This paper presents the development and implementation of a coupled electromagnetic transformer model
for the simulation of three-phase power and distribution
transformers affected by the presence of winding inter-turn
short-circuits. Generically, the coupled electromagnetic
transformer model is based on the simultaneous consideration of both magnetic and electric equivalent circuits. To incorporate the internal faults in this model a suitable equivalent
circuit of the faulty winding is proposed. Both simulation and
laboratory tests results are presented in the paper, which
demonstrate the adequacy of the proposed model

Cardoso, A. J. Marques

Oliveira, L. M. R.

Universidade do Algarve

A coupled electromagnetic transformer model for the analysis of winding inter-turn short-circuits

Sapientia

Oliveira, L. M. R. and Cardoso, A. J. M.: "A coupled electromagnetic transformer model for the analysis of winding inter-turn short-circuits", Record of the IEEE International Symposium on Diagnostics for Electrical Machines, Power Electronics and Drives (IEEE SDEMPED'01), pp. 367-372, Grado, Italy, September 1-3, 2001. A Coupled Electromagnetic Transformer Model for the Analysis ofWinding Inter-Turn Short-CircuitsLuís M. R. Oliveira (1) (2)        A. J. Marques Cardoso (1) (1) Universidade de Coimbra, Departamento de Engenharia ElectrotécnicaPólo II - Pinhal de Marrocos, P – 3030-290 Coimbra, Portugal(2) Escola Superior de Tecnologia da Universidade do AlgarveCampus da Penha, P – 8000-117 Faro, PortugalAbstract — This paper presents the development and im-plementation of a coupled electromagnetic transformer modelfor the simulation of three-phase power and distributiontransformers affected by the presence of winding inter-turnshort-circuits. Generically, the coupled electromagnetictransformer model is based on the simultaneous considerationof both magnetic and electric equivalent circuits. To incorpo-rate the internal faults in this model a suitable equivalentcircuit of the faulty winding is proposed. Both simulation andlaboratory tests results are presented in the paper, whichdemonstrate the adequacy of the proposed model.Index Terms — Transformers; modelling and simulation;coupled electromagnetic model; winding inter-turn short--circuit faults; Park's Vector Approach.I. INTRODUCTIONThe sweeping changes that have overtaken the electric-ity supply industry during the closing decade of the 20thcentury have created a powerful new commercial climate.Power utilities and supply companies must now compete inincreasingly open, de-regulated markets, on an increasinglyinternational scale. To ensure commercial survival in thisaggressively competitive environment, it is essential thatkey items of power generation plants and power transmis-sion and distribution networks not only operate efficiently,but with maximum availability and with the lowest possiblecost of ownership over – and frequently beyond – theirdesign life. Power and distribution transformers haveformed an essential part of electricity supply networks sincethe alternating current system was adopted more than acentury ago [1]. Therefore it is quite obvious the need forthe development of on-line diagnostic techniques, thatwould aid in transformers maintenance. A survey of themost important methods, actually in use, for conditionmonitoring and diagnostics of power and distributiontransformers, presented in [2], stresses the need for thedevelopment of new diagnostic techniques, which can beapplied without taking transformers out of service, andwhich can also provide a fault severity criteria, in particularfor determining transformers winding faults.Preliminary experimental results, presented in [2], con-cerning the use of the Park's Vector Approach, have dem-onstrated the effectiveness of this non-invasive techniquefor diagnosing the occurrence of inter-turn short-circuits inthe windings of operating three-phase transformers.In order to obtain a deeper knowledge in the study ofinter-turn short-circuits occurrence, and also to acquire ageneralised perspective of this phenomenon, it becomesnecessary to develop a digital model for internal faultstudies in three-phase power and distribution transformers,which is the scope of this paper.For this type of studies, an open-structure transformermodel is necessary, i.e., a model in which it would bepossible to manipulate the windings arrangement. Thecoupled electromagnetic model [3-6], allowing for themodelling and simulation of the transformer in its naturaltechnology, so that the cause-and-effect relationships canbe closely investigated [4], becomes the natural choice forthe analysis of transformer internal faults.II. TRANSFORMER MODEL FOR NORMALOPERATING CONDITIONS ANALYSIS [3]The coupled electromagnetic transformer modelconsists in the combination of both magnetic and electricalequivalent circuits, in order to obtain the flux-currentrelationships. A typical flux distribution for a three-phase,three-limb, two-winding, core-type transformer is sketchedin Fig. 1(a). Taking advantage of the vertical symmetry ofthe transformer, the magnetic equivalent circuit can bederived, as shown in Fig. 1(b), consisting of magnetomotiveforces (MMF), fi, and lumped (linear and non-linear)permeances, Pi. As a first approach, let's consider only theexcited windings and neglect the leakage fluxes associatedwith each winding. In a similar manner to that presented in[3, 6], the flux-MMF non-linear relation (φ–f) can beexpressed as:iifffaaaaaaaaafA ⋅=⇒⋅=φφφφ321332313232212131211321(1)where:( )[ ]DPPPPPa 2233322112111 1 −−= (2a)DPPPPa 3312212112 = (2b)DPPPPa 2312312113 −= (2c)( )[ ]DPPPPa 3311222122 1−= (2d)DPPPPa 2311322123 = (2e)( )[ ]DPPPPPa 2122211332133 1 −⋅−= (2f)2123322311332211 PPPPPPPD ⋅−⋅−⋅⋅== A (2g)and:014111 PPPP ++= (3a)412 PP −= (3b)0254222 PPPPP +++= (3c)523 PP −= (3d)035333 PPPP ++= (3e)Recognising that iNf 1=  and φ=λ 1N , the mainlinkage fluxes, which do not take into account the leakagefluxes, can be expressed, from (1), as:iA  ⋅= 21Nh  (4)The total linkage fluxes can now be computed, introducingthe leakage fluxes, as follows:iLiL   ⋅=⋅+= σht  (5)being Lσ the diagonal leakage inductance matrix. It isassumed that there is an individual leakage inductanceassociated with each winding, which is determined from theconventional short-circuit test, where the per-unit primaryand secondary leakage inductances are equal, lσp= lσs= lcc/2(p.u.).When the transformer's secondary-side is loaded, thereare two MMF's in each limb. It is now necessary to expand(1), as follows:+++⋅=φφφ623152214211332313232212131211321iNiNiNiNiNiNaaaaaaaaa(6)which leads to a system of six equations:t i ⋅= (7)where:1222121211−−+⋅⋅⋅⋅⋅⋅== LAAAALNNNNNN (8)The winding connections are established, in the model,from the equivalent electric circuit, which is shown in Fig.2 for the case of an Ynyn0 connection and a balancedresistive load. The equations that describe the terminalconditions at each winding can be expressed as:iRvt⋅−=dtd (9)where:[ ]Tttttttt 654321 λλλλλλ= (10)[ ]Tnnn vvv 000321=v (11)[ ]Tiiiiii 654321=i (12)= +++ )()()(diag LRsRLRsRLRsRpRpRpRR (13) (a)(b)Fig. 1: (a) Flux distribution in a three-phase, three-limb, two-winding,core-type transformer, assuming a slightly greater magnetomotiveforce in the inner windings; (b) equivalent magnetic circuit.Fig. 2: Simplified equivalent electric circuit for the case ofan Ynyn0 connection and a balanced resistive load.being Rp and Rs the primary and secondary windingresistances, respectively.In order to maintain a trade off between complexity andaccuracy, an approximation was made to include the exci-tation losses, by connecting three linear resistance branchesacross the terminals of the excited windings.The influence of the no-load losses conductances, Gfe,can now be introduced in (7):vGi fet ⋅+⋅= (14)being Gfe a 6×6 matrix, where only the first 3×3 diagonalsub-matrix has non-zero values.The electromagnetic coupled transformer model is thusbased on the combination of (9) and (14), which takes intoaccount the asymmetry and saturation effects of the core.The development, implementation and validation of thetransformer non-linear model, for normal operatingconditions analysis, was presented in a previous paper [3],where a good agreement was achieved for both simulationand experimental test results.nRST1i2i3in4in5i6iLRLRLR2315643φ2φ1φ4φ 5φ01φ 02φ 03φ01P 1P4P2P5P21f22f02P 3P 03P23fn1 2 324f25f26f01φ 02φ4φ 5φ03φ1σφ4σφ2σφ5σφ6σφ3σφ2φ1φ 3φIII. MODELLING WINDING INTER-TURNSHORT-CIRCUIT FAULTSTo incorporate the internal faults in the aforementionedmodel, the faulty winding is divided into two parts: thehealthy part (Na turns) and the faulty part (Nb turns), asshown in Fig. 3, for the case of a fault in the primarywinding (phase R), localised in the bottom part of the coil.The primary-side short-circuited turns will act as anauto-transformer load on the primary winding [7], as shownin Fig. 3, where Rsh represents the fault impedance. Theresultant magnetic equivalent circuit is shown in Fig. 4,leading to three magnetomotive forces in the limb of thefaulty phase. From the equivalent magnetic circuit, the φ–frelationship becomes:++++⋅=φφφ62315221421332313232212131211321iNiNiNiNiNiNiNaaaaaaaaa baa(15)The resultant main linkage fluxes are:φ=λφ=λφ=λφ=λ1313212111bhbhhhNNNNφ=λφ=λφ=λ326225124NNNhhh(16)from which the relation expressed by (7) can be computed,using:[ ]Tttttbtttt 654321 λλλλλλλ=  (17)[ ]Tb iiiiiii 654321=i (18)11−− ==BBBAABAALLLLL (19)where:=σ+σ+σ+σ+σ+bLabNaNbNaNbNaaNbNabNNpLaNaNaaNNabNNaNpLaNaaNNbLabNNaNaNaLaaNN11213112111131332123211311212321222112111113211221111AAL(20)=132122112332123211321232122211221132112211121aNbNaNbNaNbNaNNaNNaNNaNNaNNaNNaNNaNNaNNABL (21)=132331223121321222312221212211213121212112abNNaNNaNNaaNNabNNaNNaNNaaNNabNNaNNaNNaaNNBAL (22)=σ+σ+σ+sLaNaNaNaNsLaNaNaNaNsLaN332223221322232222221222132212221122BBL (23)Fig. 3: Equivalent circuit for a fault localised in the bottompart of the primary winding (phase R).Fig. 4: Equivalent magnetic circuit for the case of aprimary-side faulty winding (phase R).being Lσa and Lσb the leakage inductances associated withthe subwindings a and b, respectively.For the same type of fault, the corresponding electricequivalent circuit is shown in Fig. 5. The genericdifferential equation (9) still holds, but now with the fluxesvector of (17), the currents vector of (18) and:[ ]Tnnn vvv 0000321=v (24)=++++−LRsRLRRLRRbRshRshRpRpRbRaRss0000000000000000000000000000000000000000R (25)where Ra and Rb are the resistances of the subwindings aand b, respectively.With this approach, the only additional data required formodelling winding inter-turn short-circuit faults, as com-pared to the healthy transformer model, are the leakageinductances and the resistances of the subwindings a and b.Several short-circuit tests and analytical studies wereconducted to determine Lσa and Lσb. It seems, however, thata good approximation is obtained, for a generic ksubwinding, by using:pkk LNNL σσ =1(26)i1ixibv1nφσaφσb vbφ1Rshi4RLNaNbN23φ2φ1φ4φ 5φ01φ 02φ 03φ1 2 32af22f23fn24f25f26f2bf01P 1P 2P 02P 3P 03P4P 5PFig. 5: Equivalent electric circuit for the case of aprimary-side faulty winding (phase R).Additionally, the leakage inductances do not change sig-nificantly with the fault position along the affected winding.The subwindings resistances are also determined by alinear relationship based on the corresponding turns ratio:pkk RNNR1= (27)If the fault is localised in the upper part of the coil, thesame methodology applies.For a fault location in the middle part of the primarywinding, although two healthy subwindings (Nc and Ndturns) and one faulty subwinding (Nb turns) have to beconsidered (Fig. 6), the same methodology can also beapplied, assuming Na = Nc + Nd. With the approximations of(26) and (27), which results in Lσa = Lσc + Lσd and inRa = Rc + Rd, respectively, the relations (17) to (25) remainvalid.IV. MODEL VALIDATIONFor the model validation, a three-phase, three-limbtransformer, of 6 kVA, 220/127 V, was used. Thetransformer has two windings per phase on the primary andon the secondary side, having, one of each, been modifiedby the addition of a number of tappings connected to thecoils, for each of the three phases, allowing for theintroduction of different percentages of shorted turns atseveral locations in the winding, as shown in Fig. 7 for thephase R of the transformer primary winding [2].A shorting resistor was connected at the terminals of thefaulty subwinding, whose value was chosen so as to createan effect strong enough to be easily visualised, butsimultaneously big enough to limit the short-circuit currentand thus protecting the test transformer from completefailure when the short is introduced.Both harmonic magnitudes and harmonic phase anglesof the supply voltages were taken into account in theconducted simulation study.For the case of the Ynyn0 connection, a balanced resis-tor load and 10% of shorted turns in the phase R of thetransformer primary winding, both experimental and simu-lated primary-side currents waveforms are shown in Fig8(a) and 8(b), respectively, which are in relatively goodagreement.The occurrence of primary-side inter-turn short-circuitsFig. 6: Equivalent circuit of a fault in the middle partof the primary winding (phase R).Fig. 7: Location of the tappings for transformerprimary winding (phase R).leads to an increment in the magnitude of the current in theaffected winding, as compared to an healthy condition,which results in an unbalanced system of primary currents.For this reason, the magnitude of the primary neutralcurrent (in1=i1+i2+i3) is also affected.In the presence of the primary winding inter-turn short--circuits, the secondary side currents do not present anyrelevant change as compared to the transformer's healthyoperation, remaining an approximately balanced three--phase system (Fig. 9).For the same aforementioned conditions, the experi-mental and simulated fault related currents waveforms areshown in Fig. 10(a) and 10(b). The faulty subwindingcurrent, ib, is approximately in phase opposition with i1(Lenz law). The current in the short-circuit auxiliaryresistor, ix, has a higher magnitude than ib, since ix = ib - i1.For the same load conditions and transformer windingconnections mentioned above, Fig. 11 presents the experi-mental primary side current Park's Vector patterns, forseveral percentages of shorted turns in the primarywindings and for different faulty phases. The primary sidecurrent Park's Vector pattern, corresponding to the healthyoperation, differs slightly from the circular locus expectedfor ideal conditions, due to, among others, the supplyvoltage harmonic content and the minor reluctance seenfrom the central limb, with respect to the lateral limbs [2].The occurrence of primary-side inter-turn short-circuitsmanifests itself in the deformation of the supply currentPark's Vector pattern corresponding to a healthy condition,leading to an elliptic representation, whose ellipticityincreases with the severity of the fault and whose majoraxis orientation is associated to the faulty phase [2].The simulated primary-side current Park's Vectorpatterns are presented in Fig. 12, which are in closeagreement with the experimental results of Fig. 11.i1ibv1nφσcφ1ixφσb vbRshi4RLNcNbN2Nd φσdU15%UA UBUC10% 20%UD5%UEX1UF UG20% 10% 5%25%n4in5i6iLRLRLR2i3inaN bNshR1iRSTi1i2 i3in1i1i2 i3in1i5 i6i4in2i5 i6i4in2ixi1ibixi1ib(a)(b)Fig. 8: Primary-side currents waveforms for the case of an Ynyn0 connec-tion, a pure resistive balanced load and 10% of shorted turns in theprimary winding (phase R): (a) experimental; (b) simulated.(a)(b)Fig. 9: Secondary-side currents waveforms for the case of an Ynyn0 con-nection, a pure resistive balanced load and 10% of shorted turns in the primary winding (phase R): (a) experimental; (b) simulated.(a)(b)Fig. 10: Fault related currents waveforms for the case of an Ynyn0 connec-tion, a pure resistive balanced load and 10% of shorted turns in theprimary winding (phase R): (a) experimental; (b) simulated.V. CONCLUSIONSThis paper describes the development and implementa-tion of a coupled electromagnetic transformer model for theanalysis of winding inter-turn short-circuits. The model isbased on the combination of both magnetic and electriclumped-parameters equivalents circuits, which allows themodelling and simulation of the transformer in its naturaltechnology.The faults are introduced in the model by dividing theaffected winding in two parts, which represents the healthysubwinding and the faulty subwinding. With this approach,the primary-side short-circuited turns will act as anauto-transformer load on the primary winding. The openstructure of the coupled electromagnetic model is perfectlycompatible with the fault auto-transformer equivalentcircuit, which allows to study these phenomena from aphysical perspective. Both simulation and experimentaltests results demonstrate the adequacy of the model, underhealthy and faulty operating conditions.With the supply current Park's Vector pattern, it ispossible to diagnose primary winding inter-turn short--circuits faults. The on-line diagnosis is based onidentifying the appearance of an elliptic pattern, corre-sponding to the transformer supply current Park's Vectorrepresentation, whose ellipticity increases with the severityof the fault and whose major axis orientation is associatedto the faulty phase.healthy 5% 15%10%healthy15%10%5%healthy15%10%5%healthy 5% 15%10%healthy15%10%5%healthy15%10%5%(a) (b) (c)Fig. 11: Experimental primary-side current Park's Vector patterns for the case of an Ynyn0 connection and a balanced resistive load, with severalpercentages of shorted turns in the primary windings and for different faulty phases: (a) phase R; (b) phase S; (c) phase T.(a) (b) (c)Fig. 12: Simulated primary-side current Park's Vector patterns for the case of an Ynyn0 connection and a balanced resistive load, with severalpercentages of shorted turns in the primary windings and for different faulty phases: (a) phase R; (b) phase S; (c) phase T.Further work is currently in progress, concerning themodelling and simulation of the behaviour of three-phasetransformers, under the influence of the simultaneousoccurrence of winding faults, unbalanced supply voltages,unbalanced loads, or even the surrounding presence ofpower electronics equipment. Additionally, a more accuratemethod to determine the subwindings leakage inductanceswill be also investigated.ACKNOWLEDGEMENTThe authors wish to acknowledge the financial supportof the Portuguese Foundation for Science and Technology,under Project Number PRAXIS/P/EEI/14151/1998.REFERENCES[1] Ashmore, C. "Transforming technology" International PowerGeneration, Vol. 22, Nº 3, pp. 25-26, March 1999.[2] Cardoso, A. J. M. ; Oliveira, L. M. R. "Condition monitoring anddiagnostics of power transformers" International Journal ofCOMADEM, Vol. 2, Nº 3, pp. 5-11, July 1999.[3] Oliveira, L. M. R.; Cardoso, A. J. M. "Three-Phase, three-limb,steady-state transformer model: The case of a Ynzn connection"Proceedings of the IASTED International Conference, "Power andEnergy Systems", Marbella, Spain, pp. 491-472, September 2000.[4] Yacamini, R.; Bronzeado, H. "Transformer inrush calculations usinga coupled electromagnetic model" IEE Proc. Sci. Meas. Technol.,Vol. 141, Nº 6, pp. 491-498, November 1994.[5] Elleuch, M.; Poloujadoff, M. "A contribution to the modelling ofthree phase transformers using reluctances" IEEE Trans. onMagnetics, Vol. 32, Nº. 2, pp. 335-343, March 1996.[6] Chen, X. S.; Neudorfer, P. "Digital model for transient studies of athree-phase five-legged transformer" IEE Proc. Pt. C, Vol. 139, Nº4, pp. 351-359, July 1992.[7] Stigant, S. A.; Franklin, A. C., The J & P Transformer Book, 10thEdition, London, Newnes-Butterworths, 1973.

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A coupled electromagnetic transformer model for the analysis of winding inter-turn short-circuits

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