Optimal Transformer Design for an Ultra Precise Long Pulse ...

Optimal Transformer Design for an Ultra PreciseLong Pulse Solid State Modulator

S. Blume, D. Gerber and J. BielaLaboratory for High Power Electronic Systems, ETH Zurich, Email: [email protected]

Abstract—In this paper, an optimisation of a pulse transformerfor pulses in the 100µs-range as planned for the CompactLinear Collider (CLIC) is performed with the aim of minimisingthe overall losses. The investigated transformer geometry is amatrix transformer with cone windings. The leakage inductanceand the distributed capacitance are calculated analytically andverified with 2D FEM simulations. The deviation of the twomodels is smaller than 5 %. For optimisation of the system,pulse, core, winding, demagnetisation losses and losses of theprimary switches are taken into account. The overall conversionefficiency of the considered system including pulse efficiency inan optimal configuration is 97.7 %, in which the transformerhas 4 primary turns, 3 cores and a tank volume of 0.96 m3.

I. INTRODUCTION

A new type of electron-positron collider, the CompactLinear Collider (CLIC), is investigated at CERN, which willreach energy levels in the TeV range. To realize this amountof energy, more than 1600 modulators with a pulse power of24 MW each are required. The system has tight specificationsregarding the pulse flat-top stability and voltage overshoot.Due to the high number of modulators and a limited powersupply at CERN, the efficiency from grid to klystron includingthe pulse efficiency has to exceed 90 %. Selected specificationsare shown in Tab. I.

So far, pulse transformer design has mainly focused onmeeting the pulse requirements but no optimisation regardingthe overall losses was conducted.Therefore, in this paper an optimisation algorithm maximisingthe efficiency of the pulse transformer under the constraints ofelectrical and pulse requirements is proposed.At first, a short overview of the considered system is given insection II. Thereafter, in section III the optimisation algorithmis presented and each loss share is explained. Furthermore,the analytical calculations of the leakage inductance and thedistributed capacitance are described and compared to FEMsimulations in section IV. Finally, the optimisation results arediscussed in section V.

II. OVERVIEW OF SYSTEM

In this chapter, the scope of the optimisation model isintroduced and the underlying circuit diagram is explained.

1) Scope of Optimisation: In Fig. 1 the proposed CLICsystem consisting of an isolated AC/DC converter, a DC/DCcharging unit, an active bouncer, a pulse generator and apulse transformer is shown. The optimisation scope is the

TABLE ICLIC KLYSTRON MODULATORS SPECIFICATIONS[1]

DC link voltage vp 3 kV Pulse repetition trep 50HzSecondary voltage vs 150 kV Voltage overshoot Vovs 1%Pulse peak power 24MW Flat-top stability FTS 0.85 %Rise & fall times tr,f 3µs ηpulse 95%Settling time tset 5µs Flat-top length tflat 140µsModulator global efficiency ηmod global 90 %

pulse transformer. Additionally, an active premagnetisationcircuit is considered which doubles the possible magneticcore modulation amplitude. A voltage drop on the secondaryside of 0.4% is taken into account. Furthermore the pulsegenerator and the non-linear klystron load are considered.Assuming for the two-stage isolated AC/DC converter aconversion efficiency of 94,6 % (96, 5% · 98%) and for theDC/DC converter a conversion effiency of 98 %, which canbe achieved with well-known state-of-the-art systems, theconversion efficiency of the pulse forming system includingpulse efficiency has to exeed 97 % to meet the requirementfor ηmod global.

2) Pulse forming system: The considered pulse formingsystem, presented in Fig. 2, is a matrix transformer beingadvantageous in comparison to series or parallel connectedtransformers [2]. The secondary turns are arranged around thecores and primary winding in a cone shape resulting in smallerproduct of leakage inductance and the distributed capacitancethan using parallel windings [3]. The leakage inductance anddistributed capacitance have a crucial influence on the pulseshape as will be explained in section IV. Each core consists oftwo C-cores leading to two transformer legs. The secondarywindings of each leg are electrically connected in parallel.The number of cores is variable. In the considered matrixtransformer, the secondary winding of a leg encloses all cores,which leads, if equal size of the cores is assumed, to the turnratio

n =vsvp

1

nc, (1)

where nc is the number of cores, vp the primary voltage andvs the secondary voltage.The considered components for each core leg are a mainswitch Sm, the primary inductance L11, the primary resistanceR11 and an active premagnetisation circuit, consisting ofan additional switch Sr and capacitor a Cr (Fig. 2). Asload, a klystron Rklys is connected, which shows non-linearbehaviour.

AC

DC

Medium Voltage

Grid

DC

DC

ActiveBouncer

Charging Unit Pulse Generator Pulse Transformer Klystron Load

Optimisation Scope

Isolated AC/DC Converter

Active Premagnetisation

Circuit

Fig. 1. Overview of the proposed CLIC system with the scope of optimisationhighlighted in grey.

Cr

Sr

L11

Sm

Cin

R11

Lσ Cd+Csec

Rklys

Fig. 2. Overview of the matrix transformer with variable number of coresincluding an active premagnetisation circuit.

III. OPTIMISATION ALGORITHM

A. Optimisation StructureThe optimisation structure is shown in Fig. 3. The global

optimiser receives two sets of external parameters, one for thegeometric structure of the transformer and one for the pulserequirements as constraints. The cross-sectional area Ac, theminimum distance between the field shape ring and the coredw,min and the minimum secondary winding height hs,minare defined by

Ac =vp tflat

2Bmax np Fc,

dw,min = rr exp

(vs

rr Emax

),

hs,min = ns ds,min,

(2)

where Fc is the filling factor of the cut tape-wound core,tflat the flat-top length, np the number of primary turns, rr

Total Losses

Global External System ParametersPulse requirements:

eg. Rise Time, Flat-top Stability,...

Core Loss Submodel

Analytical Calculation of Lσ and Cd Geometric Setup of Transformer

Lσ , CdGeometricParameters

GeometricParameters

R11Lh , Rfe

Pulse Loss SubmodelDGL of Pulse Respecting Constraints

Con

stra

ints

Imag

Global Optimiser (Optimisation Parameters)

Active Bias Loss Submodel

Primary Switch Loss Submodel

Total Losses

Winding Loss Submodel

GeometricParameters

Losses

Losses

Losses

Optimal Design

Fig. 3. a) Optimisation procedure: The global optimiser receives externalparameters regarding pulse specifications and geometry. Losses are calculatedin each submodule.

the radius of the field shape ring, Emax = 20 kV/mm themaximum electric field strength in oil, Bmax the maximalmagnetic core modulation amplitude and ds,min = 2mm theminimum distance between two secondary turns [4].The pulse requirements and optimisation parameters lead withthe equations of (2) to the geometry of the pulse transformer.With the resulting geometry Cd and Lσ are derived with ananalytical approach proposed in section IV. Each loss sub-model calculates its specific losses for the given optimisationparameters, which are number of primary turns np, numberof cores nc, secondary winding height hs, distance betweensecondary and primary windings dw, length to width ratio ofthe core cross-sectional area rAc

and opening angle of thecone α.

B. Pulse Losses

v’1

Lσ+L’11 (Cd+Csec )

LH

R’11

RFe Rklys

2 3 4 5 6 7 8 9

140

145

150

155Resistive loadKlystron load

Time (μs)

Volta

ge (k

V)

a) b)

Fig. 4. a) Comparison of the output voltage of a resistive load and a klystronload. b) Applied circuit model as derived from [5].

Because the pulse specifications must be fulfilled,prediction of the pulse shape is crucial, which is realized bya time-domain simulation. It was shown in [4] that the risetime of the primary switches and the non-linear behaviourof the klystron influences the pulse form strongly due toa higher damping. This effect is shown in Fig. 4 a), werethe klystron load is replaced by an ohmic resistance in aconstant operating point. So far, a piecewise linearisationof the klystron load was conducted, which is not suitablefor a general optimisation. Therefore, the pulse behaviouris analysed in the time domain in this paper. The appliedcircuit model is derived from [5], based on the secondaryvoltage and shown in Fig. 4 b). The applied voltage signalv′1(t) takes the rise/fall times of switches tr,f = 200 ns anda secondary voltage drop of 0.4% during the flat-top periodinto account. The inductance on the primary side is estimatedas L11 = 100 nH. The resistance R11 is calculated in thewinding loss model (chapter III-C). The total secondary straycapacitance, including stray capacitance of the klystron, isestimated to Csec = 100 pF.

C. Further Loss Models

1) Active Premagnetisation Circuit Loss ModellThe active premagnetisation circuit is already shownin Fig. 2. It is assumed that all the stored magneticenergy can be retrieved besides losses in the switchduring premagnetisation and losses in the diode duringdemagnetisation.

2) Core Loss ModelThe core losses are calculated using the improved gener-alized Steinmetz equation (iGSE) [6]. There are two dif-ferent slopes of the magnetic field dB/dt considered. Oneduring pre- and demagnetisation which is Bmax/tpremagand one during the pulse which is 2Bmax/tflat. The con-sidered material is Metglas (amorphous alloy 2605SA1)because it offers a good compromise between low losses,

high saturation flux density and costs. The parameters forthe iGSE were obtained from [7].

3) Winding Loss ModelThe occurrence of winding losses, due to skin- andproximity effect, is described in detail in [8]. In orderto estimate the skin depth in the conductor, the pulsecurrent is estimated as a trapezoid and a FFT analysisis conducted over one pulse repetition time trep. Forcalculating the proximity effect losses, the secondarycircular windings are transformed to a sheet conductoras shown in [8]. A simplified geometry is estimated, inwhich the windings cover the entire height of the corewindow.

4) Primary Switch Loss ModelAs the pulse flat-top length tflat is relatively large andthe switching frequency 1/trep low (see Tab. I), onlyconduction losses of the primary switches are taken intoaccount.

IV. MODELLING OF LEAKAGE INDUCTANCE ANDDISTRIBUTED CAPACITANCE

Assuming a resistive constant load, the pulse shape ofa pulse transformer is mainly determined by the leakageinductance Lσ and the distributed capacitance Cd. In this casethe rise time trise and the damping coefficient σ of the pulseare described as [9]

trise ∼√LσCd, σ ∼

√LσCd

. (3)

In this paper the influence of the klystron load is taken intoaccount, which leads to higher damping of the pulse. Thereforesolutions with higher distributed capacitance still comply withthe pulse requirements. Because Lσ and Cd change duringoptimisation, they have to be described analytically, whichis realized by calculation of the stored electric and magneticenergy in the transformer geometry [4].

A. Distributed Capacitance

To calculate the distributed capacitance Cd, the pulse trans-former with its surrounding oil tank is divided into six regions(see Fig. 5). Is was previously shown, that the space betweenthe transformer and the oil tank corresponding to region R3

and R4 has to be taken into account as it contains between35 % to almost 70 % of the total electric energy dependingon the geometry [4]. In [4], analytical calculations of thestored electric energy were proposed for all six regions, but insome regions adjustment factors were added from comparisonwith FEM simulations. Additionally, assumptions such as a

Core

R1 R1 R1 R1

R5

R2

R3 R3

R4R4

R6

Fig. 5. 2D Front view of a transformer visualizing the six different regionsof interest for calculating the parasitic capacitances.

E (kV/m)5

4

3

2

1

0

20

16

12

8

4

0

H (kA/m)

Fig. 6. FEM simulation of the electric field (color scale 0-5 kV/mm) and ofthe magnetic field (color scale 0-20 kA/m).

centered field shape ring between transformer and oil tankwere made which are not given for a generalized geometry.In order to be able to compare the analytical approach withFEM simulations all calculations should be derived frommathematical calculation, especially in optimisations wherespeed is an important factor, as with the charge simulationmethod [10].The capacitances of regions R2, R3 and R4 were calculatedin the following way:

1) R2: The field shape ring in the core window can be seenas circle in a squared box, because the distances betweenthe ring and the grounded core area are equal in x- andy-direction. The capacitance of a square geometry withlength l is higher with the factor of k = 1.078 than thecapacitance of a circular geometry with equal radius Ras calculated in [11].

2) R3: The capacitance was calculated as a line charge in afree x-position between two grounded infinite plates in ahalf space with two mirrored charges.

3) R4: Each secondary turn was considered as a line chargeand was mirrored with its distance to the oil tank. Thefield shape ring was considered analogously.

B. Leakage InductanceFor calculating the leakage inductance Lσ , the approxima-

tion from [4] is adjusted for a free opening angle of the conewinding

Lσ,tot = µn2pdav

2hlR1

, dav = db +dt − db

2, (4)

where dt is the width at the top, db at the bottom and dav theaverage width between primary and secondary windings.

C. Comparison: Analytical Approach and 2D FEM SimulationIn Tab. II, the results of a comparison between the analytical

approach and FEM simulations are presented for a transformer

TABLE IICOMPARISON BETWEEN ANALYTICAL APPROACH AND 2D FEM

SIMULATIONS FOR THE SIX DIFFERENT REGIONS OF INTEREST (FIG. 5)

lx (m) FEM (pF) Calc.(pF)

C1 2.99 149.1 150.2C2 0.70 14.7 13.7C3 3.22 60.7 55.7C4 3.518 119.0 109.5C5 0.76 2.8 0.97C6 2.96 0.5 1.12Cd - 497.0 482.7

lx (m) FEM (µH) Calc.(µH)

Lσ 2.99 915.8 883.6

0 20 40 60 80 100 120 140

146

147

148

149

150

151

152Vovershoot

FTS

tflat

trise

}

}

tfall}

Ecore

Epulse

EpremagEswitch Ewinding

a) Time (μs)

Volta

ge (k

V)

b)

Fig. 7. Optimal configuration: a) Distribution of the system losses. b)Resulting pulse with pulse requirements.

geometry with 4 turns np for the primary winding and 3cores leading to 67 turns for the secondary winding ns. Bothmethods show high agreement. The geometry parameter forthis configuration are provided in Tab. III.In Fig. 6 it is shown that the considered regions comprisealmost the entire electric and magnetic energy. Due to adeviation smaller than 5 %, the proposed analytical approachis suitable for the optimisation model.

V. RESULTS

The highest efficiency could be achieved with a configura-tion of 4 primary turns and 3 cores, leading to a conversionefficiency of 97.7 % with an energy loss per pulse of 79.67 J.Fig. 7 a) shows that in this configuration the pulse losses aredominant with a share of 72 %, the core losses account for21 % whereas the other losses generate only 7 % of the overalllosses.To minimise pulse losses, the pulse has such a small overshootthat it immediately complies with the flat-top stability FTSas shown in Fig. 7 b). That is why the number of primaryturns is higher than the number of cores, leading to a higherLσ resulting in higher damping of the pulse. The importantparameters for the optimal configuration are listed in Tab. III.The matrix transformer in optimal configuration is shown inFig. 8.

VI. CONCLUSION

In this paper, a procedure designing a matrix transformeris presented maximising the conversion efficiency includingpulse efficiency while respecting the pulse specifications.

62.34 cm64

.2 cm

Secondary Winding

Primary Winding

92.1

2cm

Fig. 8. Resulting optimal configuration of the matrix transformer forspecifications of Tab. I.

TABLE IIIOPTIMISATION RESULTS

Primary turns 4 -Number of cores 3 -Secondary winding height 35.1 cmOpening angel of cone 3.21 degRatio core witdh/depth 1.1942 -Core window height 41.95 cmCore window width 11.34 cmDistance core tank 13.34 cmTank volume 0.9596 m3

Total energy loss per pulse 79.67 JTransferred energy to klystron per pulse 3342 JRise time 2.85 µsConversion efficiency 97.7 %

Lσ 883.6 µHCd 482.7 pFL′11 41.6 µH

Csec 100 pF

The optimisation algorithm has six optimisation parameters:number of primary turns, number of cores, secondary windingheight, distance between secondary and primary winding,length to width ratio of the core cross-sectional area andopening angle of the cone winding. In the optimisationpulse, core, winding, demagnetisation losses and losses of theprimary switches are considered. The leakage inductance Lσand the distributed capacitance Cd are calculated analyticallyand verified with FEM simulations with a deviation smallerthan 5 %. The pulse is analysed in the time domain to includethe influence of the non-linear klystron load on the voltageovershoot. The optimisation algorithm is applied to thespecifications of the CLIC system and an optimal geometryis derived. It can be shown that the conversion efficiency ofan optimal pulse forming system including pulse losses of97.7 % satisfies the requirement of 97 % to meet a globalefficiency higher than 90 %.

ACKNOWLEDGMENTThe authors would like to thank project the partners SNF

(project number 144324) and CERN very much for theirstrong financial support of the research project.

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