Modl Anlysis s Mens of Explining the Oscilltory Behviour of Trnsformers Modl nlysis hs proved to e n outstnding id in the clcultion nd mesurement of nturl oscill tions of trnsformers. With modl prmeters it is possile to determine y direct mens how resonnce nd impulse voltges ffect ny point in winding. It is explined, with reference to the switching of n open-circuited 85011100 MVA trnsformer, how modl nlysis is cr~ ried out nd how it differs from ppli ction of the trvelling wve theory. The rticle lso reports on the possiility of selectively influencing the oscilltory chrcteristics for relile dimensioning of the insultion. The modl prmeters re used to derive circuit digrm with which trnsformer's effect on the power system cn e ccounted for nd the voltges trnsmitted clculted. Dr. P. Glninger is former memer of the Trnsformer Division, where he ws engged in development wort< on highvoltge systems, insulting fluids nd noise emission Mnnheim, Federl Repulic of Germny Modl Anlysis s Mens ol Explining lhe Oscilllory Bel1viour of Trnnsforrners Every electricl trnsmission system cn e induced to oscillte. Typicl cuses re lightning strokes nd switching opertions. When the exciting voltge contins pronounced hrmonic component in the region of one of the system's nturl frequencies, this will induce resonnce. The sptil voltge distriution tking plce during excittion t resonnt frequency will e termed 'eigenform' in the following. 'Modl nlysis' is the nme given to the procedure wherey system cple of oscilltion is descried y the following three modl prmeters: - 'Eigenform' - Nturl frequency - Dmping 'Dmping' is the nme given to the speed with which the nturl oscilltions decy. Clculted modl nlysis is used to determine the modl prmeters from the design drwings y mens of network nlysis progrms. Experimentl modl nlysis involves crrying out mesurements on the oject itself to otin the modl prmeters. Digitl oscilltion nlyzers, of which there re mny different types ville on the mrket, re used for this. Modl nlysis is y no mens new, eing sed on network theory developed in the lst century. However, the consistent ppliction of this theory in comintion with modern dt processing methods, hs led to trnsprent representtion of complex reltionships nd to the understnding of so mny processes tht we re justified in speking tody of new method. Since the sinusoidl excittion (t resonnt frequency) of n electricl system is lmost lwys simple mtter, eigenforms cn e mesured directly. This is fr more difficult with mechnicl system, for which frequency nlysis is usully necessry to determine the modl prmeters. It therefore seems prdoxicl for modl nlysis to hve ecome such n essentil tool in mechnicl engineering, while it hs remined lrgely unknown in the electricl field. The dvntges of modl nlysis in the investigtion of electricl systems re explined in the following. Cc:mnection of Trnsformer Lightning strokes, connection nd disconnection under no-lod conditions, erth fults, etc. ll cuse overvoltges which led to trnsient oscilltions [1 ]. Figure 1 shows how n oscillting voltge t resonnt frequency ffects trnsformer internlly. Although the input voltge in Fig. 1 is periodic nd its front is not prticulrly steep, oscilltions cn e recognized in the trnsmitted voltge. The superimposed oscilltion cuses the mplitude of the trnsmitted voltge to rise (Figs. 1 to 1 d). Figure 1 e shows tht, for steep periodic impulse wve, the nturl oscilltions re very pronounced. Brown Boveri used modl nlysis to thoroughly investigte the oscilltory chrcteristics of lrge genertor trnsformers []. These trnsformers crry powers of up to 1100 MVA directly from the 7 kv terminl of the unit genertor to the 400 kv power system. Connection of such trnsformer under no-lod conditions cuses trnsient oscilltions, nd is n pproprite suject for n introduction to the prolems involved. Figure shows n exmple in which n 850/1100 MVA trnsformer on no lod is connected to the 400 kv system vi 1 km long overhed line. When the circuit-reker is closed, wve trvels t pproximtely the speed of light (v ~ 300000 km/s) towrds the trnsformer. The numer of lines leving the node determines the mgnitude of the voltge, which is n/(n+1) p.u. The end of the line reflects the wve, nd the voltge is prcticlly douled due to the high chrcteristic impednce of the trnsformer. In the Brown Boveri Review 1-86 41
\ d e Fig. 1 - Exmples of trnsformer nturl oscilltions excited y periodic nd oscillting switching impulse voltges The winding excited is the 400 kv h.v. winding of n 850/1100 MVA trnsformer (upper em, 50 V/div.). The mesurements were crried out on the 7 kv Lv. winding (lower em, 5 V/div.). Sweep 1 O µs/div. Fig. - No-lod connection of trnsformer 1 n 400 kv 850/1100 MVA 7 kv : Circuit : Voltge chrcteristic t the 400 kv terminls of the trnsformer, s function of the numer of outgoing lines n Power system voltge Time Voltge etween trnsformer terminls u UN - - - u UN u UN n-= n~3 n~1 4 Brown Boveri Review 1-86 Modl Anlysis s Mens of Explining!he Oscilltory Behviour ol Trnsformers
worst cse this is x(400/,,j3).y'. ~ 635 kv, in which 400/,,/3 is the r.m.s. vlue of the conductor to erth voltge, its pek vlue eing otined y multiplying it y y'.. The reflected wve trvels ck to the node on.the left. When the numer of lines leving this node is high (n ~oo), the wve is reflected with the opposite sign, resulting in squrewve oscilltion t the trnsformer terminls. The 50 Hz oscilltion, however, is extremely slow, nd the representtion in Fig. does not consider ny effect it might hve. As the length of the line (I~ 1 km) is trvelled four times in one period, the frequency I of the oscilltion is clculted s: I~ V/(4X /) I~ 300000/(4X 1) I~ 75000 Hz z~5 z~0,5 z~0,75 z~1 When there re three outgoing lines (n ~ 3), the nturl oscilltion decys rpidly. This dmping effect results from energy eing drwn wy over the outgoing lines. When there is only one line (n ~ 1 ), however, wve trvels with hlf of the system voltge towrds the trnsformer. Reflection cuses it to doule t the terminls, so tht it is once more equl to the full system voltge. The returning wve disppers into the power system, where its energy is susequently converted into het. It is in this point tht the trnsformer differs, since it converts the oscilltion energy into het there where it origintes. z z~0,5 z~0,5 z~0,75 l Trvelling nd Stnding Wves It is possile to interpret the chnge in voltge distriution on line with spce nd time oth s trvelling wve phenomenon nd s stedystte oscilltion. Figure 3 shows tht these two interprettions re eqully vlid. Figs. 3 nd 3 show the voltge distriution long n open-ended line nd short-circuited line fter connection of direct voltge of vlue 1, respectively. Ech of the functions h (I) shown in Fig. 3 is clled 'step response' or, since in the cse t hnd the exciting step is the unit step, 'trnsient function'. Wheres the trvelling wve theory is often pplied to clcultion of line Fig. 3 - Connecting () n openended line nd () shortcircuited line. The chnge in voltge distriution with respect to spce nd time cn e considered oth s trvelling wve phenomenon nd s stedystte oscilltion. h = Trnsient function t = Time z = Locl coordinte U = Voltge etween trnsformer terminls oscilltions [3], considertion of the trnsient function hs dvntges where trnsformers re concerned. There re three min resons for this: 1. The stndrdized 1 /50 impulse wve cn e relily simulted y step, so tht the trnsient function represents direct mesure of the stressing cused y the impulse voltge.. It is often required to know the oscilltory ehviour t only few points in the trnsformer. By mens of the llocted trnsient functions, the voltge chrcteristic cn e determined for ny given input voltge without hving to follow the trvelling wve through the network formed y the windings. The soclled 'Duhmel integrl' is used for this clcultion. Modl Anlysis s Mens o1 Explining lhe Oscilltory Behviouro!Trnsformers Brown Boveri Review 1-86 43
Fig. 4 - Trnsient function of shortcircuited line for 80 /o of the line length (z = 0 8) nd its rekdown into cosine oscilltions. The mplitude spectrum is shown on the fr left. A : Frequency rnge : Time rnge t =Time I Frequency A Amplitude spectrum 3. The originl notion of 'trvelling wve' is fundmentlly invlid for trnsformers. Due to the dependence of the trvelling wve's velocity on the frequency nd the reltively strong dmping t high frequencies, Fig. 5 - Eigenforms of short-circuited line. The eigenform indictes the mplitudes of the cosine oscilltions mking up the trnsient function. The mplitudes t z = O B correspond to Fig. 4. A 1, A, A 3, A 4 re mplitudes of the first lour prtil oscilltions. z = Locl coordinte A, -3~~ z the incoming wve chnges so much tht its originl shpe ecomes unrecognizle fter only few metres. Although use of the trvelling wve theory is fundmentlly possile, it provides no cler, visul picture of the processes tking plce. Modl Prmeters of l.ine The trnsient function cn e roken down into components y mens of hrmonic nlysis. Figure 4 shows, for exmple, the trnsient function of the short-circuited line (Fig. 3) when z ~OB, which cn e represented s superimposition of cosine oscilltions. This is represented mthemticlly s follows: h (t) ~ 1 - z+ A, (z)cos (nl 1 t) + A (z)cos (nl t) + A3(z)cos (nm where h(t) z t A 1, A, A 3 ~ Trnsientfunction t point zon the line ~ Locl coordinte in p.u. ~Time ~ Amplitudes, depending upon loction 1 1, 1, 1 3 ~ Nturl frequencies of the line The prt of the direct voltge 1-z, which contins the inductively trnsmitted component, is dispensed with in the representtion in Fig. 4. On the fr left, the mplitudes re represented s function of the frequency. This 'mplitude spectrum' descries the trnsient functions just s fully s the chrcteristic with respect to time. Depending upon the tsk t hnd, clerer picture will e provided y working either with the chrcteristic with respect to time (in the time rnge) or with the spectrum (in the frequency rnge). The mplitudes of the short-circuited line re distriuted sinusoidlly, nd re descried y the following functions (Fig. 5): A 1 (z)~ ~(/n)sin(nz) A (z) ~ -(/n) sin (nz) A 3 (z) ~ -(/3n) sin (3nz) A,(z) ~ -(/4n) sin (4nz) The oscilltions on line exhiiting losses decy with time. For reltively minor dmping, the trnsient function is pproximtely: h(t)~ 1-z + A 1 (z)exp (-d1 I) cos (nl1 I) +Adz) exp (-d t) cos (nl 1) + A 3 (z}exp (-d3t) cos (nm +. d 1, d, d 3... ~ Decy coefficients In the following, not only the decy coefficients ut lso the qulity fctors On 44 Brown Boveri Review 1-86 Modl Anlysis s Mens of Explining the Oscilltory Behviour o!trns!ormers
Fig. 6 - Comprison of the oscilltory ehviour of series resonnt circuit nd the equivlent circuit digrm of trnsformer : Circuit digrm : Amplitude-frequency response c: Trnsientfunction = Frequency t =Time t 0 = Nturl frequency t 0 = Period of the nturl oscilltion (/o ~ 1110) U = Trnsmitted voltge U 0 = Input voltge y = Cpcitive trnsformtion µ = Inductive trnsformtion Q = Qulity fctor R U!Uo L er+ 1 uu, 0 U!Uo ml",. 0 yµ;y1 Q Oly-µI y U!Uo 111 0 0 3 tit, U!Uo r 111 0 0 3 I/to c 01~nl1/d1,0~ nfld. re used to chrcterize the dmping. The nturl frequency lnd the qulity fctor Ore termed modl prmeters. The third modl prmeter is the 'eigenform', which hs lredy een defined s the voltge distriution for sinusoidl excittion t resonnt frequency. [] shows tht the voltge distriution t resonnce corresponds to pproximtely O.A(z). Since the constnt O fctor does not chnge the shpe (or 'form') of the distriution, the eigenform cn e represented eqully well y the mplitudes A(z)which vry loclly. Figure 5 illustrtes this y referring to the short-circuited line. The modl prmeters cn e used to define the condition for distortionfree propgtion of trvelling wve more precisely: A wve trvelling long line does not chnge its shpe when - the velocity of the trvelling wve is independent of the frequency, so tht 1 ~ 11' /3 ~ 311. - ll nturl oscilltions hve the sme decy coefficient (d 1 ~ d ~ d3... ), so tht 0~01, 03~301. When, s ws descried erlier, lines leving node drw energy from the oscilltion, oth conditions re pproximtely stisfied nd the wve- Modl Anlysis s Mens of Explining the Oscilltory Behviouro! Trnsformers shpe is mintined. In the cse of trnsformer windings, however, none of the nmed conditions generlly pply, resulting in mjor deformtion of the trvelling wve. Simple Equivlent Circuit Digrms In modl nlysis, the trnsient function t ny given point in structure is roken down into hrmonic components. Figure 4 shows n exmple of this. Ech of the oscilltion components cn e considered s the output signl of simple resonnt circuitto which the modl prmeters f, O nd A re llocted. Modl nlysis is therefore sed on the ide of descriing structurl ehviour y the superimposition of simple resonnt circuits. Figure 6 (top) shows simple series resonnt circuit. When direct voltge (frequency t~ O) is pplied, the entire voltge lies cross the cpcitor. The mplitude-frequency response in Fig. 6therefore egins t 1. At very high frequencies the voltge cross the cpcitor decreses stedily, so tht the frequency response ends t 0. Excittion t nturl frequency cuses resonnce mplifiction corresponding to the O fctor. With low dmping, the trnsient function (Fig. 6c) pproximtes decying cosine oscilltion with the mplitude A~ 1. When simple trnsformer equivlent circuit digrm [4] is used, conditions re different. At low frequencies (including the 50 Hz system frequency) the trnsformtion is inductiveµ, while t high frequencies the cpcitive trnsformtion y, which depends upon the rtio of the cpcitnces, tkes effect. The trnsformtion y corresponds to the initil vlue of the trnsient function t time t ~ 0 (Fig. 6c), which mkes use of trnsient oscilltions to ttin the inductive end-vlue µ. The mplitude A is here the difference etween the cpcitive nd inductive trnsformtion. y-µ~a Amplitude A ecomes especilly high when the inductive trnsformtion is negtive, condition which often occurs. According to [4], the mplifiction is yy+0.a~10.ai ( 0 ~ Qulity fctor) By djusting the cpcitive nd inductive trnsformtion it is therefore possile in the cse of trnsformers to mke oth the oscilltion of the trnsient function nd the mplifiction of the frequency response dispper completely. Reduction of the qulity fctor is difficult nd not lwys necessry. As the exmple of the trnsient functions in Fig. 6c shows, incresed dmping would only cuse the oscilltion to de- Brown Boveri Review 1-86 45
cy more rpidly, while the highest voltge vlue is more or less unffected. Surprisingly, similr process pplies lso to decying sinusoidl excittion s the increse in the qulity fctor lso lengthens the durtion of the trnsient response. Since, with decying excittion, uild-up of the trnsmitted voltge coincides with the decy, the qulity fctor hs little influence on the vlue of the voltge. Fig. 7 - Voltge distriution for homogeneous winding. Stressing due to impulse voltges nd resonnce depends upon the rtio of the erth cpcitnce to the directxis cpcitnce. z = U = Locl coordinte (turn length) erth cpcitnce direct-xis cpcitnce (cpcitive distriution fctor) Trnsmitted voltge Fig. 8 -Types of windings Uo = @~ (8) CD z Input voltge Resonnce Envelope curve of the impulse voltge Inductive Cpcitive The interleved winding fetures considerly higher equivlent cpcitnce. : Non-interleved winding : Interleved winding 9 8 7 6 5 4 3 5 14 4 13 3 1 1 13 14 15 16 17 18 16 7 17 8 18 9 Homogeneous Windings The oscilltory chrcteristics of homogeneous winding is similr to tht of line. An importnt difference, however, concerns the direct-xis cpcitnce, which cn e very high for the windings. Figure 7 shows the equivlent circuit digrm of winding. When steep-front wve is pplied, the initil distriution t time t ~ O reflects the distriution of the cpcitnces. The distriution is lso the sme for sinusoidl excittion t very high frequencies (f -oo ). This initil distriution is identified y the soclled '-vlue'. At power frequency the inductive voltge distriution long the line is lmost liner. Since the inductive distriution tkes on stedy-stte vlue t t-oo, it is designted the 'finl distriution'. The mplitude of the fundmentl wve A 1 of winding is, ccording to [5]: A 1 ~ -(/n)."/( +n) resulting in, for exmple: A1 ~ -0 1 for ~ 1.S A 1 ~ -0 30 for ~ 3 A1 ~ -0 50 for ~ 6 A 1 ~ -0 60 for ~ 1 A1 ~ -0 64 for -oo The difference etween the inductive finl distriution nd the cpcitive initil distriution t the winding centre is pproximtely equl to the mplitude A1 of the fundmentl wve when -vlues re low. If only the fundmentl oscilltion is considered, the impulse voltge stressing nd resonnce stressing in the winding centre re estimted to e UmxlUo ~ 0 5+IA1I nd Umxl Uo ~ I01. A11. respectively. The impulse voltge stressing of winding with low direct-xis cpci- 46 Brown Boveri Review 1-86 Modl Anlysis s Mens of Explining the Oscilltory Behviour o!trns!ormers
lnce ( -oo) is UmxfUo ~ 0 5+0 64 ~ 1 14. Comprehensive mesurements crried out y Brown Boveri hve shown tht qulity fctor of 10 is typicl for the first nturl frequency of windings. Consequently, the mximum resonnce mplifiction for -oo is UmxlUo ~ 10.0 64 ~ 6-4. The direct-xis cpcitnce cn e incresed y interleving winding. Figure 8 shows the interleving method most often used y Brown Boveri. The shown rrngement of the conductors increses the potentil difference etween them, resulting in n increse in the stored cpcitive energy nd therefore higher direct-xis cpcitnce [6]. -vlues of 3 nd elow re otinle y mens of interleving. At ~ 3, the impulse voltge stressing drops to Umxl U 0 ~ 0 5 + 0 3 ~ OB nd the resonnce stressing to Umxl U 0 ~ 10. 0 3 ~ 3. The stresses occurring in prctice re therefore so low tht they cn e withstood y the insultion system without ny specil mesures hving to e tken. Trnsformer Modl Anlysis s Mens of Explining lhe Oscilltory Behviour of Trnsformers Figure 9 shows highly simplified section through the winding rrngement of n 850/1100 MVA trnsformer. The equivlent circuit for this rrngement is shown in Fig. 9. Ech of the tour winding cylinders is divided into tour sections. The individul cpcitnces nd inductnces re ssumed to e suject to losses. Mutul inductnces exist etween the inductnces. Since the windings hve lrgely uniform structure, the reltively primitive ideliztion expressed y the equivlent circuit suffices for clcultion of ll importnt eigentorms of the trnsformer. Non-uniform distriution of the cpcitnces or inductnces long the winding must e tken into ccount when drwing the equivlent circuit digrm. Agreement etween mesurement results nd clcultions depends lmost exclusively upon how well the equivlent circuit is mtched to specil design fetures. A network nlysis progrm is used to determine the nturl frequencies, qulity fctors nd eigenforms. Ech node is llocted n inductive trnsformtion µ.nd cpcitive trnsformtion y. The following reltionship, wherey the sum of the mplitudes A equls the difference etween the cpcitive nd inductive distriution, is vlid for ech node: y~µ.~a,+a+as. An pproximtion of the impulse voltge stressing t the node cn e determined y mens of the trnsient function h(t)~ µ.+a 1 (cosn1 1 t) + A (cos nl t) +As (cos nfst) +. The resonnce stressing is given pproximtely y A 1. 0, (for excittion with frequency '1) or y A. 0 (for excittion with frequency 1 ) etc. Modl nlysis consequently indictes cler concept for the minimiz.. tion of internl electricl stresses in trnsformers: To otin the smllest possile mplitudes, nd therefore the lowest possile impulse nd resonnce stresses, the difference etween the cpcitive nd inductive distriutions must e smll. This is chieved y using n optimum winding rrngement nd n pproprite winding design. Relile dimensioning of the insultion is mde possile y the wide vriety of winding designs (lyer, spirl or coil windings) nd their different degrees of interleving, despite numerous oundry constrints, such s heting, which hve to e considered. Experimentl Modl Anlysis At 50 Hz trnsformer on no lod is ctive due to its no-lod inductnce. This seemingly trivil sttement no longer ppers so when it is considered tht trnsformer often egins to exhiit cpcitive chrcteristics from 00 Hz upwrds. If the frequency of the pplied voltge is incresed still further nd the input dmittnce (otined from the input current nd input voltge) is mesured, resonnce mplifictions result t the nturl frequencies of the trnsformer. Viewed from the input terminls, the trnsform- K, ; K,.. us 7 kv us..,.. ~ OS 400kV OS.., H Fig. 9 - Preprtion of n equivlent circuit digrm for the computed modl nlysis of n 850/1100 MVA trnsformer : LVwinding US (two lyers); h.v. winding OS (min nd stepped winding) : Equivlent circuit digrm i, = Mesuring points K =Core er cts in the sme wy s the circuit shown in Fig. 10. Ech point of resonnce cn e llocted nother three modl prmeters, nmely the frequency I, the qulity fctor Q nd the chnge in cpcitnce C. Thus it is possile to determine ll of the nturl frequencies nd ll qulity fctors directly y crrying out single mesurement t the input terminls. However, there is no mens of determining the eigentorms s there is no direct reltionship etween the chnge in cpcitnce C t the input terminls nd the mplitudes A of the inner nodes. The eigenforms therefore hve to e either clculted or deter.. mined y crrying out mesurements t the inner nodes. Fig. 11 shows the input dmittnce of n 850/1100 MVA trnsformer s n exmple. Four nturl frequencies chrcterize the frequency response. Figure 11 shows the mplitude-frequency response in the upper qurter of the 400 kv winding (pt. 1 in Fig. 9). Significnt mplifiction is only recognizle t frequencies 1 1 nd1. Fig. 11c shows the mplitude-frequency response in the centre of the stepped H H H Brown Boveri Review1-86 47 '
Fig.10- Circuit digrm simulting the trnsformer for power system clcultion nd for determining the trnsmitted volt~ ge for ny type of input voltge I Ro Co Lo r 0,4 H 300000 950 pf L, L, L, 60mH 190mH 54mH I L, 98mH R,, 8300 t' ]~goo I, 800 19500 c, c, c, '= ' ' c, 300 pf 10 pf 10 pf 55 pf R = Resistnce C = Cpcitnce L = Inductnce winding (pt in Fig, 9), Here, mplifiction is pronounced t the frequencies 1 3 nd 1 4. The Tle lists the modl prmeters determined from these frequency responses y modl nlyzer. If such n nlyzer is not ville, pproximte qulity fctors cn e otined from: 01 ~ lj/m1, 0 ~ l/m. M ~ Frequency nd width t 11./ (corresponding to some 70%) of the highest mplitude Synthesis of Frequency Responses The vlues given in the Tle re from the modl nlysis of the frequency responses in Fig. 11. The reversed procedure, in which modl prmeters re comined to otin frequency responses or time signls, is termed 'synthesis'. The procedure used in synthesis cn e demonstrted y mens of the simplified digrm shown in Fig. 10, with elements for which the following is vlid: R, ~ 1/(.n.11.C1.01) R~1/(.n.1.C.0) Rs~ 1/(.n.ls.Cs.Os) L1~1/((.n.11).C1) L ~ 1 /((. n.1). C) Ls~ 1/((.n.13) Cs) Modl prmeters of frequency responses in Fig. 11 Nturl oscilltion Nturl frequency t in khz Qulity fctor Q Cpcitnce C in pf (Fig. 11) Amplitude A in p.u. (Fig. 11 ) Amplitude A in p.u. (Fig. 11c) 1 180 10-4 300-09 0 007 In this figure, Lo is the open-circuit inductnce, C 0 the input cpcitnce t very high frequency nd Ro loss re- 4 9 10 8 10-0-07-05 3 47 1 11 6 10 0 03 0 05 4 68 6 1 6 55-0005 0-058 sistnce. Lo, C 0 nd Ro re derived from the minimum dmittnce, shown to occur t 8 khz in Fig. 11. The input dmittnce shown in Fig. 1 coincides exctly with the results of the mesurement in Fig. 11. Thus the circuit shown in Fig. 10 permits the trnsformers's ffect on the power system to e determined precisely. If the cpcitive trnsformtion nd mplitudes A,, A, etc. t prticulr point re known, the trnsmitted voltge t this point cn e clculted with the id of Fig. 10 s follows: U~ y-a1.u1-a.u-as.us.. U ~ Trnsmitted voltge t mesuring point y ~ Cpcitive trnsformtion t mesuring point A1, A, As, etc. ~ Amplitude t mesuring point U 1 ~ Voltge cross cpcitor C 1 in Fig.10 U ~ Voltge cross cpcitor C infig.10 Figure 1 shows the synthesis of the mplitude-frequency responses for pts. 1 nd. It is worth noting how few modl prmeters re required to descrie these complicted curves. Appliction of the lst-mentioned eqution is y no mens restricted to sinusoidl phenomen. This method cn e used to determine the curve of the trnsmitted voltge with respect to time for every phenomenon. 48 Brown Boveri Review 1-86 Modl Anlysis s Mens o! Explining the Oscilltory Behviour o! Trnsformers
y t IU/Uol j 0,8 ms 0,6 0,4 0, o.µl: ~~~~~ j 0 50 khz 100 0,8 ms 0,6 y t 0,4 0, IU/U 0 1 t 50 khz 100 c 50 khz 100 c O+-~~~~~~~~---' 0 50 khz 100 1,0 IU/U 0 1 t 0,5 1,0 IU/U 0 I t 0,5 50 khz 100 ~1 50 khz 100 Fig.11 -Amplitude-frequency responses of n 850/1100 MVA trnsformer : Input dmittnce t the 400 kv terminl : Amplitude-frequency response t point 1 in Fig. 9 c: Amplitude-frequency response t point in Fig. 9 f = Frequency Y = Input dmittnce U = Trnsmitted voltge U 0 = Input voltge Fig. 1- Modl prmeters in the Tle comined with the id of the circuit in Fig. 10 to form synthesis of the mplitude-frequency responses in Fig. 11 Symols s in Fig. 11 Summry Modl nlysis is sed on the ide of descriing structurl ehviour y superimposing simple resonnt circuits with mplitudes t defined points. A modl nlysis yields modl prmeters which cn e directly trnsformed into circuit digrm. In this digrm, ech nturl oscilltion is llocted series resonnt circuit. Such representtion is new. The circuit digrm not only permits the trnsmitted voltge to e clculted for ech point of interest nd with ny input signl, ut lso serves s n equivlent digrm which tkes ccount of the effect trnsformer hs on the grid in power system clcultions. Modl Anlysis s Meris of Explining lhe Oscilltory Behviour of Trnsformers An outstnding feture is tht the modl prmeters cn e used for direct pproximtion of the stressing cused y impulse voltges nd resonnce. Modl nlysis provides cler concept for minimizing internl electricl stressing nd thus points the wy to relile dimensioning of the insultion.!biliogrphy [i] G. Blzer: Berechnung des trnsienlen Spnnungsverluts n Trnslormtoren. Elektrizit8.tswirtschft 831984 (16) 70-75. [] P Glninger: Modle Prmeter der elektrischen Eigenschwingungen vn Trnstormtoren. etz-archiv 61984 (1) 399-405. [3] R. RUdenerg: Etektrische Wnderwellen. 4. AufL Springer-Verlg, Berlin/G6ttingen/Heidelerg, 196. [4] P. G/ninger: Ds Schwingungsverhlten eines einfchen Trnsformtor-Erstzschltildes. etz-archiv 51983 (11) 369-375. [5] K. W. Wgner: Ds Eindringen einer elektromgnetischen Welle in eine Spule mit Windungskpzit3.t. Elektrotechn. u. Mschinenu 331915 (8) 89-9. [6] H. Brechn: Stossspnnungssichere Trnsformtorwicklungen. Bulletin Oerlikon Nr. 38/39 (1958) 8901. Brown Boveri Review 1-86 49