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Calculation of Short Circuit Reactance and Electromagnetic Forces in Three Phase Transformer by Finite Element Method 1 S. Jamali2 , M. Ardebili1 , K. Abbaszadeh1 Assistant professor the electrical engineering, Department of K.N. Toosi University, Tehran, Iran 2 M.S Student of the electrical engineering, Department of K.N. Toosi University, Tehran Iran abbaszadeh@eetd.kntu.ac.ir current density of them and permeability of transformer core. Then this model is divided to triangular elements. By using magneto static analysis of the finite element method, the magnetic vector potential of three nodes of each triangular element is calculated and therefore the flux distribution over the model is obtained. Then, the flux density of each element is evaluated. Because of this reason that magnetic vector potential of each element is considered as a linear function of x and y, the flux density of each element becomes a constant value. After this step, leakage reactance of transformer can be calculated by using several post processing methods such as magnetic energy method. A model of total three phase transformer is also considered and since the magnetic vector potential of each node is evaluated, the self inductances and mutual inductance of coils is calculated by a post processing method and then leakage inductance of each winding is calculated. The accuracy of these post processing methods is verified by comparison with experimental result. For calculation of radial and axial electromagnetic forces upon the transformer coils, two components of leakage flux density in x and y directions are calculated. By analysis of leakage flux distribution over the transformer window, it can be realized that the asymmetrical LV and HV windings can result in asymmetrical forces. Abstract: A new and simple procedure for determination of leakage reactance and analysis of electromagnetic forces that acting upon the transformer coils is presented in this paper. Before manufacturing and within the design process, it is required that we model the transformer and analyze the transformer condition using this model. The analysis of electromagnetic forces is essential for mechanical considerations. This study is accomplished by using two dimensional planar models utilizing the finite element method. Just by modeling the transformer window and using FEM, the magnetic vector potential is calculated over each node. Then by using three post processing procedures, the leakage reactance of transformer coils is calculated and the results are verified by comparison with experimental result. The radial and axial electromagnetic forces are calculated over the transformer coils and the effect of asymmetrical of winding is analyzed. I. INTRODUCTION Electrical machines are usually represented, in electrical engineering, by means of their equivalent circuits. Accurate calculation of short circuit reactance of transformer is essential for modeling of transformers. However, the calculation of leakage reactance of transformer coils is performed in many papers by using different analytical and numerical methods [1] - [6], but most of the analytical methods are not accurate, especially when the axial length of HV and LV windings are not equal, and in several of numerical methods the whole or a half of a three phase transformer is modeled. In this paper we determine leakage reactance of transformer by using FEM and just by modeling the half of transformer window for a three phase core type transformer and therefore the required time for our calculation is decreased. Furthermore the forces that acting upon the transformer coils have to be evaluated before manufacturing and then required mechanical support considerations must be performed during manufacturing process. For these reasons, evaluation of magnetic field distribution of transformer is essential for calculation of electromagnetic forces and number of useful electric parameters such as leakage reactance. For our purpose, just the modeling of transformer window is adequate and therefore an appropriate model of transformer window is defined considering the construction and position of coils and the II. MODELING AND FORMULATION A. Model definition The transformer that we considered for our studying is a 30MVA, (63/20) KV, YnD three phase core type power transformer that its HV winding has 480turns and its LV winding has 264turns. A two dimensional model of the window of this transformer is defined considering construction and dimensions of coils of this transformer, as shown in Fig.1. The boundary condition of this model is homogeneous numan boundary condition over the external rectangle and means that the flux lines come to limbs and yokes vertically. This is reasonable due to this fact that the relative permeability of core and yoke is very larger than that of air and copper. 1725 υ : The reluctivity i.e. the inverse of magnetic permeability ( µ ). And, A: is magnetic vector potential. For 2D models in x-y plane, the non-zero component of A is the z component of magnetic vector potential which is function of x and y only. Therefore the equation (4) takes the following scalar form: ∂ ∂A ∂ ∂A (υ ) + (υ ) = Js (5) ∂x ∂x ∂y ∂y In [3] it is shown that the ‘core effect’ has negligible effect on the leakage reactance calculation and therefore modeling of transformer window is adequate for our purpose. 1.2 numan 1 core numan 0.8 numan boundury condition yoke 0.6 Solving equation (5), magnetic vector potential can be obtained and solving equation (3), magnetic flux density can be calculated. By using magneto static analysis of PDE TOOLBOX of MATLAB software, the flux distribution for our defined model is obtained, as shown in fig.2. 0.4 0.2 primary winding secondary winding numan 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig 1: transformer model B. Discretization of the defined model By using PDE TOOLBOX of MATLAB software, the model is discretized by triangular elements. In the following section, we determine the partial differential equation that is required for leakage reactance calculation and electromagnetic force analysis. 1.2 1 0.8 0.6 C. Formulation In this section, the partial differential equation that governs calculation of leakage reactance problem is determined. Amperes law states that: ∇× Η = J 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Fig.2. Flux distribution over the defined model for the transformer model (1) Where: H: magnetic field intensity J: total current density We will assume that H is only due to source currents i.e. no permanent magnets are present. 1) Linear magneto static analysis: Current density J in equation (1) is due to current sources i.e. current densities of the transformer’s primary and secondary winding. We have following relation between magnetic field intensity and magnetic flux density: The magnetic vector potential over the defined model is given as a three dimensional figure .3. 0.025 0.02 0.015 0.01 0.005 B = µ .H ⇒ H = υ .B (2) 0 1.5 0.4 1 And the relation between magnetic flux density and magnetic vector potential is: B = ∇× A (3) 0.3 0.2 0.5 0.1 0 0 Fig.3. Magnetic vector potential distribution Hence: ∇ × (υ .∇ × A) = J The distribution of magnetic flux density over the defined model is given in the figure.4. (4) Where: 1726 Color: -ux Height: -ux 1.2 0.12 0.12 1 0.15 0.1 0.1 0.8 0.1 0.08 0.08 0.6 0.05 0.06 0.06 0.4 0 0.04 0.04 0.2 -0.05 1.5 0.02 0.4 1 0.2 0.5 0 0 0.05 0.1 0.15 0.2 0.25 0.1 0 0.3 0.02 0.3 0 0 Fig.4. Magnetic flux density distribution (Tesla) Fig.7. Axial magnetic flux density distribution (Tesla) Figs.5, 6and 7 are the three dimensional pictures of absolute, radial and axial flux densities over the defined model, respectively. With a glance to these figures, it is clear that the value of radial leakage flux density is much smaller than that of axial leakage flux density. D. Reactance calculation The leakage reactance of a transformer arises from this fact that all the flux produced by one winding does not link the other winding. The magnitude of this leakage flux is the function of the geometry and construction of the transformer. In this stage of analysis, we able to calculate leakage reactance of transformer, by using two post processing methods. 1) Magnetic energy storage method: Due to this fact that we considered the magnetic vector potential of each triangular element as a linear function of x and y, therefore the radial and axial component of magnetic flux density and therefore absolute value of B of each triangular element become fix values, as shown in following equations. Color: abs(B) Height: abs(B) 0.12 0.2 0.1 0.15 0.08 0.1 0.06 0.05 0.04 0 1.5 0.4 1 0.3 0.02 A = C1 + C 2 x + C 3 y 0.2 0.5 0.1 0 Bx = Fig.5. Absolute magnetic flux density distribution (Tesla) 0.04 0.05 0.02 0 -0.01 -0.02 0.4 1 0.3 0.2 0.1 (7) (8) (9) Where, C1, C2 and C3 are fix coefficients. Bx and By are radial and axial component of magnetic flux densities. After the calculation of absolute value of B for each element, the magnetic energy that stored in window space can be calculated by using two following formulas. 1 W = ∫∫ B.H .dxdy (10) 2 1 W = ∫∫ J . A.dxdy (11) 2 From these two formulas, the former is more accurate than latter. In the former equation, the integration is performed on the whole window space whereas in the 0.01 0 -0.05 1.5 ∂A ∂x B = B x2 + B y2 0.03 0.5 ∂A ∂y By = − Color: uy Height: uy 0 (6) 0 -0.03 -0.04 0 Fig.6. Radial magnetic flux density distribution (Tesla) 1727 The self inductance of the primary winding, for state that the primary winding is excited by magnetizing current and the secondary winding is open circuit, as shown in Fig.8, can be calculated using the following equation: Aavg1 − Aavg 2 L p = N 2p (16) I Where: Np: is the turn number of primary winding and I is defined as following: (17) I = ∫∫ J .ds latter equation, integration must be performed only on the winding area. Once the magnetic energy is calculated, leakage reactance of transformer for each phase referred to primary can be calculated using following formula. (4 × π × f × W × t ) Xl = (12) i 2p1 + i s'21 Where: Xl: is leakage reactance of transformer referred to primary side (per phase). f: is the supply frequency. W: is the magnetic energy that calculated using (10) and (11) equations. t: is the depth of our defined model. In other words, I is the ampere-turn of primary winding. Aavg1 and Aavg 2 are the average values of magnetic i p1 : is the instantaneous current of one phase of primary vector potential of two sheets of primary winding of one phase, respectively. The mutual inductance of primary winding and secondary winding can be calculated by the following equation: Aavg 3 − Aavg 4 M ps = N p N s (18) I Where: Ns is number of turns of secondary winding. Aavg 3 and Aavg 4 are the average values of magnetic ' winding and i s1 is the instantaneous current of the same phase of secondary winding (referred to primary winding). 2) Linkage and mutual flux method: The leakage reactance of transformer can be calculated by using linkage and mutual flux calculations. In this method, the whole three phase transformer is modeled and A=0 is set as boundary condition on the rectangle that enclose the three phase transformer model. Firstly the primary winding of three phase transformer is excited and the secondary winding is open circuited. Then the secondary winding is excited and the primary winding is open circuited. In each state, the linkage flux of exited winding and the mutual flux of excited winding with other open circuit winding can be calculated using following procedure. The linkage flux of each winding can be calculated using the following formula: (13) ψ = ∫∫ B.ds = ∫ A.dl Each phase of each winding modeled as two same sheets; therefore if we apply linear integration on each winding for one phase, considering this fact that the nonzero component of magnetic vector potential is z component, we can write the linear integration as the following form: (14) ψ = A1 − A2 Where: A1 and A2 are the magnetic vector potential value of two sheets of each winding of one phase, respectively. Each phase of each winding is considered as two sheets with uniform current densities; therefore the average value of magnetic vector potential for each sheet can be calculated as following equations: ∫∫ A.ds Aavg (i ) = (15) S For each sheet, the above integration must be performed on the sheet area where S is the area of sheet. vector potential of two sheets of secondary winding of the same phase, respectively. I in this case is the product of secondary turn number and Since the Lp and Mps are calculated, the leakage inductance the primary current. of primary winding can be calculated using the following relation: l p = L p − M ps (19) 2.5 2 1.5 1 0.5 0 -0.5 0 0.5 1 1.5 2 2.5 3 Fig.8. Core flux distribution for the case that primary winding is excited and secondary winding is open circuit The leakage inductance of the secondary winding can be calculated in a same procedure. 1728 The self inductance of the secondary winding, for state that the secondary winding is excited and the primary winding is open circuit, can be calculated using the following equation: Aavg 3 − Aavg 4 Ls = N s2 (20) I The mutual inductance of secondary winding and primary winding can be calculated by the following equation: Aavg1 − Aavg 2 M sp = N s N p (21) I Since the Ls and Msp are obtained, the leakage inductance of secondary winding can be calculated using the following relation: (22) l s = L s − M sp to withstand such forces, but this is usually not validated by tests, and accurate calculations are essential. Utilities are becoming increasingly concerned with this, and now often require documentation from manufacturers that their transformers are designed to withstand short circuits, based on computer programs that they both are familiar with and have confidence in. Since the current density of the coils is known, Electromagnetic forces upon the transformer coils can be calculated, if the radial and axial leakage flux density be known. We have these quantities from (7) and (8) equations. Radial forces are dominant and due to interaction of current and axial component of leakage flux density, as shown in the following equation: r r (24) F = ∫∫ ( J × B ).l.dx.dy x The equivalent leakage reactance of transformer referred to primary can be calculated using following equation: Np 2 X l = 2πf (l p + ( ) ls ) (23) Ns y Where: J: is current density of the coils. l: is the depth of model. The above integration must be performed on the winding area. Radial forces usually produce tensile stress in the outer winding and compressive stress in the inner winding. Compressive stress can cause buckling, and the winding must be properly supported by axial spacer bars. The distribution of radial forces that acting upon the transformer coils is shown in fig (10). 3) Comparison of results: The results obtained from these methods are compared with the experimental result, as shown in the table I. The experimental leakage reactance of the under study transformer in percent is 13%. With a glance to this table, we can deduce that our modeling and post processing methods are very accurate. 5 x 10 1.5 5 x 10 Table1: Experiment and Simulation Results of Leakage Reactance Leakage reactance in method per unit Percentage error Experimental method 0.13 0% Energy storage method using 0.1227761 0.7224% equation (10) Energy storage method using 0.1227455 0.7254% equation (11) Linkage And 0.1215231 0.8477% Mutual flux method 2 1 1 0.5 0 0 -1 -0.5 -2 1.5 -1 0.4 1 0.3 0.2 0.5 -1.5 0.1 0 0 Fig.9. Distribution of radial forces that acting upon the transformer coils Axial forces due to interaction of current and radial leakage flux density are usually compressive, as shown in the following equation: r r (25) F y = ∫∫ ( J × B x ).l.dx.dy As shown in the table the results obtained from our modeling is a little smaller than experimental result. This little difference is reasonable due to this fact that the lead of winding, that coming out from the transformer, is not modeled in our modeling and thus the little of leakage flux is not considered in our modeling. In high current transformer the leakage flux of high current leads become much larger and therefore for increasing the accuracy, the high current leads must be considered in the modeling. E. Force Analysis Forces at short circuit are calculated as flux density time current time length. The windings must be designed The above integration must be performed on the winding area. These forces tend to bend the conductors in axial direction, and their sum total act on the coilclamping ring and other clamping structures. The distribution of axial forces upon the transformer coils is shown in the figure.10. The distribution of axial forces upon the LV winding of transformer is not symmetrical due to especial shape 1729 energy storage method that presented in other papers is also presented in this paper for result comparisons. Taking into account the fact that the values obtained using this modeling coincide with the experimental result, the described method establishes an industrial application to the modeling and design of three phase transformer. Electromagnetic forces that acting upon the transformer coils is also analyzed in this paper. Radial and axial electromagnetic forces on the coils is calculated and shown as the diagrams. The modeling that shown in this paper, allowing us to know the transformer behavior before manufacturing them and, thus reducing the design time and cost. of flux distribution and flux diverting in the middle of LV winding toward to core limb in the case that the axial length of HV and LV winding is not equal. Note to Fig.2. 4 x 10 4 4 x 10 6 3 4 2 2 1 0 0 -2 -1 -4 -2 -6 1.5 -3 0.4 1 0.3 -4 REFRENCES 0.2 0.5 0.1 0 0 -5 [1]. J.Wang, A.F.Witulski, J.L.Vollin, T.K.Phelps, G.I.Gardwell, Fig.10. Distribution of radial forces that acting upon the transformer coils ”Derivation, Calculation and Measurement of Parameters for a Multi–Winding Transformer Electrical Model”, IEEE Transaction, pp. 220-226, 1999. Where ampere-turns are perfectly balanced so that the leakage flux pattern is symmetrical, then the leakage field is axial over the major part of the coil height. But since the flux lines dispersing in the radial direction in the vicinity of the winding ends, the axial flux density tends to decrease, and the resultant flux density at the ends can be resolved into the radial component causing axial forces. Due to the core effect, these axial forces are unequally distributed between the outer and inner winding. The axial forces at the top and bottom are in opposite direction. In case the ampere-turns are perfectly balanced and the leakage flux pattern is symmetrical, the resultant forces on the winding would be zero. Any axial displacement between the magnetic centers of HV and LV windings will result in a net axial force, tending to increase the displacement even further. However, in practice, a complete balance of all element of winding can not be achieved entirely for a number of results like provision to tapping, dimensional accuracy and stability of windings, etc. In case of that symmetrical winding arrangement in axial direction having uniform current distribution, there is no resulting force against the yoke, and the winding tend to compressed in the axial direction only. Different yoke distances and tapping in the main windings and uneven current distribution in the axial direction can cause the force integral to reach a final value greater than zero. [2]. J.E. 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Stanton, Z.J. Cendes, L. Xu,”Numerical Modeling of Electrical Machines and Its Application”, IEEE Transaction, pp. 1936-1942, 2002. [9]. V.K. Lakhiani,” Transformers”, McGraw-Hill Publishing Company, 1987. [10]. Martin J. Heathcote,”J&P Transformer”, Reed Educational III. CONCLUSION and Professional publishing Ltd, 1998. A new model and a new post processing algorithm for calculation of short circuit reactance, within the design process, is presented in this paper. Magnetic 1730