„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 pubs-permission@ieee.org. By choosing to view this document you agree to all provisions of the copyright laws protecting it.”

Optimal Design of a DC Reset Circuit for Pulse Transformers

D. Bortis, J. Biela, J. W. Kolar

Power Electronic Systems Laboratory (PES), ETH Zurich, Switzerland E-Mail: Bortis@lem.ee.ethz.ch / Homepage: ww.pes.ee.ethz.ch

Optimal Design of a DC Reset Circuit for Pulse

Transformers

D. Bortis, J. Biela, J. W. KolarPower Electronic Systems Laboratory (PES), ETH Zurich, Switzerland

E-Mail: btis hz / Homepage: w

Abstract - Pulsed power systems often use a pulse trans-former to generate high voltage pulses. In order to improve theutilization of the transformer core material usually a dc resetcircuit is applied which results in a smaller transformer volume.However, by the use of a reset circuit additional losses are gen-erated. Usually, in the dimensioning of the reset circuit only thecopper losses are considered but these are only a small part ofthe whole losses. In this paper analytical equations for calculat-ing the losses in the pulsed power system and a method for op-timizing a dc reset circuit are presented. With the optimizationmethod a reset circuit for an existing system is designed and theresults are validated by simulation and by measurements.

I. INTRODUCTION

Pulsed power systems are used in a wide variety of appli-cations, for example in particle accelerators, radar, medicalradiation (cancer therapy), sterilization systems (drinkingwater) or ion implantation systems (semiconductor manufac-turing). In these applications pulses of several kV andMW aregenerated and the duration of the single pulse varies from afew ns to some ms. The requirements on the generated pulsesregarding for example rise/fall time, overshoot, pulse flatnessand pulse energy are high and can vary over a wide range.Therefore pulsed power systems are built in many differenttopologies [1]-[3].A lot of pulsed power systems use a pulse transformer to

step up the voltage as shown in Fig. 1. The design of a pulsetransformer is comparable to high frequency transformerswhich are widely used in power electronics. But in contrast tomany converter topologies using HF-transformers pulsepower systems only generate repeated unipolar voltagepulses. In this case the core material of the pulse transformeris not optimally utilized.

a)30

25

, 20-z 15-Z 10-

-secondary wItage

5--1 0 1 2 3 4 5 6 7 8 9

time [us] b)

Figure 1: a) Possible schematic of a pulse modulator and b) corre-

sponding 25kV pulse.

The unipolar voltage pulse leads to a unipolar flux swing inthe core. This results in a core volume approximately twiceas big as with a bipolar excitation. A bipolar operation of thetransformer could be achieved with a reset circuit whichpremagnetizes the core to a negative flux density before thepulse.

77.I]

FEDC resetcircuit

Figure 2: Block diagram of a pulsed power system with reset cir-

cuit and total flux in the transformer core.

In the literature different active [4] and dc reset circuitconcepts are discussed. The active reset circuits are morecomplex than the dc ones, because additional components arerequired. However, energy recovery from snubber circuitscan be realized with active reset circuits.The most common method to reset transformer cores is the

dc reset circuit, where a dc current is used. This current canbe supplied to the primary, secondary or to a tertiary winding.In order to keep the considerations more general, a tertiarywinding, also called reset winding, is used in this paper. Thecurrent of the dc reset is flowing in a direction that generatesa magnetic flux which is in the opposite direction than theflux induced by the voltage pulse (cf. Fig. 2).With the reset circuit a flux swing from a negative to a

positive operation point instead form zero to a positive opera-tion point (cf. Al, A2 and B1 in Fig. 5) is possible. Therefore,the total flux peak-to-peak amplitude can be increased up totwice the value Bsat without a reset circuit. This enables asignificant reduction of the core cross section because thismainly determined by the maximum flux density.

The dc current for the reset circuit is generated by an ex-ternal power supply as shown in the equivalent circuit inFig. 3. As later will be explained in detail, a large inductor isplaced between the reset winding of the transformer and thepower supply to protect the supply from high voltage pulses.The inductor, which reaches values of several mH, is one ofthe largest components of the dc reset circuit.

1-4244-0714-1/07/$20.00 C 2007 IEEE. 1171

L

In the literature [5]-[1 1] many pulsed power systems arepresented which use a dc reset circuit to improve transformerutilization. But there is no information about the design or inparticular the optimisation of the reset circuit given. A moredetailed description of the dc reset circuit for a magneticpulse compression is discussed in [12] where also no infor-mation about the optimal dimensioning is presented.

Since the reset circuit has a significant influence on thevolume and the efficiency of the pulse modulator, analyticdesign equations for the reset circuit are presented in the fol-lowing. With these equations the volume and/or the effi-ciency of a pulse modulator is optimised in this paper whichresults in a smaller pulse transformer volume and optimaldesign of the reset circuit.

First, the dc reset is described in detail in Section II. Foroptimising the volume/efficiency of the reset circuit analyticequations for the losses are required. These are presented inSection III followed by the optimal dimensioning in SectionIV. The results extracted form Section IV are verified in Sec-tion V where the analytic results are compared with simula-tions and measurements. Section VI describes how the opti-mal choke can be realized. In Section VII the systems withand without reset circuit are compared. Finally, a conclusionand topics of future research are presented in Section VIII.

II. THE DC RESET CIRCUIT

The operation of the dc reset circuit is analyzed for thepulsed power system shown in Fig. 3. The most importantpart is the non-ideal pulse transformer which matches thepulse generator on the primary side to the load on the secon-dary side. The numbers of turns are Npri for the primary andNsec for the secondary. The pulse generator itself is simplymodelled by an ideal pulse source Up,js, with inner resistanceRint and the freewheeling Diode Df with forward voltage Uf,because these do not significantly influence the design of thereset circuit.

equivaflent clircuit of the tran7sfone1r

Rint ipr L6 Npri Nsec iload

(lnse f 1iag| Cd1N RI I , t T , k. -9 u~~~~~~~~~~~~~~~reset

Nreset i ee Rco

Figure 3: Detailed circuit of a pulsed power system.

The third winding is connected to the reset circuit which isrepresented by a series connection of the voltage source Uresetand the inductor Lchoke with resistance Rchoke. The pulse trans-former is modelled with the parasitic elements like leakageinductance L,, stray capacitance Cd (calculation cf. [13]) andthe magnetizing inductance Lmag.

In Fig. 4 the input voltage Upulse during an operation cycleis shown. Neglecting the source resistance Rint this voltage isequal to the primary voltage Upri. One cycle of the voltage

could be split into the three time sections SI - S3 shown inFig. 4.

v ALiUpulse

Uf t

Ti T3 ~~~~~~~period.sSI S2 S3 time sectionls

Figure 4: Common pulse pattern for pulsed power systems.

The period T1 in time section SI is equal to the pulse width.T2 is the time after the pulse in which the freewheeling diodeis conducting and T3 is the period between the end of free-wheeling and the next pulse.

At the start up when the system is turned on, the currentsin the windings are zero and- based on amperes law (1) - themagnetic field H in the core is also zero.

H=NI (1)

Depending on the core material the flux density in the core isBrem and the starting point of the system in the B-H curve isAl as shown in Fig. 5.

B B at

Bsat- ----- -I

Figure 5: B-H curve with and without premagnetization.

Assuming UpU15. = 0, the current through the inductor andthe magnetizing inductance is rising to Ireset as soon as thereset circuit is turned on. With equation (1) the dc currentIreset leads to a constant H-field and therefore to a const~antflux density Breset in the core (cf. Fig. 5). During start up theoperating point moves from Al to A2 where Hreset and Bresetcan be expressed as

Hreset resetreet and Bre et =a 01rHreset (2)

Assuming Hreset is fixed due to the hysteresis curve of thecore material increasing the number of turns Nreset results in asmaller reset current Ireset and vice versa according to (2). Assoon as the system reaches operating point A2 an output pulsecould be generated (section SI). With the assumption of arectangular voltage pulse Faraday's law can be simplified to

ARU pul~se = NprAcore At with At=T1. (3)

1172

The voltage pulse results in a change of flux density JBwhich can be maximal 2 Bsat (with Bsat=Breset) for a systemwith dc reset circuit.A system without dc reset circuit is limited to Bsat-Brem i.e.

from Al to B1 and back (there the hysteresis curve must bescaled to the core size for a system without reset circuit). Bythis reduction of the flux swing a core cross section twice asbig as with reset must be used.

In the following, the design of the dc reset is carried outwith respect to a symmetric H-field / magnetizing currentswing from -Ireset to Ireset because this results in an optimalusage of the core material. Therefore the dc reset current Iresetis equal to half of the total change of Imagg

In section SI the voltage pulse is also transformed to thetertiary winding. Depending on the turn ratio NreselNpri thevoltage transformed to the reset circuit can be large. To pro-tect the external low voltage supply a large inductance Lreset isplaced between transformer and supply. Even though theinductance of the choke is large some energy is stored in thechoke during SI which results in an increase of the reset cur-rent Of JIreset (cf. Fig. 6).

After the pulse, in section S2, the energy which is stored inthe leakage inductance, magnetizing inductance and the addi-tional energy in the reset choke has to be dissipated throughthe freewheeling diode and the load Rl0d. The demagnetiza-tion of the magnetizing inductance and the reset choke is dueto the approximately constant voltage Uf almost linear. Onlythe leakage inductance is demagnetized much faster, becausethe major part of the energy is dissipated in the load resistorRload.As soon as the magnetizing current Imag reaches zero the

reset current starts to commutate from the diode Df to themagnetizing inductance and premagnetizes the core again.The detailed current waveforms for Imag, Ireset, Iload and If areplotted in Fig. 6.

[kA] 'r

20 'load

Ireset

S Imag4

3

2 4Ireset

1 -l 20 4 ==6 [ms]

-2

7T3 T T2 period.s

-S3 SI S2 time sections

Figure 6: Current waveforms during pulse and freewheeling period.

III. Loss CALCULATION METHODS

As explained in Section II most of the energy stored in themagnetic components is dissipated in the freewheeling pathduring section S2. A significant part of these losses, whichdepends mainly on the inductance Lchoke and the number of

turns Nreset, is caused by the reset circuit. Therefore, a lossanalysis is carried out in the following in order to optimizethe reset circuit (cf. Fig. 7) - especially the design of thechoke.To achieve a large inductance value for the reset choke a

magnetic core is needed. Therefore, the flux density in thecore must be analyzed to determine the optimal number ofturns.

In steady state, which corresponds to section S3, a DC cur-rent is flowing through the choke. The constant current gen-erates a constant H-field / flux density BchokeDC in the core asshown in Fig. 8.

In section SI the pulse which is also transformed to thetertiary winding causes a large voltage drop across the resetchoke which results in an increasing flux density BchokeAC andan increasing current Ireset+'Jreset.

L~ ~ ~

11 Lkoke| Rchoke|l, 111 1 f A I g ~~~~~~~~hwin dair,_Nreset Ureset

I I IN,~~~~~~~~~~~~~~~FAhk

a)Figure 7: a) Reset circuit and b) a realization of a choke.

The maximum allowed flux is defined by'@max = AchokeBmax (4)

and in order to avoid saturation the following equation mustbe fulfilled

.1max2 '@chokeDC + A(DchokeAC -

Using (3), (4) and (5) results in(5)

AhokeB >yAP Nchokelreset NresetUpriTpulseAchoke nax -eff choke o N Nichoke Nchoke pri

There, the effective permeability teff is defined by the air gaplength dair.

For a given pulsed power system the free parameters arethe numbers of turns Nreset and Nchoke and the core dimensionsAchoke and lchoke The dc current Ireset is determined by themagnetizing current Imag and the number of turns Nreset of thereset winding on the pulse transformer.

B X

H-Nreset Ireset - Nreset (Ireset +IJIreset )

Figure 8: Hysteresis of the inductor core Lchoke.

1173

In the next step the losses of the system are calculated.This could be done by two different approaches. With thefirst one the overall losses are calculated by summing up thelosses in the magnetic components, the copper losses and thelosses in the supplies.

The second possibility is to calculate the difference be-tween the input and the output power for one cycle. In thefollowing the equations for the first approach are presented.The equations for the second approach are presented in theappendix. In both approaches it is assumed that the inputvoltage pulse and also the diode are ideal.

A. Overall Losses (stored energies)

The total losses result of dc copper losses, losses in thepower supplies and losses due to stored magnetic energy inthe parasitic elements. The losses have to be dissipatedthrough the freewheeling path. Ac copper losses, the lossesdue to the parasitic capacitance CD and the core losses areneglected in the following, because the energy in CD iswasted in the load resistor and the other losses do not influ-ence the design of the reset circuit.

The losses of the magnetic components are calculated bycomparing the energy stored in the system at two differentpoint of time. One point in time is just before the pulse gen-eration and the second just after the pulse. The difference ofthe energy is equal to the energy dissipated by the magneticcomponents because no energy is recovered.

In Fig. 9 the energies stored in the inductances during onecycle are displayed. Before the voltage pulse is generated adc current is flowing through the choke and the magnetizinginductance. Therefore, the energy at this time is

2

Ebeforel puIse = Lchokelreset + Lmag ireset()

Now the pulse is generated and energy is transferred to theleakage inductance and the choke. If the flux swing of thetransformer is symmetric (Imag= -Ireset at the end of the pulse)the magnetizing inductance is demagnetized in the first halfof the pulse and remagnetized to the same absolute value atthe second half of the pulse (cf. Fig 9).

Hence, after the pulse the energy stored in the leakage in-ductance, magnetizing inductance and dc choke is

E _pulse =2Lk (Iet +A )+L r + 2LI2 (8)2 2 N )2~

with ipri (t) ='load (t) +imag (t)+ Nreset Aireset(t) (9)

The difference of the two energies is dissipated through thefreewheeling path and the load before the next pulse is trig-gered. Therefore, the losses in the magnetic components are

Pmag = (Eafter pulse -Ebefore pulse )rep (10)

The copper losses have to be calculated for each time sec-tion independently. These losses consist of the copper lossesin the choke, in the tertiary winding and in the connec-tors/cables of the reset circuit. Losses in the secondary wind-

ing are independent of the design of the reset circuit and arenot considered during the optimisation.

tt ~~~~~~~EchokeEmag

[J]+t -~~~~Ed

[ms]

Figure 9: Energy stored in the magnetic components.

Generally, the dc copper losses can be calculated as

Rcu = cu Acu

(1 1)

where 1,, is equal to the mean turn length of the core timesNreset / Nchoke for the tertiary winding / choke.

In section S3 the current through the winding is a dc current(cf. Fig. 6) and the copper losses can be calculated by

(12)2T3Pcu SI Rchoke reset T

rep

Because of the rectangular voltage pulse in sections S1 and S2a triangular ac current is added to the dc reset current.

This results in the following copper losses for sections SIand S2

Pcu sxI

R I +AIreset

t dtrep

choke1 reset TXre '(13)

with x = 1 or 2. Period T1 is equal to the pulse width and T2can be calculated by the equivalence of the voltage timeproduct.

U T-U'priUf

(14)

The resistance due to the cables/connectors is independentof the number of turns and is assumed to be a constant valueRcon,

In addition to the magnetic and copper losses the powersupplies also have an influence on the optimal design. If thedesign of the supply is already known, an internal voltagesource Uin, and an internal resistance Rint can be substitutedfor the power supply. Otherwise, the losses of the supply canbe modelled with an output efficiency q between 0.9 and0.95. The general losses of the former approach results in

ps-int int out int out

and the general losses of the second approach leads to

Pps5int =Uoutiout ( -)-

(15)

(16)

1174

IV. OPTIMAL DC RESET CIRcurr DESIGN

With the derived equations the losses now can be ex-pressed as a function of the number of turns Nreset, Nchoke thecross section Achoke and the window dimensions hwi, and dwi,of the choke.For the choke in Fig. 10 for example the maximal wire di-ameter would be

(17)dw _ 2hwincoNchoke

dw ire Nreset_1. ~~~4----- I|~~~~~~~~~~~~ lI ,dis,

*d _. .

2Figure 10: Quarter of the cross section of the choke.

During the optimisation additional constraints like maxi-mal current density in the wire or minimal isolation thicknessdi,, between core and coil must be considered. In order tolimit the optimization problem to two dimensions - Nre,et andNchoke - the core of the reset choke is fixed (here: AMCC1000from METGLAS). In Fig. 11 the resulting losses dependingon Nreset and Nchoke are plotted for a pulsed power system withthe specification given in Table 1.

6.8 -

6.4 -

6.0 -

Zt 5.2 -

48'4 4.8 -

4.4-

4.0 -

3.6 -

3.2 -

6.4-

i75.6--

I' 4.8-11 -

4.0

3.2

b) 5.0 10

6.4

5.6

100 N 4.8

Nch,k, 4-0-,150 3.2

2 . 55 .0 S 20 60 100 140 180

a) Nreset 1 200 C) Nchoke

Figure 11: a) 3D Plot of total losses depending on Nchoke and Nreset.Losses depending on (b) Nreset and on (c) Nchoke.

There, minimal losses of 4100W are achieved with Nreset = 50and Nchoke = 100. The reset current Ireset is therefore 50A. Theinductor has an air gap dair = 4mm per leg and a wire diame-ter dwire = 2.1mm. This results in an inductance Lchoke of3.5mH.

In Table 2 it can be seen that the major part of the losses isgenerated in the magnetic components, especially in thechoke. Usually, only the copper losses in the choke are con-sidered but these are only a small part of the overall losses.Therefore, it is important to take the overall system into ac-count.

Table 1: Specification of Power Modulator.Pulse voltage Upuise 1kV

Pulse duration Tp,,se 5us

Primary current ipri 20kA

Repetition ratefrep 500HzNumber of turns on the primary Npri 1

Magnetizing inductance Lmag 2.5uH

Magnetizing current Imag 2kA

Resistance of connectors / cables Rint rs 0.1QChoke core AMCC]000

Table 2: Individual and total losses.Total copper losses 450WLosses in power supplies, cables and con- 450WnectorsLosses due to magnetic components 3200W

Total losses 4100 W

V. VALIDATION BY SIMULATION AND MEASUREMENT

In order to validate the analytical equations different calcu-lated results are compared with simulation and measurementresults acquired with the system shown in Fig. 12.

Figure 12: Pulse generator (45cm x 30cm x 20cm, pulse powerdensity: 500kW/ltr.) and pulse transformer.

Fig. 13 shows the losses generated by the reset circuit fordifferent choke inductances Lchoke and constant copper resis-tance Rreset of the reset circuit obtained by measurement,simulation and calculation.

80 -

72 --64 -

56-

2 48-

40 -

3

C Measurement° Simulation

ICalculation

Rreset = 0.242 Ohms

0.0 2.5 5.0 7.5 10.0 12.5 15.0

Choke inductance Lchoke [mH]Figure 13: Losses in the diode and the copper resistance.

1175

2

~? _ _11

For all measurements the same choke, designed for 50 A DCand 70 A peak current, was used. To realize different induc-tance values only the air gap length 5 respectively the mag-

netic resistance Rm was changed.

Lhoke Rk where Rmm fiIo Aor

sistance and air gap length is given in [14]. Due to the fring-ing field around the air gap for bigger gap lengths the relationbetween Rm and 5 is not longer linear and therefore the airgap length a5 has to be simulated by FEM.

(18)

The results in Fig. 13 might lead to the conclusion that forhigher inductance values fewer losses are generated. But, theincrease of the choke inductance by reducing the air gap

length results in a faster saturation of the core. Contrariwisethe decrease of the choke inductance results in a worse utili-zation of the choke core.

To get the best core utilization for all choke inductancesequation (8) has to be fulfilled. With this condition for eachchoke design the magnetic resistance Rm is fixed and/or atleast limited to a realizable value. Therefore, to get higherchoke inductances considering equation (18) the number ofturns Nchoke has to be increased which results in a higher cop-

per resistance Rreset of the reset circuit. To achieve smallerchoke inductances and still fulfilling condition (8) the num-

ber of turns has to be decreased which leads to a smaller cop-

per resistance Rreet.Reasonably, considering equation (8) increasing induc-

tance values Lchoke respectively increasing number of turnsNchoke result in increasing copper losses and decreasing lossesdue to the energy stored in the choke. Therefore, the totallosses always results in an optimal number of turns Nchoke InFig. 14 the loss curve considering equation (8) is shown forcalculation and simulation.

6.5

6.0

i 5.5-I~"I 5.0

4 4.5

4.0

3.5

calculation

simulation

50 75 100 125 150 175

Nchoke

Figure 14: Losses in the freewheeling diode and the copper resis-tance considering equation (8).

As could be seen in Fig. 13 and 14 the analytical equationscorrelate really well with the measurements and simulation.

VI. REALIZATION OF THE CHOKE

The described dimensioning of the choke results in an op-

timal number of turns Nchoke of the choke, an optimal numberof turns Nreset of the reset winding and the magnetic resistanceRm of the choke. The magnetic resistance is adjusted by theair gap length a. To calculate the associated air gap length,for small magnetic resistances the correlation given in (18)can be used. A better analytic relation between magnetic re-

Figure 15: )ased calculation of the inductance value.

To reduce the ohmic losses further, the choke can also bedesigned with several layers in parallel or wire with a rectan-gular cross section could be used. To avoid overheating space

between the layers should be provided. Attention has also tobe paid to the winding arrangement because of the high volt-ages (cf. Fig. 7).

VII. SYSTEM WITHOUT RESET CIRcuIT

The pulsed power system could also be designed withoutthe presented reset circuit. The major advantage is that thesystem without reset circuit needs less components (nochoke, no additional power source, no reset winding). Be-cause of the absence of the reset choke the stress in the diodewill be smaller.

Table 3: Pulsed power system with and without dc reset circuit.

min. longer pulse rise time

On the other hand the core cross section of the transformerwill be at minimum twice as big as with reset circuit due tothe unipolar flux swing and the possible remanence of thecore material. As a result of the doubled cross section the

winding length on the transformer will be V2 times longerthan with reset circuit for a quadratic core cross section. This

causes Vi times more copper losses in the winding. Also theleakage inductance L, and the distributed capacitance Cd [13]

have to be multiplied by Vi. Considering the relation be-

1176

A Pulse power system without reset circuit compared to a

system with reset circuit needs/has ...

+ less components: choke, power supply, third winding

less power to waste in the diodes

minimum twice the core cross section of the transformer

min. ai longer windings

min. Vi bigger copper resistance in the windings

min. Vi times bigger distributed capacitance Cd

min. Vi times bigger leakage inductance L,

tween rise time and the parasitic elements L, and Cd given in[1] and [15]

trise =k CdLc , (19)

also the rise time is increasing by the factor of V . Table 3summarizes the advantages and disadvantages of a systemwith and without dc reset circuit.

VIII. CONCLUSION

In this paper a method for optimizing a dc reset circuit fora pulse modulator was presented. The analytic equations arevalidated by simulation and also by measurement results fordifferent system specifications. Based on the equations it isshown that the major part of the losses due to the reset circuitis caused by the stored energy in the magnetic componentsand therefore the optimization of the dc reset circuit requiresthe consideration of the overall system for a complete cycle.It is also shown that the system with reset circuit compared toa system without reset circuit offers some important advan-tages.

In further work also active reset circuits will be analyzedand compared to dc reset circuits regarding volume and effi-ciency.

REFERENCES

[1] N. G. Glasoe and J. V. Lebacqz, "Pulse Generators", MIT Radiation LaboratorySeries, vol. 5, McGraw-Hill Book Company, New York, 1948.

[2] Paul W. Smith, "Transient Electronics, Pulsed Circuit Technology", J. Wiley,England, 2002.

[3] S. Roche, "Solid State Pulsed Power Systems", Physique and industrie.

[4] H. Kirbie, US 2002/0186577 Al "Unified Power Architecture".

[5] E. G. Cook at al, "Inductive-Adder Kicker Modulator for Darht-2", LawrenceLivermore National Laboratory.

[6] E. G. Cook at al, "High Average Power Magnetic Modulator for Copper Lasers",Lawrence Livermore National Laboratory.

[7] E. G. Cook at al, "Solid State Modulator R&D at LLNL", Lawrence LivermoreNational Laboratory.

[8] H. Hatanaka, M. Obara, "High efficiency operation of the high-repetition- rate all-solid-state magnetic pulse compressor for KrF excimer laser".

[9] D. M. Barrett, "Magnetic Pulse Compression Techniques for Non-thermal PlasmaDischarge Applications", QM Technologies, Albuquerque.

[10] S. Ashby et al, "CLIA - A compact Linear Induction Accelerator System", PhysicInternational Company.

[11] H. C. Kirbie, G. Y. Otani and G. M. Hughes, "A magnetically switched TriggerSource for FXR", Lawrence Livermore National Laboratory.

[12] D. M. Barrett, "Core Reset Consideration in Magnetic Pulse Compression Net-works", Tetra Corporation, Albuquerque.

[13] J. Biela, D. Bortis and J. W. Kolar, "Analytical Modelling of Pulse Transformersfor Power Modulators", International Power Modulator Conference, Washington,2006

[14] N. Mohan, T. M. Undeland, W. P. Robbin, "Power Electronics: Converters, Appli-cations, and Design",John Wiley and Sons, 2nd Editon

[15] J.S. Oh et al,"Rise time analysis of pulsed klystron-modulator for efficiency im-provement of linear colliders", Nuclear Instrumentation and Methods in PhysicsResearch, 2000

APPENDIX

B. Overall Losses (input/output power)

The input power is provided by the pulse source and thereset power supply. The output power is equal to the trans-ferred power with positive current direction into the load. Allcurrent flowing in negative direction through the load is alsoconsidered to be losses. Based on this, the input power duringone cycle delivered by the two sources can be written as

input p puls e + ps reset

T1+T2+T3 T

Pps pulse = pulse f pri (t)dt = U pulse f ipri (t)dt0 0

(20)

(21)

P reet = Ureset f ireset (t)dt = U reset ((I2eset + Aireset ())dt0 0

whereas the pulse voltage UpU,,e of the source is assumed tobe rectangular as shown in Fig. 4 and the reset voltage Uresetis constant.The primary current ipri is the sum of iload(t), imag(t) andjireset(t) as defined in (9). The load current can be derivedfrom the load resistance and the load voltage which is equalto the transformed pulse voltage neglecting the source resis-tance Ri, The magnetizing current imag(t) and the AC part ofthe reset current Jireset(t) are due to the constant voltagesUpulse and Uf linear which results in a linear increasing duringthe pulse and a almost linear decreasing during the free-wheeling time section S2.

Ilo_d U pulse N

load pri

imag (t) = UX t

Lmag Tx

(23)

(24)

2

Aireset(t) = Lx Nreseti (25)Lchoke Npri )TX

whereas Ux and Tx stand for UpUle during T1 and for Uf duringT2.The total power losses can be calculated by simply sub-

tracting the effective output power from the input power. Theoutput power is given by

TI +T2 +T3

Poutput Upulse fiload (t)dt UpulseIloadTl0

and therefore the losses are

(26)

losses input output (27)

So far in this approach both sources are ideal. The internallosses resulting from the power supply have to be explicitlyadded as with approach I. Hence, the total losses are

Plosses = Pinput Poutput + Ppsmint (28)

With this, now the optimal choke can be calculated.

1177