Performance evaluation of pulse compressor based modulators with very fast rise times for plasma channel drilling

Tonis Hobejogi and Juergen Biela Laboratory for High Power Electronic Systems, ETH Zurich, hobejogi@ethz.ch

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PERFORMANCE EVALUATION OF PULSE COMPRESSOR BASEDMODULATORS WITH VERY FAST RISE TIMES FOR PLASMA CHANNEL

DRILLING

Tonis Hobejogi and Juergen BielaLaboratory for High Power Electronic Systems, ETH Zurich, hobejogi@ethz.ch

Abstract

In this paper, limitations of a compact and durable solidstate modulator, using a 4.5 kV IGBT in combinationwith a pulse compression circuit are evaluated. Theinvestigated modulator utilizes two separate paths -one for ignition voltage and one for high currents. Theignition voltage is generated with a pulse compressioncircuit based on saturable transformers, which is used tocharge a series capacitor. After saturation, the capacitoris connected to the output. Parasitics have been estimatedin order to find an optimized modulator configuration.Measurement results for validating the models as well asthe simulation performance are presented. Main limitationfactors are described and possible solutions discussed.

I. INTRODUCTION

One possible concept to improve drilling efficiency isPlasma Channel Drilling (PCD). There high voltage pulseswith very short rise times are utilized for disintegratingrocks (Fig. 1) as for very fast rising voltages the break-down field of water is higher than for rock [1], [2]. Forthe considered application pulse voltages in the range of150 kV with rise times <100 ns are required (Table I).

The most common way for generating such pulses areMarx generators based on spark gaps. However, the size ofsuch converters is one of the main disadvantages. Severalalternative topologies based on semiconductor switches incombination with a pulse compression (PC) circuit havebeen introduced [1], [3]–[9] (Fig. 2 and Fig. 3). In thispaper, the focus is on evaluating the performance limitsof circuits based on saturable transformers as shown inFig. 2.

To design and investigate the circuit performance, adetailed circuit simulation has been implemented. Thetransformer parasitics have been calculated analyticallyand in addition validated with Comsol Multiphysics. Toverify the models a test device has been built. In thefollowing, first, the concept and the design procedureis presented in section II. Thereafter, the calculation ofthe parasitics is explained in section III and a proto-type together with simulation results are introduced in

TABLE I: Pulse parameters for the considered PCD application.

Voltage Magnitude >150 kV

Voltage Rise-Time <100 ns

ElectrodeWaterRock

VHV

Trise

Ebd

Rock

~500ns

Water

Target Pointa) b)

Fig. 1: a) sketch of a PCD electrode setup and b) breakdown field vs.voltage rise time curve (Ebd = f(Trise).

LHC

C0

L0

C1 CMV CHV L HV

CL

hvTRMV TRHV

PFC

Load

Stage 1Source Stage 2

HC Stage

Fig. 2: Scheme of a two stage PC unit utilizing saturating transformersinvestigated in this paper.

LPC

LHC

C1 N2N1

1:nTR

C0L0PF

C

Source

C0L0PF

C

Load

Load

a)

b)

Fig. 3: Alternative circuits: a) circuit utilizing Pulse Transformertogether with PC and b) Marx generator utilizing semiconductor switchesinstead of spark gaps.

section IV. Finally, in section V the topology designlimitations are discussed.

II. DESIGN PROCEDURE

The considered topology is shown in Fig. 2. It consistsof two saturable transformers and two series capacitors.The basic topology idea has been introduced in [3] and[4] while the presented one has few modifications. Sofar, mainly a series inductor is used at the source sideto limit the current magnitude while the transformer

978-1-4673-5168-3/13/$31.00 ©2013 IEEE

Load Voltage

Parameters:N1, n1, N3, n2, NFe1, NFe2

Initial Values:N1, n1, N3, n2, NFe1, NFe2

Core Parameters:μ, din, dout, AFe, lFe

Voltages and Currents

Electrical Model (Gecko) Transformer Cd, LM, Lσ Calculations

Fig. 4: Design procedure block scheme.

saturates. Here, a capacitor C1 is used in order to limit thetransformer current (cf. [1]). Before saturation of TRMV

capacitor C1 has a low impact on the output voltage.Additionally, the high voltage and the high current pathsare separated [1].

For the saturable transformers, a core material with asharp saturation curve and a high magnetizing inductance(LM ) as well as low saturation inductance (LMsat) isrequired. In the considered prototype system Vacuum-schmelze (VAC) nanocrystalline material is used. Thisfulfills those requirements under low frequency as will bediscussed later (Fig. 9). In order to limit possible solutionsVAC T600006-L2160-V074 (V074) core was selected.

In the first step of the design procedure (Fig. 4) the out-put voltage is considered and compared between differentconfigurations (turns ratio, number of turns etc.). In thesecond step, the mechanical design is taken into accountand the influence of the parasitics on the output voltageis determined. The design is based on V074 cores withsingle layer windings in order to limit LMsat. In the thirdand final step the best option with the set constrains ischosen.

III. PARASITICS CALCULATIONS

In the considered PC modulator VAC V074 ring coresare used for the two transformers. In Fig. 5 a physical lay-out sketch is shown. The primary and secondary windingsare made of 2.5mm2 wire. To reduce leakage inductanceseveral primary windings are connected in parallel. Addi-tionally, two secondary windings are connected in parallelto reduce the saturation inductance. To improve electricalstrength the whole transformer is potted with epoxy.

In the following analytic calculations scripts for deter-mining the stray capacitance, the leakage inductance, themagnetizing inductance, the ohmic losses and the corelosses are presented.

A. Stray CapacitanceAs shown in [10] transformers can be modelled with

a simplified equivalent network utilizing three capacitors(Fig. 6). For comparing the measurements with calculatedresults, the equivalent capacitance for the three measure-

CorePrimary

Connections forParallelPrimary

Secondary

Fig. 5: Transformer horizontal (left) and vertical (middle) cut sketchwhile N1 = 3; N1p = 7; N2 = 9; N2p = 2. On the right builtTRHV (N3 = 3, N3p = 11, N4 = 12, N4p = 2).

ment setups given in Fig. 6 are determined:

Ca = C1 + C12 → C1 = Ca − C12 (1)

Cb = C2 + C12 → C2 = Cb − C12 (2)

Cc = C12 → C12 = Cc (3)

There C1 and/or C2 could be negative. For computationpurposes, a lumped capacitance is calculated [10] (Fig. 7):

Cd =C1

n2+ C12 · (n− 1)2

n2+ C2 (4)

In order to analytically model the capacitance severalsimplifications are made. It is assumed that each turn canbe represented as a plate, that the spacing between theturns is equal, that the core voltage is constant and that theround core with windings can be represented as a straightplane (Fig. 8).

The full capacitance for each measurement setup(Fig. 6) can be described as

C = CPC + CPS + CSC + CTT−prim + CTT−Sec (5)

The simulated voltage applied to each setup is setVsim=1 V:

Wcap =1

2· C · dV 2 → Ctotal = 2 ·Wtotal (6)

Consequently:

Ctotal =N∑i=1

Cturn (7)

Cturn = Cturn′ · dV 2 (8)

The capacitances are calculated as parallel-plate configu-ration. The assumption is valid as long several turns areused, thus the following script is suitable as long as N1>1and N2≥10. For other cases a numerical method basedon the mirroring method [11] is used for calculating theparasitic capacitances.

Cpp = h · ε0 · ε · lwd

(9)

where h is the average width of a single turn; lw is thewinding height; d is the distance between the windingsand/or core. In Fig. 8 the core layout is given. It has to bekept in mind that h=f(R) while d =f(R). Therefore, innerand outer part of the capacitance are calculated separately,nevertheless the calculations scheme is the same.

C2

C12

C1 C2

C12

C1C2

C12

C1

a) b) c)

Fig. 6: Topologies to define transformer capacitances a) Ca; b) Cb; c)Cc.

LσRsLMRP Cd

Fig. 7: Transformer model used in the simulation.

CPC =

N1∑i=1

Cpp · (V1,i − V2,j − VCore)2 ·N1p (10)

CPS =

N1∑i=1

Cpp · (V1,i − V2,j − VCore)2 ·N1p (11)

CSC =

N2∑j=1

Cpp · (V1,i − V2,j − VCore)2 ·N2p (12)

where CPC , CPS , CSC are the Primary-to-Core, Primary-to-Secondary and Secondary-to-Core capacitance; N1 isthe number of turns on the primary winding; N1p is thenumber of parallel primary windings; N2 is the numberof turns on the secondary winding; N2p is the numberof parallel secondary windings; V1,i and V2,j are thevoltages on the i-th and j-th turn of the primary and thesecondary winding, respectively; VCore is the core voltage.As the secondary winding covers much wider area thanthe primary winding, V2,j has to be computed with care.Relevant distance definitions can be seen in Fig. 8. Voltagevalues for each case can be seen in Table II. The mainerror is caused by defining the VCore value. Based onComsol analytical formulas have been derived empirically.

The turn-to-turn capacitance is calculated with the pairof parallel wire formula. Combining it with (8) results in:

CTT =π · ε0 · ε · lw · (N − 1) ·

(1N

)2

·Np

ln(

dTT

2·rwire+

√( dTT

2·rwire)2 − 1

) (13)

TABLE II: Voltages used for analytic capacitance calculations forTRHV (dPC =4 mm and dPS =10 mm). Isolation distances areunchanged during the design process.

Vi Ca Cb Cc

CPC

V1,ii

2·N30 1

V2,j 0 0 0

VCore 0.07+0.03·N3p 0.45-0.03·N3p 0.11+0.06·N3p

CPS

V1,ii

2·N30 1

V2,j 0 j2·N4

0

VCore 0 0 0

CSC

V1,i 0 0 0

V2,j 0 j2·N4

0

VCore 0.07+0.03·N3p 0.45-0.03·N3p 0.11+0.06·N3p

dPS

CPC

CPS

CSC

dPC

dSC

dTT

h

R2in

Core

Primary

Secondary

Cor

e

a) b)

Lσ’

Lσ’’

Fig. 8: a) Sketch of top view of the windings b) Definition of thecalculation parameters.

where (1/N ) is the voltage difference per turn; (N -1) isthe number of parallel turns; dTT is the distance betweenthe turns; rwire is the wire radius; Np is the number ofparallel windings.

B. Leakage InductanceThe leakage inductance Lσ (Fig. 7) is calculated with

[10]:

Emagnetic =1

2· μ ·

∫V

H2dV =1

2· Lσ · I21 (14)

| H| = N1 · I1hL

(15)

Combining (14) and (15) yields

Lσ = μ · μ0 ·N21 · lL · dL

hL· 1

N1p(16)

where lL is the height of the winding; dL is the distancebetween two windings; hL is the apparent width of theprimary winding. In the simulations it has been assumedthat one winding is represented by a single layer, nospacing between the turns (Fig. 5 and Fig. 8). In the ana-lytic calculations only the volume covered by the primarywinding is considered as empirically studies showed therethe main magnetic energy. Same constrains apply to Lσ

as for capacitance computation.

C. Magnetizing InductanceThe magnetizing inductance LM (Fig. 7) significantly

influence the modulator operation. The non-saturated in-ductance value is determined with:

LM =μ0 · μ ·N2

2 ·NFe ·AFe

lFe(17)

where μ is the permeability of the core; N2 is the numberof turns on the secondary side; NFe is the number of

TABLE III: Transformer TRHV parasitics.

Analytical Comsol Measured

LM @ 0.01 MHz (secondary) 16.2 mH - 16.1 mH

LM @ 0.10 MHz (secondary) 9.5 mH - ≈9.0 mH

LM @ 1.00 MHz (secondary) 2.5 mH - 2.2 mH

Lσ (primary) 284.9 nH 260 nH 545 nH

C1 -115 pF -111 pF -93 pF

C2 -246 pF -209 pF -236 pF

C12 309 pF 277 pF 260 pF

Rs 2.3 mΩ - 5.7 mΩ

0.01

0.1

1

10

Indu

ctan

ce (m

H)

0 0.2 0.4 0.6 0.8 1.0Current (A)

0.1kHz1kHz10kHz100kHz1MHz10MHz

Fig. 9: L dependency of current (core VAC V140, VITROPERM 500F,din =44.5 mm, dout =85.8 mm, AFe =228mm2, N = 4).

core elements; AFe is the core area of a single coreelement; NFe·AFe is the total core area; lFe is the corelength. After saturation (17) is still valid, although withfew important changes:

LMsat =μ0 ·N2

2 ·ASat

lFe(18)

where ASat is the secondary winding core area aftersaturation (in simulations LM is used in the secondaryside, Fig. 7 and Fig. 11).

The saturation current can be calculated as

ISat =NFe ·AFe ·N2 ·BSat

LM(19)

where BSat is the saturation flux density for the material.The data sheet of the core provides the information for

calculating LM and ISat for low frequency operation.Nevertheless, the core material is highly frequency de-pending (Fig. 9) what has a large impact on the modulatorperformance. The permeability is reduced largely withhigher frequencies (Table III), consequently affecting thesaturation current and performance.

For simulations, the saturating inductor is described asL=f(I). In Fig. 9 the inductance dependency of the cur-rent for the same material as used to build the transformersis described. Same normalized dependencies have beenused in the simulations.

D. Copper LossesThe winding resistance is modelled with RS (Fig. 7).

Both winding resistances are calculated and added to-gether as a lump resistance.

RS = R1 · 1

N1p+R2 ·

(N2

N1

)2

· 1

N2p(20)

where R1 and R2 is the resistance of one primary andsecondary winding. The high frequency effects can beincluded with the approach presented in [12].

E. Core LossesThe core losses are modelled with RP (Fig. 7). In [13]

it is shown that RP dependents on the core volume andtime to saturate. Using curve fitting and pulse durationsimulations RP can be estimated. However, during pulseoperation RP has relatively low effect [10].

TRMV

V0

Vload

CMV

TRHV

(CHV )

LHV

LHC

Fig. 10: Constructed pulse compression unit (LHC is added forillustration).

IV. MEASUREMENT RESULTS

The prototype transformers are designed withNFe1=NFe2=3 V074 cores in parallel. All parameters canbe found in Fig. 11. Simulation and measurement resultsfor TRHV can be seen in Table III. As can be seen,the calculated and measured capacitances have differenceless than 20%. It was noticed that the analytical scriptfor C12 tend to result in higher capacitance values thanComsol and measurements. The leakage inductancehas considerable difference. As the simulated value isrelatively low it is assumed that the difference comesfrom the measurement error.

The modulator utilizes high voltage ceramic pulsescapacitors. As can be seen in Fig. 11, resulting inCMV =9.43 nF (six 1.6 nF 50 kV capacitors in parallel) andCHV =397 pF (two 800 pF 100 kV capacitors in series).The inductor LHV is realized as an air core inductor(LHV =47μH and CLhv≈25 pF).

As can be seen in Fig. 11 the performance of thepractical unit follows well with the computed one. Inthe modelling for TRMV 100 kHz parameters are used(LM=21 mH, LMsat=15μH, Isat=1.69 A and normal-

TABLE IV: Parameters of the transformers (measured).

TRMV TRHV

Lσ1/Lσ2 350 nH 545 nH

Rs1/Rs2 7 mΩ 6 mΩ

N1/N3 3 3

N1p/N3p 13 11

N2/N4 36 12

N2p/N4p 2 2

NFe1/NFe2 3 3

RP1/RP2 5 kΩ 1.25 kΩ

LM1/LM2 (@1 kHz) 144 mH 16.1 mH

LMsat1/LMsat2 9.8μH 3.00μH

ISat1/ISat2 (@1 kHz) 0.25 A 0.74 A

Cd1/Cd2 50 pF 95 pF

−2 0 2 4 6 8 10 12 14

−60

−40

−20

0

20

40

60

0

-20

-40

-60

20

40

60

0 4 8 12

Volta

ge (k

V)

Time (μs)−2 0 2 4 6 8 10 12 14

−1

−0.5

0

0.5

1

1.5

2

−2 0 2 4 6 8 10 12 14

−20

−10

0

10

20

30

0

-10

-20

10

20

30

0 4 8 12

Volta

ge (k

V)

Time (μs)

0

-0.5

-1.0

-1.5

0.5

1.0

1.5

0 4 8 12

Volta

ge (k

V)

Time (μs)

MeasuredSimulated

MeasuredSimulated

MeasuredSimulated

C0=

80μF

C1

=2μ

F

Cd1=

50pF

CH

V=

397p

F

CH

V=

25pFV 0

=3k

V

L 0=

500n

H

L σ1

=30

0nH

L σ2

=54

5nH

R 0=

0.01

Ω

R S1

=0.

01Ω

R P1=

5kΩ

L M1

Cd2=

95pF

L Chv=

100n

H

L HV

=47

μH

CM

V=

9.43

nF

L Cm

v=50

nH

R S2

=0.

01Ω

R P2=

1.25

kΩ

L M2

N1:N2=3:36 N3:N4=3:12

1:n1 1:n2

TRMV TRHV

Fig. 11: Full simulation circuit with measured and simulated voltages.

6070

5040302010

05 15 25 35 45

Load

Vol

tage

(kV

)

n2=3n2=4n2=5

Number of Turns on the TRMV Secondary (N2)

Fig. 12: Load voltage dependency on N2 while N1 =1...3 (N3 = 3;NFe1 = NFe2 = 3).

60

80

40

20

0 2 6 10Number of Turns on the TRHV Secondary (N4)

14 18

Load

Vol

tage

(kV

)

n1=10n1=12n1=14

Fig. 13: Load voltage dependency on N4 while N3 =1...3 (N1 = 3;NFe1 = NFe2 = 3).

ized saturation curve @100 kHz from Fig. 9). Whereasfor TRHV 1 MHz parameters are used (LM=2.0 mH,LMsat=3μH, Isat=591 A and normalized saturation curve@1 MHz from Fig. 9).

60

80

100

40

20

0 6 8 10Total Number of VAC-V074 Cores

12 14 16

Load

Vol

tage

(kV

)

N1=N3=1N1=N3=2N1=N3=3

[8,9][2,3]

[6,6] [6,9][4,3]

[4,6]

Fig. 14: Load voltage dependency on total number of core elements(NFe1 +NFe2) while n1 = 12 and n3 = 4.

80

60

120

140

100

40

20

01 3 42

k

Load

Vol

tage

(kV

)

5i

LM1 MHz10 kHz0 Hz

Fig. 15: Load voltage dependency on the core material behaviour, if allother parameters are unchanged (Fig. 11, μsim = k · μ (@1 MHz)).

V. DISCUSSION

As can be seen by investigating different designs(Fig. 12 to Fig. 15) in the considered topology the mainlimiting factors are caused by the core material. Namely,as can be seen in Fig. 9 nanocrystalline material is highly

frequency dependent. The simulations are performed intime domain and therefore the frequency dependencyis challenging to count. Good modulator performancerequires LMLMsat. As one can see in Fig. 11 TRMV

operates in a range of a few hundred kHz and TRHV inmuch higher range (>1 MHz). Consequently, inductancechange from LM to LMsat is low and slow if Fig. 9is considered. Moreover, one could see in Fig. 11 highvoltage drop after CMV caused by high LMsat resultingin a lower input voltage of the next stage.

For reducing high LMsat one could use less turns (18).In Fig. 12, Fig. 13 and Fig. 14 one can see the change inthe output voltage combined with number of turns. Withincreasing number of turns voltage increase, until a pointwhen the output voltage starts to reduce due to voltagedrop on LMsat.

Other option to reduce LMsat could be by using othermaterial, e.g. ferrites [8], [9]. However, the latter one hasmuch lower permeability values. Moreover, saturation isnot as fast as with nanocrystalline materials (@0 Hz).

In Fig. 15 theoretical voltages for improved material isplotted. In the simulations, LM (@1 MHz) values weremultiplied by a constant k to see the change of outputvoltage. The LM value itself has an impact, but muchmore important is to use the correct saturation behaviourcurve. If one could use DC material properties the outputvoltage is doubled compared to the 1 MHz case. Conse-quently, in the very high frequency range the change ofthe LM limits the performance.

As shown with a two stage topology higher volt-ages are possible if more cores and different wind-ing arrangement is used (N1/N2=2/24; N3/N4=3/9;NFe1/NFe2=8/7). If adding an extra stage evenhigher voltages are possible (N1/N2=1/7; N3/N4=1/5;N5/N6=1/3; NFe1/NFe2/NFe3=14/12/12). Neverthe-less, if to use topology described in [1] (Fig. 3a) similarvoltages are achieved (TR: AFe=2400mm2, lFe=875 mm,N1=1, N2=50, m=24 kg; LPC : NPC=8, NPC=35). InTable V are shown possible output voltages and requirednumber of V074 cores together with total core weight. Ascan be seen by using pulse transformer setup the requiredvoltage is achieved with similar iron weight as with 3stage arrangement.

VI. CONCLUSION

Plasma channel drilling is a promising method to im-prove drilling technology. In order to perform efficientdrilling compact, efficient and reliable generators arerequired. In this paper limitations of a topology utilizingsaturable transformers is investigated.

In order to better understand the modulator performance

TABLE V: Possible output voltages for optimal design.

Peak Vload Nr of V074 Weight (kg)

2 Stage 92 kV 15 14 kg

3 Stage 151 kV 38 35 kg

Pulse TR 160 kV (TR)+8 32 kg

parasitics have been evaluated. With the parasitics thelimits of the investigated modulator can be estimated. Par-ticularly, transformer capacitances and leakage inductancehave minor impact to the performance. In contrast, thedependency of the core permeability on frequency and arelatively high saturated magnetizing inductance are majorlimiting effects.

Due to the frequency dependency, transformers havelow main inductance at high frequencies. Thus, differencebetween the un- and saturated inductance is not as highas desired. Consequently, after saturation of the corethe ”switching” action is not as big as desired. Addi-tionally, the high saturated magnetizing inductance causehigh voltage drop, therefore output voltage is limited. Apossible solution would be to use 3 stages or a pulsetransformer topology combined with a pulse compressionstage (Fig. 3a).

ACKNOWLEDGMENT

The Authors would like to thank CTI for their financialsupport (10856.1 PFIW-IW) and Vacuumschmelze forproviding the magnetic cores.

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