Design Procedure For Compact Pulse Transformers With .
2011 IEEEIEEE Transactions on Dielectrics and Electrical Insulation, Vol. 18, No. 4, pp. 1171-1180, August 2011.Design Procedure for Compact Pulse Transformers with Rectangular Pulse Shape and Fast Rise TimesD. BortisG. OrtizJ. W. KolarŋJ. BielaThis material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEEendorsement of any of ETH Zurich‘s products or services. Internal or personal use of this material is permitted. However,permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works forresale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view thisdocument, you agree to all provisions of the copyright laws protecting it.
IEEE Transactions on Dielectrics and Electrical InsulationVol. 18, No. 4; August 20111171Design Procedure for Compact Pulse Transformerswith Rectangular Pulse Shape and Fast Rise TimesD. Bortis, G. Ortiz, J.W. KolarPower Electronic Systems Laboratory, ETH Zurich, Switzerlandand J. BielaLaboratory for High Power Electronic Systems, ETH Zurich, SwitzerlandABSTRACTMicroseconds range pulse modulators based on solid state technology often utilize apulse transformer, since it could offer an inherent current balancing for parallelconnected switches and with the turns ratio the modulator design could be adapted tothe available semiconductor switch technology. In many applications as e.g. radarsystems, linear accelerators or klystron/magnetron modulators a rectangular pulseshape with a fast rise time and a as small as possible overshoot is required. In reality,however, parasitic elements of the pulse transformer as leakage inductance andcapacitances limit the achievable rise time and result in overshoot. Thus, the design ofthe pulse transformer is crucial for the modulator performance. In this paper, a step bystep design procedure of a pulse transformer for rectangular pulse shape with fast risetime is presented. Different transformer topologies are compared with respect of theparasitic elements, which are then calculated analytically depending on the mechanicaldimensions of the transformer. Additionally, the influence of the core material, thelimited switching speed of semiconductors and the nonlinear impedance characteristicof a klystron are analyzed.Index Terms - Pulse transformer, rise time, transformer topology, transformerdesign, solid state modulator.1 INTRODUCTIONIN many application areas, the required output power levelof test facilities in laboratories or in industry is rising and inmore and more applications solid state modulators deployingfor example IGBT modules, with a constantly increasingpower handling capability, are utilized. In contrast to the sparkgap switches, which can only be turned-on and have a limitedlife time and switching frequency, available fastsemiconductor switches have a limited power handlingcapability, so that a parallel and/or series connection of theswitches is required. The parallel connection of thesemiconductors basically offers a more robust design due tothe better capability of the switches to handle over-currentscompared to over-voltages. In [1] it has been shown, that amodulator based on pulse transformer is the most suitabletopology for pulses in the Ɋs-range, since it could offer aninherent current balancing in parallel connected powersemiconductors. Additionally, the turn ratio of the pulsetransformer offers a degree of freedom that allows adaptingthe modulator design to the current and voltage ratings ofavailable switch technology.Manuscript received on 11 October 2010, in final form 24 January 2011.In applications like radar systems, linear accelerators orklystronand magnetron modulators, where a nearlyrectangular pulse shape is needed, also the requirements withrespect to rise times, overshoot or voltage droop are high. InFigure 1a the schematic waveform of a typical powermodulator’s output voltage and in b) a power modulator withthe specifications given in Table 1 are shown. Since thetransformer parasitic limit the achievable rise time and definethe resulting overshoot, the design of the transformer iscrucial. On the one hand, due to non-ideal material propertieslike the limited permeability (ȝ ) or the limited maximumflux density Bmax of the core material, the maximum voltagetime-product respectively the minimum cut-off frequency fu ofthe pulse transformer is defined.Figure 1. a) Typical pulse waveform; b) 20 MW pulse modulator.1070-9878/11/ 25.00 2011 IEEE
1172D. Bortis et al.: Design Procedure for Compact Pulse Transformers with Rectangular Pulse Shape and Fast Rise TimesTable 1. Specification of the 20 MW, 5ȝs pulse modulator.DC Link Voltage Vin1000VOutput Voltage Vout200kVPulse Duration Tp5ȝsOutput Power Pout20MWRise Time Tr 500nsOvershoot οVmax 3%Turns ratio1:170On the other hand, in transformers no ideal magneticcoupling between windings can be achieved, which results ina certain leakage inductance Lɐ. Additionally, parasiticcapacitances of the transformer define the transient voltagedistribution and result in combination with Lɐ in an upper cutoff frequency fo of the transformer. Often the parasiticcapacitances are summarized in one single lumped capacitorCd as will be shown later.The output voltage with an almost rectangular pulse shape,however, exhibits a wide frequency spectrum. In order totransfer the voltage pulse with a minimum pulse distortion,especially during the rise time, a maximum bandwidth has tobe achieved, which means that the mentioned parasitic of thetransformer must be minimized.Consequently, the pulse transformer is one of the keycomponents of pulse modulators, which mainly defines theachievable rise time Tr and overshoot ǻVmax of the outputvoltage pulse.In this paper, a general step by step design procedure of apulse transformer is presented. In section 2 the influence of theparasitic elements Lɐ and Cd is analyzed with a standardizedpulse transformer model. During the rise time this model canbe simplified, which allows to derive basic design equationsconcerning rise time and overshoot of the pulse transformer.Based on this, in section 3 different transformer topologies arecompared with respect to the fastest achievable rise time. Inorder to define the mechanical dimensions, the leakageinductance Lɐ and the capacitance Cd are calculatedanalytically in section 4. In order to achieve faster rise time,transformers with multiple cores can be used, as described insection 5. In section 6 the influence of the core materialproperties like permeability Ɋ, maximum flux density B andthe core losses during pulse excitation is evaluated based onexperimental results.In section 7 the influence of the limited switching speed ofsemiconductors and the nonlinear impedance characteristic ofa klystron is evaluated. Experimental results of the built pulsemodulator are shown in section 8.2 PULSE TRANSFORMER'SEQUIVALENT CIRCUITIn literature numerous electrical equivalent circuitsconsidering LF and HF properties of pulse transformershave been proposed and IEEE standardized the equivalentcircuit of pulse transformers [3] as shown in Figure 2a. Inorder to simplify the analysis of the transient behavior foroperation with rectangular pulse voltages, the standardizedequivalent circuit can be reduced to the equivalent circuitFigure 2. a) IEEE standardized equivalent circuit of a pulse transformer andb) simplified equivalent circuit during the leading edge.shown in Figure 2b during the leading edge if n ب 1 [4].There, all impedances and the input voltage Vg aretransferred to the secondary and, hence, the idealtransformer can be neglected. If nothing mentioned, in thispaper, also all measured impedances are referred to thesecondary. Since the pulse rise time Tr is in the range ofsome 100 ns and there is - due to very small voltage-timeproduct – no excitation of the core, the influence of the corematerial, i.e. RFe and Lmag, can be neglected during the risetime. Even if the core resistance RFe would be considered, itwould not have an influence on the rise time, since it isconnected in parallel to the load resistance, which in thiscase is Rload 1500 ё if referred to the secondary or Rload;pri 0.052 ё if referred to the primary. There, the loadresistance Rload is much smaller than the resistance of thecore material, which was calculated to RFe ൎ 4 ё on theprimary.Therefore, the rise time and the overshoot of the outputvoltage, are mainly defined by the leakage inductance Lɐand the capacitance Cd. Assuming an ideal step voltage atthe primary, the output voltage vout(t) can be calculated withthe Laplace-transform as described in [4].where the damping coefficient ߪ of (1) is given byIf it is assumed, that the output pulse shape is mainlydefined by the transformer characteristics,, the modulator’simpedance Rg can be neglected. Thus, the dampingcoefficient ߪ considering only the influence of thetransformer can be simplified toAs will be shown later, however, depending on the turnsratio of the pulse transformer, the pulse generator’sparasitic inductance Lgen and capacitance Cgen - resultingfrom the dc-link capacitors, the switches and theinterconnections – as well as the parasitic capacitance ofthe load Cd have to be considered for the calculation of the
IEEE Transactions on Dielectrics and Electrical InsulationVol. 18, No. 4; August 20111173vload(t) rises from 10% to 90% (cf. Figure 3). For ߪ 0.75the factor is T10%-90% 0.365.Since the rise time Tr is proportional to LߪήCd, theparasitics have to be minimized in order to achieve thefastest possible rise time. For example, to keep the rise timebelow Tr 500 ns, LߪήCd has to be smaller than 4.75x10-14if ߪ 0.75.Since the load impedance is Rload 1500 ё, the ratio of Lߪand Cd is fixed and the maximum values for thespecifications in Table I are: Lߪ 490 ɊH, Cd 97 pF.Figure 3. Transient behavior of the normalized output voltage for differentdamping coefficients ɐ.overshoot and the rise time. For these cases, the inputimpedance should be changed from a resistance Rg to animpedance Zg. If the step up ratio of the pulse transformer ishigh, the parasitic capacitance Cgen can be neglected, sinceit is transferred to the secondary, the capacitance is dividedby n2, which is much smaller than the parasitic capacitanceof the transformer Cd or the load Cload. In Figure 3 thetransient behavior of the normalized output voltage duringbTt is illustrated. A decreasing damping coefficient ߪ 2Sresults in a faster rise time Tr. Starting from ߪ 1 atradeoff between Tr and overshoot is found. Therefore, toachieve a minimum rise time Tr, the damping coefficient ߪhas to be selected as small as possible while the resultingovershoot has to be still below the maximum allowed value(Figure 1a and Table 1).2.1 OVERSHOOTConsidering equation (3), ߪ depends on Lߪ and Cd, i. e. onthe pulse transformer’s mechanical dimensions and on theload impedance Rload. In general, Rload, for example of aklystron, is defined by the application. Therefore, the pulsetransformer’s mechanical dimensions must be adjusted inorder to fulfill the specifications of the pulse shape.Assuming a klystron load of Rload 1500 ё, for a maximumovershoot of 3% a damping coefficient of ߪ 0.75 isneeded (equation (3)). Consequently, with a given Rload andߪ, the ratio of leakage inductance Lߪ and capacitance Cd isfixed by2.3 DESIGN CRITERIAIn order to fulfill the requirements for the maximumovershoot and the maximum rise time, both a given ratio ofLߪ to Cd and a maximum product of Lߪ and Cd have to beguaranteed. In general, the pulse modulator connected tothe transformer’s primary as well as the load connected tothe secondary winding have a certain inductance Lgen /capacitance Cload, which also have to be considered. For therealized pulse generator a parasitic inductance of Lgen 260ɊH was measured. Typical capacitance values ofklystrons are in the range of Cload 40 -120 pF [18] forthe considered application. This means that the leakageinductance and the distributed capacitance of thetransformer must be small to meet the pulse specifications.Therefore, equations (4) and (5) have to be extended to3 TRANSFORMER TOPOLOGYThe ratio of Lߪ and Cd can be varied by the mechanicaldimensions of the transformer, i.e. the distances, the heightsand the lengths of the windings. The product of Lߪ and Cd,however, is defined by the transformer topology and can beassumed to be approximately constant [4]. Therefore, firstthe transformer topology resulting in the smallest LߪCdproduct has to be selected. Afterwards, the mechanicaldimensions must be calculated to achieve the needed LߪCdratio.In the following, the LߪCd-product of three differenttransformer topologies is analyzed. The leakage inductanceLߪ and the capacitance Cd are calculated with the energystored in the magnetic & electric field.2.2 RISE TIMEIn addition to the overshoot, the rise time Tr of the outputvoltage can be derived from equation (1). As shown inequation (5), Tr is proportional to the product of Lߪ and Cd.Factor T10%-90% depends on the selected dampingcoefficient ߪ and equals the time in which the voltageFigure 4. a) Picture of a pulse transformer with parallel winding and b) 2Ddrawing of one leg with simplified run of the magnetic and electric fieldlines.
1174D. Bortis et al.: Design Procedure for Compact Pulse Transformers with Rectangular Pulse Shape and Fast Rise TimesFinally, the LߪCd-product of the transformer topology withparallel windings isTo simplify the comparison, only the energies betweenthe windings are considered. Finally, for the transformertopology with the smallest LߪCd-product a more detailedcalculation of the parasitic is presented.3.1 PARALLEL WINDINGDue to the simple construction, the parallel windingtopology is widely used. The primary and secondary arewound on two parallel bobbins, whose distance is definedby the required isolation. In Figure 4a picture and 2Ddrawing of the parallel winding are shown.The leakage inductance is mainly defined by the volumeand the magnetic field strength between the bobbins(equation (7)). According to Ampere’s law and assumingan ideal core material (ߤ λ), the magnetic field strengthJGH in the core window is given by the primary current timesthe number of turns NpriήIpri and the height of the core hk.3.2 CONE WINDINGSince the distance between the windings of thetransformer with parallel winding is constant but thevoltage is increasing linearly in y-direction, the electricfield between the windings also increases linearly. In orderJGto achieve a constant electric field E (y) the distancebetween the windings dw has to be linearly decreased forsmaller voltage differences, which results in a cone winding[4], [19] as shown in Figure 5.Compared to the parallel winding, the volume betweenthe windings and therefore also the leakage inductance Lߪcan be reduced by a factor of two. However, due to thesmaller distance between the windings Cd is increases.To calculate the leakage inductance Lߪ of the conewinding, again, a constant magnetic field in y-direction isassumed (Figure 5), which was confirmed by FEMsimulation as long as dw ا hw.Considering equation (7), the stored magnetic energy Emagand the resulting leakage inductance Lߪ,cone areUsing equations (7) and (9), the stored magnetic energyEmag between the windings Wpri and Wsec can beapproximately calculated byand the resulting leakage inductance Lߪ,parallel isTo calculate the capacitance Cd, a linear voltagedistributionVpri(y) and Vsec(y) is assumed across the windings.Therewith, the voltage difference between the primaryand secondary winding depending on the vertical position yis οV(y) Vsec(y)-Vpri(y).Due to the voltage difference between the windings Wpriand Wsec, the electric field lines run approximatelyJGhorizontally (Figure 4b). Thus, the electric field E (y)depending on the y-position isDue to the linearly increasing distance dw(y) and thevoltage distribution οV (y) in y-direction, the electric fieldJGE between the winding is constant and runs approximatelyparallel to the x-direction (Figure 5b). Hence, the storedelectric energy (equation (8)) for a cone winding and thecapacitance Cd areFinally, the resulting LߪCd-product of the cone windingisCompared to the parallel winding, the LߪCd-product canbe reduced by 25%, which results in a rise timeimprovement of13.4%.Considering equation (8), the stored energy between thewindings Wpri and Wsec and therewith the capacitance Cd arecalculated.Figure 5. a) Picture of a pulse transformer with cone winding and b) 2Ddrawing of one leg with simplified run of the magnetic and electric fieldlines.
IEEE Transactions on Dielectrics and Electrical InsulationVol. 18, No. 4; August 20113.3 FOIL WINDINGFinally, for the primary Wpri and secondary Wsec foilwindings are considered. The secondary is directly woundon the primary winding as shown in Figure 6. For theisolation of the turns a material with a low permittivity isused.The thickness diso of the isolation can be kept small, sincethe voltage difference between two consecutive turns is justVw,w Vsec/Nsec. However, due to the increasing voltagedifference between the turns and the core, the winding’sheight is linearly decreased from hw,1 to hw,2 (Figure 6b).The total thickness dw of the winding is defined by thethickness of the isolation diso and the foil dcu times thenumber of turns.The leakage inductance Lߪ of the foil winding iscalculated again with the stored magnetic energy (equation(7)). Based on Ampere’s law, the magnetic field isgradually increasing with the number of turns nL, since theenclosed amount of current is increasing gradually (Figure6).The total magnetic energy is the sum of all energiesbetween two consecutive turns, which isThus, the resulting Lߪ,foil isCapacitance Cd can be calculated as a series connection ofparallel plate capacitors between consecutive turns Cw,w.The distance of the plates equals diso, which can beexpressed by the total winding thickness.1175Considering only the stored magnetic and electric energybetween the windings Wpri and Wsec, the smallest LߪCdproduct and therefore the fastest Tr can be achieved for thetransformer with a cone winding. Since the consideredvolume contains the major share of the magnetic andelectric energy, the calculated LߪCd-product is a reliableindicator for selecting the best transformer topology.4 PARASITICS CALCUALTIONIn a next step, also the magnetic and electric fieldsbetween the winding and the core as well as the electricfields between the windings and the enclosing wall of atank are considered in order to obtain a more precise modelfor designing the transformer. For example, in Figure 7a,JGthe resulting electric field E for a transformer placed in atank is shown.As it was proposed in [4], in Figure 8a a measured and acalculated waveform considering only the energy storedbetween the windings are shown. It clearly indicates themismatch between measurement and simplified calculationof the parasitics. Since only the electric energy between thewindings is considered, Cd is too small and results in a toosmall overshoot predicted by the transformer model.Therefore, a more detailed calculation procedure, whichconsiders all stored electric and magnetic energies, isneeded.4.1 DISTRIBUTED CAPACITANCETo improve parasitics calculation the energy outside thewindings is considered in the following. As shown in Figure7b, the space around the transformer is divided into sixrelevant regions R1 to R6. With geometric approximations, thestored energy in each region can then be calculatedanalytically. In [2] the detailed calculation of the distributedcapacitances depending on the mechanical dimensions of thetransformer is investigated. There, the calculated values havebeen compared with measured and simulated impedancevalues determined by FEM-simulation. The output voltagepredict
Consequently, the pulse transformer is one of the key components of pulse modulators, which mainly defines the achievable rise time T r and overshoot ûV max of the output voltage pulse. In this paper, a general step by step design procedure of a pulse transformer is presented. In se
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