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IEEE Industry Applications SocietyAnnual MeetingNew Orleans , Louisiana, October 5-9, 1997

Identif ication of High Frequency Transform er Equivalent Circuit Us ingMatlab from Frequenc y Dom ain DataSyed Mofizul Islam' Kathryn 'M. Coates' Gerard Ledwich'Department of Electrical & Computer Engineering

'Curtin University of TechnologyPerth, WA 6001'The University of Newcastle

Callaghan, NSW 2308AUSTRALIA

Ph: 61) 049) 1 6084 Fax: 61) 049) 1 6993 Email: eeislam@ee.inewcastle.edu.au

Abstract High frequency modelling is essential inthe design of power transformers, to study impulsevoltage distribution, winding integrity and insulationdiagnosis. In this paper, a PC based, fully automatedtechnique for identifying parameters of a highfrequency transformer equivalent circuit is proposed.At each discrete measurement point, voltage ratio andphase between input and output is calculated to obtainthe frequency response. A parametric systemidentification technique is utilised to determine thecoefficients of an appropriate transfer function tomodel the measured frequency response from 50 Hzto MHz, divided into low, medium and highfrequency ranges. The proposed technique is simpleto implement, fully computerised and avoids timeconsuming measurements reported earlier. Testresults on several transformers indicate that themethod is highly reliable, consistent and is sensitive totransformer winding faults.

(ii)core with respect to frequency, and,frequency upon the parameters of the transformer.

they fail to consider the changing effects of the

(iii) they fail to account for the effects of varying

Addressing these shorteomings a transformerequivalent circuit was proposed by Douglass [l] asshown in Fig 1.

Figure 1. TransformerGeneral E q u i v o L e n t C i r c u i t

11 INTRODUCTION.Every transformer has a unique transfer function andcan be modelled by a string of inductances tapped bytheir parallel capacitances to earth and shunted bytheir stray capacitances between windings. At veryhigh frequencies, the magnetic fluxes cannotpenetrate the core, and the core acts effectively as anearthed boundary. It may also be assumed thatmutual inductances can be lumped into the seriesinductances.Constant power frequency models provide a goodbasis for the development of variable frequencymodels, however, there are three major shortcomings:(i) they fail to recognise the insulation systems of thetransformer as electrical entities,0-7803-4067-1 97/ 10.00 1997 IEEE.

In this model the following parameters are in theirlumped form:(i) an ideal transformer for ratio purposes only,(ii) Cp and Cs, whiclh represent the primary andsecondary winding capacitances to groundrespectively. This is a representation of the winding toground insulation (with IR = m ,(iii) Cps, the capacitance between primary andsecondary windings,(iv) Lw is the equivalent leakage inductance of theprimary and secondary windings,(v) Rw is the equivalent resistance of the primary andsecondary windings (referred to the secondarywinding), and(vi) Zh is the core exciting impedance composed of aresistance and an inductance.

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In [l] four different time consuming experiments weresuggested to obtain the equivalent circuit parametersvalid up to 10 kHz. Vermeulen et al [2] roposed adiscrete parameter estimation technique to obtain thelumped parameters of Fig. 1valid up to 25 kHz. In[3], a parameter estimation technique was proposedbased on measurements of both admittance andvoltage transfer function obtained from square waveexcitation. The technique was reported to be valid upto several hundred kHz and was applied to a small (Le20/0 11 V 30 VA) potential transformer. In reality,however, most power transformers are tens ofmegavoltamperes at hundreds of kilovolts in rating. Inaddition, the method in [3] also failed to identify C theprimary capacitance. In this paper, a fully automated,simple and fast technique is proposed which identifiesall parameters, needs only voltage ratiomeasurements from realistic sinusoidal excitation andis valid up to 1 MHz.

I I TRANSFORMER MODELS.The model presented in Fig. is an overview model ofa real transformer, however, given that it is composedprimarily of reactive components, it is appreciated thatthe various elements of the circuit will make differentcontributions at different frequencies - some may evenbecome unnecessary. As a result, the above generalmodel may be appropriately divided into low, mediumand high frequency range models, reflectingparameter significance.A. Low Frequency Model.At frequencies below about 500 Hz [3], Cps, has littleinfluence and appears as an open circuit. Conversely,flux penetration of the core is significant and henceZh, the core excitation impedance, is included. Theresultant circuit appears in Fig. 2below. Note that Zhis a parallel combination of Rh and Lh.

Figure 2Low Frequency M o t h Model

This model was used for frequencies up to 2 kHz. Itstransfer function is

1

displaying no zeroes and two poles.B. Medium Frequency Model.In the mid frequency range (approximately2 kHz to 80kHz) the model used is shown in Fig. 3. Jensen et al,in modelling an instrument transformer, exclude theeffects of the core exciting impedance at frequenciesabove 500 Hz. However, it was found in course of thisresearch that for power transformers, the transformerfrequency responses were better emulated by using ahigher order model - i.e retaining the core excitingimpedance in the medium frequency range.

Figure 3. T r a n s f o r m e rMedium Frequency Model

The circuit transfer function appears below, exhibitingone zero and three poles. Note that Rh and Lh arethe core exciting impedances, but referred to thesecondary side.

C. High Frequency Model.At high frequencies, the circuit capacitances dominate,thus the high frequency model appears as in Fig. 4.

Figure 4.High Frequency M a t h Model

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This model is proposed for frequencies above 80 kHzand is frequency independent with a transfer function:

Ill. TEST AND MEASUREMENT SET UP.The transformers were prepared for test by beingremoved from service and electrically isolated from thetransmission system. For those windings which weredelta connected, the delta points were disjointed. Forstar windings, the neutral points were earthed, and thetests were conducted on one phase pair at a time.Transformer tap positions were noted. Theinstruments used to conduct the tests were anArbitrary Wave / Function Generator, a Cathode RayOscilloscope and a PC with a portable GPlB (generalpurpose interface bus) card. The method ofconnection of these instruments is shown infigure5.

Figure 5Connect ionD i a g r o n

GPlB

Conputer

W o v eGenero o r

I

Essentially, the test methodology consisted of usingthe arbitrary wave generator to inject a signal into oneof the phase windings, then using the CRO tomeasure this input voltage, its frequency, thecorresponding output voltage and the time lagbetween the output and input signals. The tests wereconducted over a range of frequencies from 50 Hz to1 MHz maximum.The information measured by the CRO wastransferred via a C++ program (with the use of a GPlBcard) to the computer and logged in an ascii file. Theprogram gets the maximum and minimum of each ofthe waveforms from the oscilloscope. To determinethe zero point of the waveforms (an imaginary zeroline), the program averages the maximum andminimum values for each of the waveforms. The timedelay between the signals is then the time differencebetween each of the signals crossing this newimaginary zero line.

IV. PARAMETER IDENTIFICATION.Data manipulation was executed to derive transferfunctions and transformer parameters from the rawdata collected by the IC++ program. The data wasdivided into low frequency, mid frequency and highfrequency ranges. The ranges were adjusted slightlydepending on the frequency steps used in datacollection, the range of frequencies over whichcollection took place and the degree to which a bettercomparison between data and model could beachieved by slight alterations.The data was entered into Matlab programs in asciiformat where the Mlatlab routine inv f reqs wasinvoked. This commarid is able to find a continuoustime transfer function that corresponds to the complexfrequency response (ie it can convert the magnitudeand phase data collected in the field, into a transferfunction). lnvfreqs uses a non-linear least square(NLS) method [5] to rninimize the squared norm oferror between the estimated model and the measuredfrequency response data G,as shown below:

where, Gk = G ejek = Gk + 17kq k being the measurement error.The routine returns the coefficients of the numeratorand denominator in vectors A and B such that atransfer function is yielded in the form:

B s ) b l)sflb b 2)snb-1+.. .+b nb+)s ) = ($) ~ 1)sa 2)s-+...+a na +1)Invoking the invfreqs routine requires the specification,by the user, of the orders of the A and B polynomials,i.e the number of poles and zeroes associated with themodel in question.Once the routine has delivered a suitable transferfunction, it is possible to plot the frequency responseof the fitted transfer function along with the frequencyresponse of the origiinal data. In this way, acomparison of the two1 curves may be made, and anassessment made of the degree of accuracy of thederived model.The actual parameters of the transformer werecalculated using the models developed previously andthe transfer function coefficients delivered by invfreqs.The coefficients for each frequency range were

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delivered in two matrices: B - the numerator and Athe denominator.

I

transformers. All four transformers (Tl, T2, A1 andA2) were commissioned in the mid-1960's at variouslocations in the state of New South Wales. Averageparameters obtained appear inTable1.

Frequency radlsec

V. RESULTS.A. 30MVA Transformers.Using the technique discussed, multiple tests wereconducted on each phase of a set of four identical132/66/11 kV 30 MVA YyNOdl vector grouping power

Transformer T1 failed in early 1996 due to a windingmovement resulting from a through fault. Inspectionshowed that 'a' phase had developed an earth fault,which is demonstrated by a large variation in 1Transformer A2 exhibited minimal variation in

TABLE 1ESTIMATED PARAMETERSOFTHE FOUR 30 MVA TRANSFORMERS.

Iig 6. 72 Tranrbmer Actual Estimated rawler FunclionsI 0'I

10'

parameters. Results for T2 transformer indicatesome variation in parameters among the phaseswhilst more significant changes are noticeable forthe A1 transformer. However, standard deviation ofparameters obtained for any single phase resultingfrom multiple estimation among the fourtransformers was within 2 . Figure 6shows theactual and estimated transfer functions for T2transformer in the medium/high frequency range.B. 390MVA Transformers.

The second set of tests were conducted on eachphase of two identical 23/350 kV 390 MVA Yndlvector grouping power transformers. Averageparameters obtained appear in Table 2and plots ofthe actual and estimated transfer functions for themedium frequency range appear in Figure 7.Themedium frequency model response is not ideallyaccurate due to the complex shape of themeasured frequency response, which depicts anumber of resonant points. Nevertheless, given theconstraints on the order of the model (one zero,three poles) the modellingof the data is quite good.

Fig. 7 5A Transformer Actual Estimated Transfer Funclions.10

1os 1oFrequency radlsecLO ' '

I0

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Two sets of tests were conducted on the 5Btransformer, before and after the windings weremechanically reclamped. The windings arecomprised of a number of flat horizontal disks inwhich turns are wound in the form of a spiral, thedisks being mounted on the transformer core.Winding reclamping exerts force on the windings inthe axial direction, meaning that the disks are

compressed (slightly) in the axial direction. Thiscauses the space between adjacent disks todecrease slightly tlhe overall change in verticalheight in the B transformer case was only a fewmillimetres (in about two metres). The effect,however, for a transformer in which the primary andsecondary windings are wound on the one core, isthat the distance between windings decreases.

TABLE 2.ESTIMATED PARAMETERS OF THE FOUR 390 MVA TRANSFOIRMERS.

In addition, the closer proximity between windingsresults in the gap between them being made up ofmore primary insulation material (paper, tape etc) andless secondary insulation (oil, moisture, contaminantsetc).The overall result of these effects is that thecapacitance between windings increases. This can beseen clearly from the basic equation of a capacitor:

C = - & , Ad

where C is the capacitance, E, , is the permittivity, Ais the area and d is the distance between conductors.After reclamping the permittivity is increased and thedistance is decreased, whilst the area changes little.This phenomena is reflected in the results obtainedfrom the tests for 5B.From Table 2, it can be seen that C for each phasehas become substantially larger after reclamping ofthe transformer. It is notable that the values for Cthe secondary capacitance to ground shows littlechange after reclamping, for whilst the windingsthemselves are forced into closer proximity, theirrelative distance to ground (ie the transformer tank)remains practically unchanged.

Also notable is the fact that no observable trends inchanges of other parameters are recognisable in theevent of reclamping. This indicates that theresistances and inductances of the windings areunaltered, as they should be in all but extreme cases.Some negative entries irr Table 2are attributed to thefact that invfreqs uses an unconstrained optimisationalgorithm.(iii) Determination of CWhen testing transformers, the signals were injected,initially, only into the lower voltage windings, causingproblems with the evaluation of C (the lumpedprimary winding capacitance to ground. Given theshape of the high frequency model circuit (figure 4) itcan be seen that it is impossible to calculate the valueof C without injecting into the higher voltage windingof the transformer (Reverse Injection) as well asinjecting in the low voltage side. Then, using thesame labelling as in figure 4, C, can be calculatedfrom:

Results obtained from a 500V/33kV 350 kVA mobileTest truck Transformer using reverse injection aregiven inTable 3.

' Pre winding reclamp.All te st conducted with transformer at tap 1 except this tap 7.Post winding reclamp.361

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Table 3 :Parameters of 05/33 kV Mobile Test TruckTransformer Using Reverse Injection

InjectLV

Rw(C2) Lw(mH) CPS S CP34 73.88 1.09nF 0.28 uF

8x10. 0.15 2.3 mF 0.18mF

The value of Cp obtained was 0.18 mF which initiallyseems high (in comparison with other capacitancesand other transformers), but is not so given thefollowing:(a) The transformer was not really a powertransformer. Its voltage rating was 500 V 33 kV(with the secondary current rated at only 7A) andphysically the transformer was very much smaller.(b) Because of the large transformation ratio (66:l)and compact size, the high voltage winding consists ofmany turns of small diameter wire. As a result, theresistance of the secondary was quite high. Whenadded to the referred LV winding resistance, the valuewas about 34 Q (compared to about 1 to 2 Q forpower transformers).(c) The winding inductance was calculated to be quitehigh (74 mH for LV injection, 0.150 mH for HVinjection) in comparison to the power transformers.This again can be attributed to the large number ofturns on the high voltage winding.

VI. COMPARISON OF RESULTS.Authors in [3,4] quote values for parameters oftransformers upon which they have conducted theirresearch. Although in the case of [3] the transformerunder test was rated at only 30 VA, their resultingvalues served as a general guide in terms ofmagnitudes to be expected. Bearing in mind thedifferences between transformers, there was goodcomparison with both the 30 MVA and 390 MVAtransformers.One noticeable difference between the model in[I ,3,4] and the current research is the treatment of thecore exciting impedance Zh. In [3], the impedance istreated as a series combination of Rh and Lh, whilst inthis paper, it is treated as a parallel combination, thatis: Zh = Rh // Lh. The decision to deviate from theBak-Jensen/Douglass model was taken becausepower frequency models invariably depict themagnetising branch as a parallel combination of anadmittance and a conductance.

Given the configuration adopted, it was initiallyexpected that the value for Rh would be very high,particularly n the light of the results of an open circuittest conducted on one of the 390 MVA transformerswhich yielded a magnetising branch resistance of2204 per phase, referred to the primary. The valuerealised from the frequency sweep was far less thanthis , however, this can be explained.Open circuit tests are conducted with rated voltageapplied to the LV winding (here 23kV). Also,generator power transformers are designed such thatthey operate at flux densities where the core steel isdriven a fair way into saturation. On the other hand,the frequency sweeps tests were conducted with verylow voltages (2V to 15V) applied to the LV sideresulting in less current flow and a smaller resultantflux density. Consequently, the open circuit test wouldresult in an operating point in the saturation region ofthe B-H curve, whilst the frequency response testswould result in operation in the low flux density region.Thus the core exciting parameters (which are alreadyfrequency dependent) will be dependent on the levelof excitation of the core - minor excitation, as in thiscase, will result in lower values being obtained. Thisconfirms the result that at high frequencies the corepresents very low impedance.Verification of some results was also achieved bycomparison of values obtained using methodsdescribed in [4]. The authors in [4] calculatedparameters for an 8 MVA transformer with n=116winding sections using a 36Q termination resistance(Ro). Adopting their method for the 390 MVA 5Atransformer A phase,

Ro-oVin 2n Lw--so Rovo-*2nf nVin

L, =

Substituting values from the 5A transformer lowfrequency plot, VoNin = 9.89, f = 2kH2, Ro = 36(impedance of measuring instruments) yields a valueof 1 = 0.29 mH. This compares extremely well withthe experimental value of 0.267 mH.At medium frequency, the calculation for Cs is asfollows:The phase lag between two resonant peaks is 50 (0.87 rad) at 77 kradfs [6]. Thus,

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p = 21~f /L,C,y = 0.87

This value compares adequately well with theexperimental result of 6.4 pF.

For the high frequency response, again following thecalculation method in [4]:vuVzn

y

where n is the number of winding sections (n = 1 forthe lumped parameter case). The magnitude ofVoNin for the high frequency range (from the 5ATransformer high frequency range plot) is 0.15 [6].Thus:From

y = 0.15

So using the calculated value of Cs,= 2nF

This value again is in adequate proximity to the valueobtained experimentally, to give a degree ofconfidence to the experimental results.VII. APPLICATIONS IN CONDITION MONITORING.ATwo (5A and 5B) identical 390 MVA generatortransformers that have been in service since the early1970s were tested to determine the mechanicalintegrity of their windings. The 5A transformer hadbeen reclamped and was in service prior to testingwhilst 5 8 transformer was tested before and afterreclamping. Table 2shows the results of the proposedparameter estimation method for the 5B transformerbefore and after the reclamping. Figure 8 shows thetransfer function plots for both cases far the C phaseof the 5B transformer. The resonant frequencyreduction of about 20 kHz is clearly visible in thetransfer function plot in the 100 to 200 kHz range. Thedecrease in resonant frequency may be correlated toincreased capacitances due to increase in relativepermitivity and reduction in axial space betweenwindings as a result of increased clamping pressure.This is also clear from Table 2which shows that both

C and C, has increased in magnitude afterreclamping.

-i Figure 8. Transfer Function Plot of C phase 5BTransformer Before arid After Reclamping.B.Four 132/66/11 kV 30 MVA transformers (A1 A2,T1and T2) placed in service since mid 1960s wereinvestigated for winding integrity using the proposedparameter identification technique. Transformer T1failed in service in eilrly 1996. This failure occuredwhilst the transformer was supplying an external faulton the LV side. It is believed that the mechanicalforces arising from the fault contributed to the failure.An internal inspection revealed that there had beensufficient tertiary winding movement to resultultimately in an earth fault. The phase frequencyresponse was significantly different from the other twophases as shown in Fig. 9. Results presented inTable1 show that the phase inductance, 1 hasincreased significantly from the remaining phases to978 pH and C and C, decreased to 132 and 24 pFrespectively. Higher value of 1 and lower value of Cand C are direct clonsequences of large windingdeformations which are the cause of multipleresonances in the phase response as shown inFig. 9.

Figure 9. Failed T I Transformer Phase A , B , and C Frequency Response.

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Reduction in resonant peak damping can be explainedby the fact that increased winding inductances lead toreduced main flux and therefore reduction in V Aphase plot [6] also clearly indicated dominance ofinductance in the low frequency part of the frequencyresponseAS demonstrated above, the estimatedparameters served as an excellent confirmation tool incondition monitoring of transformers using transferfunction method.

VIII. CONCLUSIONS.Results indicate that the method developed in thisresearch is a successful tool for determination ofbroadband equivalent circuit parameters for conditionmonitoring of power transformers. It is particularlyuseful for the detection of faults of a mechanicalnature such as winding displacement and distortionfaults not easily detected by conventional electricalmethods.The method requires the development of modelsbased on a compromise between the frequency rangeof signals injected and the shape of the frequencyresponse achieved. From such electrical models,mathematical models are derived from which thespecific parameters of the winding are extracted.Despite having fairly small data sets with which towork, the method has been able to provide reasonablevalues for transformer parameters. These parametersallow more direct comparisons of transfer functionsthan do graphs of the frequency response alone formodel based diagnosis of results to determine thenature of transformer faults.The lumped parameters obtained may be employed toconstruct an analog distributed parameter model of awinding [7] uch a model may then be used forsliding fault analysis as discussed in [8] to locate afault position.The proposed method has the benefits of being apotentially quick, relatively simple, easily implemented,effective method of non intrusive transformer conditionmonitoring.

ACKNOWLEDGEMENTSThe authors greatly acknowledges the help and co-operation of Mr. Mark Stace of Advanced TechnologyCenter of Pacific Power International, Newcastle,Australia, in supplying experimental data andinformation on power transformers.

REFERENCES

[2] H.J. Vermeulen, R.G. Koch, and W.T. Rawlins,Broadband modelling of magnetic voltagetransformers through application of parameterestimation techniques, 6th International Symposiumon High Voltage Engineering, New Orleans 1989, Vol.3, paper no. 50.1 2.[3] J. B. Jensen, B. B. Jensen, S.D. Mikkelsen andC.G. Jensen,Parametric Identification in PotentialTransformer Modelling. IEEE Transactions onPower Delivery, Vol7, No.1, January 1992.[4] E.P. Dick and C.C. Erven, Transformer Diagnostictesting By Frequency Response Analysis, IEEETrans. PAS, Nov/Dec 1978, pp.2144-2153.[5] J.O. Smith, Techniques for Digital Filter Designand System Identification with Application to theViolin, Ph.D. Dissertation, Dept. of ElectricalEngineering, Stanford University, June, 1983, p. 50.[6] K.M. Coates, Condition Monitoring of PowerTransformers Using Transfer Functions, B.Sc.E.EHonours Thesis, Vols. 1 2, Dept. of Electrical andComputer Engineering, The University of Newcastle,NSW 2308, Australia.[7] R. Malewski and B. Poulin, Impulse Testing ofPower Transformers Using the Transfer FunctionMethod. IEEE Transactions on Power Delivery, Vol3, No. 2, April 1988, pp. 476-489.[8] S. M. Islam and G. Ledwich, Locating Faults in aTransformer Winding Through Sliding SensitivityAnalysis of Analog Modeling Using Transfer FunctionMethod. Proceedings of the IEEE Symposium onElectrical Insulation, Montreal, Canada, June 16-19,1996, Vol. 1, pp.38-41.

[ l ] D.A. Douglass , Potential Transformer Accuracy at60 Hz Voltages above and Below rating and atFrequencies above 60 Hz. IEEE Transactions onPower Apparatus and Systems, Vol. PAS-100, No. 3,March 1981.

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