍCH SỰ PHÂN BỐ NHIỆT TRONG MÁY BIẾN ÁP KHÔ LÕI THÉP 
SỬ DỤNG VẬT LIỆU VÔ ĐỊNH HÌNH 
BẰNG PHƯƠNG PHÁP PHẦN TỬ HỮU HẠN 
Tóm tắt: Máy biến áp khô lõi thép sử dụng vật liệu vô đình hình là một loại máy 
biến áp tiết kiệm năng lượng, trong đó tổn hao không tải lõi thép sử dụng vật liệu vô 
định hình nhỏ hơn từ 60% đến 70% so với lõi thép sử dụng loại vật liệu là thép kỹ 
thuật điện thông thường. Trong bài báo này, nhiệt trong lõi thép và cuộn dây của máy 
biến áp khô được mô phỏng/phân tích bằng phương pháp phần tử hữu hạn. Việc phân 
tích đặc tính nhiệt độ trong lõi thép và các cuộn dây của máy biến áp là một bài toán 
khá quá trọng đối với các nhà nghiên cứu để chỉ ra điểm phát nóng cục bộ và độ tăng 
nhiệt lớn trong máy biến áp. Điều này giúp cho nhà thiết kế-chế tạo nhận biết được 
bức tranh phân bố nhiệt trong lõi thép và cuộn dây. Trên cơ sở đó, các nhà thiết kế có 
thể đưa ra được giải pháp cách điện phù hợp cho máy biến áp. 
Từ khoá: Máy biến áp vô định hình; Tổn hao có tải; Dòng điện xoáy; Mô hình nhiệt; Phương pháp phần tử 
hữu hạn. 
Received, 10th January, 2020 
Revised, 05th February, 2020 
Published, 17th February, 2020 
Author affiliations: 
 School of Electrical Engineering, Hanoi University of Science and Technology. 
 *Corresponding author: tung.leduc1@hust.edu.vn.