Analytical Modelling of Pulse Transformers for Power Modulators

Transcription

1 nalytical oelling of Pulse Transformers for Power oulators J. Biela,. Bortis, J.. Kolar Power Electronic Systems Laboratory (PES, ETH Zuric, Switzerlan E-ail: / Homepage: bstract Te parasitic capacitances of transformers significantly influence te resulting pulse sape of a power moulator system. In orer to preict te pulse sape an optimize te geometry of te pulse transformer before builing te transformer an equivalent circuit an analytic epressions relating te geometry wit te parasitic elements are neee. Terefore, a moel consisting of 6 equivalent capacitors an a simplifie circuit as well as te belonging equations are presente. Te equations are verifie by measurement results for a pulse transformer an a soli state moulator esigne for linear accelerators. Keywors: Transformer moel, parasitic capacitance, soli state moulator, analytic moel kV 1 8.8kV 6 4.4kV kv I. INTROUTION.E-6 4.E-6 6.E-6 8.E-6-1.E-6 1us 3.E-6 3us 5.E-6 5us 7.E-6 7us 9.E-6 9us Figure 1. omparison of measurement an te two ifferent calculations kV kV us pulse measurement calculation previous moel 1.9us.3us.7us 1.5E-6 1.7E-6 1.9E-6.1E-6.3E-6.5E-6.7E-6.9E-6 Figure. Soli state moulator (1kV/4k / transformer for measurement. Hig an ig power pulses are use in a wie variety of applications, for eample in accelerators, raar, meical raiation prouction or ionization systems. In many of tese applications te requirements on te generate pulses regaring for eample rise/fall, oversoot, pulse flatness an pulse energy are ig. Te pulses for tese applications are usually generate wit pulse moulators, wic often use a pulse transformer for generating ig output s. Tere, te parasitic elements of te transformer significantly influence te acievable sape of te pulse. For preicting te pulse sape of te moulator system, for esigning pulse forming networks an for optimizing te geometry of te pulse transformer before builing te transformer an appropriate equivalent circuit of te transformer is neee. Tis equivalent circuit must on te one an preict te transfer function of te transformer an on te oter an te parameters of te circuit soul be analytically calculable wit te geometric an electric parameters of te transformer. Terefore, an equivalent moel of te pulse transformer an te analytic equations for calculating te parameters are presente in tis paper. In [1, ] a simple L-L- moel of te transformer is use to preict te resulting pulse sape. Te value of te capacitance in tis moel is calculate by applying te equation for parallel-plate capacitor [1] an for non parallel-plate capacitor [] on te / seconary wining interface wat results in εlgg N 1 εlgg N + N + 1 \/ = = (1 g N g N In bot, only te istribute electric energy in te volume between te an te seconary wining is consiere, wat results in a relatively poor accuracy. Tis coul be seen in figure 1 were a measure an two calculate pulse responses of a transformer are sown. Te input of te calculation moels was te measure output of te soli state switc. In contrast to te preiction of te simple moels te result of te etene moel matces te measurement results muc better. In tis moel all te regions wic are relevant wit respect to te istribute energy are consiere. Tus, in section II of te paper first te energies wic are store in te ifferent relevant regions are calculate by analytic approimations. In te net step te calculate energies are compare wit te energies store in te equivalent circuit in section III. By tis comparison te parameters of te equivalent circuit of te pulse transformer are etermine. Te suggeste moel comprises si capacitors an coul be use in any connection of te transformer. If bot winings are groune te moel coul be simplifie to an equivalent circuit wit just one capacitor. it te consierations of section II an equation is erive wic allows te calculation of te equivalent capacitance by means of te transformer geometry. Base on tis equation a goo preiction of te pulse sape is possible as sown in figure 1. noter possibility to obtain te parameter of te suggeste equivalent circuit is to use FE-simulations. For tis reason te setup of te simulations is eplaine in section IV. Te escribe setups also coul be use to parameterise te moel by impeance measurements. In section V te propose equations are valiate by comparing te calculate an te measure pulse sape for ifferent operation an loa conitions for a soli state pulse moulator. Finally, a conclusion is presente in section VI. II. LULTION OF PRSITI PITNES For etermining te equivalent capacitances of te pulse transformer s equivalent circuit te istribute energies in all regions must be calculate. In orer to be able to calculate te store energies te 3 imensional istribution of te electric fiel strengt must be known. Te fiel istribution, owever, generally only coul be calculate wit consuming numeric FE-simulations. Since in most regions te run of te electric flu lines approimately lies witin a plane wic is parallel to te wining ais, te per unit energies for tese planes are consiere in te following. In figure 3 a cut of te transformer wit surrouning tank is given. Tere, si ifferent planes/regions are sown wic are consiere in te following for calculating te store energy. Since te calculations are performe for planes per unit energies result. In orer to obtain te value of te store energy of te respective region te per unit energy must be R3 R4 R6 R R5 tank R6 R3 R4 seconary Figure 3. Regions for te istribute capacitance of te pulse transformer.

2 V 4 V + l V l R4 Oil( V( 1 l R l R5 seconary tank iso E-saping ring Figure 5. alculation of te energy by means of te parallel plate capacitor l R6 l R3 Figure 4. efinition of te lengts for te istribute capacitances. multiplie by te lengts sown in figure 4. Te areas/volumes wic are not covere by a region are neglecte in te following, since te energy ensity an terewit te sare in te total equivalent capacitance is relatively small. Te presente calculations are performe for a pulse transformer wit non parallel-plate winings. However, te presente proceure can analogically applie to oter wining arrangements, e.g. transformer wit parallel winings. Furtermore, it is assume tat te an also te tank is groune wat is usually true in practice. In te following paragraps te store energies for te si regions are calculate separately. Tere, te presente equations always represent te part of te energy wic is store in te wining of one leg, tat means tat for eample te energies must be multiplie by two for te setup sown in figure 3.. Energy between te winings In region R 1 te area between te an te seconary wining is summarise. Tis is te only area wic is consiere for etermining te equivalent circuit of te pulse transformer in te approac presente in [1, ] cf. equation (1. For ing te calculation of te energy between te an te seconary it is assume in te following tat bot winings consist of a conuctive plate wit a linear istribution. Te wining is groune at te lower sie an te at te upper en is, wereas te istribution of te seconary wining starts at te offset an ens at V + as sown in figure 5. In case te seconary wining is also groune is zero. it te assume istribution te ifference between te two plates an te istance between te two plates coul be written as 1 V( = ( V V + V ( = ( + = + ( w Oil Iso 1 Tere, 1 is te istance at te lower en an at te upper en. In orer to te calculations furter, te electric flu lines between te an te seconary wining in region R 1 are approimate by straigt lines wic are ortogonal to te wining. In tis case te energy store in te ifferential element coul be calculate by (( V4 V3 V3 (( 1 1 ε ( ε + V = = (3 ( Oil ( + Iso + Integrating tis epression along te wining yiels te total per unit energy wic is store in te plane (R 1 between te two plates. ultiplying te per unit energy by te lengt l (cf. fig. 4 yiels te energy store in region R 1. R (( V4 V3 V3 (( w ε + = l (4 In general, tis energy epens on te ifference between te two plates an also on te offset of te seconary wining. So far it as been assume tat te materials between te two plates ave te same permittivity ε Iso = ε Oil =ε. Usually, te space between te two winings is fille wit oil an te coil former of te seconary wining. If tese two materials ave a ifferent permittivity an equivalent value for te permittivity coul be calculate by εiso εoil ( Iso + Oil ( ε eq( =. (5 εiso Oil ( + εoil Iso Tis equivalent permittivity is a function of since te pat lengt of te fiel line in te oil varies wit. Substituting te permittivity in equation (3 by tis epression an calculating te energy results in 1 lεisoεoil (( V4 V3 + V3 = ( εiso ( 1 + ( 1 Iso + εoil Iso. (6 gain, te coul be set to zero if te seconary winings is also groune an te V 4 coul be substitute by V 4 = V -. B. ining winow: above seconary R Region R consists of te area above te seconary wining witin te wining winow. Te borer of tis region as a comple sape wic is etermine by te, te wining an te saping ring. In orer to be able to calculate te istribute capacitance of tis region analytically te geometry is simplifie as sown on figure 6. First, it is assume tat te wining consists of a metal plate wic is groune, i.e. is neglecte since V = N >>. Furtermore, tis plate is etene so tat te upper en touces te. Secon, te seconary wining is neglecte an te saping ring is replace by a circle. Tis simplifications result in a coaial structure wic energy is approimately (±1% te same as te one of te original structure as coul be proven by FE-simulations. Te energy store in te coaial structure in figure 7(a coul be calculate wit te equation for te cylinrical capacitor, wat results in te per unit equation 1 πε πεoil V+ V Oil 3 R= V+ V3 = wit k= 1.8. (7 4ln r r 4ln k r ( ( ( ( o i i Tere, only one alf of te energy is calculate since te equations represent te part of te energy wic is store in te wining of one leg. i.e. alf te region R. Te factor k is empirically etermine by FE-simulations so tat te resulting ifference between te energy of te original structure an te equivalent one is minimal. In [3] a similar transformation is escribe but te factor k is set equal to For te consiere setup, tis coice resulte in a larger error for te equivalent energy tan k=1.8. In orer to obtain te energy R te per unit energy R must be multiplie by l R.. bove seconary wining outsie wining winow R 3 Region R 3 is te equivalent of region R outsie te wining winow, but tere te run of te borer in te upper region is more comple. R R R Figure 6. Simplification of te region R to a coaial structure.

3 a r o = k original approimation saping ring r i omogeneous oil tank b approimation saping ring r o= k ri original omogeneous Figure 7. (a Equivalent structure for calculating te energy of region R. (b Original an simplifie geometry for region R 3. For ing te setup it is assume tat te wining consists of a groune plate (i.e. neglect << V wic is stretce to te cover of te tank. Te influence of te cover itself is neglecte since its istance to te saping ring is relatively large (cf. fig. 3. Furtermore, it is assume tat te saping ring is in te mile of te wining an te wall of te tank, wat is approimately fulfille for a compact esign, were te istance between te saping ring an te tank is equal to te minimum possible one. Te resulting rectangular borer is again approimate by a coaial structure as sown in figure 7(b. it tis approimation te store energy for tis structure coul be calculate by lr3 πεoil R3 = V + V3 wit k = 1.75, (8 ln k r ( ( i were k is empirically aapte by FE-simulations again. For tis structure te value wic results in a minimum error is 1.75, wic is te same as propose in []. Remark: In case te transformer is not insie a groune tank te plate on te left an sie in figure 7(b is be omitte. Tere, te geometry coul be simplifie by assuming tat te wining is groune an etene to infinity so tat te energy in tis region coul be calculate wit te equations for te capacitance of two wire lines. Tis results in πεoil R3, II = lr3 (9 ln ri + ( ri 1 for te energy store in region R 3.. Between seconary an tank R 4 Te energy between te seconary an te wall of te tank below te saping ring is partially inclue in region R 4. Tere, te wall of te tank an te seconary wining form a non-parallel plate capacitor wit a approimately linear istribution again. Te an te istance between te plates are given as a function of by 1 V( = V+ V3 ( = + 1,. (1 were te variables are efine as sown in figure 8(a. Using tese epressions te energy coul be calculate by ε ( V + V Oil 3 R4 = lr4 (11 + (( 1 1 as escribe in section II.. ssuming a compact system, te istances coul be set to = an 1 = - 1 as also as been one for region R 3. In figure 8(b a plot of te simulate electric flu lines between te seconary wining an te tank is sown. Tere, it coul be seen tat especially at te upper an te lower en of te wining (blue circles te run of te simulate flu lines eviates from te one wic as been assume in te calculation moel for region R 4. t te upper en te reason for te eviation is te saping ring. t te lower en te fiel is mainly istorte by te a V + V 4 seconary V( X V + a 1 b Figure 8. (a efinition of te variables for region R 4. (b Electric flu lines between te seconary wining an te wall of te tank. istribution on te seconary an also by te proimity of te an te wining. Tese eviations result in a reuce accuracy of te eplaine calculation meto for region R 4. Te eact run of te flu lines, owever, just coul be calculate wit consuming FE-simulations. Furtermore, ue to te small sare of te energy store in tese parts of region R 4 in te overall store energy te resulting overall error for calculating te effective capacitance is acceptable. Remark: In case te transformer is not insie a groune tank te eviations at te lower en of te seconary wining sown in te blue circle in figure 8(b increase. Tat means tat te electric flu lines at te left sie of te seconary wining in figure 8(b ten to start at te upper en of te wining an en at te lower en. Unfortunately, te energy store witin tis fiel istribution coul not be easily calculate wit te approaces use ere. Instea of tat te energy is calculate by FE-simulations. E. ining winow: below seconary R5 In te net step te energy store in region R 5, i.e. below te seconary wining in te wining winow, is calculate. Tere, te electric flu lines are approimate by straigt lines again. Tese lines start at te seconary wining an are ortogonal to te wining winow of te groune as sown in figure 9(a. it tis approimation te store energy coul be calculate as escribe in section II. for region R 1. Te resulting equations are V( = V V3 ( + = +. (1 an R5 R5 1 1 εoil (( 1 V3 V ( 1 ( ( = l, (13 were te evaluate integral is given in te appeni again. ue to te limite volume an te low average energy ensity te store energy usually is relatively small an coul be neglecte in b P Oil iso V Figure 9. (a Variables / simplifie run of electric flu lines for region R 5. (b efinition of variables for region R 6

4 many cases. F. rea between an - R 6 : Finally, te energy store between te wining an te is calculate. Tis structure acts like a parallel plate capacitor wit two ifferent ielectrics te oil an te coil former of te wining as sown in figure 9(b. ssuming a linear istribution again, te istribution, te istance an te permittivity are V( = V1 ( = Oil, P + Iso, P (14 εisoεoil ( Iso, P + Oil, P ε eq = ε + ε Iso Oil, P Oil Iso, P an te energy coul be calculate by 1 εisoεoil ( Iso, P + Oil, P R6 = lr6 V1. (15 6 ε + ε (,, Iso Oil P Oil Iso P P G. ining capacitance So far, only te store energy/capacitances between te winings or between one wining an te /tank ave been consiere. But also between te singe turns of one wining electric energy is store. Tis energy coul be calculate by approaces presente in [5-7]. ue to fact tat te winings are usually implemente wit only one or two layers an te turn to turn is relatively small as well as te istance between te single turns is relatively large tis part of te store energy coul be neglecte. III. EQUIVLENT IRUIT OF PULSE TRNSFORER In te last preceing section te energies store in te ifferent regions of te pulse transformer/tank setup ave been calculate. In te net step te parameters of te equivalent circuit of tis setup are calculate. Tis is performe by comparing te energy store in te equivalent circuit, wic is a function of -, wit te calculate store energy, wic is also a function of -. For etermining te energy store in te equivalent circuit, first an appropriate equivalent circuit must be cosen. s coul be sown te electrostatic beaviour of an arbitrary transformer coul be moelle by a tree input multipole ( an seconary an te between te winings [3]. In te linear working area an as long as propagation s can be ignore, te electrostatic energy / beaviour of tis multipole coul be moelle by si inepenent capacitors as sown in figure 1. Te energy store in te equivalent circuit is given by 1 1+ V + V ( V+ V3 V1 Eq =, (16 + 5( V + V3 + 6( V1 V3 wat results from 1V. In te same manner te calculate energy coul be written al = ( + R + R3+ R4 + R5 + R6 (17 = f( V1, V, V3, were V 4 as been replace by V 4 = V + - an te factor results from te fact tat te energies ave been calculate for eac leg separately. 1 V Since bot energies must be B 3 equal Eq = al te equations of Figure 1. General equivalent circuit of pulse transformer. be erive te capacitors can by a b 1 + L σ L m + V B B Figure 11. (a Simplifie equivalent circuit (cf. fig.1. (b pproimate simplifie circuit wit capacitances transferre to seconary sie. setting te coefficients of te variables/ terms, V,, V, an V equal. Tis results in si inepenent equations wic can be solve for te capacitances 1 -. In contrast to te results publise in [3] te capacitors 1 /, / an / are not interepenent since te winings are not arrange in parallel an terefore te wining construction is not symmetric wit respect to te low an te ig sie. it te escribe moel te transfer beaviour of te pulse transformer an terewit te influence on te transferre pulse sape coul be calculate an/or simulate for arbitrary connections. oreover, wit te equations relating te geometry of te transformer irectly wit te capacitances of te equivalent moel te construction of te transformer coul be optimize for te require transfer beaviour. L σ L m. Simplifie circuit wit new equations In many pulse power applications te pulse transformer is not use for galvanic isolation an te low sie of te as well as te low sie of te seconary wining are groune, i.e. = in figure 1. In tis case capacitor is replace by a sort circuit an 1 / as well as / are in parallel. oreover, te s across all capacitors coul be erive from te an/or seconary by using te turns ratio N. it te s known te energy wic is store in te capacitors coul be calculate as a function of te seconary (or ( N 1 + ( + 5 N = V = V (18 Furtermore, te equivalent circuit coul be simplifie to te circuit sown in figure 11(a. Neglecting te parallel resonance between te leakage an capacitor tis circuit coul be furter simplifie to te circuit sown in figure 11(b, were only one capacitor is use wic is transferre to te seconary sie. Te capacitance value for te equivalent capacitor referre to te seconary sie is ( N 1 = + + ( + 5 N N (19 ~ + + for large N 4 5 Te circuit of figure 13(b is te same as use in [1, ] but in tose publications no equation for calculating te equivalent capacitance from te geometry of te transformer was given ecept for te simple parallel plate approac for te region between te winings. IV. ETERINTION OF THE EQUIVLENT IRUIT BY FE-SIU- LTION OR ESUREENT Besies te presente possibility to calculate te values of te si capacitors of te general equivalent circuit (cf. fig. 1 by means of te transformer geometry it is also possible to obtain te values by measurement or by FE-simulation. Since tere are si inepenent capacitors in te equivalent circuit, si inepenent simulations / measurements must be carrie out. Te belonging measurement setups are sown in figure 1. For te measurement results following below te values of te capacitances ave been etermine by using resonance peaks in te V

5 Te equivalent circuit calculate for te transformer sown in figure is given in figure 13. V Pri Tere, also te leakage inuctance B an te magnetising inuctance are sown, wic can be calculate from te magnetic fiel istribution in te winow /. 145pF 1.74µH 191µH -64.5pF pf 17.5pF 1:1 5.5pF -1pF V Sec Figure 13. alculate equivalent circuit for te transformer in figure. In figure 14 measure pulse responses are sown. itionally, curves resulting from te moel wit si capacitors were te capacitance values ave been etermine by analytic calculations, simulations an impeance measurements are sown. Tere, it coul be seen tat te pulse sape incluing ringing coul be preicte very well by means of te presente set of new equations. Figure 1. easurement for etermining te equivalent capacitances. impeance plot. Te require inuctance values are irectly measure wit te impeance analyzer gilent 494 an ten te capacitances are calculate wit te frequency of te resonance peak. it te measure capacitances te values of te equivalent capacitors of te circuit in figure 1 coul be calculate by te following equations. 1= ( = ( = ( = ( ( 5= ( = ( Instea of measuring te capacitances wit an impeance analyzer, te same setups coul be use for etermining te capacitances by FEsimulations. Tere, eiter 3 simulations, wic are quite accurate but very consuming, or simulations, wic are muc faster but less accurate, coul be performe. Te equivalent capacitors coul be calculate wit te same equations ( as use for te measurements. In table 1 measurement, simulation an calculation results for a transformer are given. Tere, it coul be seen tat te values correspon very well. Table 1 - Values for 1-6 for measurements, simulation an analytic calculation for te transformer in air witout tank. No. easure Simulate alculate pf 446 pf 48 pf 11 pf 138 pf 17 pf pf 394 pf 393 pf 4 58 pf 53pF 49 pf 5 19 pf 167 pf 141 pf 6 15 pf 133 pf 153 pf V. ESUREENT RESULTS In orer to verify te presente equations measurements at te pulse transformer sown in figure ecite by te soli state moulator also sown in figure ave been carrie out. Te measurements ave been conucte at relatively low (<kv so tat very fast an accurate probes coul be use an measurement errors relate to iviers coul be avoie. Furtermore, uring te measurements bot winings were groune, i.e. =, an te transformer was not insie a tank since none was available at te of measurement. Furter measurement results wit tank an also loss equations will be presente in a future paper. 1.4kV 1.kV us 1.9us.3us.7us 1.5E-6 1.7E-6 1.9E-6.1E-6.3E-6.5E-6.7E-6.9E-6 VI. ONLUSION In tis paper a general moel for pulse transformer wic allows for preicting te pulse sape of a moulator system before builing te transformer is presente. Te parameters of te moel can be analytically calculate from te geometry of te transformer or can be etermine by simulation or measurement. If bot winings are connecte to groun te presente moel coul be simplifie to te known L-L-. Furtermore, from te equations of te general moel, analytic epressions for etermining te parameters of te simple moel base on te energy istribution in all relevant regions of te transformer are erive. Bot, te general an te simplifie moel base on te new equations sow a goo corresponence wit te presente measurement results. REFERENES pulse measurement impeance measurement simulation calculation previous moel Figure 14. easure, simulate an calculate pulse for 15Om loa. [1] N. G. Glasoe an J. V. Lebacqz, Pulse Generators, IT Raiation Laboratory Series, vol. 5, cgraw-hill Book ompany, New York, 1948 []. kemoto, S. Gol,. Krasnyk an R. Koontz, Pulse Transformer R& for NL Klystron Pulse oulator, SL, Stanfor University [3] Harol. eeler, Transmission-Line Properties of a Roun ire in a Polygon Siel. [4] B. ogitore, J.-P. Keraec, J. Barbarou, Te two-wining transformer: an eperimental meto to obtain a wie frequency range equivalent circuit, Trans-action on Instrumentation an easurement, Vol. 43, pril, 1994, pp [5]. assarini,.k. Kazimierczuk, oelling te Parasitic apacitance of Inuctors, 16 apacitor an Resistor Tecnology Symposium 96, 1996, arc 11-15, pp [6]. assarini,.k. Kazimierczuk, G.Gani, Lumpe Parameter oels for Singlean ultiple-layers Inuctors, 7 t Power Electronics Specialists onference PES '96, Vol. 1, June 3-7, 1996, pp [7] J. Biela an J..Kolar, Using Transformer Parasitics for Resonant onverters Review of te alculation of te Stray apacitance of Transformers, onference Recor of te IEEE Inustry pplications onference, Hong Kong, Oct. -6, 5.