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Comparative the effect of distribution transformer coil shape on electromagnetic forces and their distribution using the FEM

  • Kassim Rasheed Hameed EMAIL logo , Ahlam Luaibi Suraiji and Abduljabbar O. Hanfesh
Published/Copyright: January 18, 2024
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Open Engineering
From the journal Open Engineering Volume 14 Issue 1

Abstract

During the normal life of the transformers, they are subjected to different electromagnetic stresses. One of these stresses is the electromagnetic forces resulting from the passage of short-circuit current in the transformer coils due to internal or external faults and may lead to the failure of the transformer when this electromagnetic force exceeds the threshold level. This work deals with the computation and analysis of leakage magnetic flux and electromagnetic forces when the worst-case fault (symmetrical short-circuit current) occurs in distribution transformer (DT) coils using the finite element method (FEM). The three types of DTs that were adopted in this work are similar in capacity and voltage transformation ratio (250 kVA and 11,000/416 V), but they are different in the shape of coils (oval, cylindrical, and rectangular coils).

ANSYS software was used to build two-dimensional models of the three transformers, which were different in coil shapes and in the type of iron core. The objective of this work is to compare the effect of coil shape on the distribution of electromagnetic forces and their value, in order to find out which coil shape is the best to withstand electromagnetic forces when short-circuit current is passed in the coils. The results of the simulation of the finite element models were approximately equivalent to the results of the design calculations that depend on the classical method (the analytical method), but the FEM is more accurate, due to the accuracy of calculating the magnetic flux and its distribution, which cannot be calculated using the classical method. One of the most important contributions of this research is the analysis and calculation of electromagnetic forces for three types of DTs with different coils by applying the same design parameters to all the transformers. The research also contributed to determining the best coil shape by comparing simulation results, and it was found that transformers with cylindrical coils are the best to withstand electromagnetic forces due to the homogeneity of the cylindrical coil structure from all sides.

Keywords: electromagnetic force; transformers; short-circuit; finite element analysis

Nomenclature

A

magnetic vector potential, Wb/m

Am

ampere-turn density, AT/m

B

flux density, T

B a

axial flux density, T

B r

radial flux density, T

e r

percentage resistance

e x

percentage reactance

F em

electromagnetic force, N/m

F a

axial electromagnetic forces, N/m

F r

radial electromagnetic forces, N/m

I SC

short current, A

I m

peak value of short-circuit current, A

J

current density

L

winding length, mm

ℓ eff

length of leakage magnetic path, mm

Um

mean turn of coil, mm

J

current density

K R

asymmetry factor

Abbreviations

DT

distribution transformer

FEM

finite element method

FEA

finite element analysis

FE

finite element

2D

two dimensional

3D

three dimensional

1 Introduction

Distribution transformers (DTs) are among the most expensive and important units in electric power networks. The transformer failure results in high costs for repair or replacement and may often lead to temporary loss of power supply. The most important failure in transformers is when the transformer is exposed to an internal or external fault and the passage of high short-circuit current in its coils. These high currents lead to the creation of large electromagnetic forces that are directly proportional to the square of the short-circuit current [1], which causes the deformation of the winding conductors’ arrangement and destruction of electrical insulations and raises the temperature or destroys the transformer’s fixed parts, which leads to the transformer being out of service [2]. Failure in transformers under short-circuit conditions is a major problem for users and manufacturers. Statistics show that the failures resulting from short-circuit faults are a major cause of transformers being out of service. Figure 1 shows the percentage of faults that DTs are exposed to when working in the electrical network. The percentage of faults caused by faulty winding is (33%) [3,4].

Figure 1 Percentages of different failures of DTs.
Figure 1

Percentages of different failures of DTs.

The operation of many electromagnetic devices depends on their ability to withstand electromagnetic forces in coils or in magnetic parts. So one of the most important topics is to increase the reliability of transformers by accurate calculation of the leakage magnetic field and the resulting short-circuit forces in transformer windings, which is very important at the design stage [5].

Nowadays, the designers of transformers use different methods to calculate the short-circuit forces. The most common is the classical method (analytical method) and numerical method. In the early decades of the last century, the well-known Rogowski’s method (the old method) has been used for establishing an approximate leakage field analysis [6,7]. Bill (1943) has given mathematical formulae and curves for the calculation of radial and axial short-circuit forces in transformers. Forgestad (1958) has given some empirical formulae for calculating the effective length of the path for the radial flux, based on his experiments [8]. Waters in 1966 presented mathematical formulas and constants for calculating the radial and axial forces using the method ampere-turn, to find out the inhomogeneous distribution of ampere turn, and introduced certain empirical constants that are used to calculate axial forces [9]. The analytical (classical) method is often used by transformer designers/manufacturers to calculate short-circuit forces because of its simplicity and fast calculation. However, this method is inaccurate because it only gives an approximate global value of the forces without any distribution of the magnetic flux density along the windings [10].

Since the 70s of the last century, a number of researchers have used various numerical methods to solve electromagnetic problems (e.g., finite differences, finite elements, variational methods, and finite element method; FEM) [11]. In the past two decades, several studies appeared that used the FEM for calculating the transformer’s leakage magnetic field and electromagnetic forces in transformers. Silvester and Kourad presented a FEM to calculate the leakage magnetic field and electromagnetic force of the transformer using the high-order triangle finite elements [12]. Arturi analyzed the electromagnetic axial force distribution in a step-up transformer winding under out-of-phase synchronization. Due to magnetic motive force (MMF) unbalance, the nonlinear finite element was used in his research [13]. Steurer and Frohlich calculated the electromagnetic forces in the transformer winding under the short-circuit current and inrush current and compared them using three-dimensional (3D) finite elements [14]. De Azevedo et al. [15] presented an FEM to obtain electromagnetic forces on the transformer concentric windings under external short-circuit conditions, Najafi and Iskender [2] investigated the leakage flux and electromagnetic forces due to short-circuits and inrush currents on the windings. Kumbhar and Kulkarni [16] presented a short-circuit analysis for a split winding transformer using coupled field circuit. Faiz et al. [17] presented the 3D and 2D finite element analysis (FEA) of short-circuit force for core-type power transformers. Behjat et al. [18], in this work, the FEM is used to obtain the distribution of electromagnetic forces on transformer windings under electrical and mechanical fault conditions.

Relevant previous literature that used the FEM in calculating the electromagnetic forces in transformers focused mostly on stacked core type transformers with cylindrical coils, but this study used the FEM in building models for three types of DTs different in the shape of the coils and the type of iron core. In this study, the FEM is used to compute the electromagnetic forces and their distribution on transformer windings which occurs under worst-case fault (symmetrical short-circuit current). The main problems of this research can be summed up in two important points:

First, the manufacturers of DTs in Iraq suffer from the inability to prove that the transformers can withstand the effects of short-circuit forces due to the lack of a special test line to achieve the withstanding short-circuit strength test.

Second, it is difficult for manufacturers of DTs to obtain accurate results of short-circuit forces when designing the transformer using the classical method (conventional formula).

In this study, 2D FEM based on magnetic vector potential formulation (A) has been used in order to model and compute leakage flux and electromagnetic forces due to short-circuit in three types of DTs that have the same capacity (250 kVA, 11,000/416 V) and a connection group (D/Y), but they are different in the shape of the coils and the type of iron cores. The forces were calculated under the worst fault condition of a three-phase symmetrical short-circuit current and the results are compared with classical method results. The main objectives of this study are:

  1. To compute the electromagnetic forces by building 2D FEM models for DTs with different coil shapes

  2. To study the effect of transformer coil shape on the values of the electromagnetic force (F em) and the behavior of the distribution of those forces

The significance and contribution of this research can be summarized as follows:

The analysis of 2D FE transformer models gives the accurate distribution of magnetic flux and electromagnetic forces and it can help the transformer manufacturers/design engineers to compute the forces at any local point in any region of the coil.

This work can help the engineer to improve the design of transformers to be manufactured. Also, this work gives the transformer factories in Iraq the ability to overcome the problem of the lack of a special short-circuit test line by implementing simulations of the DT models before manufacturing. This saves time, effort, and cost in addition to the test risks.

2 Theoretical background of electromagnetic forces in transformers and its classifications

The origin of electromagnetic force (F em) arises in a conductor carrying a current in a magnetic field.

(1) F em = B · I · L ,

where B is the leakage flux density, I is the current, and L is the winding length.

During normal operating conditions, the windings and its surrounding parts are strong enough to withstand actions of these electromagnetic forces. But the sudden increase in current when a short-circuit occurs leads to an increase in the electromagnetic forces arising in the coils in proportion to the square of the short-circuit current. The current passing through the concentric coils in the transformer leads to the production of magnetic flux, this magnetic flux consists of two components, a radial component, and the other, axial component [19]. The interaction of the axial flux with the winding current leads to the production of a radial force (F r), and the interaction of the radial flux with the winding current leads to the production of an axial force (F a) [20]. Figure 2 shows the axial direction that refers to the direction parallel to the central axis of the iron core and coil windings, while the radial direction refers to the path along the coil winding radius.

Figure 2 Magnetic flux and the associated electromagnetic forces [20]. (a) Magnetic flux plot, (b) radial force, and (c) axial force.
Figure 2

Magnetic flux and the associated electromagnetic forces [20]. (a) Magnetic flux plot, (b) radial force, and (c) axial force.

2.1 Radial forces

The radial electromagnetic forces (F r) are generated by the interaction of the axial component of the leakage flux density and current and act in the radial direction. Due to the passage of currents in opposite directions in the high- and low-voltage windings, the radial forces act as an inward pressure on the inner coil and an outward tensile force on the outer coil, as shown in Figure 3, so the coil must be able to withstand such pressure [21].

Figure 3 Leakage flux and radial electromagnetic forces.
Figure 3

Leakage flux and radial electromagnetic forces.

2.2 Axial forces

The interaction of the radial magnetic flux density and the current passing through the coil results in the generation of axial electromagnetic forces (F a), which act in an axial direction. Due to the fact that the coil windings carry current in the same direction, the axial forces exert pressure in axial direction and cause the conductors to bend. The total of those forces acts as pressure on the coil-clamping. The resultant sum of the axial forces is very small when there is a uniform ampere-turn distribution of the coils and when there is no axial displacement between the centers of the coils, and also, the coils have a symmetric distance from the yoke of the iron core. This makes the net resultant axial forces in the coil almost small as shown in Figure 4(a). But in reality, there is no such homogeneity and equilibrium, so any small displacement in the axial direction between the magnetic centers of coils leads axial electromagnetic forces to bend conductors in the axial direction and to move one coil vertically upward and the other coil downward as shown in Figure 4(b) [22].

Figure 4 Axial electromagnetic forces. (a) Without axial displacement and (b) with axial displacement.
Figure 4

Axial electromagnetic forces. (a) Without axial displacement and (b) with axial displacement.

2.3 Methods of electromagnetic force calculations

There are two methods that can be applied to calculate the leakage field and force in transformer windings under short-circuit, they are the analytical method (conventional method) and the numerical method.

2.3.1 Analytical methods

Until now, analytical methods are often used in calculating electromagnetic forces, where designers rely on mathematical equations and relationships as well as the help of graphs, curves, and tables derived from experiments. There are several theories involved in calculating the leaking magnetic field and electromagnetic forces at the time of a short-circuit.

Analytical methods can provide rapid calculation and a general conception of the electromagnetic forces, but the calculation process requires simplification and approximation. Due to this simplification and approximation, the analytical method cannot accurately calculate the electromagnetic forces and their distribution behavior in coils or in some other parts and components [10,23].

2.3.1.1 Electromagnetic forces calculation in transformer coils

The calculations of the symmetrical short current (I SC) that occurs in the transformer windings can be done by using the classical mathematical relationships as follows [20,24]:

(2) I sc = I n ⋅ 100 e z ,

where (I n) is the rated current A, and e Z is the transformer’s impedance expressed in percentage (%).

(3) I m = 2 ⋅ I sc ⋅ ( K R ) ,

where I m is the peak value of short-circuit current and, K R is the asymmetry factor calculated by equation (4) [23,25] or from Figure 5.

(4) ( K R ) = 1 + sin φ ⋅ e − e r e X π 2 + φ 1 − e r e X ⋅ cos φ ⋅ e − e r e X π 2 + φ ,

where e r is the percentage resistance, e x is the percentage reactance, and φ = tan − 1 e r e x .

Figure 5 Relationship between (e r/e x) and (K R) [23].
Figure 5

Relationship between (e r/e x) and (K R) [23].

The value K R is 1.8 and the maximum instantaneous short short-circuit current can be estimated to be around 2.55 times the I SC.

The ampere-turn density can be considered to be the ampere-turn divided by the coil height or the AT per unit height of coil (AT/m).

(5) A .T = I m h ⋅ N × 10 − 3 [ KA/mm ] ,

where h is the coil height and N is the number of turns.

The average of the axial magnetic flux density (Ba) in the center of concentric coil winding is [20,25]

(6) Ba = 1 2 μ o I m h ⋅ N ,

(7) Ba = 2 π I m h ⋅ N × 10 ` − 7 T ,

where Um is the mean turn of the coil, N is the no. of turns.

The radial forces (F r) can be calculated from equations (1) and (7).

(8) F r = 2 π × ( N ⋅ I m ) 2 × Um h × 10 − 7 N .

Calculating the axial component of electromagnetic forces (F a) is difficult, because of the difficulty of calculating the radial component of magnetic flux (B r). The value of the radial magnetic flux and axial force can be calculated from equations (9) and (10) [25,26,27].

(9) B r = μ 0 N ⋅ I m h × a ℓ eff T ,

(10) F a = 2 π ( N ⋅ I m ) 2 × Um h × a ℓ eff × 10 − 7 N,

(11) ℓ eff = 1.3 ( a 1 + d + a 2 ) ,

where ℓ eff is the length of leakage magnetic path, a is the length of gap due to displacement (mm), a1 is the width of internal coil (mm), a2 is the width of external coil (mm), and d is the H.L gap insulation between HV coil and LV coil (mm).

The dimensions (a, a1, a2, and d) are illustrated in Figure 6.

Figure 6 Illustrates the dimensions (a, a1, a2, d).
Figure 6

Illustrates the dimensions (a, a1, a2, d).

2.3.2 Numerical methods

With the advent of the digital computer and the development of numerical techniques, numerical methods became very suitable to analyze electromagnetic problems. In recent years, the calculation of electromagnetic force using numerical methods is a widely important topic. FEM is the most common of the numerical methods in solving the electromagnetic field as well as its ability to deal with complex geometric shapes [27]. The analysis of the electric and magnetic field in electromagnetic devices depends on Maxwell’s equations, and the DT is one of the electromagnetic devices whose behavior can be described by Maxwell’s equations. The formulations of magnetic field analysis and the calculation of electromagnetic forces using Maxwell’s equations have been presented in many research works [28,29,30].

(12) ∇ ⋅ B = 0 ,

(13) ∇ × H = J ,

(14) B = μ H ,

where B is the magnetic flux density, H is the magnetic field strength, J is the current density, and σ is the electrical conductivity.

The relation between magnetic vector potential (A) and magnetic flux density (B) is:

(15) ∇ × A = B .

From relations (15), (12), and (14), the field equation describing the vector potential [26,31] is

(16) ∇ × ( ν ∇ × A ) = J .

From equations (16), and (15), magnetic vector potential (A) and magnetic flux density (B) can be calculated. The electromagnetic forces (F em) in the transformer winding are generated due to the interaction between the flux density (B) vector and the current density (J) vector, according to the Lorentz equation, which is calculated as follows [32]:

(17) F em = J × B .

When taking a current density in the Z-axis in the 2D analysis, the leakage flux density (B) at any point can be resolved into two components, one in the radial direction (Bx) or (B r) and the other in the axial direction (By) or (B a).

(18) F y = ∬ ( J z B x ) d x d y ,

(19) F x = ∬ ( J z B y ) d x d y .

3 Methodology

3.1 FEM

The FEM is a numerical analysis method for finding solutions to differential and integral equations that describe a wide range of physical problems in many disciplines, e.g., electromagnetism, thermal conductivity, solid and structural mechanics, fluid dynamics, and acoustics.

The basic idea of the FEM is to subdivide the overall complicated problem into a finite series of simple sub-problems of solving differential equations that correspond to the linear system of equations in order to facilitate their solution. The main step involved in finding the solution usually begins with the sub-division of the problem domain into well-defined simple subdomains called elements. Different finite element shapes can be used in the same solution area, such as triangular or rectangular elements. The corners of the finite element contain the mesh nodes. The basic procedures for solving any physical problem using FEA are creation of a geometry model, assigning material properties and the element type, meshing the solid model geometry, applying boundary conditions and loads, and solving the physical problem.

3.2 Transformer modeling using the FEM

In this research, the ANSYS program, which is one of the software for the FEM, was used to build three models of DTs with different coil shapes and different types of iron cores. The DTs that we adopted in this work are:

  1. Stack core type DT with a cylindrical coil.

  2. Stack core type DT with a oval-shaped coil.

  3. Wound-core type DT with a rectangular coil

The dimensions and technical specifications of the transformers were taken from their manufacturers which are General Company for Electrical and Electronic Industries/Iraq, (Diyala Distribution Transformers Company/Iraq, and Al-Tahadi Company for Distribution Transformers/Iraq.

The specifications and main parameters of transformers are illustrated in Table 1, and Table A1 shows the technical specifications and dimensions for coils and iron cores for three transformers.

Table 1

Main design specification of the transformers

Design specification of the transformers Wound-core-type and rectangular coil Stack core type and cylindrical coil Stack core type and oval coil
Rating capacity 250 kVA 250 kVA 250 kVA
Rating voltage 11/0.416 kV 11/0.416 kV 11/0.416 kV
Rating current 13.1/347 A 13.1/347 A 13.1/347 A
Frequency 50 Hz 50 Hz 50 Hz
Type of core Wound-core Core type Core type
Core material M5 M5 M5
Nominal flux density 1.77 T 1.73 T 1.71 T
Net cross section of core 254.02 cm2 189 cm2 210.7 cm2
HV copper wire ∅ 2 mm ∅ 2.2 mm ∅ 1.9 mm
No. of turn 1155 1587 1443
Current density 2.41 A/mm2 1.99 A/mm2 2.67 A/mm2
Winding type Cross over Continuous Continuous
LV copper conductor Copper strip (0.6 × 210) mm Copper strip (0.4 × 290) mm Copper strip (0.58 × 240) mm
No. of turns 24 33 30
Current density 2.67 A/mm2 2.99 A/mm2 2.49 A/mm2

Figures 7–9 show three finite element models of DTs that differ in the shape of the coils and the type of iron core. In order to check the validity of the 2D transformer models, three-phase voltages are applied to the FE transformer models at the no-load condition, using the voltage source circuit element (external circuits element). Then, the obtained results are verified with those of the practical test values, and the design values, as shown in the table below (Table 2).

Figure 7 2D model of stack core type transformer with the oval coils.
Figure 7

2D model of stack core type transformer with the oval coils.

Figure 8 2D model of stack core type transformer with the cylindrical coils.
Figure 8

2D model of stack core type transformer with the cylindrical coils.

Figure 9 2 D model of wound-core type transformer with the rectangular coils.
Figure 9

2 D model of wound-core type transformer with the rectangular coils.

Table 2

Comparison of the FE results with practical test and design values at no load condition

Terminal voltage (r.m.s value) (V)
Design value FEM value Practical test value
Primary coil 11,550 11,550 11,550
Secondary coil 249 239.85 240

3.3 Electromagnetic force calculation using FEM

After the transformer models are validated, the basic procedures for calculating the electromagnetic force under the worst-case fault (symmetrical short-circuit current) are started by energizing the LV and HV windings with short-circuit current density. By using the finite element magnetic static analysis method, the magnetic vector potential is calculated in each element within the regions of the coils, then the magnetic flux distribution is obtained on the transformer model, and after that, the magnetic flux of each element and the corresponding electromagnetic forces are computed.

In order to know the effect of the shape of coils of the transformer on the electromagnetic force and its distribution, it is required to perform the analysis of the transformer models in two cases:

Case 1: Energizing LV and HV coils with short-circuit current densities, according to their percentage impedance value of each transformer, and based on the actual design parameters and actual technical data of each transformer such as the number of turns, size of coil conductors, current density, magnetic flux density, and coil resistance. The actual design parameters and actual technical data were taken from the design documents of the transformer manufacturers (Table 3).

Table 3

Short-circuit current density applied to LV and HV coils of each transformer

Shape of coil Short-circuit current density on windings ( J s ) (A/m2)
HV LV
Oval 78252238.8 78578478.5
Rectangular 69387031.71 81728610.93
Cylindrical 42156059.77 73487978.98

Case 2: Energizing the HV and LV coils of each transformer with the same short-circuit current density values J HV = 69387031.71 A/m2 and J LV = 81728610.93 A/m2, respectively, and applying some design parameters and technical data of wound-core type DT with rectangular coil on all transformers, in order to make all transformers work under similar operating conditions.

The steps of calculation of electromagnetic force using FEM are shown in the flowchart given in Figure 10.

Figure 10 Flowchart of implementing steps of electromagnetic force calculation using FEM.
Figure 10

Flowchart of implementing steps of electromagnetic force calculation using FEM.

4 Results and discussion

In this section, the leakage flux and electromagnetic forces under worst-case fault (symmetric short-circuit current) are calculated and analyzed for three types of transformers which differ in the shape of the coils and the iron core. The transformer models were solved by energizing the coils with short- circuit current density in two cases

Case 1: Energizing LV and HV coils with short-circuit current densities, according to their percentage impedance value of each transformer and based on the actual design parameters of each transformer,

Case 2: Energizing the HV and LV coils of each transformer with the same short-circuit current density values J HV = 69387031.71 A/m2 and J LV = 81728610.93 A/m2, respectively, and applying the same values of some design parameters and technical data on all transformers, in order to make all transformers work under the same operating conditions

4.1 Magnetic flux density calculation and its distribution based on the actual design parameters and actual technical data

4.1.1 Results of axial flux density calculation and its distribution in coils

It is clear from Figure 11 that the axial flux density distribution of the (B a) along the width of the coil (towards the x-axis) is highly concentrated on the outer edges side of the LV coil and gradually decreases on the inner edges of the LV coil. And also, the axial leakage flux is concentrated on the inner edges of the HV coil and gradually decreases towards the outer edges of the HV coil. Figure 10 also shows that the distribution of the B a along the coil height (y-axis) is highly concentrated in the middle of the coils and gradually decreases towards the ends of the coils.

Figure 11 Contour plot of axial flux distribution of HV coil and LV coil. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.
Figure 11

Contour plot of axial flux distribution of HV coil and LV coil. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.

The B a results illustrated in Table 4 show that the value of B a in the cylindrical coil (2.32 T) is lower than the values of B a in the oval coil and the rectangular coil, because the height of the cylindrical coil, 278 mm, is greater than the heights of the other two coils (the oval coil and the rectangular coil), at 201 and 187.2 mm, respectively, despite the cylindrical coil having more turns than the other two coils. Additionally, it was found that the value of B a calculated by the classical method according to equation (6) is less than the values calculated by the FEM. Because the FEM method calculates the B a values based on the magnetic flux distribution in all directions of the coil, while the classical method calculates the magnetic flux approximatively because it is based on the assumption that the axial flux direction is parallel to the height of the coil.

Table 4

HV and LV coils’ axial flux density values

Axial flux density (B a) (T)
Rectangular coil Oval coil Cylindrical coil
Conventional calculations 2.4 2.5 2.2
FEM calculations 2.59 2.63 2.32
Error% −4.16 −4 −4.5

To find out the behavior of the B a distribution along the width of the coils, a number of nodes can be chosen on a central line path in the middle of the coil area as shown in Figure 12. The behavior of the B a distribution along the centerline path of coil width is shown in Figure 13.

Figure 12 The location of the center line in the middle region along the coil width.
Figure 12

The location of the center line in the middle region along the coil width.

Figure 13 Distribution of axial flux density along the coil width.
Figure 13

Distribution of axial flux density along the coil width.

In order to know the distribution of (B a) along the path of the central line of the height of the HV coil and the LV coil, the nodes along the path of the central line along the height of the coil can be selected as shown in Figure 14. The highest B a values appear in the central region of the coils and the lowest B a values appear at the ends of the two coils as shown in Figure 15.

Figure 14 The location of the center line for the coil height along the middle area.
Figure 14

The location of the center line for the coil height along the middle area.

Figure 15 Distribution of axial flux density along the centerline of HV coil height.
Figure 15

Distribution of axial flux density along the centerline of HV coil height.

4.1.2 Results of radial flux density calculation and its distribution in coils

The results of the analysis showed that the values of the radial flux density (B r) in the coils differed from the axial flux values (B a), and the distribution behavior also differed. It can be seen that the radial flux (B r) values are large in the coil end regions and gradually decreases toward the middle region of the coils as shown in Figures 16 and 17. It is also evident from the analysis results that in the place where B r values are at their maximum, B a values are at their minimum, and vice versa. The results in Table 5 show that the value of B r in coils with an oval shape (1.145 T) is greater than the values of B r in coils with cylindrical and rectangular shapes. The reason for this is that the axial displacement distance (a) between the HV and LV coils of the oval coil is greater than that of the cylindrical and rectangular coils. It can also be observed that the radial flux values calculated using FEM are larger than the value calculated based on the classical method in equation (13) because the classical formula did not take into account the radial flux distribution along the width of the coil, which makes it difficult to obtain an accurate value.

Figure 16 Radial flux density distribution on HV coils. (a) Cylindrical coil, (b) oval coil, and (c) rectangular.
Figure 16

Radial flux density distribution on HV coils. (a) Cylindrical coil, (b) oval coil, and (c) rectangular.

Figure 17 Radial flux distribution along the height of HV coil.
Figure 17

Radial flux distribution along the height of HV coil.

Table 5

Radial flux density in HV coils

Radial flux density (B r) (T)
Rectangular coil Oval coil Cylindrical coil
Conventional calculations 0.339 0.565 0.127
FEM calculations 0.858 1.145 0.592

4.2 Electromagnetic force calculation when applying the actual design parameters and actual technical data for each transformer

The accurate calculation of the axial and radial magnetic flux and their distribution behavior using FEM certainly leads to an accurate calculation of the radial and axial electromagnetic forces in the coils.

According to Figures 18 and 19, which show the contour plots of the axial and radial force distributions of the HV and LV coil, it can be observed that the shape of the pattern (B a) is similar to the shape of the pattern (F r), and the shape of the pattern (B r) is similar to the shape of the pattern (F a).

Figure 18 Contour plot of Radial force distribution of HV-coil and LV-coil. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.
Figure 18

Contour plot of Radial force distribution of HV-coil and LV-coil. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.

Figure 19 Contour plot of axial force distribution of HV-coil and LV-coil. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.
Figure 19

Contour plot of axial force distribution of HV-coil and LV-coil. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.

It can be seen from the results of the analysis in Figure 20 that the radial forces push the inner coil inward and pull the outer coil outward, and the axial forces bend the conductors in an axial direction and may lead to moving one coil vertically upwards and moving the other coil downwards, the reason for the opposite directions of the electromagnetic forces is due to the opposite passage of currents in the internal and external coils.

Figure 20 Electromagnetic force vectors on windings region. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.
Figure 20

Electromagnetic force vectors on windings region. (a) Cylindrical coil, (b) oval coil, and (c) rectangular coil.

According to the results of the analysis in Figure 20, it is clear that the distribution of electromagnetic force vectors in cylindrical coils is uniform and regular, while the electromagnetic force vectors in oval and rectangular coils are inhomogeneous and irregular. The reason for this is that the structure of the cylindrical coil is symmetrical on all sides, while the structures of the oval and rectangular files are not the same on all sides, as there are regions of various shapes and areas in their coil structure.

4.2.1 Radial forces component calculation

In order to obtain an accurate calculation of the radial forces and their distribution in the coils, it is required to calculate the axial magnetic flux with high accuracy. According to Figure 21, the values of the radial forces are low at two ends of the coil, but high in the middle region of the coil. This is due to the fact that the axial flux values (B a) are high in the coil’s midsection and low on the ends of the coil, so it can be seen that the axial magnetic flux distribution patterns in Figure 11 are similar to the radial force distribution patterns in Figure 18.

Figure 21 Distribution of radial forces in an HV coil.
Figure 21

Distribution of radial forces in an HV coil.

Tables 6 and 7 show the results of the electromagnetic forces analysis in the three transformer models. The results of the radial electromagnetic forces (F r) in the rectangular coils have values of 491,818 N/m in the low-voltage coils and 498,275 N/m in the high-voltage coils. These values are less than the radial forces in the oval and cylindrical coils. The reason is that the number of turns in the rectangular coils of high voltage (1,155 turns) is the lowest number of turns with the oval coils being 1,443 turns and cylindrical coils being 1,587 turns. Despite the fact that the height of the rectangular winding in wound-core transformers is less than the height of the oval and cylindrical coil as shown in Appendix 1. It is clear from tables 6 and 7 that the values of F r obtained from FEM are higher and more accurate than the values of radial forces calculated by the classical method because the FEM method calculates the magnetic flux distributed over all regions of the coils, while the classical method almost assumes that the axial flux is parallel to the height of coils only and in a specific area of the coils.

Table 6

Radial force in HV coils

Radial force in HV coils (N/m)
Rectangular coil Oval coil Cylindrical coil
Conventional calculations 490,507 541,271 567,948
FEM calculations 498,275 554,525 575,026
Error% −1.5 −2.4 −1.24
Table 7

Radial force in LV coils

Radial force in LV coils (N/m)
Rectangular coil Oval coil Cylindrical coil
Conventional calculations 490,756 540,376 566,996
FEM calculations 491,818 541,048 576,642
Error% −0.2 −1.2 −1.7

4.2.2 Axial forces component calculation

The accuracy of calculating the radial flux (B r) and its distribution in the transformer coils makes the calculation of axial forces (F a) more accurate. So, it can be seen that the radial magnetic flux distribution patterns in Figure 16 are similar to the axial force distribution patterns in Figure 19. The values of axial forces (F a) from the FEM analysis of transformer models are shown in Table 8. According to Figure 22, it is clear that the F a produced in the HV and LV coils have low values in comparison to the values of the F r, because the resultant amount of applied axial forces on the coil is very small. Since the forces on the lower half of the winding are in the positive direction and vice versa, the forces on the upper half of the coil are in the negative direction which makes the net value of the axial forces very small. The obtaining of small net values for the axial forces occurs as a result of the uniform distribution of the MMF in the coils in addition to the presence of a symmetrical axial displacement between the height of the high voltage coil and the height of the low voltage coil.

Table 8

Axial force in LV and HV coils

Axial force in LV and HV coils (N/m)
HV side LV side
Rectangular coil 3.1 4.5
Oval coil 4 5.4
Cylindrical coil 3.5 3.9
Figure 22 Axial force distribution on HV coil.
Figure 22

Axial force distribution on HV coil.

4.3 Electromagnetic force calculation when applying the same design parameters and technical data on all three types of transformers

In order to choose the best coil shape for the DT that can best withstand the electromagnetic forces when a short-circuit current occurs, as well as in order to identify the distribution behavior of those forces and compare them, it is required that the finite element model analysis be done for the three transformers under identical operating conditions, and the same design parameters were applied to each transformer. These identical design parameters are (number of coil turns of 1,155, short-circuit current density of J HV = 69387031.71 A/m2 and J LV = 81728610.93 A/m2, same coil resistance values, same type of magnetic material for iron core grade M5, same number of finite elements in all coils). The main design parameters of the core-wound transformer with the rectangular coil were taken as a basic reference, and these design parameters were applied to the other two types of stacked-core transformers, which have oval and cylindrical coils. Table 9 shows the comparison of the radial forces values for different coil shapes, and Figures 23 and 24 show the distribution of radial and axial forces on high-voltage coils. It can be seen from Table 9 that the radial forces in the cylindrical coil are less than that in the oval and rectangular coils, and the reason is due to the fact that the height of the cylindrical coil is 278 mm, which is greater than the height of the rectangular coil (187.2 mm) and the oval coil (201 mm), respectively. It can also be noted that the distribution of radial forces in the cylindrical coil is more regular than the other two shapes (oval coil and rectangular coil) because the structure of the cylindrical coil is symmetrical on all sides.

Table 9

Radial force comparison for various coil shapes

HV coil radial force (N/m)
FEM calculations at similar design parameters FEM calculations at actual design parameters
Cylindrical coil 351,505 575,026
Oval coil 430,172 554,525
Rectangular coil 498,275 498,275
Figure 23 Radial force distribution on HV coil at similar design parameters.
Figure 23

Radial force distribution on HV coil at similar design parameters.

Figure 24 Axial force distribution on HV coil at similar design parameters.
Figure 24

Axial force distribution on HV coil at similar design parameters.

5 Conclusion

This study is concerned with the analysis and calculation of magnetic flux (B) and electromagnetic forces in DT coils. The FEM was used to build three models for three DTs, with different coil shapes and iron cores, each transformer has a 250 kVA capacity, 11,000/416 V voltage ratio, and a D/Y connection group. The most important conclusions obtained from the research are as follows:

  1. The maximum radial force occurs at the middle of coil and the maximum axial force occurs at two ends of the coil.

  2. Transformers with cylindrical coils are better than transformers with oval or rectangular coils in withstanding the radial electromagnetic forces when identical design parameters are applied to all the transformers under study, because the radial electromagnetic forces in the cylindrical coil are smaller than that in the rectangular and oval coils, as a result of the symmetry of the structure of the cylindrical coils from all sides.

  3. Although the radial forces in rectangular coils are greater than the radial forces in oval and cylindrical coils when applying the same design parameters for all transformers, the wound-core type transformers with rectangular coils have a good ability to withstand electromagnetic forces during short-circuits due to the presence of a tight iron frame, which holds and supports the active part of the transformer (coil + iron core).

  4. The electromagnetic force values calculated using the FEM are more accurate than those calculated using the classical method, because the representation of the transformer model in the FEM is closer to the real transformers’ geometry configuration.

  5. The use of simulations of electromagnetic forces under short-circuit conditions by FEM allows for obtaining a better design of the transformer, and increasing its reliability.

  6. This work gives assistance to design engineers in DT factories in building FE models for transformers and conducting special short-circuit testing for the purpose of improving the design without the need to manufacture prototypes, and this saves effort, time, and cost.

6 Future work

  1. The methodology of this research can be applied to other types of transformers such as power transformers and current limiting reactors to calculate electromagnetic forces using the 3D FEM method.

  2. It is possible to benefit from the FEM models of the transformers in conducting the analysis of the electric field under the phenomenon of transient voltage to test the durability of the isolation system in the transformer.

  1. Conflict of interest: Authors state no conflict of interest.

  2. Data availability statement: Most datasets generated and analyzed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

Appendix

Table A1

Dimensions of the coils and iron core

Type of transformer Net cross section of core (mm2) Flux density (T) No. of turns Height of coil (mm) Width of coil (mm)
HV coil LV coil HV coil LV coil HV coil LV coil
Wound-core with rectangular coil 254.02 1.77 1,155 24 187 210 32.1 24.3
Stack core with oval coil 210.7 1.71 1,443 30 201 240 30 25
Stack core with cylindrical coil 189 1.73 1,587 33 278 290 45.5 25

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Received: 2023-06-11
Revised: 2023-07-23
Accepted: 2023-07-30
Published Online: 2024-01-18

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/eng-2022-0503
Keywords for this article
electromagnetic force; transformers; short-circuit; finite element analysis
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