Lumped parameter electromagnetic modelling approach for transient analysis in EHV transformers

Transcription

1 ISSN , England, UK World Journal of Modelling and Simulation Vol. 8 (1) No. 3, pp Lumped parameter electromagnetic modelling approach for transient analysis in EHV transformers Dilip Debnath 1, Abhinandan De, Abhijit Chakrabarti 3 1 School of Electrical Engineering, VIT University, Vellore-6314, India Department of Electrical Engineering, Bengal Engineering & Science University, Howrah 71113, India 3 Jadavpur University, Kolkata 73, India (Received January 1 11, Accepted March 6 1) Abstract. This paper presents a novel lumped parameter high frequency circuit model of transformer using EMTP analysis of transients within windings. The model which has been developed based on transformer geometry, configuration and design parameters, can adequately reproduce frequency response characteristics of original transformer. The response characteristics of EMTP model presented in this work has been validated with original transformer s laboratory test results. The behavioural responses of windings were studied under 1./5 µs standard lightning impulse and fast-front switching surge. It has been observed that developed model successfully reproduced windings response under se transients and results indicate that voltage stresses developed on winding insulation are marginally higher under fast-front switching surge as compared to standard lightning impulse. Keywords: EHV transformer, EMTP, fast-front switching surge, frequency response 1 Introduction The rate of internal insulation failure of grid connected EHV transformers is quite high. This has prompted authors to investigate possible reason for insulation failure in EHV transformers connected to EHV power grids. The studies got more relevance when a series of dielectric failure in several large EHV transformers were reported on American Electric Power (AEP) System [9, 11]. The matter received considerable attention within IEEE Transformer Committee and a working group was formed to investigate phenomena involved. Diagnostic investigations inferred that failures were initiated eir by switching actions in grid or due to system faults which triggered internal natural resonance frequencies of windings of transformer. Thereafter many researchers have tried to develop high frequency models of EHV transformer to study frequency response and distribution of transient high voltages in transformer windings to assess accurate voltage stresses that are likely to occur under surge conditions. Accordingly, insulation structure for windings is decided. High-voltage large power transformer windings are much complicated systems than simple single layer coil systems. Mamatical modelling of such a complicated system is difficult without simplifying assumptions. Moreover, transient response of transformers depends on many factors, like type of winding, manner in which windings are subjected to surge, how or windings are connected, wher grounding of windings are solid or through resistance, degree of non-homogeneity in windings due to local reinforcement of turn insulation, etc. In this paper, a modelling technique for transformer windings has been developed with concentrated resistances, inductances and capacitances that simulate electrical properties and a suitable numerical method of analysis utilizing circuit solution principles has been developed. The major contribution of this paper is simple but accurate approach towards high frequency modelling of EHV transformer windings which can faithfully reproduce Corresponding author. Tel.: address: d.dn@ovi.com. Published by World Academic Press, World Academic Union

2 3 D. Debnath & A. De & A. Chakrabarti: Lumped parameter electromagnetic modelling approach transient characteristics of transformer. The techniques developed in computing lumped inductance, capacitance and resistance of winding disk coils are original contributions of authors and are simple to adopt. The or important contribution of paper is thorough analysis and comparison of winding insulation stresses under standard system transients like lightning and switching. The results of this study should be useful for transformer designers to assess and evaluate magnitude of insulation stresses in transformers under such operating conditions. This paper is organised as follows: In Section, development of equivalent circuit model for transformer has been explained with design data. The consideration for computation of model parameters has been given in Section 3. In Section 4, development of EMTP model of transformer has been discussed. The developed EMTP model has been validated with experimental frequency response in Section 5. Frequency response of 4 kv main and tap changer windings have been analysed in Section 6. In Section 7, voltage stress under standard lightning impulse and fast-front switching surge has been compared. Finally, in Section 8, conclusion has been drawn about accuracy and validity of model. Future scope of work is also included in Section 8. Development of equivalent circuit model for transformer The equivalent circuit is developed on assumption that winding terminals are connected as per IEC specification during impulse tests [1]. Based on transformer geometry and configuration, a lumped parameter high frequency circuit model [5, 15, 17] of concerned 31.5 MVA, 4//33 kv transformer has been developed. The developed equivalent circuit model contains 78 elementary sections, representing 46 main and 3 tap winding disks. The following design data of grid connected three-phase, 31.5 MVA, 4//33 kv; Ynd11 transformer with tap changer winding has been used to calculate lumped parameters for model. Table 1. Design data: 4 kv Winding details Main winding Tap winding Type of winding: Interleaved continuous disk Type of winding: Interleaved continuous disk Number of parallel paths: 1 Number of disks: 46 Number of disks: 3 Turns per disk: 4 Turns per disk: 18 Separation between disks: Separation between disks: (i) 8mm between interleaved disk pairs (i) 6mm between interleaved disk pairs (ii) 1mm between two successive interleaved groups (ii) 8mm between two successive interleaved groups 3 Computation of model parameters Computation of model parameters is based on available design data and relevant dielectric and or properties of insulating materials used in design. Detailed computation methods followed in determination of various model parameters have been presented in Appendix section of paper. Only a summary of se computational methods are discussed in this section. (1) Determination of Series capacitance of each coil Inter-turn as well as inter-disk series capacitances [3, 1] between adjacent coils have been taken into consideration. However, small stray capacitances between one coil to or distant coil have been neglected. () Determination of Shunt capacitance between coils and neighboring eard bodies Shunt capacitance between coils and neighboring eard bodies, that include both metal tank as well as non-impulse windings, have also been calculated. (3) Determination of Shunt capacitances between different winding sections Electrostatic coupling between main as well as tap windings has been considered separately to determine shunt capacitances between different winding sections. WJMS for contribution: submit@wjms.org.uk

3 World Journal of Modelling and Simulation, dissipation Vol. losses 8 (1) in No. 3, capacitor pp. 31-4dielectrics, and eddy current losses formed 33 by low voltage conducting cylinder have been neglected. Table. Design data: Disc coil dimensions 4. Development of EMTP Model for 4 kv Winding 4 kv Main winding Tap winding Mean radius After computation of 65parameter mm values, EMTP 83 mm(electromagnetic Transient Programming Radial width used to simulate and analyze 87.7 mm frequency response 3.9 mmcharacteristic of transformer. Also, Axial height standard 1./5 µs lightning 16. mm impulse and fast-front 14.6 mm switching surges on windings have be Conductor and width compared. A part.4 of mmdeveloped EMTP. model mm containing n number of identical section Conductor shown height in Fig mm 1.6 mm Thickness of paper insulation.75 mm.6 mm capacitor dielectrics, and eddy current 5. Validation losses formed of by Developed low voltage EMTP winding Model in ave been neglected. An example has been added here to demonstrate accuracy of proposed modelling techniq (4) Determination of Self and establish Mutual inductance validity of of results coils presented subsequently in this paper. The incident lightnin MTP Model for Self-inductance 4 kv Winding of disk voltage coils has and been corresponding calculated using neutral rover s current extension wave of to Rosa s transformer formula for were computation of self inductance of laboratory circular disk during coilslightning of rectangular impulse cross-section test on [] actual. Mutual transformer. inductance These between data have been utilize recorded in th parameter disk values, coils has EMTP been calculated (Electromagnetic frequency by Lyle method responses Transient of Equivalent of Programming) whole Filament transformer [1] has, and been (neutral tables current/applied presented by rover impulse voltage). T lyze frequency [8]. In a lumped response parameter characteristic transformer obtained of from model, transformer. developed representation EMTP Also, of model evenly impact closely distributed match of numerous with mutual response inductances and formed fast-front by each switching coil with transformer, surges or on showing coils is windings similar virtuallyresonant impossible. have peak been and Such studied cross distributed over, as mutual shown inductances Fig.. characteristics of t ing impulse f developed have been EMTP lumped model toger containing to form an single number equivalent of identical mutualsections inductance has [6]. Then been after considering inductive The agreement between computed and actual frequency response characteristics, both in coupling between windings, mutual inductances between different distinct winding sections (which are and frequency, establishes accuracy of developed model. not too far away e.g. main and tap windings) has been calculated. eveloped EMTP Model ded here to demonstrate accuracy of proposed n... modelling technique 1 and to results presented subsequently in this paper. C D C D The incident C D lightning C D C D impulse R L R L R L ing neutral current wave of transformer were recorded in HV test ng impulse test on actual transformer. These data have been utilized to obtain C t C t C t whole transformer (neutral current/applied impulse voltage). The results C D C D C D C D C ped EMTP model closely match with response characteristics of D original ilar resonant peak and cross over, as shown in Fig.. C D C D C D C D C D R L R L R L computed and actual frequency response characteristics, both in amplitude s accuracy of developed model. C t C t C t C D C D C D C D C D 6. Frequency response of 4 kv Main and Tap changer Windings The 5 frequency responses of main and tap winding disk coils on 4 kv side of tr have been determined by analyzing developed circuit model using Electromagnetic 4 Programming (EMTP). actual To determine frequency responses of windings, each phase of t has 3 been excited computed at its terminals by 4/ 3 kv (rms) variable frequency sinusoidal voltage s Amplitude (p.u.) actual computed Frequency (khz) 1 Fig. 1. Part of n section Fig. circuit 1. Part model of of n 4//33kV section circuit transformer model of developed for studyfig.. Transfer function (wi C D C D C D 4//33kV transformer developed for study admittance) frequency resp R L R L C t C t C D C D C D C D C D C D R L R L Amplitude (p.u.) 1 C t C t C D C D C D Frequency (khz) section circuit model of er developed for study Fig.. Transfer function (winding admittance) frequency response Fig.. Transfer function (winding admittance) frequency response WJMS for subscription: info@wjms.org.uk

4 34 D. Debnath & A. De & A. Chakrabarti: Lumped parameter electromagnetic modelling approach All inductance components have finally been lumped toger to an easily representable form of equivalent inductance per disk coil using method suggested by K. A. Wirgau [18]. The validity and accuracy of proposed modelling technique has been established in [3, 4]. 5. Determination of Equivalent series resistance (R) of coils Effective resistance of coils has been calculated, considering non-uniform distribution of current in conductors due to skin effect and eddy currents [13, 14, 16]. However parallel resistances, representing dissipation losses in capacitor dielectrics, and eddy current losses formed by low voltage winding in conducting cylinder have been neglected. 4 Development of EMTP model for 4 kv winding After computation of parameter values, EMTP (Electromagnetic Transient Programming) has been used to simulate and analyze frequency response characteristic of transformer. Also, impact of standard 1./5 µs lightning impulse and fast-front switching surges on windings have been studied and compared. A part of developed EMTP model containing n number of identical sections has been shown in Fig. 1. responses responses obtained obtained from from ATP ATP simulation have have been been shown shown in Fig. in Fig. 3 and 3 and Fig. Fig. 4. The 4. The voltage voltage amplification in 5 in Validation coils coils has has been of been presented developed presented in per in EMTP per unit unit (pu) model (pu) values values as multiples as multiples of of nominal nominal coil coil voltage voltage under under power power frequency frequency operation. The The Anfrequency example response has response been of added of group here group of to of demonstrate disk disk coils coils of accuracy of 4 of 4 kv kv main proposed main winding winding modelling of of transformer technique and to shows establish shows one validity one single single major of major resonance results presented around around 7. subsequently khz 7. khz with with voltage in voltage thisamplification paper. The incident factor factor of lightning about of about 4.5 impulse (Fig. 4.5 (Fig. 3). voltage 3). andthis signifies corresponding signifies that that excitation neutral excitation current at transformer wave of terminal transformer terminal at this at were this frequency frequency recorded would in would force HV force internal test internal laboratory resonance resonance during in lightning in main impulse main winding winding test oncoils and actual and oretically transformer. produce produce These internal data internal have over over been voltage voltage utilized of magnitude of to obtain frequency as high as high as responses 4.5 as 4.5 times of times whole nominal nominal transformer voltage voltage (neutral experienced current/applied by se by se coils impulse coils under under voltage). normal normal Thepower results frequency frequency obtained operation. from The developed The tap tap winding EMTP winding model on on or closely or hand match hand exhibits exhibits withfour four major response major resonances characteristics at 14.4 at 14.4 khz, of khz, original khz, khz, transformer,.3.3 khz khz and showing and khz similar khz with resonance with still still higher peak higher andvoltage cross amplification over, as shown factors factors in Fig. in in. range range (Fig. (Fig. 4). 4). 6 Voltage Amplification Factor Voltage Amplification Factor Frequency Frequency (Hz) (Hz) Voltage Amplification Factor Voltage Amplification Factor Frequency Frequency (Hz) (Hz) Fig. 4. Frequency response of a group of disk-coils Fig. Fig. Fig Frequency Frequency 3. Frequency response response response of of a a of group group a group of of of disk-coils disk-coils disk-coils Fig. 4. Frequency response of a group of disk-coils of Fig. 4. Frequency response of a group of disk-coils of 4kV of 4kV winding winding of tap of winding tap winding 4kV winding of tap winding Comparison The agreement of voltage between of voltage stress stress computed under under Standard actual frequency Lightning response Impulse Impulse characteristics, and and Fast-Front both in amplitude andswitching frequency, Surge establishes Surge accuracy of developed model. Transformers are are subject subject to standard to standard high high voltage voltage dielectric dielectric insulation insulation tests tests during during ir ir manufacture. Such 6 Such Frequency tests tests replicate replicate response aperiodic aperiodic of 4 voltage kvvoltage main wave wave and forms, tap forms, which changer which windings transformers are are likely likely to experience to during during ir ir operation. Standard Standard aperiodic aperiodic waveforms do not do not necessarily represent represent most most disastrous disastrous voltage voltage The stress frequency stress condition condition responses which which of transformer main and tap could winding could actually actually diskexperience coils on during 4during kvservice. side service. ofit is It transformer quite is quite possible have possible been that determined that voltages voltages by generated analyzing generated within within developed system circuit system could model could produce using produce even Electromagnetic even more more unfavorable Transient voltage Programming stress condition (EMTP). condition to Tosome to determine some transformers frequency in certain certain responses systems. systems. of It is windings, It thus is thus necessary necessary each phase to ascertain to of ascertain how transformer how winding winding has been voltage stress stresses excited stresses under at its under terminals system system originated by 4/ voltages 3 kv voltages (rms) compare variable compare with frequency with voltage sinusoidal voltage stresses stresses voltage produced produced source. by The standard by standard responses aperiodic obtained aperiodic laboratory from ATPtest simulation test waves. waves. have been shown in Fig. 3 and Fig. 4. The voltage amplification in coils i) WJMS Response i) Response for under contribution: under Standard Standard submit@wjms.org.uk Lightning Lightning Impulse Impulse Response Response of of transformer s 4kV 4kV winding winding under under 1./5 1./5 μs standard μs standard lightning lightning impulse impulse [18] [18] has has been been shown shown in Fig. in Fig. 5. The 5. The peak peak voltage voltage across across group group of of disk disk coils coils of of 4 4 kv main kv main and and tap windings tap windings are are observed observed to be to 4% be 4% and and 9% 9% of of applied applied terminal terminal voltage voltage respectively. No No sign sign of resonance of resonance is is

5 World Journal of Modelling and Simulation, Vol. 8 (1) No. 3, pp has been presented in per unit (pu) values as multiples of nominal coil voltage under power frequency operation. The frequency response of group of disk coils of 4 kv main winding of transformer shows one single major resonance around 7. khz with voltage amplification factor of about 4.5 (Fig. 3). This signifies that excitation at transformer terminal at this frequency would force internal resonance in main winding coils and oretically produce internal over voltage of magnitude as high as 4.5 times nominal voltage experienced by se coils under normal power frequency operation. The tap winding on or hand exhibits four major resonances at 14.4 khz, 17.7 khz,.3 khz and 1.5 khz with still higher voltage amplification factors in range 1-9 (Fig. 4). 7 Comparison of voltage stress under standard lightning impulse and fast-front switching surge Transformers are subject to standard high voltage dielectric insulation tests during ir manufacture. Such tests replicate aperiodic voltage wave forms, which transformers are likely to experience during ir operation. Standard aperiodic waveforms do not necessarily represent most disastrous voltage stress condition which transformer could actually experience during service. It is quite possible that voltages generated within system could produce even more unfavorable voltage stress condition to some transformers in certain systems. It is thus necessary to ascertain how winding stresses under system originated voltages compare with voltage stresses produced by standard aperiodic laboratory test waves. ii) (1) Response Response under under Fast-Front StandardLong-Tail LightningSwitching Impulse Surge The Response authors of have investigated transformer s how 4kV winding same transformer under 1./5 would µs behave standard under lightning 1./4 impulse µs steep-front [7] has been shown long-tail in Fig. switching 5. The peak surge. voltage The across results of group study of shown disk coils in Fig.6 of indicate 4 kv that main steep and front-long tap windings tailed are observed surge can to be stress 4% windings and 9% of to higher applied levels terminal compared voltage to lightning respectively. and No slow sign rising of resonance switching is surges. visible The from peak diagram. voltage This developed should be across of great group concern of because disk coils line end of coils 4kV are generally main and tap provided windings with under highest steep degree front of surge insulation, are observed as maximum to be stress 4% on and dielectric 11% of is anticipated applied terminal at line voltage end under respectively lightning which impulse are or steep-front marginally aperiodic greater switching than surge. corresponding Due to non-linear lightning distribution impulse responses of surge (Fig.6). voltage in It It is seen windings, that due middle to andbroader end coils spectrum in general of frequency experience content, muchsteep lower front stress surges and are can generally excite windings provided at ir withnatural lower frequencies insulation to optimize and subject cost. m to high stresses for longer duration of time. 1 Voltage in (%) Incident lightning impulse Response of of main winding group Response of of tap winding group Time (sec) Fig. 5. Response of windings under Fig. 5. Responsestandard of windings lightning under impulse standard lightning impulse Voltage in (%) Incident steep front switching surge Response of of main winding group Response of of tap winding group Time (sec) Fig. 6. Response Fig. 6. Response of windings of windings under steep-front under switching surge steep-front switching surge 8. Conclusion () Response under Fast-Front Long-Tail Switching Surge A The technique authors for have modelling investigated grid how connected same EHV transformer transformers would has behave been suggested under 1./4 for µs analysis steep-front of long-tail winding s switching response surge. to transients. The results A of 315 study MVA, shown 4//33 in Fig. kv 6 indicate EHV transformer that steep-front has been long-tailed modelled surge in can EMTP stress windings using to proposed higher levels modelling compared technique to lightning and and simulation slow rising results switching were surges. compared The peak with voltage actual developed response across of transformer group of collected disk coils from of 4kV field main for and validation tap windings purpose. under The steep-front developed surge model are observed successfully to be 4% reproduced and 11% of winding s applied frequency terminal response voltage respectively characteristics. which The are behavioural marginally response greater than of corresponding transformer lightning under standard impulse 1./5µs responses lightning (Fig. 6). impulse It is seen as that well due as under to broader fast-front spectrum switching of frequency surge have been studied and compared. The results indicate that voltage stresses developed on windings are marginally higher under fast-front switching surge as compared to standard lightning impulse. The WJMS for subscription: info@wjms.org.uk results presented in paper should be valuable for transformer designers as it it carries important information regarding voltage stresses on winding insulation under transients. To make results more exhaustive, future research work will be appreciated to assess windings response under or types of

6 36 D. Debnath & A. De & A. Chakrabarti: Lumped parameter electromagnetic modelling approach content, steep-front surges can excite windings at ir natural frequencies and subject m to high stresses for longer duration of time. 8 Conclusion A technique for modelling grid connected EHV transformers has been suggested for analysis of winding s response to transients. A 31.5 MVA, 4//33 kv EHV transformer has been modelled in EMTP using proposed modelling technique and simulation results were compared with actual response of transformer collected from field for validation purpose. The developed model successfully reproduced winding s frequency response characteristics. The behavioural response of transformer under standard 1./5 µs lightning impulse as well as under fast-front switching surge have been studied and compared. The results indicate that voltage stresses developed on windings are marginally higher under fast-front switching surge as compared to standard lightning impulse. The results presented in paper should be valuable for transformer designers as it carries important information regarding voltage stresses on winding insulation under transients. To make results more exhaustive, future research work will be appreciated to assess windings response under or types of transient wave-forms like oscillatory switching transients, chopped lightning impulse wave etc. and results can be compared to decide upon an effective and adequate insulation design for EHV transformers. References [1] Power transformer insulation levels and dielectric tests, 3, vol. 76. IEC Publication, 197. [] P. Blanken. A lumped winding model for use in transformer models for circuit simulation. IEEE Transaction Power Electron, 1, 16(3): [3] A. De, N. Chatterjee. Part winding resonance: Demerit of interleaved high-voltage transformer winding. in: IEE Proceedings-Electric Power Applications, vol. 147,, [4] A. De, D. Debnath, A. Chakrabarti. A study on impact of low-amplitude oscillatory switching transients on grid connected ehv transformer windings in a longitudinal power supply system. IEEE Transaction on Power Delivery, 9, 4(): [5] R. Degeneff, M. Vakilian. Modelling power transformers for transient voltage calculations. Council on Large Electric Systems, 199, [6] F. rover. Inductance Calculation: Working Formulas and Tables. Dover Publications, Inc., 196. [7] M. Jayaraju, I. Duat, M. Adzman. Impulse voltage generator modelling using matlab. World Journal of Modelling and Simulation, 8, 4(1): [8] T. Lyle, P. Mag. Philosophical Magazine, 3, vol. 31. Taylor and Francis Ltd, 19. [9] M. Elroy. On significance of recent ehv transformer failures involving winding resonance. IEEE Transactions on Power Apparatus and Systems, 1975, PAS-94(4): [1] A. Morched, L. Marti, J. Ottenvangers. A high-frequency transformer model for emtp. IEEE Transaction on Power Delivery, 1993, 8(3): [11] J. Phelps, A. Carlomagno. Experience with part-winding resonance in EHV auto-transformers: Diagnosis and corrective measures. IEEE Transactions on Power Apparatus and Systems, 1975, PAS-94(4): [1] E. Rosa. Calculation of self-inductance of single-layer coils. Bulletin of Bureau of Standards, 196, : [13] A. Sawheny. A Course on Electrical Machine Design. Dhanpat Rai & Sons Publishing Company, India, [14] A. Singh, J. Marti, K. Srivastava. Circuit reduction techniques in multiphase modelling of power transformers. IEEE Transaction Power Delivery, 1, 5(3): [15] A. Soyal. A method for wide frequency range modelling of power transformer and rotating machines. IEEE Transactions Power Delivery, 1993, 8(4): [16] R. Stoll. The Analysis of Eddy Currents. Clarendon Press, Oxford, [17] P. Vaessen. Transformer model for high frequencies. IEEE Transactions Power Delivery, 1988, 3(4): [18] K. Wirgau. Inductance calculation of an air-core disk winding. IEEE Transactions on Power Apparatus and Systems, 1976, PAS-95(1): WJMS for contribution: submit@wjms.org.uk

7 World Journal of Modelling and Simulation, Vol. 8 (1) No. 3, pp Appendix Details of computational methods used in calculating model parameters 1. Self Inductance of Disk Coils The formula used here applies to thick coils of disk shape for which radial dimension is considerably greater than axial dimension [, 1]. The equivalent self-inductance is given by: L = L s.4 π N a ( 1 + H 1 ), where, and, and, L s =.1 N a P µh(a is in cm), 8a 1 c P = 4π [(ln 8ac 8a ) ( P = 4π ln.5 + ln c...[3] ) (ln 8a )] + c8a 41 a c c 8a P = 4π ln.5 + ln a...[3] c c 4 a c All or All nomenclatures or nomenclatures are given are given along along with with Fig. Fig All or nomenclatures are given along with Fig. 7. a C a C B B p N turns c p a = Mean N radius turns of a disk coil c p = Winding pitch C Thickness of insulation between adjacent turns a = Mean radius of a disk coil B Height of insulated conductor p = Winding pitch N Number of turns in a disk coil c C = = N Thickness p = Approximate of insulation radial width between of a adjacent disk coil turns B = Height of insulated conductor N = Number of turns in a disk coil c = N Fig. 7. Schematic of disk coil p Fig. = Approximate 7. Schematic radial of width diskof coil a disk coil b a 1 a 1' 1 ' 3 4 A 4' 3' b c D c 1 ' D c A 4' 3' c Fig.8. Reduction of coils to equivalent filaments Fig. 8. Reduction of coils to equivalent filaments b 1 1' b 1 Fig. 7. Schematic of disk coil Fig.8. Reduction of coils to equivalent filaments. Mutual-inductance ii) Mutual-inductance between between elements elements The The mutual mutual inductance inductance between between two two co-axial co-axial circular circularfilaments filaments of of negligible negligible cross-sectional cross-sectional area, area, and and radii a andii) radii AMutual-inductance respectively, a and A respectively, separated between separated by distance elements by distance d between d between ir ir planes planes is is found to to be be dependent upon two parameters: two a/a parameters: and d/a, a/a and is d/a, given and by is given Eq. by (1). equation [4]. The mutual inductance between two co-axial circular filaments of negligible cross-sectional area, and radii a and A respectively, separated M = f Aa = fa a / A μh...[4] M = by f distance Aa = fa d between a/aµh, ir planes is found to be dependent upon (1) two parameters: a/a and d/a, and is given by equation [4]. in which f is obtained from [1]. in which f is obtained from [1]. The formulae M = f Aa given = faabove a / Ahowever μh...[4] apply only to circular filaments of negligible cross-section. LyleThe method formulae of equivalent given above filaments however apply only to circular filaments of negligible cross-section. Thisin iswhich very accurate f is obtained method from for [1]. co-axial coils of dimensions such that 4 th order and furr higher order differential Lyle method co-efficients of equivalent in Taylor s filaments: series expansion are negligible. The dimensions of equivalent filaments in athe general formulae casegiven is illustrated above however by Fig. apply 8, which only to shows circular two filaments circular of coils negligible This is very accurate method for co-axial coils of dimensions such that 4 th of rectangular cross-section. cross-sections order and furr higher of mean radii order a Lyle method differential & A, axial of equivalent co-efficients dimensions filaments: in btaylor s 1 & b, series radialexpansion dimensions are negligible. c 1 & c, having The dimensions number of turns equivalent N 1 & N and spacing filaments of median in a general planescase D. is Lyle illustrated methodby replaces Fig. 8, which shows coils by two 4circular equivalent coils filaments. of rectangular Each crosssections This to have is very of half mean accurate radii number a method & A, of axial for turnco-axial dimensions of its coil. coils b 1 Ifof & dimensions b axial, radial cross-sectional dimensions such that c 1 dimension & 4 th c filament is assumed order, having and b 1 number isfurr greater of higher than radial c turns order N differential 1 & N co-efficients spacing of median in Taylor s planes series D. Lyle expansion method are replaces negligible. The coils dimensions by 4 equivalent 1, filaments 11 and will have equivalent radius r 1 slightly larger than mean radius of equivalent a, and filaments. Each filament is assumed to have half number of turn of its coil. If axial cross-sectional two filaments in a general case is illustrated by Fig. 8, which shows two circular coils of rectangular crosssections of mean radii a & A, axial dimensions b dimension are b 1 is located greater at than an radial axial cdistance 1, filaments β on 11' eir and ' side will of have an median equivalent plane. radius Ther defining 1 slightly larger equations for r than mean radius a, and two filaments are located 1 & b, radial dimensions c at an axial distance β on 1 & c eir, having number of 1 and β are: side of turns N 1 & N and spacing of median planes D. Lyle method replaces coils by 4 equivalent median plane. The defining equations for r 1 and β are: filaments. Each filament is assumed ( to have half ) number (b of turn of its coil. If axial cross-sectional dimension b 1 is greater than r 1 = radial a 1 c 1 +, filaments c 1 11' and ' will 1 have an equivalent radius r 1 slightly larger than mean radius a, and two filaments 4a, β = ) c 1, are located at 1 an axial distance β on eir side of median plane. The defining equations for r 1 and β are: WJMS for subscription: info@wjms.org.uk

8 38 D. Debnath & A. De & A. Chakrabarti: Lumped parameter electromagnetic modelling approach If, on or hand, second coil has its radial dimension c greater than axial b, coil is to be replaced by two co-planer circular filaments 33 and 44 located at median plane but having equivalent radii (r + δ) and (r δ) respectively, where: ( ) r = A 1 + b A. The mutual inductance between two coils is n given by formula: ( ) M13 + M 14 + M 3 + M 4 M = N 1 N, 4 The mutual inductance between filaments is calculated by Eq. (1). The table, used for calculation of mutual inductance between two coils is as follows: Table 3. Design data: Disc coil dimensions Filaments Product of turns Radii Axial spacing 11 & 33 (N 1 N )/4 r 1 and (r + δ) D + β 11 & 44 (N 1 N )/4 r 1 and (r δ) D + β & 33 (N 1 N )/4 r 1 and (r + δ) D β & 44 (N 1 N )/4 r 1 and (r δ) D β Determination of equivalent inductance of a coil Assuming a section of winding have n number of identical coils each of self-inductance L and mutual inductances Mi between two pairs of coils, where i indicates number of coils away from reference coil, equivalent inductance of whole winding section is given by formula: n 1 L eq = n L + (n i) M i. Moreover, for two distinct winding sections, which are not too far away from each or, (for e.g. Main and Tap changer winding of a transformer) mutual inductances among elements of two windings should also be taken into consideration. This method originally proposed by Wirgau [4] has been used in present case. It replaces all elements within one winding section by an equivalent lumped element. The proposed method has been illustrated in Fig. 9. Here two winding sections have been considered. The first having X number of elements with each of N 1 number of turns, and second one having Y number of sections with each of N number of turns have been replaced by equivalent lumped elements of XN 1 and Y N number of turns respectively. Distance D between elements is same as distance of separation of central elements of two winding sections. 3. Series Capacitance of Interleaved disk coils In calculation of series capacitance, inter-turn as well as inter-disc capacitances between adjacent coils have been taken into consideration [6, 18]. However small stray capacitances between one coil to or distant coils have been neglected. Series capacitance of a disk coil is composed of two parts, being resultant of: Inter-turn Capacitance and Inter-disc Capacitance. 1. Inter-turn Capacitance C t Since C t depends upon common area between two adjacent turns, which again depends upon diameter of concerned turns, refore, C t does not have a constant value and changes with turn diameter. So, for practical calculations, value of C t at mean turn of a disk coil has to be calculated. Value of C t at mean turn is given by: i=1 C t = πd m (h + δ t ) ε ε paper δ t, WJMS for contribution: submit@wjms.org.uk

9 pends upon common area between two adjacent turns, which again depends upon diameter where (refer Fig.1), erned turns, refore, C t does not have a constant value and changes with turn diameter. tical calculations, value of C t at mean turn of H a = disk Bare coil conductor has to height, be calculated. δ t = Thickness of paper insulation, D m = Diameter of mean turn, ε paper = Relative permittivity of paper = 3.5 World Journal of Modelling and Simulation, Vol. 8 (1) No. 3, pp t at mean turn is given by: ( h + δ ) t δ t ε ε paper W 1 h 1 N 1...[1] W 1 N 1 h 1 XN 1 r Fig.1), N 1 (X=3) D D conductor height, δ t = Thickness of paper insulation, D m = Diameter of mean turn, ε paper = rmittivity of paper = 3.5 W 1 N 1 N 1 h 1 W 1 XN 1 h N W (Y=) N h YN W Fig. 9. Mutual Inductance between two distinct winding sections Fig. 9. Mutual Inductance between two distinct winding sections H D m Fig. 1. Conductor arrangement in a disk coil N 1 (X=3) where (refer Fig. 1), H = Bare conductor height, t = Thickness of paper insulation, D m = Diameter of mean turn, ε paper = Relative permittivity of Dpaper = 3.5. D N h YN N W (Y=) W. Mutual Inductance between two distinct winding sections H D m Fig. 1. Conductor arrangement in a disk coil Fig. 1. Conductor arrangement in a disk coil It is to be noted that in calculation of common area between turns, twice thickness of paper insulation (δ t ) has been added to bare conductor height to take into account effect of flux fringing and stray capacitances. Inter-disk Capacitance C D Inter-disk capacitance is calculated using formula for capacitance between two parallel plate electrodes separated by composite dielectric of paper, press-board and oil, in form as shown in Fig. 11. Assuming 35% of common area between coils being covered by press-board insulation and remaining 65% of area being covered by oil, resultant inter-disk capacitance between two adjacent disk coils is expressed as: C D = π 4 [( ( D Di ) ε.35 δ t ε paper + δ ε p.b ) + (.65 δ ε paper + δ ε oil where, D o = Outer diameter of disk coil, D i = Inner diameter of disk coil, δ t = Thickness of paper insulation, δ = Thickness of dielectric between coils, i.e. inter-disk separation, ε paper = Relative permittivity of paper = 3.5, ε p.b = Relative permittivity of press-board = 4., ε oil = Relative permittivity of oil =.. 4. Equivalent high frequency resistance of disk coils Fig. 1 shows a case of subdivided (laminated) conductors placed in iron slot. It is assumed that conductor is divided in N layers, each of height h 1, width b and length L. Total height of all layers is h = Nh 1. The average loss ratio for N layers is given by: K e(average) = R ac R dc = 1 + (αh 1 ) 4 N 9, )], WJMS for subscription: info@wjms.org.uk

10 4 D. Debnath & A. De & A. Chakrabarti: Lumped parameter electromagnetic modelling approach where, R d.c is d.c. resistance and R a.c is equivalent resistance at high frequency, considering skin effect, and πµ bf α = ρw, where, f is frequency and ρ is resistivity of conductor. b b Metal electrode Metal electrode paper H w H w L c L c p.b p.b CC D D Oil δδ paper paper N layers N layers b=radial thickness of one conductor H=Height b=radial of transformer thickness winding of one conductor L c = Axial H=Height height of transformer a winding winding L c = Axial height of a winding Fig. 11. Fig.11. Composition Composition of of Inter-Disk Inter-Disk Capacitance Capacitance Fig. Fig Transformer Transformer winding winding inside inside tank tank Fig.11. Composition of Inter-Disk Capacitance Fig.1. Transformer winding inside tank The above generalized expression can be extended for a transformer winding, if each winding of axial The above generalized expression can be extended for a transformer winding, if each winding of axial height L C The height can be above L C considered generalized can be considered to be located expression to be located in a can be in slot extended a slot of width of for width H w a transformer H, w, which which is is winding, height height if of of each transformer transformer winding of axial window. window. height L C can be considered to be located in a slot of width H w, which is height of transformer window. WJMS for contribution: submit@wjms.org.uk

11 Lumped parameter electromagnetic modelling approach 41