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Arrangements of transformer winding with respect to impulse stress
Journal of Electrostatics 71 (2013) 533e539
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Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Arrangements of transformer winding with respect to impulse stress Jan Mikes a, *, Dalibor Kokes b a b
Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague 6, Dejvice, Czech Republic Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Br ehová 7, 115 19 Prague 1, Czech Republic
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 August 2012 Received in revised form 10 December 2012 Accepted 10 December 2012 Available online 21 December 2012
Danger and stress posed to transformer winding through overvoltage still represent a hotly debated and hitherto unresolved technical issue the designers and operators of high-performance equipment have been grappling with. The actual impossibility of accepting all the real parameters of a transformer in its substitute model leaves considerable space for its constant improvement and modifications. Accepting the surrounding phenomena and properties of the transformers gives rise to complex situations and difficulties in the process of solving the model. Models tackling some of the issues pertaining to circuit models or electromagnetic field models have been developed on a long-term basis. Another issue in hand is the very complexity of the process of solving a model. This study introduces a model accepting solely the capacitance influences of transformer components, using the methods derived from the theory of LaxeWendroff’s and LaxeFriedrich’s approximation of differential equations of the hyperbolic type for the solution of the respective equations. It does not represent solutions for all the parameters of a transformer, but provides an overview of the size of the initial impulse stress of transformer winding, doing so with adequate accuracy. Ó 2012 Elsevier B.V. All rights reserved.
Keywords: High voltage transformers Power transformer protection Transformer windings Atmospheric discharge Voltage distribution Longitudinal capacitance of transformer winding Lateral capacitance of transformer winding
1. Introduction Dynamic interactions in transformer winding follow either the distribution of the electric field and overvoltage phenomena in the winding, at the entry of the surge, or the distribution of the power field and the mechanical behaviour of transformer winding during various types of short-circuit. The first type is designated as fast, the other one as slow. The study deals with the first type of interactions. Examination of overvoltage relations in the transformer winding has been the subject of innumerable studies. Modern findings in the field of mathematical analysis and numerical mathematics have facilitated specification of the physical model under scrutiny. The problems concerning a substitution transformer diagram cover its discrete model analogous, in circuit models, to electric wiring where longitudinal capacitance, eventually resistivity of the conductor used (that is, however, frequently ignored) operates between the turns. This fundamental model was published in 1915 by K. W. Wagner and all the subsequent theories proceed therefrom [1]. The first works stemmed from the model of a single-layer coil without iron, which made it possible to perform certain predictions
* Corresponding author. E-mail addresses: [email protected] (J. Mikes), kokesdal@fjfi.cvut.cz (D. Kokes). 0304-3886/$ e see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.elstat.2012.12.015
analytically, further studies stemmed from the gradually more complex physical models, with numerical methods being incorporated into their behaviour step by step. In ref. [2] authors explained that the role played by the iron core in the response of an impulse-stressed winding is negligible. However, even modern studies proceed from a relatively heavily simplified configuration of the physical model. In methodological terms, two approaches may be distinguished in the physicalemathematical description of the overvoltage phenomena. The first consists in the construction of the so called field model, i.e. in the formulation of the electromagnetic field in the sphere of winding as a marginal assignment for the partial differential equations of the type of wave equations in which the vector magnetic potential, less often the vector electric potential, figures most frequently as the unknown quantity. In numerical terms, the solution of this 3D, eventually 2D, assignment can be managed quite satisfactorily by applying a suitable commercial program (evidently based on FEM e Finite Element Method), but it is not easy to determine the appropriate boundary conditions. The other concept is based on the layout of the so called circuit model, i.e. formulation of a system of ordinary differential equations for a locally discretized circuit comprising elements R, L, C for the numerical solution of which one can use some standard numerical method. This type of solution poses the problem of precisely
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determining the parameter R, L, C, and also the process of setting up a 3D circuit model runs up against certain difficulties. However, in both types the numerical solution itself is accompanied by the necessity of coping with the reliability of the obtained results (especially stability, convergence etc.). 2. Justification of the choice of the problems under scrutiny and current status of solution Considerable attention has been devoted to impulse stress of electric machines on a long-term basis. More precise, more computationally complex as well as less challenging and also less accurate mathematical models have been developed, based on the available mathematical theories (since the early 20th century). Owing to what is a frequently inaccurate and less credible way of stipulating the input parameters of the winding in the model there have emerged diverse simplified notions of voltage distribution in a transformer. Quite frequently, this issue is resolved on the basis of single-layer coil. The resulting solutions are then extended to cover the entire winding as well as multiple-winding transformers. In terms of voltage distribution, transformer’s multiple-layer coil e without adequately thorough knowledge of its parameters e constitutes such a complex task that very small congruence may be anticipated if such parameters are neglected. Former Czechoslovakia (from the 1950s until 1989) figured among the leading countries generating valuable mathematical models of single- and Mate na. multiple-layer coils e see works by A. Veverka, B. Heller, S. Since these studies could not be published abroad before 1989 they now represent a valuable biography for the topic that has not yet been reflected abroad. The most important specialist studies on the subject are as follows [3e5]: (in Czech). In addition, a large number of contributions was published in the journal Elektrotechnický obzor (active in 1910e1989) by A. Veverka and B. Heller. There are details that were not possible to publish in ref. [3]. However, both academic, business and manufacturing centres have been returning to the subject of impulse stress in the past decade. For their part, the major manufacturers of transformers and reactance coils strive to have the possibility of predicting computations through which they could declare e prior to designing a transformer e its resistance to impulse stress (most frequently an impulse of 1.2/50 ms). Up to now resistance has been verified solely by recording oscilloscopic response from the taps of the transformer winding when leading the impulse in to its input clamp. This gives rise to temporalespatial behaviour of the voltage wave in which the spot of maximum coil stress is monitored. In view of the limitations of the possibility of calculating all the parameters of the winding, of reckoning precisely with own and mutual inductance in the model and of respecting the impact of the iron core, we have decided to verify the validity of the numerical solution solely while respecting the capacitance reserve model. With this particular model we want to single out the necessity of devoting great care when making conclusions from incomplete or very little valid circuits which represent only limited properties transferred to the entire winding, eventually to multiple-winding systems e threephase transformer, while respecting the impact of all the interlinks involved. As for the entry of voltage impulse to the transformer winding, we can state that voltage distribution along the winding is dependent solely on the winding’s capacitance conditions since inductance in a time interval close to zero may be neglected. The design parameters of the coils of the transformer itself have a highly decisive influence on the initial voltage distribution. Assuming that we know initial and terminal voltage distribution on the transformer winding, it is quite easy to determine the free oscillations envelope, which represents a theoretical maximum
voltage of the insulation in any point of the winding. Much smaller attention has been given to the calculations of the parameters of the longitudinal and earth capacitance themselves, and own and mutual inductance of the coils of transformers than, for instance, to the theories of calculation themselves, and to the proposed numerical solution of wave phenomena in the winding. Such a model yields more objective results than endeavours to capture simultaneously all the influences within a transformer. Among the key studies for the computation of C, K, L and M we can mention, for instance, the following refs. [6e12]. It is possible to trace in the literature two approaches to the studied issues of impulse voltage distribution in a transformer. One of them is the design of a model with concentric parameters; the other consists in observing the significance of distributed parameters. Authors in ref. [13] tend to distinguish models into Fast transient overvoltages (FTO) and Very fast transient overvoltage (VFTO). As for the FTO models, in which a frequency range from 10 kHz f 1 MHz is assumed, many models have already been published, based on the theory of quadrupole [14,15] which are cascaded and calculated by means of the respective computational instruments. Subsequently, most [16,17] of other models are based on the solution of the telegraph equation, solved either analytically or numerically. For the VFTO models, hence models with a frequency over 1 MHz, it is no longer possible to neglect the wavelength of the input high-frequency impulse, and the circuits are resolved by means of distributed parameters e the most frequently used methods are then the hybrid calculation methods where parts of the winding are calculated as concentric e for instance for lower frequencies where the influence of conductivity prevails, and as distributed ones for moments when capacity is of explicit importance within a circuit. The models with concentric parameters reduce the calculation solely to predetermined points in the winding; it is impossible to monitor voltage behaviour in any random spot of the winding. The types of the used windings themselves, too, have considerable impact on the design of the relevant models e the most frequently used is the simplified method via simple single-layer coils, but the transformer winding tends to be much more complex; multiple-layer disk and cylindrical windings are often used, or special adjustments, for instance interlaced windings, are utilized. Relevant studies dealing with these subjects may be found in: refs. [18,19]. The last approach to modelling resistant transformers is the application of various limiting elements, overvoltage arresters or the use of an older method involving voltagedependent varnishes [20,21]. Transition from the transformer’s real winding (single-layer) to its discrete model is described in Fig. 1.The classical theory of transformer model for the effects of overvoltage has been discussed in great detail, for instance, in refs. [3,4,22]. For the purpose of our study, we proceed from a simplified solution of the equation describing the overall diagram given in Fig. 2. This set of equations originated as a result of solving the Kirchhoff laws in the described Fig. 2.
v2 u v4 u v2 u þ LK 2 2 LC 2 ¼ 0 2 vx vx vt vt
(1)
vi vu ¼ C vx vt
(2)
v2 u vxvt
(3)
i ¼ K
In case of initial voltage distribution at voltage impulse, a singlelayer transformer winding may be approximated by means of
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method or rather the implicit Wendroff differential approximation set up according to refs. [23,24]. We will intersperse the continuous definition area U of the set of equations (2) and (3) evenly with a temporalespatial network with steps Dx and Dt so that we will discretize a winding d long with an equivalent step Dx, obtaining a one-dimensional geometric network containing N elements, delineated by N þ 1 nodes where N ¼ d/Dx. We discretize the semi-delineated time coordinate t with equivalent step Dt, obtaining a set of discrete time levels t1 ¼ lDt, where l ¼ 0, 1, 2, ... All the spatial derivations contained in the equations (2) and (3) will be replaced with central differences in the following forms.
Fig. 1. Capacitance location in transformer winding.
a substitute capacitance diagram, given specifically in Fig. 2. Inductivities from Wagner’s diagram have no impact on either initial or terminal voltage distribution. See refs. [3,4,22]. Under the assumption of knowing initial and terminal voltage distribution on the transformer winding it is easy to establish free oscillations envelope, which constitutes a theoretical maximum voltage stress of the insulation at any random point of the winding. This procedure is suitable for simple verification of the winding’s resistance to impulse stress. When accepting only capacitance links, as assumed by Fig. 2, the problems are simplified into two hyperbolic differential equations [5,23,24]. Application in the Matlab programming environment thus offers a tentative idea for checking the correctness of any designed winding.
vi l 1 ¼ ik;lþ1 þ ikþ1;l ik;l i vx k 2Dx kþ1;lþ1
(4)
vu l 1 ¼ uk;l þ ukþ1;lþ1 ukþ1;l u vt k 2Dt k;lþ1
(5)
Equation (3) is a mixed partial derivation. For its approximation we have used the following procedure according to the relations (4) and (5), mentioned, for instance, in refs. [24,25].
v2 u v vu v vu ¼ ¼ vxvt vx vt vt vx
(6)
vu l 1 ¼ ukþ1;l1 u vt kþ1 2Dt kþ1;lþ1
(6a)
vu l 1 uk1;jþ1 uk1;l1 ¼ vt k1 2Dt
(6b)
vu l vu l 1 1 l vt kþ1 vt k1 2Dt ukþ1;lþ1 ukþ1;l1 2Dt uk1;jþ1 uk1;l1 v2 u 1 ukþ1;l1 uk1;lþ1 þ uk1;l1 z z u ¼ 2Dx 2Dx vxvt 4DxDt kþ1;lþ1 k
(7)
3. Simulations and results The equations (2) and (3) can be solved analytically, but under the assumption that a nominal impulse is formed by input voltage. Solution in an analytical manner is difficult to attain for other input impulses. That is why methods of numerical calculation respecting, for instance, impulse in the form of 1,2/50 ms have been sought. Analytical solution is derived in great detail in ref. [3]. To solve the set of equations (2) and (3) we will create a temporalespatial network with spatial step Dx and temporal step Dt by means of the method of finite differences. There are several numerical methods to calculate partial differential equations of the hyperbolic type, a case in point may be provided by the FTCS, Laxe Friedrichs, LaxeWendroff and the more complex method Upwinding, which, however, best succeeds in responding to a dramatic change in the function gradient. To calculate the following set of equations we have employed the LaxeWendroff
After substituting (4), (5), (6) and (7) into the equations (2) and (3), we obtain the following relations (8) and (9), and adjust the given expressions into the form (10) and (11).
vu 1 vi 1 uk;l þ ukþ1;lþ1 ukþ1;l u ¼ z vt C vx 2Dt k;lþ1 1 1 ik;lþ1 þ ikþ1;l ik;l i ¼ C 2Dx kþ1;lþ1
i v2 u 1 ¼ z i þ ikþ1;l þ ik;lþ1 þ ikþ1;lþ1 K vxvt 4K k;l 1 ukþ1;l1 uk1;lþ1 þ uk1;l1 ¼ u 4DtDx kþ1;lþ1
(8)
(9)
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1 i þ ikþ1;l þik;lþ1 þ ikþ1;lþ1 4K k;l 1 ukþ1;lþ1 ukþ1;l1 uk1;lþ1 þ uk1;l1 ¼ 4DtDx 1 þ ikþ1;lþ1 i K k;lþ1 1 1 ik;l þ ikþ1;l ¼ ukþ1;l DtDx K
¼
1
DtDx
ukþ1;lþ1
þ
(11)
Sign change corresponds recurrent LaxeWendroff transformation and underlined elements correspond: Boundary value conditions Fig. 2. Initial substitute model for numerical solution.
t ¼ 0; x ¼ 0 : Fðu; i; tÞ ¼ 0; x ¼ d : Fðu; i; tÞ ¼ 0 Initial value conditions
1 uk;l þukþ1;lþ1 ukþ1;l u 2Dt k;lþ1 1 1 i þi i i C 2Dx kþ1;lþ1 k;lþ1 kþ1;l k;l 1 1 1 þ ukþ1;lþ1 i ¼ u i 2Dt k;lþ1 C 2Dx kþ1;lþ1 k;lþ1 1 1 1 ¼ uk;l þ ukþ1;l þ ikþ1;l ik;l 2Dt C 2Dx
t ¼ 0; 0 x d : iðx; 0Þ ¼ 0; uðx; 0Þ ¼ 0 (10)
It will be written as equations (10) and (11) for all k elements of the differential network (k ¼ 1, ., N), obtaining a set of equations with 2.(N) unknowns. We will supplement these equations with algebraic approximation of equations proceeding from marginal conditions. We will obtain a set of 2.(N þ 1) linear algebraic equation (12) [24,25].
Fig. 3. Obtained behaviour of voltage distribution along the axis of winding for a change in K (F/m) and C (F m) e values C and K in order a) K ¼ 20 1012 and C ¼ 17 1010 b) K ¼ 20 1012 and C ¼ 17 1011 c) K ¼ 20 1012 and C ¼ 17 1012 d) K ¼ 20 1011 and C ¼ 17 1012.
J. Mikes, D. Kokes / Journal of Electrostatics 71 (2013) 533e539
AX lþ1 ¼ BX l
537
(12)
where vector Xlþ1 contains elements uk,lþ1, ik,lþ1 and vector Xl contains elements uk,l, ik,l pro k ¼ Nþ1. By solving equation (12) in software Matlab we will obtain voltage values and flow of time level on the basis of known quantities from the previous time level. Calculation commences from the time level for t ¼ 0, i.e. at the moment of the entry of voltage impulse on the winding. Matrix A and B without boundary conditions were generated from Matlab software. The charts in Fig. 3 contain initial voltage distribution (in red), terminal voltage distribution (in green) and approximate free oscillations envelopes (in blue). An earthed end of the winding is involved in case of selected following courses. The values C and K given in the description of Fig. 3 were gradually inserted, and, using a program written in the Matlab code, we have obtained the relevant graphic outputs. In all the cases, the normalized impulse 1.2/50 ms, modelled by means of exponential curves, was used as an input signal. In our instance, we modelled the impulse according to the exponential relation confirmed by the standards:
uðtÞ ¼ 1; 03$U0 $ ea$i$dt eb$i$dt ; where constants a ¼ 14400 and b ¼ 3500000
(13)
similarly it is possible to model impulses in the form: a) for the leading edge, b) for the trailing edge
Umaxð1 cos2; 6$tÞ a uðtÞ ¼ for 2
0 t tforehead
(14)
and
b uðtÞ ¼ Umax $e½0;014$ðttforehead Þ
(15)
It is evident in Fig. 3 that if the value of longitudinal capacitance K is lower than the value of the earth capacitance C by less than an order, a virtually problem-free initial voltage distribution occurs, and thus the shape of the free oscillation envelopes will be highly favourable. The model was constructed for depicting initial and terminal voltage distribution for homogenous winding at the entry of nominal voltage impulse amounting to 500 V on the winding with axial length of l ¼ 1.5 m. The program serves for tentative verification of maximum possible overvoltage generated within the winding. This is highly demanded for practical purposes, even though it is only an approximate method. Parameters of longitudinal and earth capacitance may be established either by measurement (with high performance transformers this is possible only very approximately indeed) or numerically on the basis of works derived, for instance, from such authors as Massarini and Kazimierczuk [26], who determine longitudinal and earth capacitance by means of the relation derived on the basis of the magnetic field theory. This particular approach is frequently employed in determining parasite transformers [7] and thanks to this procedure values with an accuracy of up to 20% may be obtained [7,9]. One of the most widely used methods of ensuring linear voltage distribution along the winding is to compensate the impact of earth capacitance C by means of metal shields and shades. The function of a capacitance shield consists in mutual compensation of the discharge on the earth capacitance C and discharge on the inserted external capacitance Cex. These discharges then have no impact on the distribution of the residual discharge in the chain of lateral capacitances K/dx. Under the assumption that the lateral capacitance K/dx is equally large in all the high-voltage points of the
Fig. 4. Behaviour of voltage distribution when using a shield at a) 30%, b) 60%, c) 90% of the winding’s axial length for K ¼ 20 1012 and C ¼ 17 1010.
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winding, voltage distribution along the winding will be linear and free oscillations will be eliminated. This shielding may be performed by means of a beam trap. The most suitable behaviour of the initial voltage distribution can be obtained under the conditions whereby the shield is placed around the entire circumference of the winding. However, in a real situation, the beam trap usually does not surround the winding’s entire circumference, but only its part, and that is primarily due to reasons of dimension. Such a shield located along the entire length of the winding must be insulated at the maximum voltage. This condition leads to an increase in the dimension of transformers to such an extent that only parts of the winding are normally shielded. As a rule, a shield is placed at the entry of the winding to increase the longitudinal capacitance K in view of the earth capacitance C. If the shield eliminates the influence of the capacitance C, or the winding’s earth capacitance, then there occurs a linear voltage distribution in the shielded part. Fig. 4 depicts simulated shielding of the earth capacitance C at 30, 60 and 90% of the winding’s axial length from the entry (values K are given in F/m and C in F m). Fig. 5 describes the case when, in the first part of the wiring, we markedly increase the value of the lateral capacitance K by means of external capacities, for instance by a chain of condensers. The first part of the winding’s insulation is then stressed by constant voltage of 500 V and in the remaining section of the winding insulation stress will be lower than in case of the same winding without an external capacity (see Fig. 4a). It is evident that this is a little efficient mode of protecting insulation against puncture. 4. Objective facts The generated model can be used for monitoring initial distribution of a single-layer winding, which e according to prerequisites e would correspond to the relation (16) at the evenly distributed capacitance K and C [3].
u0 ¼
sinhaðl xÞ sinhal
(16)
When reducing the winding’s earth capacitance in a real transformer by using a capacitance shield, it is possible also by increasing the capacitance K in the individual segments of the winding to linearize initial voltage and thus prevent emergence of oscillations. If a single condenser is used the capacitance increases
Fig. 5. Behaviour of voltage distribution when using increased lateral capacitance K at 30% of the winding’s axial length from the entry for K ¼ 20 1012, Kex ¼ 20 1010 and C ¼ 17 1010.
but in terms of the free oscillations envelope the terminal voltage distribution simultaneously deteriorates. The resulting free oscillations envelopes do not make it possible to determine the ideal capacitance connected between the winding’s input and winding’s branch. However, the program created in the Matlab programming environment allows e for the purpose of designing a transformer resistant to overvoltage impulse e to stipulate an optimum value of the inserted capacitance (capacitance chain). 5. Conclusion In our article we have concentrated on an alternative electrical model of the transformer which is easier for calculating, however, it respects all important phenomena which during approaching impulse overvoltage stress the winding. Such a model itself is analytically very complicated to solve. For the solution we have applied and verified generally known numerical mathematical method. In the Matlab programming environment we have created, out of the capacitance model, graphic outputs of dependence of voltage at a given time and position in a VN transformer winding. Owing to disregarding inductivity it is impossible to establish from the model at which particular point maximum stress of longitudinal insulation occurs. This results from the fact that it is precisely inductivity which affects the oscillation period, or rather the amplitude of voltage oscillation. This means that voltage behaviour at time levels between the first and last level will be considerably affected by the absence of inductivity in the approximate equivalent circuit. Their patterns thus do not correspond with the real status, or rather no free oscillations will occur. With this model we have stipulated initial and terminal voltage distribution, which is not dependent on induction, and the impact of secondary winding is minimal for these cases. Thanks to that we are in a position to stipulate the free oscillations envelope of the so-called values of maximum possible voltage, which may appear in the individual spots of the winding during transitional process depending on the capacitance relations. References [1] K.W. Wagner, Das eindringen einer elektromagnetischen Welle in eine Spule mit Windungskapazität, Elektrotechnische Maschinen-Bau, 1915 (in German). [2] A. Miki, T. Hosoya, K. Okuyama, A calculation method for impulse voltage distribution and transferred voltage in transformer windings, IEEE Transaction on Power Apparatus and System PAS-97 (1978) 930e939. [3] A. Veverka, B. Heller, Surge Phenomena in Electrical Machines, London Ilife, London, UK, 1968. [4] Eugeniusz Jezierski, Transformátory: Teoretické základy, Academia, Praha, 1973 (in Czech). [5] Bachelor thesis: Kokes Dalibor, Modification Winding Transformers with Respect to Impulse Stress, CTU, Praha, 2012, Supervisor Jan Mikes (in Czech). [6] M. Bagheri, M. Vakilian, A. Hekmati, R. Heidarzadeh, Influence of electrostatic shielding of disc winding on increasing the series capacitance in transformer, Power Tech, 2007 IEEE Lausanne (2007) 1780e1784. [7] A. Massarini, M.K. Kazimierczuk, Self-capacitance of inductors, IEEE Transactions on Power Electronics 12 (4) (1997) 671e676. [8] G. Grandi, M.K. Kazimierczuk, A. Massarini, U. Reggiani, Stray capacitances of single-layer solenoid air-core inductors, IEEE Transactions on Industry Applications 35 (5) (1999) 1162e1168. [9] L. Dalessandro, F. Da Silveira Cavalcante, J.W. Kolar, Self-capacitance of highvoltage transformers, IEEE Transactions on Power Electronics 22 (5) (2007) 2081e2092. [10] L.M. Popovic, New method for calculation of series capacitance for transient analysis of windings, MELECON 98., 9th Mediterranean Electrotechnical Conference, 2 (1998) 1042e1046. [11] V. Pankrác, The algorithm for calculation of the self and mutual inductance of thin-walled air coils of general shape with parallel axes, IEEE Transactions on Magnetics 48 (5) (2012) 1875e1889. [12] V. Pankrác, Generalization of relations for calculating the mutual inductance of coaxial coils in terms of their applicability to non-coaxial coils, IEEE Transactions on Magnetics 47 (11) (2011) 4552e4563. [13] S.M.H. Hosseini, M. Vakilian, G.B. Gharehpetian, Comparison of transformer detailed models for fast and very fast transient studies, IEEE Transactions on Power Delivery 23 (2) (2008) 733e741.
J. Mikes, D. Kokes / Journal of Electrostatics 71 (2013) 533e539 [14] Y. Li, G. Liu, L. Zhang, Z. Lin, Transformer Winding Deformation Diagnosis Using Middle Band Frequency Response Analysis Solid Dielectrics, ICSD ’07, IEEE International Conference on Solid Dielectrics, UK (2007) 677e680. [15] L. Guishu, S. Haifeng, Z. Xile, C. Xiang, Modeling of transformer windings under very fast transient overvoltages, IEEE Transactions on Electromagnetic Compatibility 48 (4) (2006) 621e627. [16] Y. Yang, Z. Wang, Ch. Shao, A circuit model in a wide frequency range for power transformer and analysis of its characteristics, APPEEC 2009, AsiaPacific Power and Energy Engineering Conference (2009) 1e5. [17] M. Popov, L. Sluis, R.P.P. Smeets, L. Roldan, Analysis of very fast transients in layer-type transformer winding, IEEE Transactions on Power Delivery 22 (1) (2007) 238e247. [18] G.B. Gharehpetian, H. Mohseni, K. Moller, Hybrid modelling of inhomogeneous transformer winding for very fast transient overvoltage studies, IEEE Transactions on Power Delivery 13 (1) (1998) 157e163. [19] G. Hoogendorp, M. Popov, L. van der Sluis, Application of hybrid modeling for calculating interturn voltages in transformer windings, IEEE Transactions on Power Delivery 24 (3) (2009) 1742e1744.
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[20] G.R. Gurumurthy, M.Z.A. Ansari, J. Amarnath, Reduction of voltage stresses in power transformer windings due to surges, CEIDP ’05, 2005 Annual Report Conference on Electrical Insulation and Dielectric Phenomena (2005) 51e54. [21] C.L. Bak, K.E. Einarsdottir, E. Andresson, J.M. Rasmussen, J. Lykkegaard, W. Wiechowski, Overvoltage protection of large power transformersea reallife study case, IEEE Transactions on Power Delivery 23 (2) (2008) 657e666. [22] K. Karsai, D. Kerényi, L. Kiss, Large Power Transformers, Académiai Kiadó, Budapest, 1987, ISBN 963-05-4112-2. [23] Mayer Daniel, Elektrodynamika V Energetice, BEN e technická literatura, Praha, 2005, ISBN 80-7300-164-0 (in Czech). [24] K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, Cambridge University Press, Cambridge, 1994. [25] Finite Differences Technische Universität Wien: Institute for Microelectronics [online] [cit. 2012.07.13]. Available from:, http://www.iue.tuwien.ac.at/phd/ heinzl/node27.html. [26] A. Massarini, K. Kazimierczuk, G. Grandi, Lumped parameter models for single- and multiple-layer inductors. Power Electronics Specialists Conference, 1 (23e27) (1996) 295e301.
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Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Arrangements of transformer winding with respect to impulse stress Jan Mikes a, *, Dalibor Kokes b a b
Czech Technical University in Prague, Faculty of Electrical Engineering, Technická 2, 166 27 Prague 6, Dejvice, Czech Republic Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering, Br ehová 7, 115 19 Prague 1, Czech Republic
a r t i c l e i n f o
a b s t r a c t
Article history: Received 31 August 2012 Received in revised form 10 December 2012 Accepted 10 December 2012 Available online 21 December 2012
Danger and stress posed to transformer winding through overvoltage still represent a hotly debated and hitherto unresolved technical issue the designers and operators of high-performance equipment have been grappling with. The actual impossibility of accepting all the real parameters of a transformer in its substitute model leaves considerable space for its constant improvement and modifications. Accepting the surrounding phenomena and properties of the transformers gives rise to complex situations and difficulties in the process of solving the model. Models tackling some of the issues pertaining to circuit models or electromagnetic field models have been developed on a long-term basis. Another issue in hand is the very complexity of the process of solving a model. This study introduces a model accepting solely the capacitance influences of transformer components, using the methods derived from the theory of LaxeWendroff’s and LaxeFriedrich’s approximation of differential equations of the hyperbolic type for the solution of the respective equations. It does not represent solutions for all the parameters of a transformer, but provides an overview of the size of the initial impulse stress of transformer winding, doing so with adequate accuracy. Ó 2012 Elsevier B.V. All rights reserved.
Keywords: High voltage transformers Power transformer protection Transformer windings Atmospheric discharge Voltage distribution Longitudinal capacitance of transformer winding Lateral capacitance of transformer winding
1. Introduction Dynamic interactions in transformer winding follow either the distribution of the electric field and overvoltage phenomena in the winding, at the entry of the surge, or the distribution of the power field and the mechanical behaviour of transformer winding during various types of short-circuit. The first type is designated as fast, the other one as slow. The study deals with the first type of interactions. Examination of overvoltage relations in the transformer winding has been the subject of innumerable studies. Modern findings in the field of mathematical analysis and numerical mathematics have facilitated specification of the physical model under scrutiny. The problems concerning a substitution transformer diagram cover its discrete model analogous, in circuit models, to electric wiring where longitudinal capacitance, eventually resistivity of the conductor used (that is, however, frequently ignored) operates between the turns. This fundamental model was published in 1915 by K. W. Wagner and all the subsequent theories proceed therefrom [1]. The first works stemmed from the model of a single-layer coil without iron, which made it possible to perform certain predictions
* Corresponding author. E-mail addresses: [email protected] (J. Mikes), kokesdal@fjfi.cvut.cz (D. Kokes). 0304-3886/$ e see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.elstat.2012.12.015
analytically, further studies stemmed from the gradually more complex physical models, with numerical methods being incorporated into their behaviour step by step. In ref. [2] authors explained that the role played by the iron core in the response of an impulse-stressed winding is negligible. However, even modern studies proceed from a relatively heavily simplified configuration of the physical model. In methodological terms, two approaches may be distinguished in the physicalemathematical description of the overvoltage phenomena. The first consists in the construction of the so called field model, i.e. in the formulation of the electromagnetic field in the sphere of winding as a marginal assignment for the partial differential equations of the type of wave equations in which the vector magnetic potential, less often the vector electric potential, figures most frequently as the unknown quantity. In numerical terms, the solution of this 3D, eventually 2D, assignment can be managed quite satisfactorily by applying a suitable commercial program (evidently based on FEM e Finite Element Method), but it is not easy to determine the appropriate boundary conditions. The other concept is based on the layout of the so called circuit model, i.e. formulation of a system of ordinary differential equations for a locally discretized circuit comprising elements R, L, C for the numerical solution of which one can use some standard numerical method. This type of solution poses the problem of precisely
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determining the parameter R, L, C, and also the process of setting up a 3D circuit model runs up against certain difficulties. However, in both types the numerical solution itself is accompanied by the necessity of coping with the reliability of the obtained results (especially stability, convergence etc.). 2. Justification of the choice of the problems under scrutiny and current status of solution Considerable attention has been devoted to impulse stress of electric machines on a long-term basis. More precise, more computationally complex as well as less challenging and also less accurate mathematical models have been developed, based on the available mathematical theories (since the early 20th century). Owing to what is a frequently inaccurate and less credible way of stipulating the input parameters of the winding in the model there have emerged diverse simplified notions of voltage distribution in a transformer. Quite frequently, this issue is resolved on the basis of single-layer coil. The resulting solutions are then extended to cover the entire winding as well as multiple-winding transformers. In terms of voltage distribution, transformer’s multiple-layer coil e without adequately thorough knowledge of its parameters e constitutes such a complex task that very small congruence may be anticipated if such parameters are neglected. Former Czechoslovakia (from the 1950s until 1989) figured among the leading countries generating valuable mathematical models of single- and Mate na. multiple-layer coils e see works by A. Veverka, B. Heller, S. Since these studies could not be published abroad before 1989 they now represent a valuable biography for the topic that has not yet been reflected abroad. The most important specialist studies on the subject are as follows [3e5]: (in Czech). In addition, a large number of contributions was published in the journal Elektrotechnický obzor (active in 1910e1989) by A. Veverka and B. Heller. There are details that were not possible to publish in ref. [3]. However, both academic, business and manufacturing centres have been returning to the subject of impulse stress in the past decade. For their part, the major manufacturers of transformers and reactance coils strive to have the possibility of predicting computations through which they could declare e prior to designing a transformer e its resistance to impulse stress (most frequently an impulse of 1.2/50 ms). Up to now resistance has been verified solely by recording oscilloscopic response from the taps of the transformer winding when leading the impulse in to its input clamp. This gives rise to temporalespatial behaviour of the voltage wave in which the spot of maximum coil stress is monitored. In view of the limitations of the possibility of calculating all the parameters of the winding, of reckoning precisely with own and mutual inductance in the model and of respecting the impact of the iron core, we have decided to verify the validity of the numerical solution solely while respecting the capacitance reserve model. With this particular model we want to single out the necessity of devoting great care when making conclusions from incomplete or very little valid circuits which represent only limited properties transferred to the entire winding, eventually to multiple-winding systems e threephase transformer, while respecting the impact of all the interlinks involved. As for the entry of voltage impulse to the transformer winding, we can state that voltage distribution along the winding is dependent solely on the winding’s capacitance conditions since inductance in a time interval close to zero may be neglected. The design parameters of the coils of the transformer itself have a highly decisive influence on the initial voltage distribution. Assuming that we know initial and terminal voltage distribution on the transformer winding, it is quite easy to determine the free oscillations envelope, which represents a theoretical maximum
voltage of the insulation in any point of the winding. Much smaller attention has been given to the calculations of the parameters of the longitudinal and earth capacitance themselves, and own and mutual inductance of the coils of transformers than, for instance, to the theories of calculation themselves, and to the proposed numerical solution of wave phenomena in the winding. Such a model yields more objective results than endeavours to capture simultaneously all the influences within a transformer. Among the key studies for the computation of C, K, L and M we can mention, for instance, the following refs. [6e12]. It is possible to trace in the literature two approaches to the studied issues of impulse voltage distribution in a transformer. One of them is the design of a model with concentric parameters; the other consists in observing the significance of distributed parameters. Authors in ref. [13] tend to distinguish models into Fast transient overvoltages (FTO) and Very fast transient overvoltage (VFTO). As for the FTO models, in which a frequency range from 10 kHz f 1 MHz is assumed, many models have already been published, based on the theory of quadrupole [14,15] which are cascaded and calculated by means of the respective computational instruments. Subsequently, most [16,17] of other models are based on the solution of the telegraph equation, solved either analytically or numerically. For the VFTO models, hence models with a frequency over 1 MHz, it is no longer possible to neglect the wavelength of the input high-frequency impulse, and the circuits are resolved by means of distributed parameters e the most frequently used methods are then the hybrid calculation methods where parts of the winding are calculated as concentric e for instance for lower frequencies where the influence of conductivity prevails, and as distributed ones for moments when capacity is of explicit importance within a circuit. The models with concentric parameters reduce the calculation solely to predetermined points in the winding; it is impossible to monitor voltage behaviour in any random spot of the winding. The types of the used windings themselves, too, have considerable impact on the design of the relevant models e the most frequently used is the simplified method via simple single-layer coils, but the transformer winding tends to be much more complex; multiple-layer disk and cylindrical windings are often used, or special adjustments, for instance interlaced windings, are utilized. Relevant studies dealing with these subjects may be found in: refs. [18,19]. The last approach to modelling resistant transformers is the application of various limiting elements, overvoltage arresters or the use of an older method involving voltagedependent varnishes [20,21]. Transition from the transformer’s real winding (single-layer) to its discrete model is described in Fig. 1.The classical theory of transformer model for the effects of overvoltage has been discussed in great detail, for instance, in refs. [3,4,22]. For the purpose of our study, we proceed from a simplified solution of the equation describing the overall diagram given in Fig. 2. This set of equations originated as a result of solving the Kirchhoff laws in the described Fig. 2.
v2 u v4 u v2 u þ LK 2 2 LC 2 ¼ 0 2 vx vx vt vt
(1)
vi vu ¼ C vx vt
(2)
v2 u vxvt
(3)
i ¼ K
In case of initial voltage distribution at voltage impulse, a singlelayer transformer winding may be approximated by means of
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method or rather the implicit Wendroff differential approximation set up according to refs. [23,24]. We will intersperse the continuous definition area U of the set of equations (2) and (3) evenly with a temporalespatial network with steps Dx and Dt so that we will discretize a winding d long with an equivalent step Dx, obtaining a one-dimensional geometric network containing N elements, delineated by N þ 1 nodes where N ¼ d/Dx. We discretize the semi-delineated time coordinate t with equivalent step Dt, obtaining a set of discrete time levels t1 ¼ lDt, where l ¼ 0, 1, 2, ... All the spatial derivations contained in the equations (2) and (3) will be replaced with central differences in the following forms.
Fig. 1. Capacitance location in transformer winding.
a substitute capacitance diagram, given specifically in Fig. 2. Inductivities from Wagner’s diagram have no impact on either initial or terminal voltage distribution. See refs. [3,4,22]. Under the assumption of knowing initial and terminal voltage distribution on the transformer winding it is easy to establish free oscillations envelope, which constitutes a theoretical maximum voltage stress of the insulation at any random point of the winding. This procedure is suitable for simple verification of the winding’s resistance to impulse stress. When accepting only capacitance links, as assumed by Fig. 2, the problems are simplified into two hyperbolic differential equations [5,23,24]. Application in the Matlab programming environment thus offers a tentative idea for checking the correctness of any designed winding.
vi l 1 ¼ ik;lþ1 þ ikþ1;l ik;l i vx k 2Dx kþ1;lþ1
(4)
vu l 1 ¼ uk;l þ ukþ1;lþ1 ukþ1;l u vt k 2Dt k;lþ1
(5)
Equation (3) is a mixed partial derivation. For its approximation we have used the following procedure according to the relations (4) and (5), mentioned, for instance, in refs. [24,25].
v2 u v vu v vu ¼ ¼ vxvt vx vt vt vx
(6)
vu l 1 ¼ ukþ1;l1 u vt kþ1 2Dt kþ1;lþ1
(6a)
vu l 1 uk1;jþ1 uk1;l1 ¼ vt k1 2Dt
(6b)
vu l vu l 1 1 l vt kþ1 vt k1 2Dt ukþ1;lþ1 ukþ1;l1 2Dt uk1;jþ1 uk1;l1 v2 u 1 ukþ1;l1 uk1;lþ1 þ uk1;l1 z z u ¼ 2Dx 2Dx vxvt 4DxDt kþ1;lþ1 k
(7)
3. Simulations and results The equations (2) and (3) can be solved analytically, but under the assumption that a nominal impulse is formed by input voltage. Solution in an analytical manner is difficult to attain for other input impulses. That is why methods of numerical calculation respecting, for instance, impulse in the form of 1,2/50 ms have been sought. Analytical solution is derived in great detail in ref. [3]. To solve the set of equations (2) and (3) we will create a temporalespatial network with spatial step Dx and temporal step Dt by means of the method of finite differences. There are several numerical methods to calculate partial differential equations of the hyperbolic type, a case in point may be provided by the FTCS, Laxe Friedrichs, LaxeWendroff and the more complex method Upwinding, which, however, best succeeds in responding to a dramatic change in the function gradient. To calculate the following set of equations we have employed the LaxeWendroff
After substituting (4), (5), (6) and (7) into the equations (2) and (3), we obtain the following relations (8) and (9), and adjust the given expressions into the form (10) and (11).
vu 1 vi 1 uk;l þ ukþ1;lþ1 ukþ1;l u ¼ z vt C vx 2Dt k;lþ1 1 1 ik;lþ1 þ ikþ1;l ik;l i ¼ C 2Dx kþ1;lþ1
i v2 u 1 ¼ z i þ ikþ1;l þ ik;lþ1 þ ikþ1;lþ1 K vxvt 4K k;l 1 ukþ1;l1 uk1;lþ1 þ uk1;l1 ¼ u 4DtDx kþ1;lþ1
(8)
(9)
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1 i þ ikþ1;l þik;lþ1 þ ikþ1;lþ1 4K k;l 1 ukþ1;lþ1 ukþ1;l1 uk1;lþ1 þ uk1;l1 ¼ 4DtDx 1 þ ikþ1;lþ1 i K k;lþ1 1 1 ik;l þ ikþ1;l ¼ ukþ1;l DtDx K
¼
1
DtDx
ukþ1;lþ1
þ
(11)
Sign change corresponds recurrent LaxeWendroff transformation and underlined elements correspond: Boundary value conditions Fig. 2. Initial substitute model for numerical solution.
t ¼ 0; x ¼ 0 : Fðu; i; tÞ ¼ 0; x ¼ d : Fðu; i; tÞ ¼ 0 Initial value conditions
1 uk;l þukþ1;lþ1 ukþ1;l u 2Dt k;lþ1 1 1 i þi i i C 2Dx kþ1;lþ1 k;lþ1 kþ1;l k;l 1 1 1 þ ukþ1;lþ1 i ¼ u i 2Dt k;lþ1 C 2Dx kþ1;lþ1 k;lþ1 1 1 1 ¼ uk;l þ ukþ1;l þ ikþ1;l ik;l 2Dt C 2Dx
t ¼ 0; 0 x d : iðx; 0Þ ¼ 0; uðx; 0Þ ¼ 0 (10)
It will be written as equations (10) and (11) for all k elements of the differential network (k ¼ 1, ., N), obtaining a set of equations with 2.(N) unknowns. We will supplement these equations with algebraic approximation of equations proceeding from marginal conditions. We will obtain a set of 2.(N þ 1) linear algebraic equation (12) [24,25].
Fig. 3. Obtained behaviour of voltage distribution along the axis of winding for a change in K (F/m) and C (F m) e values C and K in order a) K ¼ 20 1012 and C ¼ 17 1010 b) K ¼ 20 1012 and C ¼ 17 1011 c) K ¼ 20 1012 and C ¼ 17 1012 d) K ¼ 20 1011 and C ¼ 17 1012.
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AX lþ1 ¼ BX l
537
(12)
where vector Xlþ1 contains elements uk,lþ1, ik,lþ1 and vector Xl contains elements uk,l, ik,l pro k ¼ Nþ1. By solving equation (12) in software Matlab we will obtain voltage values and flow of time level on the basis of known quantities from the previous time level. Calculation commences from the time level for t ¼ 0, i.e. at the moment of the entry of voltage impulse on the winding. Matrix A and B without boundary conditions were generated from Matlab software. The charts in Fig. 3 contain initial voltage distribution (in red), terminal voltage distribution (in green) and approximate free oscillations envelopes (in blue). An earthed end of the winding is involved in case of selected following courses. The values C and K given in the description of Fig. 3 were gradually inserted, and, using a program written in the Matlab code, we have obtained the relevant graphic outputs. In all the cases, the normalized impulse 1.2/50 ms, modelled by means of exponential curves, was used as an input signal. In our instance, we modelled the impulse according to the exponential relation confirmed by the standards:
uðtÞ ¼ 1; 03$U0 $ ea$i$dt eb$i$dt ; where constants a ¼ 14400 and b ¼ 3500000
(13)
similarly it is possible to model impulses in the form: a) for the leading edge, b) for the trailing edge
Umaxð1 cos2; 6$tÞ a uðtÞ ¼ for 2
0 t tforehead
(14)
and
b uðtÞ ¼ Umax $e½0;014$ðttforehead Þ
(15)
It is evident in Fig. 3 that if the value of longitudinal capacitance K is lower than the value of the earth capacitance C by less than an order, a virtually problem-free initial voltage distribution occurs, and thus the shape of the free oscillation envelopes will be highly favourable. The model was constructed for depicting initial and terminal voltage distribution for homogenous winding at the entry of nominal voltage impulse amounting to 500 V on the winding with axial length of l ¼ 1.5 m. The program serves for tentative verification of maximum possible overvoltage generated within the winding. This is highly demanded for practical purposes, even though it is only an approximate method. Parameters of longitudinal and earth capacitance may be established either by measurement (with high performance transformers this is possible only very approximately indeed) or numerically on the basis of works derived, for instance, from such authors as Massarini and Kazimierczuk [26], who determine longitudinal and earth capacitance by means of the relation derived on the basis of the magnetic field theory. This particular approach is frequently employed in determining parasite transformers [7] and thanks to this procedure values with an accuracy of up to 20% may be obtained [7,9]. One of the most widely used methods of ensuring linear voltage distribution along the winding is to compensate the impact of earth capacitance C by means of metal shields and shades. The function of a capacitance shield consists in mutual compensation of the discharge on the earth capacitance C and discharge on the inserted external capacitance Cex. These discharges then have no impact on the distribution of the residual discharge in the chain of lateral capacitances K/dx. Under the assumption that the lateral capacitance K/dx is equally large in all the high-voltage points of the
Fig. 4. Behaviour of voltage distribution when using a shield at a) 30%, b) 60%, c) 90% of the winding’s axial length for K ¼ 20 1012 and C ¼ 17 1010.
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winding, voltage distribution along the winding will be linear and free oscillations will be eliminated. This shielding may be performed by means of a beam trap. The most suitable behaviour of the initial voltage distribution can be obtained under the conditions whereby the shield is placed around the entire circumference of the winding. However, in a real situation, the beam trap usually does not surround the winding’s entire circumference, but only its part, and that is primarily due to reasons of dimension. Such a shield located along the entire length of the winding must be insulated at the maximum voltage. This condition leads to an increase in the dimension of transformers to such an extent that only parts of the winding are normally shielded. As a rule, a shield is placed at the entry of the winding to increase the longitudinal capacitance K in view of the earth capacitance C. If the shield eliminates the influence of the capacitance C, or the winding’s earth capacitance, then there occurs a linear voltage distribution in the shielded part. Fig. 4 depicts simulated shielding of the earth capacitance C at 30, 60 and 90% of the winding’s axial length from the entry (values K are given in F/m and C in F m). Fig. 5 describes the case when, in the first part of the wiring, we markedly increase the value of the lateral capacitance K by means of external capacities, for instance by a chain of condensers. The first part of the winding’s insulation is then stressed by constant voltage of 500 V and in the remaining section of the winding insulation stress will be lower than in case of the same winding without an external capacity (see Fig. 4a). It is evident that this is a little efficient mode of protecting insulation against puncture. 4. Objective facts The generated model can be used for monitoring initial distribution of a single-layer winding, which e according to prerequisites e would correspond to the relation (16) at the evenly distributed capacitance K and C [3].
u0 ¼
sinhaðl xÞ sinhal
(16)
When reducing the winding’s earth capacitance in a real transformer by using a capacitance shield, it is possible also by increasing the capacitance K in the individual segments of the winding to linearize initial voltage and thus prevent emergence of oscillations. If a single condenser is used the capacitance increases
Fig. 5. Behaviour of voltage distribution when using increased lateral capacitance K at 30% of the winding’s axial length from the entry for K ¼ 20 1012, Kex ¼ 20 1010 and C ¼ 17 1010.
but in terms of the free oscillations envelope the terminal voltage distribution simultaneously deteriorates. The resulting free oscillations envelopes do not make it possible to determine the ideal capacitance connected between the winding’s input and winding’s branch. However, the program created in the Matlab programming environment allows e for the purpose of designing a transformer resistant to overvoltage impulse e to stipulate an optimum value of the inserted capacitance (capacitance chain). 5. Conclusion In our article we have concentrated on an alternative electrical model of the transformer which is easier for calculating, however, it respects all important phenomena which during approaching impulse overvoltage stress the winding. Such a model itself is analytically very complicated to solve. For the solution we have applied and verified generally known numerical mathematical method. In the Matlab programming environment we have created, out of the capacitance model, graphic outputs of dependence of voltage at a given time and position in a VN transformer winding. Owing to disregarding inductivity it is impossible to establish from the model at which particular point maximum stress of longitudinal insulation occurs. This results from the fact that it is precisely inductivity which affects the oscillation period, or rather the amplitude of voltage oscillation. This means that voltage behaviour at time levels between the first and last level will be considerably affected by the absence of inductivity in the approximate equivalent circuit. Their patterns thus do not correspond with the real status, or rather no free oscillations will occur. With this model we have stipulated initial and terminal voltage distribution, which is not dependent on induction, and the impact of secondary winding is minimal for these cases. Thanks to that we are in a position to stipulate the free oscillations envelope of the so-called values of maximum possible voltage, which may appear in the individual spots of the winding during transitional process depending on the capacitance relations. References [1] K.W. Wagner, Das eindringen einer elektromagnetischen Welle in eine Spule mit Windungskapazität, Elektrotechnische Maschinen-Bau, 1915 (in German). [2] A. Miki, T. Hosoya, K. Okuyama, A calculation method for impulse voltage distribution and transferred voltage in transformer windings, IEEE Transaction on Power Apparatus and System PAS-97 (1978) 930e939. [3] A. Veverka, B. Heller, Surge Phenomena in Electrical Machines, London Ilife, London, UK, 1968. [4] Eugeniusz Jezierski, Transformátory: Teoretické základy, Academia, Praha, 1973 (in Czech). [5] Bachelor thesis: Kokes Dalibor, Modification Winding Transformers with Respect to Impulse Stress, CTU, Praha, 2012, Supervisor Jan Mikes (in Czech). [6] M. Bagheri, M. Vakilian, A. Hekmati, R. Heidarzadeh, Influence of electrostatic shielding of disc winding on increasing the series capacitance in transformer, Power Tech, 2007 IEEE Lausanne (2007) 1780e1784. [7] A. Massarini, M.K. Kazimierczuk, Self-capacitance of inductors, IEEE Transactions on Power Electronics 12 (4) (1997) 671e676. [8] G. Grandi, M.K. Kazimierczuk, A. Massarini, U. Reggiani, Stray capacitances of single-layer solenoid air-core inductors, IEEE Transactions on Industry Applications 35 (5) (1999) 1162e1168. [9] L. Dalessandro, F. Da Silveira Cavalcante, J.W. Kolar, Self-capacitance of highvoltage transformers, IEEE Transactions on Power Electronics 22 (5) (2007) 2081e2092. [10] L.M. Popovic, New method for calculation of series capacitance for transient analysis of windings, MELECON 98., 9th Mediterranean Electrotechnical Conference, 2 (1998) 1042e1046. [11] V. Pankrác, The algorithm for calculation of the self and mutual inductance of thin-walled air coils of general shape with parallel axes, IEEE Transactions on Magnetics 48 (5) (2012) 1875e1889. [12] V. Pankrác, Generalization of relations for calculating the mutual inductance of coaxial coils in terms of their applicability to non-coaxial coils, IEEE Transactions on Magnetics 47 (11) (2011) 4552e4563. [13] S.M.H. Hosseini, M. Vakilian, G.B. Gharehpetian, Comparison of transformer detailed models for fast and very fast transient studies, IEEE Transactions on Power Delivery 23 (2) (2008) 733e741.
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[20] G.R. Gurumurthy, M.Z.A. Ansari, J. Amarnath, Reduction of voltage stresses in power transformer windings due to surges, CEIDP ’05, 2005 Annual Report Conference on Electrical Insulation and Dielectric Phenomena (2005) 51e54. [21] C.L. Bak, K.E. Einarsdottir, E. Andresson, J.M. Rasmussen, J. Lykkegaard, W. Wiechowski, Overvoltage protection of large power transformersea reallife study case, IEEE Transactions on Power Delivery 23 (2) (2008) 657e666. [22] K. Karsai, D. Kerényi, L. Kiss, Large Power Transformers, Académiai Kiadó, Budapest, 1987, ISBN 963-05-4112-2. [23] Mayer Daniel, Elektrodynamika V Energetice, BEN e technická literatura, Praha, 2005, ISBN 80-7300-164-0 (in Czech). [24] K.W. Morton, D.F. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, Cambridge University Press, Cambridge, 1994. [25] Finite Differences Technische Universität Wien: Institute for Microelectronics [online] [cit. 2012.07.13]. Available from:, http://www.iue.tuwien.ac.at/phd/ heinzl/node27.html. [26] A. Massarini, K. Kazimierczuk, G. Grandi, Lumped parameter models for single- and multiple-layer inductors. Power Electronics Specialists Conference, 1 (23e27) (1996) 295e301.