# Finite Element Calculation of Leakage Reactance in

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Journal of Engineering and Development, Vol. 16, No.3, Sep. 2012 ISSN 1813- 7822
Finite Element Calculation of Leakage Reactance in Distribution Transformer Wound Core Type Using Energy Method
Lecturer Dr. Kassim Rasheed Hameed Department of Electrical Engineering
University of Al-mustansiriya
Abstract
This paper presents the accurate energy method for calculation of leakage reactance of wound core transformers. It takes into consideration the curvature of the winding. The energy technique procedures for computing the leakage reactance are based on finite element analysis. This method is very efficient compared with the classical design methodology which based on magnetic circuit theory. The electromagnetic stored energy is obtained by leakage magnetic flux computing based on finite element method (FEM), and then leakage inductance can be computed. The Finite Element model of distribution transformer with non-linear magnetic characteristic for iron core is built using software "ANSYS". Two dimensional (2D) and three dimensional (3D) finite element modeling of distribution transformer have been used to analyze the leakage field, The obtained results have shown that the 3D model provides higher accuracy in the prediction of the leakage reactance than 2D model, with respect to the test value, due to the better representation of the transformer geometry especially the portions of the coil out of the core window(End and Curvature region),Therefore, the adoption of this energy technique during the design phase is able to enhance transformer manufacturer ability to predict the transformer leakage reactance, thus resulting to better performance and reducing the design time and cost before manufacturing . Two types of analyses are performed, including static and transient analysis. Finally, the transformer leakage reactance is calculated and compared with the value obtained from the actual test. The results obtained are very similar with the test values.
:‫الخلاصة‬
‫يقدم هذه البحث طريقة الطاقة الدقيقة لحساب مفاعلة التسرب في محولات التوزيع ذات القلب الحديدي الملفوف وتاخذ‬ ‫بنظرألاعتبار المناطق المنحنية في الملف. وتعتمد إجراءات تقنية الطاقة لحساب مفاعلة التسرب على طريقة تحليل‬ ‫ ) .هذه الطريقة فعالة جدا بالمقارنة مع منهجية التصميم الكلاسيكي التي تعتمد على نظرية‬FEA ) ‫العنصرالمحدود‬
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Journal of Engineering and Development, Vol. 16, No.3, Sep. 2012 ISSN 1813- 7822
‫الدائرة المغناطسي يتم الحصول على الطاقة المغناطسية المخزونة من خلال حساب الفيض المغناطسي المتسرب ومن ثم‬ ‫( وبأستخدام‬FEM( ‫حساب المحاثة. بناء موديل المحول بخصائص لاخطية للقلب الحديدي بطريقة العناصر المحدودة‬ ‫. في هذا البحث أستخدم موديلات ثنائية الابعاد وثلاثية الابعاد لمحول التوزيع لتحليل الفيض‬)ANSYS) ‫برنامج هندسي‬ ‫المغناطيسي المتسرب. وأظهرت النتائج التي تم التوصل أليها ,ان موديل الثلاثي الابعاد يوفر أعلى قدرة من الدقة في‬ ‫التنبوء بقيمة المفاعلة التسربية من موديل ثنائي الابعاد نسبة الى القيمة الفحص, ويرجع ذلك الى التمثيل الجيد للشكل‬ .)‫الهندسي للمحول وخاصة لاجزاء الملفات التي تقع خارج نافذة القلب الحديدي )مناطق نهاية الملف والمناطق المنحنية‬ ‫لذلك فأن أعتماد طريقة الطاقة المخزونة أثناء مرحلة التصميم تساعد الشركة المصنعة للمحولات على التنبوء بقيمة‬ ‫المفاعلة التسربية وتحسين الأداء وتقليل الوقت والكلفة قبل تصنيع المحول. تم إجراء نوعين من التحليلات، بما في ذلك‬ ‫تحليل )الساكن( و )العابر( واخيرا تم حساب مفاعلة التسرب ومقارنتها مع قيمة التي تم الحصول عليها من الفحص‬
.‫الفعلي للمحول وكانت النتائج متطابقة جدا مع قيمة الفحص‬
1-Introduction
In power systems, the transformer is one of the essential elements, and the distribution transformers are the heart of every electrical distribution system, and its failures can cause serious problems in electric utility operation. Furthermore as transformer remain energized for all the 24 hours of the day, whether they are supplying any load or not, constant losses occur in it for the whole day, and copper losses are different during different period of the day. Therefore both efficiency and the voltage drop in a transformer on load are chiefly affected by its leakage reactance, which must be kept as low as design manufacturing techniques would permit [1]. For this purpose it is crucial to be able to calculate leakage reactance. The transformers manufacturing industry improve transformer efficiency and reliability. Transformer efficiency is improved by reducing load and no-load losses and transformer reliability is improved mainly by the accurate evaluation of the short-circuit reactance and the resulting forces on transformer windings under short-circuit, since these enable the avoidance of mechanical damage and failures during short-circuit tests and power system faults[2]. The Leakage reactance is one of the important characteristic parameters of transformers. It is necessary and very helpful to calculate the reactance’s accurately in the transformer design before making it, and Leakage reactance calculations play an important role in designing geometry of transformers. The design parameters may be varied as such that the required short circuit Leakage reactance is determined [3]. The calculation of leakage reactance is performed in many papers by using different analytical methods [4],[5],[6] and numerical methods[7],[8],[9], but most of the analytical methods are not accurate, especially when the axial length of HV and LV windings are not equal. There are different techniques for the leakage-reactance evaluation in transformer; the most common technique is the use of the flux leakage elements and estimation of the flux in different parts of the transformer [2], [10]. The images technique (Rogowski method) [7, 11] which have been established in the first half of the last century .The base of this method is considering the image of every turn of the winding with the effect of iron core taken into account. The Main weaknesses of image
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method are Incapability to calculating reactance when the axial lengths of HV and LV windings are not equal and with unbalanced windings, and assume that μ= ∞ in all calculations [12]. In 1928 Roth [11], [13] introduced a considerably more advanced method of calculation. He extended Rogowski's analysis by using a double-Fourier-series solution to calculate the leakage reactance for irregular distribution of windings. The advantage of this method is that it is applicable to uniform as well as non-uniform ampere-turn distributions of windings. The ampere-turn distribution was transformed into a double Fourier series axially and radically, which could be solved analytically and the disadvantage of this method is failed to take into account the field curvature. In 1956 L. Rabin’s [11] presented a solution for axi-symmetric fields, more suitable for numerical calculations. He also used Fourier series representation of the ampere-turn distribution, but only in the axial direction. The field was considered to be unbounded in the radial direction. In this method the effect of winding curvature is taken into account and became more suitable. During recent decades the development of the philosophy of transformer design has been a logical extension of the use of computers and numerical techniques enabling one to model accurately the geometrical complexities as well as the nonlinear material characteristics for problem analysis. Numerical modeling techniques are now-a-days well established for transformer analysis and enable representation of all important features of these devices [14]. Among the numerical techniques, the most popular method for the solution of electromagnetic field problems is the finite element method (FEM). The main advantage of the FEM is its ability to deal with complex geometries, as well as properties of the materials and it yields stable and accurate solutions [15]. Finite element analysis (FEA) is now very important tool during the transformer design phase, when the manufacturer needs to check the correctness of the transformer leakage reactance or short-circuit impedance. The transformer leakage reactance determination using FEA had already been done in [3], [16], [17], [18]. In the present paper, finite element techniques are used for the magnetic field analysis of three phase, wound core, distribution transformers. The analysis focuses in the leakage field evaluation for calculation of leakage inductance in transformer using the electromagnetic stored energy in the winding and surrounding air volumes. The proper modeling and postprocessing operation are of great importance. In this paper two dimensional and three dimensional finite element modeling of the three phase distribution transformer have been used to analyze the leakage field. Just one half of the whole model of the three phase transformer was modeled for 2D modeling and quarter of the whole model for 3D modeling. By using magneto-static analysis, the magnetic vector potential of the model nodes was calculated, and then the flux distribution over the model was obtained. Then, in the postprocessing stage, by using the energy storage method, the leakage reactance of the transformer windings was calculated.
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The transformer that was considered in this paper is a 400 kVA, (D/Y) connected, rated voltages (11000/ 416 V), three-phase, wound core, oil-immersed, distribution transformer. The main design parameters of this transformer were taken from the design documents from the manufacturing company (Diyala Company of Electrical Industries)[19]. The transformer models are analyzed with ANSYS 11 software electromagnetic packages that solves problems of electromagnetic fields in two and three dimensions based on the FEA. The "ANSYS" Package provides an excellent and accurate analysis tool. The calculation of Leakage reactance in the transformer winding were studied by using two types of analysis (static and transient) for the non-linear transformer models. For the validation of the model, the obtained results of the solution are compared with the results obtained from the actual routine tests performed to the transformer at the factory.
2- Leakage Reactance
The definitions of leakage inductance is based on an academic consideration of the electromagnetism, that not all the magnetic flux generated by AC current excitation on the primary side follows the magnetic circuit and link with the other windings complete. Some flux leaks from the core and returns to the air, winding layers and insulator layers. This flux exists in the spaces between windings and in the spaces occupied by the windings. The magnitude of this leakage flux is the function of the number of turns in the windings, the current in the windings, and the geometry of the core and windings. [20] [21].
3- Leakage reactance calculation
The leakage reactance of a transformer is one of the most important specifications that have significant impact on its overall design and the Leakage reactance calculations play an important role in designing geometry of transformers. There are different techniques for the leakage-reactance evaluation in transformer using different analytical and numerical methods. But most of the analytical methods are not accurate, especially when the axial length of HV and LV winding are not equal.
3-1 Analytical methods
Several methods have been applied to determine the leakage field distribution and the leakage reactance in transformer. Most of them are based on magnetic field calculations for simplified configurations. Among analytical methods, the most popular method for the leakage-reactance evaluation in transformer is the classical method. In the classical method the leakage flux can be calculated by using the concept of equivalent magnetic circuits and this method was based upon simplifying assumptions of the leakage field being unidirectional and without curvature.
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Journal of Engineering and Development, Vol. 16, No.3, Sep. 2012 ISSN 1813- 7822 This method has certain limitations: the effect of core is not taken into account. It is also not take into account axial gaps in windings and asymmetries in ampere-turn distribution. The transformer manufacturers are often employed this method in order to simplify the time and complexity of the calculations required in automated design process. The classical method is first approach for reactance calculation is based on the fundamental definition of inductance in which inductance is defined as the ratio of total leakage flux ( to a current (I) and the leakage flux for a two-winding transformer, based on the above assumptions is [11]:
------ (1)
------ (2)
------(3)
All considered parameters in above equations shown in Fig.1 which shows a part section of a transformer taken axially through the Centre of the wound limb and cutting the primary and secondary windings. The principal dimensions are marked in the figure, as follows: Lmt is Mean length of primary and secondary turns LC is axial length of windings (assumed the same for primary and secondary) a is the radial spacing between windings ds is the radial depth of the secondary winding next to the core dP isthe radial depth of the primary winding (outer winding) NP is the number of turn of the primary NS is the number of turn of the primary
Using the following equation: Fig.1 part section of a transformer
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------ (4) And reflecting leakage reactance between windings to the primary side yields
------ (5)

Equation 5 will be as follows

------- (6)

If it is assumed that Lmtp = Lmts (meaning that the length of each turn of primary and secondary windings are equal). Equation 6 can be simplified as follows:
------ (7)

The magnitude of this leakage flux is a function of the geometry and construction of the transformer. This is the conventional equation used in References [1], [8],[11]. Furthermore, in the engineering applications, the value of leakage reactance can show the percentage of leakage reactance voltage and rated voltage, which is written as
------- (8)

In case of rectangular winding of “wound core” transformer shown in Fig. 2, the calculation of leakage reactance in the axial and radial direction as follows[15 ][22].
------- (9)

------- (10)

-------- (11)

-------- (12)

----- (13) -------(14)

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And the total per cent leakage reactance -------- (15)

Iron core Secondary winding Primary winding
gap spacing between windings

Where

Fig.2 Part section of wound core transformer

%IXa , %IXr: Leakage reactance in the axial direction and radial direction respectively. f: Rated frequency , IP : Phase current of primary winding , TP : Turn of primary winding
La :Leakage flux length in axial direction. , Lr :Leakage flux length in radial direction AP: Cross section area of primary winding. , AS: Cross section area of secondary winding Ag: Cross section area of gap spacing between windings Sr : Equivalent leakage area of winding in radial direction Sa : Equivalent leakage area of winding in axial direction hP ,hS : Height of primary and secondary windings Lmtp :Average mean turn of primary &secondary windings

3-2 Numerical method

Transformers involve magnetostatic problems. These problems can be solved by analytical and numerical techniques. The limitation of the analytical techniques as well as the progress of computers has facilitated the development of numerical techniques for the solution of electromagnetic field problems. The most important numerical techniques are the following: (Finite difference method), (Boundary element method), Finite element method Among the numerical techniques, the most popular method in the solution of magnetostatic problems is the Finite Element Method (FEM). The main advantage of FEM is that any complex geometry can be analyzed since the FEM formulation depends only on the class of problem and is independent of its geometry. Another advantage is that it yields stable and accurate solutions [15] [23].

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3-2-1 Finite Element Method:
The finite element method (FEM) is a numerical technique for obtaining approximation solutions to boundary value problems of mathematical physics, which can be described by partial equations .The basic step involved in finding the solution usually begins with the sub division of the problem domain into well defined simple sub domains called element. A variety of element shapes may be used, and different element shapes may be employed in the same solution region. The corners of the finite element are called grid points or nodes. These nodes are assigned to each element and then the interpolation function chosen to represent the variation of the field variable over the element [24]. The finite element model contains information about the device to be analyzed such as geometry (sub divided into finite elements), material, excitations, and constraints. The material properties, excitations and constraints can often be expressed quickly and easily but geometry is usually difficult to be described. There are generally two types of modeling that are used in analysis: 2D and 3D modelling. While 2D modeling conserves simplicity and allows the analysis to be run on relatively normal computer, the 3D modeling, however, produces more accurate results, and run on the fastest computers. (FEM) is the most commonly used numerical method for reactance calculation of non-standard winding configurations and asymmetrical/ non-uniform ampereturn distributions, which cannot be easily and accurately handled by the classical method. Early work on FEA of transformers was presented over four decades ago by P.Silvester and Andersen [13], [25], focused on 2D modeling, due to the restricted performance abilities provided by the early development of personal computers. The 3D solution becomes necessary, due to nature of the transformer structure (asymmetrical), and 3D analysis is essential for more accurate calculations even though it may be computationally very time consuming. Many commercial 2-D and 3-D FEM software packages are now available [26] and many manufacturers develop their own customized FEM programs for optimization and reliability enhancement of transformers.
4- Electromagnetic filed in transformers
Transformer is one of the electromagnetic devices whose behavior can be described by field equations. The electromagnetic fields inside the transformer at low frequencies, with displacement current ignored, are described by a subset of Maxwell's equations. A general formulation of electromagnetic field problems in electrical machine has already been presented by many authors [11], [24]. In this section the partial differential equations of the vector and scalar potentials are derived from Maxwell’s equations that is required for leakage reactance calculation
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Journal of Engineering and Development, Vol. 16, No.3, Sep. 2012 ISSN 1813- 7822
(Derived from Ampere law) ------ (16)
(Derived from Gauss law) ------ (17)
(Derived from Faraday’s law) ------ (18)
Where H: the magnetic field strength. , J: the current density. , B: the magnetic flux density E: the electric field strength.
The equations that describe the material properties are:
------- (19)
------- (20)
Where υ: the magnetic reflectivity (reciprocal of magnetic permeability μ), σ: the electrical conductivity
And the relation between magnetic flux density (B) and magnetic vector potential (A) is:
------ (21)
Substitution of (21) into (16) using relation (19) gives the field equation describing the vector potential [38], [39].
------- (22)
Solving equation (22), magnetic vector potential (A) can be calculated and solving equation (21), magnetic flux density (B) can be calculated.
5-The transformer configuration
The transformer under consideration is a 400 KVA,( delta / star) connected ,rated primary voltages 11 kV, rated secondary voltage 416V, three-phase, wound core, oil-immersed, distribution transformer. Fig.(3) shows the active part of the three-phase, wound core, distribution transformer considered .The secondary winding comprises 19 layers (per phase) of copper sheet, while the primary consists of 914 turns (per phase) of insulated copper wire. In a typical rectangular wound core type transformer, the low voltage winding (secondary) is mounted about the vertical axis of a core leg, the high voltage winding (primary) is located around the outside of the low voltage winding and separated form it by the high-low space insulation. Fig.(4) illustrates the perspective view of LV and HV winding one-phase.
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The transformer magnetic circuit is of shell type and is assembled from two small and two large irons wound cores, Fig.5 shows the small and large iron wound cores. The main design parameters and the dimensions of this transformer under the study were taken from the design documents from the manufacturing company (Diyala Company of Electrical Industries) [19] as shown in Table (1).

Fig.3 Active part configuration of the Wound core distribution transformer

Table. (1) Design parameters of the Transformer

Rating Core HV-Coil LV-Coil

Capacity: 400 KVA Voltage :11000 ±5% / 416 V
Current :21 / 555.14 A Frequency : 50 Hz Phase : 3-Phase Type : "Wound Core" Materials: M5 Nominal Flux Density:1.76 T Cross Section: 161.28×2 mm2 Winding Type : Cross Over Materials : Cu.Wire φ 2.5 mm No. of Turns : 914 Current Density : 2.46 A/mm2 Winding Type: Concentric Winding Materials : Cu. Strip(0.9×250)mm No. of Turn : 19 Current Density :2.47 A/mm2

Fig.4 LV and HV winding of one phase

Fig.5 Small and Large Iron wound cores

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TransformerLeakage ReactanceMethodWindingsDevelopment