optimal transformer design for ultraprecise solid state ... · ieee transactions on plasma science,...

Optimal Transformer Design for Ultraprecise Solid State Modulators

Sebastian Blume and Juergen Biela, Member, IEEE „This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of ETH Zürich’s products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promo-tional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document you agree to all provisions of the copyright laws protecting it.”

mailto:[email protected]

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 10, OCTOBER 2013 2691

Optimal Transformer Designfor Ultraprecise Solid State Modulators

Sebastian Blume and Juergen Biela, Member, IEEE

Abstract— In this paper, a procedure for optimal transformerdesign with a variable set of constraints for pulse transform-ers with a pulse range of 3–140µs is presented. During theoptimization procedure, the pulse shape is analyzed in the timedomain, ensuring that the pulse constraints, such as rise time andovershoot are met. For accurate prediction of the pulse shape,analytical approaches are proposed to estimate the distributedcapacitance and leakage inductance of the transformer. Theanalytical approach is verified by 2-D-FEM simulations andmeasurements. The optimization procedure considers pulse, core,winding, demagnetization losses, and losses of the primaryswitches. First, the procedure is applied to an existing pulsetransformer with specifications for SwissFEL. An improvementof 16.6 % in conversion efficiency is achieved in comparison withthe existing design. In a second step, the procedure is appliedto specifications of the compact linear collider, which demandshigh conversion efficiency. The resulting optimal transformerconsists of three cores with five primary turns and requires atank volume of 0.915 m3. In an optimal configuration, an overallconversion efficiency of 97.7% is achieved for the consideredsystem including pulse losses.

Index Terms— Core losses, distributed capacitance, highvoltage, leakage inductance, optimization procedure, pulsetransformer, transformer design.

I. INTRODUCTION

PULSE power converter systems are used in a wide rangeof applications and have to comply with various system

requirements, such as output power, flat-top length, and pulserepetition. In addition, the requirements for the pulse shape,rise time, pulse repetition accuracy, flat-top stability (FTS), andovershoot are highly application dependent as can be observedin two different sets of selected requirements listed in Table I,one for a short-pulse system, the SwissFEL, and one for along-pulse system, the compact linear collider (CLIC).

In general, pulse power conversion systems can be dividedinto four main groups: 1) line-type modulators, such aspulse-forming networks [1]; 2) transformer-free systems basedon a single high voltage switch or on the Marx-generatorprinciple [2]; 3) concepts based on pulse transformers [3],often realized with solid state switches [4]; and 4) resonanttopologies for pulses in the millisecond range [5], [6].

Manuscript received December 3, 2012; revised March 13, 2013 and July 26,2013; accepted August 28, 2013. Date of publication September 23, 2013; dateof current version October 7, 2013. This work was supported in part by thepartners SNF under Project 144324 and in part by CERN.

The authors are with the Laboratory for High Power Electronic Sys-tems, ETH Zurich, Zurich 8092, Switzerland (e-mail: [email protected];[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2013.2280429

TABLE I

SPECIFICATIONS FOR TWO DIFFERENT MODULATOR SYSTEMS

This paper focuses on systems with a pulse transformerand solid state switches for a pulse range 3 –140 μs. Thistechnology provides high reliability due to its simple structureand reliable short-circuit protection.

Solid state modulators with matrix transformers have beeninvestigated in [7]– [9].

Due to the varying specifications, the pulse power conver-sion system must be adapted for its individual purpose. Toensure the optimal conversion system for every specificationset, optimization procedures can be applied. To allow a rea-sonable overall computing time, one optimization cycle shouldnot exceed 1 s. Thus, analytical approximations are used tocalculate the transformer parasitics. Analytical approximationshave already been conducted in [4], but they are not applicablefor arbitrary geometry.

Therefore, improved analytical approximations are proposedin this paper, which are suitable for a broader range of trans-former geometries. They are then integrated in an optimizationprocedure maximizing the efficiency of the pulse transformerunder the constraints of electrical and pulse requirementsincluding switching unit.

At first, an overview of the investigated system is givenin Section II. Subsequently, the optimization procedure ispresented in Section III, which analyzes the pulse shape in thetime domain and controls the pulse specifications. In addition,the algorithm contains several models estimating pulse, core,winding, active reset circuit losses, and losses of the primaryswitches.

Thereafter, in Section IV, an analytical approach for esti-mation of the leakage inductance and the stray capacitance isproposed, which is suitable for a variable pulse transformergeometry. Both the approaches are compared with 2-D-FEMsimulations and in addition validated with a measurement ofan existing pulse transformer.

0093-3813 © 2013 IEEE

2692 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 10, OCTOBER 2013

Fig. 1. Simplified schematics of the solid state modulator, including matrixtransformer with varying number of cores and an active bias circuit.

Finally, in Section V, the optimization procedure is con-ducted for specifications of SwissFEL. The resulting pulseshape is compared with the pulse shape with parametersderived from an existing pulse transformer with equal spec-ifications. In a second step, the procedure is applied for thespecifications of CLIC and the results are presented.

II. OVERVIEW OF INVESTIGATED SYSTEM

The investigated system is a matrix transformer [8], asshown in a simplified schematics in Fig. 1. In a matrixtransformer, the secondary winding encloses all the coresleading to a turns ratio n, which is defined by

n = vs

v p

1

nc(1)

where nc is the number of cores, v p the primary voltage, and vs

the secondary voltage. A cone winding arrangement is chosen,because it outperforms a parallel winding arrangement in termsof leakage inductance [10]. Each core has two legs leading totwo identical transmission systems, which are connected inparallel. The number of cores is variable. For each core leg,a switching unit, composed of the main capacitor bank Cm

and the main switch Sm , is considered. Additionally, an activereset circuit per core leg is considered (Cr , Sr , and Dr ), whichwill be described further in Section III-F. Due to a high ratioof vs /v p , only the primary leakage inductance Lσ1, due tothe loop comprising main capacitor and switching unit, theprimary copper resistance R11 of this loop and the secondarycapacitance Csec are considered for this transformer setup.Additionally, the leakage inductance Lσ and the distributedcapacitance Cd of the transformer are considered as well asthe nonlinear klystron load Rklys.

III. OPTIMIZATION PROCEDURE

The aim of the proposed optimization procedure is toprovide a transformer design, which suits the selected require-ments. This could be a limited overshoot, a limited risetime, a fixed number of primary windings, or other certain

Fig. 2. Optimization procedure: the global optimizer receives systemspecifications and material limits. The transformer parasitics are calculatedanalytically. Losses are determined for each model, which are transformed inequivalent components for pulse analysis in the time domain.

geometry constraints. As shown in Table I, the CLIC systemhas demanding specifications regarding modulator global effi-ciency, from grid to the klystron load. Therefore, in this paper,the optimization procedure is not only designed to meet thepulse requirements but also to find a solution with highestsystem efficiency.

In the following sections, the optimization procedure isdescribed briefly, followed by the detailed description of eachloss model.

A. Structure of the Optimization Procedure

The optimization procedure consists of several models, asshown in Fig. 2. The global optimizer runs with a set of sixoptimization variables: number of primary turns n p , numberof cores nc, secondary winding height hs , distance betweenthe secondary and primary windings dw, width of the corecross-sectional area wAc , and opening angle of the cone α.

In addition, a set of fixed parameters are given such asconstants or predefined geometric distances, e.g., the distancebetween the core and tank. These parameters and the opti-mization variables define the geometry of the transformerand its surrounding tank. Once the geometry is determined,its parameters are used to calculate all required componentsfor the analysis of the pulse shape. Initially, the distributedcapacitance Cd and the leakage inductance Lσ are calculated,as explained in Section IV. Second, the winding losses arecomputed and the ohmic copper resistance R11 of the primaryside is transferred to the pulse shape analysis. Then, thecore losses are calculated and the ohmic resistance RFe is

BLUME AND BIELA: OPTIMAL TRANSFORMER DESIGN 2693

Fig. 3. Geometric setup of the transformer. (a) Geometric variables in thecore window. (b) Geometric variables between core and tank wall includingcooling pipes.

defined as well as the magnetizing inductance Lh . With Lh ,the magnetizing current at the beginning of the pulse Ipremagis derived.

With all these parameters, the pulse shape is analyzed inthe time domain. During this step, the optimizer comparesthe shape with a set of external constraints such as rise timeor overshoot. If all the constraints are met, the pulse losses(as described in Section III-D.2) are calculated. The magne-tizing current Idemag at the end of the pulse is fed back to theactive bias loss model, which then calculates the demagneti-zation losses. Finally, losses of each model are combined toobtain the total losses of the given parameter set.

At the end of the optimization procedure, the algorithmsupplies a transformer configuration, where all demandedpulse requirements are met and highest efficiency is achieved.

B. Geometric Transformer Setup

In the first step of the optimization procedure, the geometryof the transformer depending on the set of optimizationparameters is defined as a basis for the analytical calculations.The correlation of geometric parameters is shown in Fig. 3.The entire geometry is defined by a few parameters to builtit as compact as possible while avoiding field enhancement.Therefore, the minimal distance dw,min between the conductorwith the highest voltage potential of the secondary and theprimary windings must be derived. The same distance isapplied between this conductor and the grounded core. Theconductor with the highest secondary voltage potential isusually realized as a field shape ring with radius rr . Toestimate the resulting electrical peak field, the geometry isapproximated as a cylindrical capacitor (Fig. 3), where thehighest electrical field occurs at the inner radius rr and isdefined by

E(rr ) = vs

rr

1

ln( dw

rr

) (2)

where vs is the applied secondary voltage, dw the outer, and rr

the inner radius of the cylinder. Therefore, the minimal alloweddistance dw,min between the windings can be calculated as

dw,min = rr exp

(vs

rr Epeak

)(3)

where Epeak is the maximal tolerable peak electric field in oil.This peak field occurs only at the surface of the field shapering. An average electrical field between the windings Eav,when a homogenous field is assumed, is defined by

Eav = (vs − v p)

(dw − rr). (4)

Calculating the average electrical field Eav of the analyzedtransformer geometries would result in much lower val-ues than the peak electric field Epeak, which is shown inFigs. 10 and 11. Because the electrical field in case of theanalyzed geometries is inhomogeneous, only the peak electricfield is considered and set as constraint. A value of Epeak =20 kV/mm is assumed for short pulses of a few microsecondsand for longer pulses in the 100 μs range the value is set toEpeak = 12 kV/mm.

The cross-sectional area of each single core can be definedby

Ac = v ptflat

2Bmaxn p Fc(5)

where Fc is the filling factor of the cut tape-wound core andBmax the desired flux density.

The other geometry parameters, shown in Fig. 3, are

bc = 2rr + 2dw + 2dtp + 2dprim (6)

hs,min = nsds,min cos(αmax) (7)

hc = dw + hs + dtop (8)

dAc = Ac

wAc

. (9)

The area outside the core is equally structured as the corewindow, except that the distance between the field shapering and oil tank is enlarged by dadd, leading to smallercapacitance values of the transformer, but to higher tankvolume. Therefore, the value of dadd is set externally.

1) Transformer Cooling: The transformer is considered tobe in a tank filled with standard transformer oil, to allow amore compact design. The cooling is realized by groundedcooling pipes. The positioning of these pipes is important,as they contribute to the distributed capacitance of the trans-former. To minimize their influence, they are positioned at thelevel of the lower voltage turns. To allow an effective cooling,the transformer is placed upside down, as shown in Fig. 3into the tank. Consequently, the cooling pipes are positionedat the top close to the tank wall, where the heated oil flowsdue to convection. Even though the ground plate needs to beenforced to carry the transformer weight, this configuration ischosen as it has the additional advantage of wiring being ledthrough the top, thereby avoiding oil leakage.


Fig. 4. Simplified circuit diagram transferred to the secondary side, whichis applicable from beginning of the pulse until demagnetization.

C. Winding Losses

The winding losses, due to skin and proximity effectswere subjected in a number of publications, e.g., [11]. Toestimate the skin depth in the conductor, the pulse current isapproximated as a trapezoid and a FFT analysis is conducted.For calculating the proximity effect losses, the secondarycircular windings are transformed to a sheet conductor, asshown in [11]. A simplified geometry is assumed, in whichthe windings cover the entire height of the core window.

D. Pulse Shape Analysis

For the design of a pulse transformer, it is crucial to predictthe resulting pulse shape. Therefore, in this paper, the time-domain circuit model for pulse prediction is introduced, fol-lowed by the pulse constraint implementation and descriptionof the algorithm’s penalty arrangement.

1) Time Domain Circuit Model: Because the klystron loadRklys is nonlinear and influences the pulse shape signifi-cantly [10], the pulse shape is evaluated in the time domain.The applied circuit model is derived from [12], based on thesecondary voltage potential and shown in Fig. 4.

This circuit model includes the mentioned parameters Cd ,Lσ , Lh , RFe, Rklys, Lσ1, and R′

11. On the secondary side,a capacitance Csec = 100 pF, due to klystron load andmeasurement equipment, is considered.

Due to the time domain analysis, the model is able tocope with any given nonlinear load function. However, forthe klystron the load function is defined by

ik(t) = kv1.5k (t) and k = Ik,nom

V 1.5k,nom

(10)

where k is the perveance, Ik,nom and Vk,nom are the nominal,ik(t) and vk(t) are the time-dependent values of current andvoltage of the klystron.

To consider the rise and fall times of the primary switchesas well as the voltage drop in the primary capacitances �vcapduring flat top, the voltage signal v ′

1(t) is applied as a time-dependent function. This circuit model is not applicable duringdemagnetization of the transformer. However, it is a validassumption that the energy stored in the transformer can beretrieved by the active bias circuit, except losses occurringin its components [4]. Therefore, losses during the pre- anddemagnetization periods are considered in the active biascircuit loss model.

2) Pulse Losses: Because the klystron load cannot be ini-tiated until the voltage complies with the FTS criteria, a partof the transferred energy of the pulse is lost. In [13], the ratiobetween the ideal and real pulses has been defined as pulseefficiency. Since in this paper, the pulse shape is analyzed in

Fig. 5. Pulse shape with the constraints of rise time, settling time, maximalovershoot voltage, and the voltage band for FTS. Once the pulse remainswithin the voltage band, the flat-top period begins.

the time domain, the energy lost before the klystron can beactivated and the energy lost during the fall time of the pulsecan be computed. These losses will be referred to as pulselosses in the following.

3) Implementation of the Pulse Constraints: During theoptimization procedure, the pulse constraints are surveyed. Ifthe pulse exceeds a constraint, a penalty will be set.

The pulse behavior is analyzed in the time domain witha nonlinear differential equation system, which is solvedanalytically. A possible pulse shape with constraints is shownin Fig. 5.

The resulting pulse signal is compared in every time stepwith the predefined overshoot voltage and the voltage band.If the signal crosses the overshoot voltage boundary, thecalculation will be terminated. As long as the signal exceedsthe upper voltage or falls below the lower voltage of thevoltage band, the time vector tref is increased. Thus trefindicates the beginning of the flat-top period tflat. To minimizethe computation time, the analysis is only conducted for theperiod of

tcontroll = trise + tsettle + tadd (11)

where trise is the rise time, tsettle the settling time and tadd =2 μs. If tref is higher than trise + tsettle, a penalty will be set.This will also be the case, if trise, which is derived from thepulse signal, is exceeded.

To ensure that the algorithm converges to the predefineddesign space, the penalties increase with their difference fromthe set point value.

E. Core Losses

The considered material for the pulse transformer is Metglas(amorphous alloy 2605SA1) because it offers a good compro-mise between the low losses, high saturation flux density, andcosts. For a rectangular voltage, resulting in different gradientsof the magnetic flux density d B/dt , losses can be predictedby the improved generalized Steinmetz equation (iGSE) [14].In case of a pulse transformer with active bias circuit, whichallows to double the magnetic flux density swing, the energy


TABLE II

PARAMETER SETS FOR IGSE DEPENDENT ON PULSEWIDTH

Fig. 6. Measured core losses for a pulse with pre- and demagnetization independence of the flux density swing �B and the pulselength. Due to theactive reset circuit, the possible maximal flux density swing is doubled. Thepremagnetization time is higher by the factor of 7.5 compared with the pulselength. The test core of amorphous alloy 2605SA1 has an effective cross-sectional area of 5.29 cm2 and a magnetic length of 43.78 cm.

loss per pulse Ev is described by

Ev = k1 Ac,eff leff (2 Bmax)β−α

·(

tpremag

∣∣∣∣

Bmax

tpremag

∣∣∣∣

α

+ tflat

∣∣∣∣2Bmax

tflat

∣∣∣∣

α

+ tdemag

∣∣∣∣

Bmax

tdemag

∣∣∣∣

α)

(12)

where α, β, and k1 are the Steinmetz parameters, Ac,eff isthe effective cross-sectional area, leff is the effective magneticlength, tpremag and tdemag are the pre- and demagnetizationtime, tflat is the flat-top length, and Bmax is the desiredmagnetic flux density, which was set for the design procedureto Bmax = 1.2 T.

Steinmetz parameters are based on curve fitting and theconsidered pulse range from 3 to 140 μs is wide. Therefore,parameter sets for three different ranges, as shown in Table IIare derived from the measurements. The test core is an AMCC367S core, which has an effective cross-sectional area of5.29 cm2 and a magnetic length of 43.78 cm. The core is cutto consider the losses, which also occur in the original-sizedcore. The schematics of the measurement setup and a photoare shown in Fig. 7. The losses were calculated, as describedin [15] and are shown in Fig. 6. The premagnetization periodwas chosen by a factor of 7.5 larger than the pulsewidth.

To estimate the magnetizing inductance Lh , as proposedin [4], the mean relative permeability μr must be estimatedin each measuring point, which is dependent on flux density

Fig. 7. (a) Simplified schematics of core loss measurement setup. (b) Pictureof core loss measurement setup.

swing and pulselength, leading to [4]

μr = �B

μ0�H. (13)

F. Active Reset Circuit Loss Model

An active reset circuit as realized in [4] was already shownin Fig. 1. This circuit on the one hand allows to double the fluxdensity swing, which decreases the core volume, on the otherhand contributes to the short-circuit handling of the convertersystem. In case of klystron gun arcing, the short-circuit currentflows through the antiparallel diode Dr . Because of the inversevoltage of Cr , this current is reduced faster than if it was onlydischarged over the diode.

It is assumed that all the stored magnetic energy in thetransformer can be retrieved by the active reset circuit, exceptfor losses in the switch Sr during premagnetization and lossesin the diode during Dr demagnetization.

G. Losses of Main Switches

Because of high reverse voltage and high current ratingsare required, there are only insulated gate bipolar transistors(IGBTs) as main switches considered. The loss energy EIGBTof an IGBT is defined by

EIGBT = VCE,sat Iprim tflat + Eon + Eoff (14)

where EON is the energy loss during turn-ON, EOFF duringturn-OFF and VCE,sat the saturation collector emitter voltage.Typical values from datasheet are assumed [16].

IV. CALCULATION OF PARASITICS

In Section III-D, it is pointed out that for analyzing thepulse shape, the distributed capacitance Cd and the stray induc-tance Lσ are critical parameters. In [4], simplified analyticalequations were proposed for both parameters. These equationsare, however, only valid for certain geometries and containassumptions such as a centered field shape ring between thetransformer and oil tank. Therefore, these are not suitable fora flexible geometry in an optimization procedure.

Consequently, in this section analytical methods are pro-posed to calculate the distributed capacitance and the leakageinductance for a matrix transformer with flexible geometry.The only assumption made is that the core as well as theprimary and secondary windings is grounded at one end. Thisis the case for most of the pulse transformers to obtain adefined potential between the windings [4].


Fig. 8. (a) Resulting capacitances of a geometry of four wires with individualpotential and charge in relation to a potential, which is situated at infinity.(b) Three wires in a geometry limited in each direction with grounded surfaces.These grounded surfaces are considered by applying mirror charges in eachdirection. The mirror charges have alternately positive and negative signs.

First, the method for calculating Cd is introduced, followedby the calculation method of Lσ . Second, these methods arecompared with 2-D finite element method (FEM) simulations.In addition, the simulations are verified with measurements ofan existing pulse transformer.

A. Calculation of Distributed Capacitance

To estimate the distributed capacitance Cd of a pulse trans-former, the geometry is analyzed in the 2-D space to obtainthe capacitance per length C ′

d , which is then multiplied withits associated length.

The ns secondary turns are considered as line conductorswith a voltage potential vk of the kth conductor of

vk = vsk − 1

ns − 1, 1 ≤ k ≤ ns (15)

where vs is the secondary voltage potential of the transformer.The primary windings are realized as foil conductors and areconsidered for the analytical calculation as entirely grounded.This simplification leads to a slightly higher capacitance, butthe error is negligible (e.g., for Fig. 10 smaller than 2%)because

Vs/Vp � 1. (16)

In a multiconductor system in 2-D space, which is shownin Fig. 8(a) for four conductors, the relation between thepotential �′ and charge Q is described by

[Q] = [p]−1 · [�′] = [c] · [�′]. (17)

In case of the pulse transformer, the conductors are surroundedby the grounded core and the oil tank. The influence ofthese surfaces is considered applying the charge simulationmethod [17]. Each conductor is mirrored in each direction toconsider the influence of the grounded surfaces. To describethe resulting electrical field correct, the mirror charges havealternately positive and negative signs, as shown in Fig. 8(b).For fast computation time, only N = 24 mirrored charges areconsidered for each conductor.

In a 2-D space, the potential coefficient pij between theconductor i and j can be described based on the superposition

Fig. 9. (a) Approximation of transformer geometry with round conductors.(b) Magnetic flux per length for two round conductors.

principle [18] as

pij = 1

2πε0εr

⎛

⎝ln(rij

) +N/2∑

mp j =mn j =1

ln

(ri,mp j

ri,mn j

)⎞

⎠ i �= j

(18)where rij is the distance between the two conductors, ri,mp j isthe distance of conductor i to all positive mirror charges andri,mn j to all negative mirror charges of conductor j .

In case of i = j , the potential coefficient is obtained by

pii = 1

2πε0εr

⎛

⎝ln (rr ) +N/2∑

mpi =mni =1

ln

(ri,mpi

ri,mni

)⎞

⎠ (19)

where rr is the radius of the conductor, ri,mpi the distance ofconductor i to all its positive mirror charges and ri,mni to allits negative mirror charges.

To obtain the partial capacitances between the conductors,the potential coefficient matrix [p] has to be inverted toobtain the capacitance coefficients [c]. The partial capacitanceper unit length between the two conductors C ′

i j and thecapacitance per unit length to the surrounding potential C ′

i∞,can be obtained by

C ′i j = −ci j and C ′

i∞ =n∑

j=1

ci j . (20)

The total distributed capacitance per unit length C ′d based

on the secondary voltage potential vs is derived from thetotal stored electric energy Wel,tot of the geometry, which iscalculated by summing up the electric energy of each partialcapacitance:

W ′el,tot =

n∑

j=1

0.5C ′i, j (vi − v j )

2 and C ′d = 2W ′

el,tot

v2s

. (21)

B. Calculation of Leakage Inductance

The leakage inductance Lσ is derived by multiplying ofthe inductance per unit length L ′

σ in the 2-D space with itsassociated length. The approach used in this paper is basedon a multiconductor system. The geometry is simplified, asshown in Fig. 9, where the primary winding is approximatedby ns circular conductors, equally distributed over the primarywinding height h p . The field shape ring is approximated by asmaller circular conductor. The secondary winding conductors


Fig. 10. FEM simulation for a long pulse transformer geometry with apeak electrical field of Epeak = 12 kV/mm and an average field of Eav =3.2 kV/mm.

are then rearranged, to keep the secondary winding height hs

constant.In a multiconductor system, the magnetic flux per length is

defined by�′

i =∑

j

L ′i j · I j . (22)

Due to the 2-D analysis, it is not of interest how the conductorsare connected as turns [19]. Therefore, one primary and onesecondary turns can be combined to a double circuit line. Ifthe radius rrr is small compared with the distance d1 betweenthe two conductors, the self-inductance per unit length L ′

11 ofa double circuit line can be defined by [20]

L ′11 = 2�′

1

i1= μ0

πln

(d1

rrr

). (23)

The mutual inductance per unit length, which is caused bycurrent i1 of circuit line l1 in l2, as shown in Fig. 9(b) can bedescribed by [20]

M ′ = L ′21 = �′

21

i1= μ0

2πln

(r12r21

r11r22

). (24)

Because in the chosen arrangement it is assumed that all theconductors carry an equal current, the total inductance per unitlength of the geometry can be obtained by

L ′σ =

∑

i

∑

j

L ′i j . (25)

The core has a very high permeability μr and therefore canbe observed as a magnetic mirror, where the conductors aremirrored, as shown in [15]. For fast computation, only eightmirrored conductors are considered for each turn. Outside thecore, the conductors are only mirrored in one direction at theaxis of the core, because the surrounding oil tank has a lowerpermeability and the distance to the windings is much higherthan between the windings and core.

C. Validation by 2-D FEM Simulations

To validate the analytical approaches of C ′d and L ′

σ , theresults are compared with a 2-D-FEM simulation. In Fig. 10,a matrix transformer geometry for long pulses and in Fig. 11a geometry for short pulses is shown, each with the resulting

Fig. 11. FEM simulation for a short pulse transformer geometry with a peakelectrical field of Epeak = 16 kV/mm and an average field of Eav = 4 kV/mm.

TABLE III

COMPARISON BETWEEN ANALYTICAL APPROACH AND 2-D FEM

SIMULATIONS FOR THE CORE WINDOW AND THE AREA

BETWEEN CORE AND TANK (FIGS. 10 AND 11)

magnetic and electric field. 2-D-FEM simulations of these twodifferent geometries are compared in Table III with C ′

d and L ′σ

for the core window and the space between the core and tank.It can be observed that the difference between the 2-D-FEM

simulations and calculation is smaller than 6%. This indicatesthat although simplifications are applied, the accuracy is stillhigh. For further validation, analytical calculation results arecompared with the measurement data in the following section.

D. Verification With Measurements

To validate the proposed analytical calculations of Cd

and Lσ , the calculations are compared with measurementsof an existing pulse transformer. The pulse transformer isrealized for the specifications in Table I(b) and corresponds toFigs. 11 and 13.

At first, it is described in detail how Cd and Lσ of thepulse transformer are obtained from analytical calculations,followed by a comparison of the approach with 2-D FEM andmeasurements.

1) Parasitics in Dependence on Geometry: Because thedistances between the core and tank wall differ at the front,the rear, and the side of the pulse transformer, the distributedcapacitance per unit length C ′

d is calculated for three differentregions and the core window, which are shown in Fig. 13.These values are then multiplied with the average length ofthe corresponding region.

The stray inductance does not depend on the distancebetween the core and tank wall, because almost of the mag-netic energy is comprised in the area between the primaryand secondary windings. Therefore, only two different regionsare considered: 1) inside the core window and 2) outsideof the core. As an average length of the second region, the


TABLE IV

DISTRIBUTED CAPACITANCE AND STRAY INDUCTANCE

IN DEPENDENCE ON THE AREA

Fig. 12. Measured pulse shape in comparison with predicted pulse shape byanalytical calculations and 2-D-FEM simulations.

central point between the primary and secondary windings ismultiplied with the corresponding length, which is indicatedwith lL ′σ2.

The results for Cd and Lσ in dependence on the area areshown in Table IV.

2) Comparison: To validate the proposed analytical calcu-lations further to pulse measurements of an existing pulsetransformer were performed.

In a first measurement, the primary inductance of a switch-ing unit is determined to be L11 = 90 nH, which can betransferred to the secondary side with respect to the numberof cores and the turn ratio to L ′

11 = 119 μH.The transformer is then measured with a resistive load

of Rl = 1000 at a pulse voltage of Vs = 189 kV. Themeasured pulse shape is then compared with a second-orderdelay element, which can be used to describe the pulse shape atthe beginning of the pulse [4]. The second-order delay elementconsiders the calculated parasitics and in addition the rise timeof the switches of Tr = 200 ns.

In Fig. 12, the measurement data are compared with thepulse shape with results of 2-D-FEM analysis and calculation.Both results show high correspondence to measurements andto each other.

Fig. 13. Dimensions in millimeter of the investigated pulse transformer forshort pulses. R1: area of the core window. R2: area at the front of tank.R3: area at the sides of tank. R4: area at the back of tank. lL ′σ2: averagelength of central point between primary and secondary windings.

TABLE V

COMPARISON BETWEEN OPTIMIZATION CONSTRAINED TO PARAMETERS

OF EXISTING PULSE TRANSFORMER AND OPTIMIZATION

WITH UNCONSTRAINED PARAMETERS

V. APPLICATION OF THE PROPOSED OPTIMIZATION

PROCEDURE

A. Specifications of SwissFEL

The proposed optimization procedure is conducted withspecifications of the SwissFEL (Table I). At first, the pulseshape is predicted using the geometry parameters of theinvestigated pulse transformer, shown in Fig. 13. The parasiticsare taken from Section IV-D.2, assuming a klystron loadand an additional secondary capacitance of Csec = 50 pF.In a second step, the algorithm is conducted with equalassumptions, but with unconstrained optimization parameters.Only two optimization parameters, nc and wAc are preset:Because of the high required power of 120 MW at least12 pulse switches are required. Therefore, nc = 6 is usedin the algorithm, since there are two switching units per core.The minimum core width is preset to have an equal width asthe used IGBT modules wAc = 140 mm, which is also thecase for the existing transformer, to allow a direct connectionof the modules and to minimize the primary inductances.

The results of the comparison are shown in Table V. Thefirst column lists the results when the optimization parameterscorrespond to the investigated pulse transformer. In the second


Fig. 14. Section of pulse shape with specifications of SwissFEL with para-meters of the investigated pulse transformer and pulse shape resulting from theoptimization procedure with unconstrained parameters. Both displayed pulseshave an equal starting point and a pulsewidth of 3 μs.

TABLE VI

OPTIMIZATION RESULTS FOR MATRIX TRANSFORMER

WITH SPECIFICATIONS OF CLIC

column, the results are displayed, when the parameters areoptimized to high conversion efficiency.

It can be observed that the optimization procedure results ina similar core geometry, but increases the secondary windingheight, reduces the distance at the bottom of the windings andincreases the number of primary turns to two. Due to a highernumber of turns, the leakage inductance and therefore the risetime as well as the damping of the pulse is increased. Onthe contrary, the time to flat top is reduced, because the pulsewith parameters of the investigated transformer overshoots andcomplies later with the flat-top criteria, as shown in Fig. 14,where a section of both pulse shapes is displayed. Thus, byapplying the optimization procedure, an increase in conversionefficiency including pulse losses from 53% to 61.8% can beachieved corresponding to an relative improvement of 16.6%.

The proposed optimization procedure is therefore wellsuited to improve the design process of a pulse transformer.

B. Specifications of CLIC

The proposed optimization procedure is in a second stepexecuted for the specifications of the CLIC system (Table I).The optimization results are shown in Table VI. A sectionof the pulse shape is shown in Fig. 15(a). In an optimalconfiguration, the pulse does not exceed the voltage band ofthe FTS and therefore immediately reaches the flat-top criteria.

Fig. 15. Optimal configuration of matrix transformer with specifications ofCLIC. (a) Resulting pulse with pulse requirements. (b) Distribution of thesystem losses.

Fig. 16. Matrix transformer in an optimal configuration for specifications ofTable I.

The distribution of the losses per pulse is displayedin Fig. 15(b), showing the dominance of the pulse losses with61%, followed by the core losses with 22% and a share ofthe other loss components of 18%. The 2-D-FEM analysis isshown in Fig. 10, and the resulting transformer is shown inFig. 16.

VI. CONCLUSION

In this paper, a general procedure for pulse transformeroptimization is presented, considering a given set of pulsespecifications.

The optimization parameters include number of primaryturns, number of cores, secondary winding height, distancebetween the secondary and primary windings, width of thecore cross-sectional area, and opening angle of the windingcone. In the procedure, the total losses consisting of pulse,core, winding, active bias circuit losses, and losses of theprimary switches are minimized.

Due to the nonlinear klystron load, the pulse shape isanalyzed in the time domain, ensuring that the given pulseconstraints are met. For core loss estimation, measurementson a test core are performed for pulselengths in the range of3−300 μs.

In addition, analytical calculations of the transformer para-sitics are proposed and verified with 2-D-FEM simulations aswell as pulse transformer measurements.


The procedure is then applied to investigate an existingpulse transformer with the specifications of SwissFEL, tovalidate the method and to optimize the existing design. Theresulting transformer parameters of the optimization procedurelead to a shorter time to flat top and improve the systemconversion efficiency by 16.6%. Finally, the procedure isconducted for a long-pulse system with the specifications ofCLIC optimizing the overall efficiency. An overall conversionefficiency of 97.7% including pulse losses is achieved.

REFERENCES

[1] H. Li, H.-J. Ryoo, J.-S. Kim, G.-H. Rim, Y.-B. Kim, and J. Deng,“Development of rectangle-pulse Marx generator based on PFN,” IEEETrans. Plasma Sci., vol. 37, no. 1, pp. 190–194, Jan. 2009.

[2] G. Dale, H. Kirbie, W. Haynes, C. Heath, T. Lopez, F. Romero,and R. Wheat, “Design and application of a diode-directed solid-state Marx modulator,” in Proc. IEEE Pulsed Power Conf., Jun. 2005,pp. 1211–1214.

[3] P. Viarouge, D. Aguglia, C. A. Martins, and J. Cros, “Modeling anddimensioning of high voltage pulse transformers for klystron modula-tors,” in Proc. 20th ICEM, Sep. 2012, pp. 2332–2338.

[4] D. Bortis, “20 MW halbleiter-leistungsmodulator-system,” Ph.D. disser-tation, Dept. Electron., Power Electron. Mechatron., Univ. ETH Zürich,Zürich, The Switzerland, 2009.

[5] W. Reass, R. Gribble, and J. McCarthy, “Polyphase resonant converter-modulators for the KAERI proton engineering frontier project accelera-tor klystron RF amplifier system,” in Proc. IEEE Int. Power ModulatorsHigh Voltage Conf., May 2008, pp. 231–234.

[6] M. Jaritz and J. Biela, “Optimal design of a series parallel resonantconverter for a solid state long pulse modulator,” in Proc. 4th Euro-Asian Pulsed Power Conf., 2012.

[7] J. S. Bakken, “Study and development of solid state based long pulseklystron modulators for future linear accelerators at CERN,” Ph.D. dis-sertation, Dept. Electr. Power Eng., Norwegian Univ. Sci. Technol.,Trondheim, Norway, 2007.

[8] D. Bortis, J. Biela, and J. Kolar, “Transient behavior of solid-statemodulators with matrix transformers,” IEEE Trans. Plasma Sci., vol. 38,no. 10, pp. 2785–2792, Oct. 2010.

[9] D. Gerber and J. Biela, “Bouncer circuit for a 120 MW/370 kV solidstate modulator,” in Proc. PPC, Jun. 2011, pp. 1392–1397.

[10] D. Bortis, G. Ortiz, J. Kolar, and J. Biela, “Design procedure for compactpulse transformers with rectangular pulse shape and fast rise times,”IEEE Trans. Dielectr. Electr. Insul., vol. 18, no. 4, pp. 1171–1180,Aug. 2011.

[11] J. Biela. (2011). Wirbelstromverluste in Wicklungen Induktiver Bauele-mente [Online]. Available: http://www.hpe.ee.ethz.ch

[12] IEEE Standard for Pulse Transformers, ANSI/IEEE Standard 390-1987.[13] D. Aguglia and E. Sklavounou, “Klystron modulators capacitor chargers

design compromises for ac power quality increase of the compact linearcollider (CLIC),” in Proc. Power Electron. Int. Symp. Electr. Drives,Autom. Motion, Jun. 2012, pp. 1535–1541.

[14] K. Venkatachalam, C. Sullivan, T. Abdallah, and H. Tacca, “Accurateprediction of ferrite core loss with nonsinusoidal waveforms usingonly Steinmetz parameters,” in Proc. IEEE Workshop Comput. PowerElectron., Jun. 2002, pp. 36–41.

[15] J. Muhlethaler, J. Kolar, and M. Ecklebe, “Loss modeling of inductivecomponents employed in power electronic systems,” in Proc. IEEE 18thICPE ECCE, Jun. 2011, pp. 945–952.

[16] (2012). Datasheet 5SNA 1200G450350 [Online]. Available:http://www05.abb.com/global/scot/scot256.nsf/veritydisplay/7f0947b55a3d4720c12579fa0049e277/$file/5sna%201200g450350_5sya%201415-02%2004-2012.pdf

[17] H. Singer, H. Steinbigler, and P. Weiss, “A charge simulation methodfor the calculation of high voltage fields,” IEEE Trans. Power ApparatusSyst., vol. 93, no. 5, pp. 1660–1668, Sep. 1974.

[18] N. Malik, “A review of the charge simulation method and its applica-tions,” IEEE Trans. Electr. Insul., vol. 24, no. 1, pp. 3–20, Feb. 1989.

[19] S. Voss, “Semianalytische modellierung elektrischer ersatzschaltbilderfür drahtgewickelte spulen und transformatoren in der leistungse-lektronik,” Ph.D. dissertation, German Nat. Library Sci. Technol.,Univ. Erlangen-Nuremberg, Erlangen, Germany, 2006.

[20] M. Albach, Grundlagen Der Elektrotechnik 1. Rancho Cordova,CA, USA: Pearson, 2004.

Sebastian Blume was born in Frankfurtam Main,Germany, in 1985. He received the Diploma Degreein electrical engineering from the Karlsruhe Instituteof Technology, Karlsruhe, Germany, in 2001, withspecialization in renewable energies focusing onpower electronics. In his thesis he investigated a newapproach determining the harmonic emissions of PVinverters at the Fraunhofer ISE, Freiburg, Germany.

He has been a Ph.D. Student with the Laboratoryfor High Power Electronic Systems since May 2012,and works on the development of ultra high precision

klystron modulators for compact linear colliders.

Juergen Biela (S’04–M’06) received the Diploma(Hons.) degree from Friedrich-Alexander UniversitätErlangen–Nürnberg, Nuremberg, Germany, in 1999,and the Ph.D. degree from the Swiss Federal Insti-tute of Technology (ETH) Zurich, Zurich, Switzer-land, in 2006.

He dealt in particular with resonant dc-link invert-ers with the University of Strathclyde, Glasgow,U.K., and the active control of series-connectedIGCTs with the Technical University of Munich,Munich, Germany. In 2000, he joined the Research

Department, Siemens A&D, Erlangen, on inverters with very high switchingfrequencies, SiC components, and EMC. In July 2002, he joined the PowerElectronic Systems Laboratory (PES), ETH Zurich. He is focusing on opti-mized electromagnetically integrated resonant converters. From 2006 to 2007,he was a Post-Doctoral Fellow with PES and a Guest Researcher with theTokyo Institute of Technology, Tokyo, Japan. From 2007 to 2010, he wasa Senior Research Associate with PES. Since August 2010, he has been anAssociate Professor of high-power electronic systems with ETH Zurich. Hiscurrent research interests include the design, modeling, and optimization ofPFC, dc–dc and multilevel converters with emphasis on passive components,the design of pulsed-power systems, and power electronic systems for futureenergy distribution.

optimal transformer design for ultraprecise solid state ... · ieee transactions on plasma science,...

Documents