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- 1. New approach to the analysis of impulse voltagedistribution in transformer windings0.HonoratiE. SantiniIndexing terms: Transformers, Modelling and their degree of flexibility is rather limited. Inversion Abstract: A new calculation method for the of the inductance matrices can produce marked errors, analysis of transient voltage phenomena in HV while poor reliability in the assumed values for the induc- power transformers is presented. Consideration is tive and resistive parameters can lead to uncertain given to the electric, magnetic and current fields in results. the windings. An equivalent circuit is derived; this In this paper a new approach is presented which may consists in the connection of two networks, one be considered an alternative to those described above. magnetic the other electric, whose components The model is again of the lumped parameters variety, but with their numerical values are deduced directly is derived through the theory of magnetic networks, from the geometry of the transformer and from its extended here to account for the capacitive phenomena. electrical parameters. Application of the ensuing The model provides the transient voltages within the mathematical model to a real case of a 132 kV, windings, and its main advantage is that there are no 50,000 kVa power transformer is discussed. mutual inductances in the equivalent circuit. The model’s inductive and capacitive parameters can be calculated through the geometry of the windings and the magnetic structures. The approach is suitable for any configuration1 Introduction of the windings and core, and permits analysis of the transient voltage inside the windings (open or short-The matter of overvoltage in windings has been a basic circuited) resulting from any waveform overvoltageproblem in the design of transformers for many decades applied at the terminals.[l]. Many important studies have been carried out with The approach adopted is first defined, then the algo-the aim of determining the distribution of the impulse rithms for studying the electrical and magnetic pheno-overvoltages and the resonance frequencies of the wind- mena are presented. A mathematical description is givenings [2, 3, 41. The question is still far from being solved. of the problem in the state space, and finally an applica-The relevant phenomena are so complex that accurate tion to a real case is detailed.modelling is very difficult. As power and voltage increase,the electrical stresses become so high that greater accu-racy is required in the analysis of the space and time 2 Derivation of the modelvariation of the impulse overvoltages, as well as the volt-ages induced in the other windings [S, 61. 2.1 Approach and schematisarion of the system The analysis of impulse overvoltages has usually been The following points are the basis for establishing theconducted on a distributed capacitance model, so that mathematical model:transmission line theory is applicable [7]. This allows (a) The phenomena considered include a current fieldqualitative evaluation of the phenomena under consider- within the windings, an electric field within the dielectricsation; series or derived capacities being the parameters space and a magnetic field throughout the entire space.which upgrade or degrade the non-uniform distribution (b) The continuous systems represented by the fieldsof the voltage in the windings. This model provides are reduced to lumped parameters systems, consisting ofsimple formulas that are used for the design of many interlinked networks of electrical and magnetic two-transformers. However, this method suffers from a dis- terminal circuits: the relevant approximations are left to advantage in that it cannot represent the voltage dis- the designer’s ingenuity, depending on the required accu- tribution at times following the application of the racy. impulse, and therefore can only provide the initial dis- (c) Each winding is divided into sections; each section* tribution. is replaced by a two-terminal electrical circuit, made up More sophisticated approaches have recently been by a resistance and an emf produced by the time change introduced. In these, turns or groups of turns are rep- of the magnetic flux in the branches linked with the resented by finite cells, each defmed by the values of its section. The sections should be chosen so that a linear resistance, self inductance and mutual inductance to the variation of the voltage from the beginning to the end of other cells, as well by the capacity to earth and to the the section may be assumed. The two-terminal electric other cells [8, 91. The solution programs are very large circuits constitute the electric network. Paper 7201C (P7), received 10th October 1989 * Here ‘section’ stands for ‘element’: one or more pairs of discs, one or The authors are with the Dipartmento di Ingegneria Elettrica Uni- more layers or some turns of a helix can be considered as a winding versitl di Roma ‘La Sapienza’,Via Eudossiana 18,00184 Rome, ltaly section. IEE PROCEEDINGS, Vol. 137, PI. C,No. 4, JULY I990 283
- 2. (4 magnetic field is replaced by branches of flux The (ii) nonlinearity and the hysteresis of the magneticpaths, each of which is considered as a two-terminal mag- materialsnetic circuit characterised by its reluctance. These two (iii) losses due to eddy currents in the conductorterminal magnetic circuits constitute the magnetic materialsnetwork, where the mmf generators due to the currentsflowing in the windings are to be inserted. In order to simplify the model, the losses causing (e) The electric and magnetic networks are interlinked. damping were taken into account by an adequateThe linkage determines the mmf in the magnetic network increase in the ohmic resistances of the windings. For theand the emf (or magnetic flux linkages) in the electric transformer examined in the example, calculation of thenetwork. The generators need not be present in both net- frequency variation and the experimental pointers indi-works and the networks consist of resistances and reluc- cated that a one hundredfold increase would be advis-tances. able. cf)The phenomena due to the electric field: (i) those within each winding section (capacitive coup- 2.3 Capacitance networklings between internal turns) For the phenomena associated with the electric field, the (ii) those between each section and the adjacent sec- continuous system is replaced by a set of capacitorstions which is equivalent within the prescribed approximation (iii) those between the core and the facing sections limits. On the basis of point (c) above, it can be assumedcan be replaced by capacitances derived from the termin- that the voltage distribution is linear in each windingals of each two-terminal circuit. This results in a capac- section, so the potentials of each two terminal electric cir- itance network the nodes of which are the same as the cuits corresponding to the section unequivocally define nodes of the electric network the field distribution within the section. (g) In accordance with well known theorems, the mag- To analyse the influence of the electric field between netic network can be replaced by its dual network, with the different sections, the electric field is replaced by a set parameters interpreted as inductances (each correspond- of capacitors shunted between the two terminals of each ing to the permeance of a branch of flux path). This electric circuit and between adjacent circuits (electrically network is hooked up to nodes of the electric network coupled) according to the following procedure. through ideal transformers. Electric field within each winding section and series In this manner a single network of resistances, induc- capacitancetances and capacitances (without mutual inductances) is If there is a voltage difference between terminals A and Bobtained; this can be analysed by means of a specific of a winding section, then there is an electric field withinapproach. it and an energy WEassociated with it; this can be calcu- lated on the basis of the geometry of the system assuming a linear distribution of the potential. The value of the2.2 The magnetic network and its equivalent series equivalent capacitance C , to be shunted between inductance network the terminals A and B can be determined on the basis ofModelling a magnetic field through time-invariant flux the equation:paths may seem impossible, but in the case of trans-former structures (presence of ferromagnetic core andconcentric windings with small separation spaces) some npaths can be identified for the magnetic flux and theproblem becomes easier. where V, and V, are the voltages of the terminals A and It is known that an accurate analysis of the magnetic B, E the permittivity and E the electric field within thefield through numerical methods (finite differences or winding region (instead of the volume integral it isfinite elements) is equivalent to replacing the field with an common practice to add the energies in the capacitiveinvariant network of permeances. It is therefore reason- couplings between adjacent turns).able to assume that a grid with a limited number ofnodes can represent the magnetic field with sufficient Electric field towards earth and capacitances shunted toaccuracy, automatically taking into account the different earthdistribution of the flux among the paths, as magneto- The coupling between the two terminal circuits represent-motive forces vary in time. ing a winding section and the core is modelled as illus- On the basis of the principle of duality, a magnetic trated in Fig. 1, where it is assumed that the earth is atnetwork can be replaced by an equivalent circuit(consisting of inductances and accessible on the outsidethrough the insertion of ideal transformers). This consti-tutes the model for the flux-linkage/current relations tobe introduced into the electric network so as to represent the emf and to account for the effects of the magneticfield. The primary terminals of the ideal transformers, with the resistances in series, must then be connected in series or in parallel, exactly like the considered winding sec- tions. Reference 10 should be consulted for a complete ‘L L/ I discussion of the subject. This approach allows the intro- o duction into the model of: Fig. 1 Circuit sirnulatiom (i) effects of the partial linkage which occur within the a Capacitive coupling of section lo earth windings b Voltage distribution across section 284 IEE PROCEEDINGS, Vol. 137, Pt. C, No. 4, JULY 1990
- 3. zero potential and the terminals A and B have different causes no alteration in the circuit simulation or in thepotentials. solution of the final system. The numerical value of the capacitances C,, C, and C,can be determined by imposing equality between the 3.1 Mathematical model of the capacitive networkenergies stored and is obtained by means of analysis of The capacitive network and its N constituent nodes arethe electric field. considered. Once the reference node has been chosen Let CG denote the capacity to earth of the whole (normally the earth), the matrix equation describing thesection under uniform voltage conditions between the equilibrium for the remaining N - 1 nodes is:terminals, and let V and V denote the potentials at the , ,section terminals: the energy associated with the electric pcv= I + U (4)field is then . where D = dldt: , , C is a square matrix of order (N - 1, N - I), con- W = tcG(fv: E + + fv v,) (2) structed according to the classic rule of formation forfor the capacitors of Fig. 1 this is nodal matrices; V i s an N - 1 vector whose components represent the = + WE tc,v: tc, v: tC,(V - V,)2 + (3) node voltages:By imposing equality of the two previous expressions for I and ri are N - 1 vectors and represent the knownany value of V and V ,then , terms of eqn. 4, i.e. the currents in the primaries of the ideal transformers and those injected by current gener- c , = c, = t c , ; c, = - fc, ators outside the transformer, respectively.No physical meaning is attributed to the negative capac-itance C, since it is the result of a process of synthesis. 32 Mathematical model of inductive network The mesh-method algorithm is used for writing the equa-Electricfield between two winding sections and capacitance tions of the inductive network. To apply this method, thebetween sections number and composition of the independent meshesThe electric field in the insulation space between two must be determined; from them the number of the N ,winding sections is approximated by the circuit shown in mesh currents is obtained. The meshes are then chosen inFig. 2. Through a procedure analogous to the one above, such a way that the secondaries of the ideal transformers appear in one mesh only. The descriptive equation is of the type: pLIM = EM (5) where p = d/dt; L is a square matrix of (N,, N,) order, formed in accordance with the classic rule of mesh-method analysis; I is the (N,) vector of the mesh currents; , E, is the ( N M )vector of the known terms (the second- U 1 b ary voltages of the ideal transformers).Fig. 2 Circuit simulations By transferring the resistances of the winding sections(1Capacitive coupling between two sections to the secondaries of the ideal transformers, the meshb Voltage distribution across two sections equation is modified as follows: pLIM + RI, = E, (6)equality of the energy of the electric field and that storedin the capacitors yields the following capacitance values: where matrix is a diagonal quare matrix Of Order ( N ? , N M ) ,which only has positive values in the elements C1 - c - - 2 - - i C D which refer to meshes where the secondary of an ideal transformer is present. c-C- 5 - 6-fCD 4 Global mathematical modelwhere CD denotes the shunted capacitance between two At this point eqns. 4 and 6 written for the capacitive andadjacent winding sections, in uniform potential condi- inductive networks must be assembled. The componentstions in each section. representing the connection between the two networks are ideal transformers, which are described by the equa-3 Integral model tions: EL = KEM (7)The final network consists of the coupling of the induc-tance, capacitance and resistance networks, through the IM = KILideal transformers described in the construction of the where E , and IL are the primary quantities, and theinductive network. The structure of the final network transformation matrix K is diagonal of order ( N , , N M ) .suggests the elaboration of a mathematical model which The linkage between the nodal voltages of the capac-allows the networks to be modelled separately and subse- itive network and the voltages on the primaries of thequently joined up. ideal transformers is expressed by the relationship : The parameters representing the winding resistancesshould be inserted in series with the primaries of the ideal Et= -M,V (9)transformers. To facilitate the numerical solution it is in which the matrix M is a connection matrix of orderadvisable to consider the resistive elements as being (N - 1, N M ) ;the values of its coefficients can be 1, 0, orinserted in the secondary of the ideal transformers; this - 1. IEE PROCEEDINGS, Vol. 137, Pt. C,No. 4, JULY 1990 285
- 4. A similar equation can be written for the currents: The schematisation involves two different levels of approximation : I = MIL (10) (a) every whole winding of the transformer is con-Starting from eqns. 4 and 5 and taking into account eqns. sidered as a section9-12 then: (b) windings are divided into three sections, to take account of the presence of flux paths which transverse to pCV = MIL (1 1) the windings pKLKrM = -M , V - KRKr, (12) For approximation a the previously presentedand letting: approach and the modelling with electric and magnetic networks leads to the Fig. 4 circuit; with appropriate KLK = L , KRK= Rthe system can be written in the final form: A%= BX + U; %= d X / d t (13)For the time integration of this system of ordinary differ-ential equations, a fourth-order multi-step method basedon Runge-Kuttas algorithm has been set up; this allowsa check of local error based on the distance in the state Fig. 4 Division of windings in single sectionspace between the tips of the vector solution calculatedwith step h and that calculated with step h/2. simplification, this is derived by schematisation of the magnetic field indicated in Fig. 3. The constraints inher-5 Application of method ent in the configuration of the winding connections must be duly introduced. The ensuing circuit has beenThe method described has been applied to calculating the analysed by means of a computer program for thetime behaviour and internal distribution of the voltage in various connection configurations. In all the tests, thea transformer subjected to an impulse voltage, using a free terminal of the regulating winding was supplied withdedicated computer program. The features of the 3-phase the impulse voltage described by the functiontransformer under consideration are given in the Appen-dix. v,(t)= V2,(e-" - e-@) (14) Computer simulation has permitted determination of with a = 1.72 x lo4 and fi = 1.54 x lo6.the configuration which is most affected by the over- In the following, V, ( t ) denotes the voltage difference atvoltages, as well as the time variation of the voltage gra- the terminals of the HV, and V,(t) the voltage across thedient in the windings. LV terminals (if not short-circuited). The transformer has concentric windings. Fig. 3 shows The simulation was carried out up to a maximum timethe column of the core around which they are wound. of 80 ps, for all the cases described above.Starting from the iron core, there is the low-voltagewinding (LV), the high-voltage winding (HV) and the 5.1 Division of windings into one sectionhigh-voltage regulating winding (HVR). The study is con-fined to just one phase, since this does not lead to any Configuration with H V R in addition and LVsignificant errors. short-circuited Figs. 5-11 illustrate the voltage in the windings for several conditions as a function of time. Fig. 5a illustrates the voltage of nodes 7 and 9 of the I circuit in Fig. 4. These nodes represent the terminal of the regulating winding directly impulsed by the overvoltage and the other terminal of the regulating winding, coincid- ing with the HV terminal, respectively. I In Fig. 5a it can be seen that the voltage on the HV is I characterised by an alternating component with a fre- II ~~ $" - quency of about 50KHz superimposed on a unidirec- tional damped component. The ratio between the peak value of the voltage across the terminals of HV, V,, ,and the peak value of the input voltage, V,,, is about 1.36. t Fig. 5b shows the traces of VI and V, in the experimental I test. I I Confguration with H V R in addition and LV I open-circuited I Fig. 6 illustrates the results obtained in the analysis of this case. It is noted that the ratio between the impulse Fig. 3 Transformer schematisation maximum and the maximum value of the voltage reached 286 IEE PROCEEDINGS, Vol. 137, Pt. C , No. 4, JULY 1990
- 5. by the end of the main winding attains a value of 1.40 in 40 kHz. Fig. 7 b shows the traces of VI and V, in thethis case. The ratio between the peak value of the voltage experimental test.applied to the line terminal and the maximum valuereached on the LV, V,, , is about 1 1 , which is thus lessthan the transformer turns-ratio (about 20). I . 0 IO 20 30 40 50 60 70 80 t . P a b Fig. 7 Comparison o uoltages f HVR in subtraction,LV short-circuited a Calculated b Expenmental ( I hor. div. = IO ps) Configuration with H V R in subtraction and LV b open-circuitedFig. 5 Comparison o voltages f Fig. 8 gives the results of the numerical simulation. TheHVR in addition,LV short-circuitted(1Calculated frequency of the oscillation is about 44 kHz in this case.b Experimental (1 hor. div. = lops) The V,,/V,, ratio is about 1.76, which is the highest value. The ratio between V I , and V,, has also increased with respect to the case of HVR in addition, reaching a value of about 12. N 0 i - 50 80 t.15 t.P Fig. 8 Calculated voltage to earthFig. 6 Calaculated uoltage to earth HVR in subtraction,LV open-circuitedHVR in addition,LV open-circuited 5.2 Division of windings into three sections: ConJguration with H V R in subtraction and LV configuration with HVR in addition, LV short-circuited short-circuited Fig. 7 a shows the time variation of the voltages. The By dividing each winding of the transformer into three ratio V,,/V,, in this case attains a value of 1.67, and the sections, the magnetic field is schematized as indicated in frequency of the alternating component of VI is about Fig. 9a and the inductive network of 9b is obtained. The 1EE PROCEEDINGS, Vol. 137, Pt. C, No. 4, JULY 19W 287
- 6. inductances deriving from branches of flux paths in iron be noted that the V,,/V,, ratio in Fig. 11 is approx-have been shown in black, the inductances which come imately 1.39, the difference being about 2% comparedfrom flux paths in air are in white and the inductances with the same ratio determined by considering eachwhich come from flux paths inside the windings are indi- winding as a single section. Fig. 12 shows the voltage gra-cated by hatched lines. N 0 > a 11 1 0 t.PS Fig. 11 Calculated voltage to earth HVR in addition, LV short-circuited Winding divided into three sections 18 17 17 16 16 15 bFig. 9 Diuision ofwindings(I Flux paths b Inductive network Fig. 10 shows the capacitive network, which must beconnected to the inductive network of Fig. 9b at nodesbearing the same number. The constraints deriving from winding connectionsmust be imposed on the network. For the case of HVR inaddition and LV in short circuit, Fig. 11 shows the timevariation of the voltages at the HVR and the HV termin-als, practically coinciding with those of Fig. 5a. It should 501 Fig. 12 Calculated uoltages in H V R D Section a e Smtian cFig. 10 Capacitive network b Section b 288 IEE PROCEEDINGS, Vol. 137, Pi.C, No. 4, JULY 1990
- 7. dient in the three HVR sections, and Fig. 13 the voltage sents greater difficulty, but its permeance has less influ- gradient in the HV sections. The magnitude of the greater ence ; voltage gradient in the first section of the main winding is (c) The capacitance network has parameters which particularly evident in the latter. depend on the geometry of the transformer, in particular on the distance between windings and between turns; (4 Damping elements, which are very important for analysing the time behaviour of the phenomena, can be introduced into the model, the easiest way being to ensure an adequate increase in the winding ohmic resist- ance. The approach also allows a more accurate simula- tion of losses by introducing other damping parameters; (e) The model can be used for any case of rapidly varying electromagnetic transients, so as to include capacitive phenomena, core magnetisation phenomena N (non linearity) and effects due to the other windings (open or short circuited); for these windings the model deter- > v mines the internal voltage distribution. The computer program proposed to solve the network was designed to suit the structure itself and proves to be -25- particularly eficient, providing results very close to those experimentally measured. An examination of the results shows how the schema- tisation of the windings in one single section allows the time variation of the voltage at the terminals to be deter- mined with sufficient accuracy. If the variation of the voltage gradients in the most stressed zones is required, these must be divided up, albeit into only a limited number of sections. I 7 Acknowledgments -25 The authors would like to express their thanks to O.E.L. for the experimental tests and the features of the trans- former presented in the example. 50 I 8 References 1 HELLER, B., and VEVERKA, A.: ‘Surge phenomena in electrical machines’ (London, 1968) 2 McWHIRTER, J.H., FAHRNKOPF, C.D., and STEELE, J.H.: ‘Determination of impulse stresses within transformer windings by5 20 10 20 30 40 50 60 70 80 computer’, AIEE Trans., 1957, PAS-75, pp. 1267-1273 3 McNUTT, W.J., BLALOCK, T.J., and HINTON, R.A.: ‘Response of transformer windings to system transient voltage’, IEEE Trans., 1974, PAS-93, (2), pp. 4 5 7 4 6 4 DEGENEFF, R.C., McNUTT, W.J., NEUGEBAUER, W., Fig. 13 Calculated Doltages in H V PANEK, I. McCALLUM, M.E., and HONEY, C.C.: ‘Transformer LIscctiona - - resuonse to system switchinn voltaaes’. IEEE Trans.. 1982, PAS- b Section b lOi, ( ) pp. 1457-1465 6. e Seciionc 5 MUSIL, R.J., PREININGER, G., SCHOPPPER, E.., and . ~. WENGER. S.: ‘Voltage stresses oroduced bv amriodic and oscil- lating system overvoltages in transformer windings’, IEEE Trans., 0 Conclusions 1981, PAS-100, pp. 431438 6 ADIELSON, T., CARLSON, A., MARGOLIS, H.B., and HALL- The main advantages offered by the approach described ADAY, J.A.: ‘Resonant overvoltages in EHV transformers’, IEEE Trans., 1981, PAS-100, pp. 3563-3570 are : 7 STEIN, G.M.: ‘A study of the initial surge distribution in concentric (a) The number, type and dimensions of the sections transformer windings’, IEEE Trans., 1964, PAS-83, pp. 877-892 into which the windings are divided can be varied as 8 DEGENEFF, R.C. : ‘A general method for determining resonances desired, depending on the ingenuity of the designer, in in transformer windings’, IEEE Trans., 1977, PAS-%, pp. 423-430 order to increase the precision of the calculation and con- 9 MIKI, A., HOSOYA, T., and OKUYAMA, K.: ‘A calculation method for impulse voltage distribution and transferred voltage in sequently the complexity of the model; transformer windings’, IEEE Trans., 1978, PAS91, pp. 93&939 (b) The network of inductances derived through the 10 HONORATI, 0.. and SANTINI, E.: ‘Response of transformer theory of magnetic networks avoids introducing mutual windings to transient voltages: The magnetic-electric network inductances; the inductive parameters depend on the per- method, modeling and application’, to be published. 11 CREPAZ, S., DOGLIO, G., PESCU, G., and UBALDINI, M.: meance of the branches of flux paths and can therefore be ‘Ripanizione della tensione ad impulso nei trasformatori con avvol- directly computed on the basis of the geometric features gimento di regolazione di linea’, L’Energia Elettrica, 1982, 11, pp. of the windings and of the core; the flux in the tank pre- 455473 IEE PROCEEDINGS, Vol. 137, Pt. C , No. 4, JULY I990 289
- 8. 9 Appendix Capacitances to earth, pF LV 3250 HVR 800Features of the transformer on which the simulation was Binary short circuit HVR-HV 0.6867carried out. inductances, H HVR-LV 1.2655Power, KVA 50 OOO HV-LV 0.4269Rated voltage, KV 132 10.1% 140 Number of turns HVRSeries capacitances, pF HVR 400 HV 1372 HV 370 LV 12 LV 10Capacitances between HV-HVR 1200 The experimental test was obtained by applying waveswindings, pF HV-LV 1300 from a recurrent surge generator to the transformer. IEE PROCEEDINGS, Vol. 137, Pt. C , No. 4, JULY 1990

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