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Welding Transformer Analysis and Optimization by Finite Element Method István Király and Nándor Burány Subotica Tech – College of Applied Sciences [email protected] Abstract - The development of computers made possible the magnetic circuit analysis with finite element method (FEM). This methodology is suitable for transformer inductance evaluation and lumped model developement for further analysis with simulation software. In this paper, the modeling, simulation and optimization of welding transformers based on FEM is discussed. This kind of transformer differs from the classical power transformer where good coupling of windings and low magnetization current is mandatory. By welding transformers, the aims are acceptable open circuit voltage and short circuit current at the secondary to obtain good welding arc stability. To fulfill these requirements, large flux leakage have to be tolerated. Load current regulation is achieved by magnetic shunt which changes the leakage, or by thyristor phase control. The transformer currents can be calculated by use of step by step numerical integration methods (for example Runge - Kutta) based on lumped parameters obtained from FEM modeling and circuit modeling of the welding arc. Determined values of transformer currents can be substituted back in the finite element model to recalculate the magnetic field distribution. The solution of FEM model gives a visual view of the magnetic field, allows the determination of flux via appropriate areas. Different core forms and winding techniques are analyzed to optimize transformer parameters. Figure 1. Change of current by constant voltage welding with the length of arc Keywords: welding transformer, finite element method, numerical simulation, leakage inductances. I. INTRODUCTION In our highly industrialized world, welding is the most used binding method since the first patent issued in the late nineteenth century. Many types of welding are known. The common in every method is that heat is generated at the point of binding, which melt the base material. Heat can be generated by electric arc. Two types of the arc welding apparatus can be distinguished, they behaves approximately as a constant current (CC) source, or as a constant voltage (CV) source. In both cases electrical arc exists, but the arc behavior is different. The voltage drop on arc is proportional to its length, and slightly changes with arc current Fig 1 [1]. By constant voltage welding sources, the arc current changes dramatically with arc voltage change, Fig. 2 [1], so current changes abruptly with arc length. This fluctuation of current has a great influence on the quality of welding. The behavior is very different by machines, which provide constant current during the welding. This Figure 2. Change of current by constant voltage welding Figure 3. Change of current by constant current welding way the influence of the voltage change, which occurs by change of the arc length is damped Fig. 3 [1]. Constant current welding apparatus have to include a self regulation mechanism to maintain constant current. The CV welding machines need a quick external regulation, because their current changes abruptly with distance between the electrode and the base metal. The CC welding machines with appropriately designed welding transformers have a current limiting feature. This behavior is achieved by abrupt change of the leakage inductance of the transformer by current change. This fact made difficult their mathematical modeling and simulation, which differs significantly from the approach used by ordinary power transformers. By power transformers the lines of leakage flux close in air, while at the welding transformers the leakage flux lines go mainly trough the magnetic shunt). In the forthcoming part of this work the welding transformer will be discussed trough calculation of its leakage inductance and by simulation of its behavior. Taking into account the advantages of the magnetic shunt, the magnetic circuit of the welding transformer can be modeled in two-dimensions to calculate the leakage inductance by Finite Element Method (FEM), as the leakage flux lines mainly close through the magnetic shunt. The results obtained this way, can be used to determine the characteristics influenced by leakage inductance by use of other mathematical software packages. II. THE TWO-DIMENSION MODEL OF MAGNETIC CIRCUIT One of the main steps of the electrical machine desing is the determination of the characteristics of theirs magnetic circuit. Earlier analytical models were used, which did not consider the exact distribution of magnetic field and the core saturation. Nowadays, instead of analytical methods, the inductivities are determined by FEM software. In this paper FEM calculations were made by FEMM 4.2 [2] software. This software includes graphical pre- and post-processors, which can be manipulated by the LUA script language [3]. The program is designed so that axis-symmetrical and twodimensional magnetic problems can be solved. The scope of problems which can be treated by this program is restricted to the cases, where the lines of flux density are parallel to a plane. This condition is realized at welding transformers, which FEM model is shown in Fig. 4., and Figure 5. Distribution of the magnetic field in the welding transformer a solution of magnetic filed distribution for this case is shown in Fig. 5. The primary winding (denoted in the Fig 4. by Primary + and -) generates a Magneto Motive Force (MMF) which develops the flux shown in Fig. 5. The greater part of flux (resultant mutual flux) links the secondary winding, while the other part of flux closes through the magnetic shunt and the air gap and links only primary winding (it is the primary leakage flux). The resultant mutual flux depends on the difference of the MMF of primary and secondary windings. The leakage flux depends on the sum of the two MMF, it leads to the saturation of the shunt, and the abrupt fall of leakage inductance of primary and secondary windings as it is shown in Fig 6. for a given magnetic shunts and air gap. III. CHARACTERISTICS OF LEAKAGE INDUCTANCE By solving of FEM model of magnetic circuit, the flux linkages of the primary and secondary windings can be determined. They are relevant quantities to the determination of leakage inductance, which varies with the magnetic saturation of the shunt, and can be determined by following expression: L (1) Where the: - L - is the leakage inductance, - - the difference between two calculated flux linkages, - I - the difference of two currents which were used to excite the windings by calculation of the two flux linkages. The inductance values between two calculated values can be determined by a linear function which is defined by the known end points of the interval. L ( I ) Figure 4. The configuration of FEM model I L ( I1 ) L ( I 2 ) ( I I1 ) L ( I1 ) I1 I 2 (2) - N1 / N 2 - winding turns ratio. After the determination of the transformer parameters, the reduced differential equation can be written. During the simulation the primary and the transformed secondary currents to be equal. This assumption simplifies the whole process of calculation. In the equation (5) the leakage inductance LK is a function of the current, it decreases by increase of the current. di u p (t ) RK i p u s (t ) dt LK (i p ) Figure 6. The characteristic of leakage inductance for various magnetic shunt These continuous curves make possible further numerical simulations of the welding process. Here: L ( I 1 ) and L ( I 2 ) are the calculated values of leakage inductivities, while the current I is in the interval, determined by currents I 1 and I 2 . The leakage - inductances can be adjusted by change of air gap at the magnetic shunt. By increase of the air gap the leakage inductance decreases, at the same time the saturation current increases. IV. THE VOLTAGE DIFFERENTIAL EQUATION OF THE WELDING TRANSFORMER By simulation of ordinary power transformers theirs differential voltage equations are used [3]. In this case it is not so relevant. The leakage inductance changes abruptly, which dictates the decrease of the actual time step during the numerical simulation. This increases the duration of modeling but the result will not differ significantly from the result calculated by the simplified model. The simplified model consists of a reactance and an inductance, their values can be determined by a short circuit test. If the parameters of the transformer are known, then by the transforming the resistance and leakage inductance of the secondary to the primary, the components of the equivalent circuit will be obtained as follows: RK R1 ( N1 / N 2 ) 2 R2 (3) LK L1 ( N1 / N 2 ) 2 L2 (4) The low value of leakage inductance is very useful to decrease the DC current, which appears by switching transients [5]. V. OPTIMIZATION OF WELDING TRANSFORMER After the definition of the magnetic problem, the conditions, which define the purpose of design, have to be defined. These are the maximum and minimum values of the secondary current. The maximum appears when the shunt is pulled out, while the minimum value can be measured, when the shunt is entirely pulled in on its place. The first operation point helps to determine the number of turns of the secondary winding, while the second point is used to evaluate the air-gap length. A. Determination of the number of turns of secondary This process starts with the definition of excitations. At this stage, the best solution is to presume the same number of turns at the primary and the secondary windings (for example: 150), and the current increases (for instance from 0 A to 100 A). The directions of currents are assumed so, that their MMF-s point in the opposite direction in the magnetic shunt. This way, as equal MMF forces were assumed on the primary and the secondary side, the magnetizing MMF is neglected, because it makes only 1-2% of the whole excitation. When this analysis is finished, from the calculated flux leakages, the leakage inductances of primary and secondary winding can be evaluated. This is almost constant, as the flux density lines Where: - R K - resistance obtained by short circuit test, - LK - inductance obtained by short circuit test, - R1 - resistance of primary winding, - R2 - resistance of secondary winding, (5) Figure 7. The voltage versus current characteristics close in the air, and it results a constant leakage inductance. The voltage to current ratio can be calculated at the given number of turns by use of leakage inductance and reactance. The maximum value of current can be determined from the allowed current density and the whole area reserved for conductors. The product of he winding window cross section and the filling factor divided by the number of turns gives the maximum conductor cross section and the maximum working current. The necessary welding current, and the presumed number of turns make possible to determine the number of turns of secondary winding. From this result, the number of turns of the primary windings can be evaluated by use of the ratio of the secondary and primary voltage and the number of turns of secondary. B. Determination of air gap length The determination of air gap length consists of two steps which are repeated until they do not give an adequate result. The magnetic shunt is now filling the whole space between the primary and secondary windings. The used excitation is same as in the previous case, only the current changes from 0 A to the maximum value of previously determined current. When the magnetizing curve is determined, the next step follows. It is the circuit simulation by use of the determined leakage inductance. If the current at zero secondary voltage is less than the prescribed value, then the air gap has to be increased, otherwise it has to be decreased. These steps have to be repeated, with new air gap length, until the result does not correspond to minimum value of secondary current. VI. THE SIMULATION OF THE TRANSFORMER The FEM analysis allows to use functions determined by previous circuit simulation. This way that processes, which are described by non traditional mathematical functions can be investigated. The results of simulation help to understand the behavior and working of welding transformers, and allow their optimization. By the substitution of numerical values of the current obtained by circuit simulation back to the FEM transformer model, the changes of magnetic field can be observed visually. The results of simulation, besides the optimization of magnetic circuit, are useful for developing the regulators, which are used in welding apparatus. The current-voltage characteristic of the transformer Figure 9. Secondary voltage or arc voltage waveforms for different arc lengths, a simplified model. shown in Fig 7, is created by use of RMS values of voltage and current. It describes the dependence of voltage on current. As can be seen, the voltage versus current static characteristics show the same features as usual constant current welding transformers reported in the literature [1] The validity of these static characteristics can be proven also by analysis of arc voltage and arc current waveforms. To this purpose the instantaneous values of current, leakage inductance and the secondary windings voltage are shown in Fig. 8, 9 and 10 respectively. By this analysis the arc is simply modeled by a bidirectional constant voltage source. The voltage polarity is the same as the current polarity in each moment. The evaluated characteristics clearly describe the behavior of welding transformers. As it is shown in Fig 8. the transformer works as an alternating current generator. The current waveform is almost sinusoidal. The current amplitude or the current RMS value changes by arc voltage but the relative changes of current in Fig 8 are much damped compared to the voltage changes in Fig 9. This way the arc current will be almost independent of arc length. Another critical factor is the secondary voltage, which has a great influence on the welding quality. Its polarity changes rapidly which helps to minimize the low voltage period, and to provide enough voltage to maintain the arc. Figure 10. The leakage inductance variations during time. Figure 8. Current waveforms for different arc voltages and arc lengths This special performance of the leakage inductance is ensured by its rapid increasing and decreasing. This helps the commutation and forces the current to an adequate value. This great leakage inductivity at small secondary and primary currents, helps the open circuit current to restrict together with magnetizing inductivity to an adequate value. The result of simulation compared to the parameters of real transformer proves the closeness of model to the real welding transformer. These parameters are: - the open circuit secondary voltage varies from 43 V to 49 V. - the welding current varies form 40 A to 120 A The arc voltage during the welding varies from 25 V to about 30 V. These parameters can be read from Fig 7. The neglectiof of magnetizing current and iron losses cause some difference between the simulation results and the real transformer behavior. CONCLUSION In this paper the FEM analysis approach to a traditional, line frequency, welding transformer is shown. The transformer core form allows 2D modeling. The FEM analysis results are incorporated in circuit analysis to show the transformer behavior under different load conditions. Further work in this field will include the use of FEM for three phase welding transformer analysis and design, the analysis of rectifier influence on transformer voltage and current waveforms and also 2D and 3D modeling of high frequency transformers incorporated in modern inverter based welding apparatus. REFERENCES [1] [2] [3] [4] [5] [6] [7] T. Bánki, I. Baránszky-Jób, F. Béres, Hegesztési Kézikönyv. Budapest, Műszaki Könyvkiadó, 1985. D. Meeker, Finite Element Method Magnetics Version 4.2 User’s Manual, 2009. R. Ierusalimschy, L.H. de Figueiredo, W. Celes, Lua 5.1 Reference Manual, Lua.org, 2006. Gy. Retter, Villamosenergia – Átalakítók, Budapest, Tankönyvkiadó, 1992. M. Cundeva-Blajer, S. Cundeva, and Lj Arsov”Nonlinear Elecromagnetic Transient Analysis of Special Transformers”,Advanced Computer Tchniques in Applied Electromanetics, pp 167-174, IO Pres, 2008 R. Escarela-Perez, S. V. Kulkarni, and E. Melgoza, “3D FiniteElement Determination of Multi-Port Network Models forTransformer Calculations,” Paper PA1-8, Proceedings of the 16th Conference on the Computation of Electromagnetic Fields, COMPUMAG 2007, June 24-28, Aachen, Germany R. Doebbelin, C. Teichert, M. Benecke, and A. Lindemann, Computerized Calculation of Leakage Inductance Values of Transformers, Piers Online, Vol. 5, No. 8, 2009.