Water-dielectric-breakdown relation for the design of large-area multimegavolt pulsed-power systems

W. A. Stygar, T. C. Wagoner, H. C. Ives, Z. R. Wallace, V. Anaya, J. P. Corley, M. E. Cuneo, H. C. Harjes, J. A. Lott, G. R. Mowrer, E. A. Puetz, T. A. Thompson, S. E. Tripp, J. P. VanDevender, and J. R. Woodworth Sandia National Laboratories, Albuquerque, New Mexico 87185, USA Ktech Corporation, Albuquerque, New Mexico 87123, USA EG&G, Albuquerque, New Mexico 87107, USA (Received 24 March 2006; published 26 July 2006)

A number of useful water-dielectric-breakdown relations have been presented in the literature [24 -33].The relation developed by Eilbert and Lupton [26], which presently appears to be the most commonly used for the design of water-insulated systems, suggests that the probability of water breakdown in a uniform-electric-field system is 50% when eff A 0:058 0:230: The quantity E p V p =d is the peak value in time of the spatially averaged electric field between the anode and cathode (in MV=cm), V p is the peak voltage across the electrodes, d is the distance between the anode and cathode, eff is the temporal width (in s) of the voltage pulse at 63% of peak voltage, and A is the electrode area (in cm 2 ).
Equation ( 1) is based on measurements performed by Smith and colleagues and reported in Refs.[24 -26,31], and an additional measurement by Shipman [34] which is reported in [26].These measurements were made at 7 different values of A, which range from 50 to 5520 cm 2 .For 6 of these, the peak voltage is 1:5 MV; for 3 of these, the voltage is <0:5 MV.
Equation ( 1) is similar to uniform-field waterbreakdown relations developed by the Atomic Weapons Research Establishment (AWRE) at Aldermaston, England [24,25,30,31].The AWRE relation that is presented in Refs.[30,31] can be expressed as follows: eff A 1=10 0:3: Although Eqs. ( 1) and ( 2) have the same exponent for eff , the exponents of A differ.Equation ( 2) is based on the same set of measurements as is Eq. ( 1), with the exception of the measurement by Shipman.
As is well known [31], Eqs. ( 1) and ( 2) must cease to be applicable at sufficiently large values of A. When eff is held constant, these equations predict that as A ! 1, the value of E p required for complete dielectric failure approaches 0. Similarly when E p is held constant, Eqs. ( 1) and (2) predict that as A ! 1, the time required to achieve dielectric failure approaches 0.
Consequently, Eqs. ( 1) and ( 2) must cease to be applicable to the design of a pulsed-power accelerator when the area of its water-insulated system is sufficiently large.We consider here a possible example of such a situation.We estimate that a future 1000-TW accelerator might require that the total area of the accelerator's intermediate-store capacitors be on the order of 5 10 7 cm 2 [23].Assuming that the effective pulse width of the voltage across such capacitors were to be 0:5 s, Eq. ( 1) predicts that the breakdown electric field would be 0:104 MV=cm.If the capacitors were to be designed to have a 20% safety factor, this would require that the field be limited to 0:083 MV=cm.Equation (2) predicts that breakdown would occur at 0:064 MV=cm; applying a 20% safety factor would require that the field be limited to 0:051 MV=cm.
To determine whether the peak field in such a system of capacitors would, in fact, need to be as low as suggested by either Eq. ( 1) or (2), we develop in this article a waterbreakdown relation that estimates, for a given pulse width, the minimum value of E p that would be required to complete a dielectric breakdown.Thus we adopt the approach described in [31]; i.e., we develop an empirical waterstreamer transit-time relation.The relation, which is developed in Sec.II, is based in part on ideas developed in Chapter 7c of Ref. [31], and uses data that became available after that chapter was written.
In Sec.III, we use Eq. ( 1) and the relation developed in Sec.II to develop a design criterion for large-area waterinsulated systems.As we show in Sec.III, the criterion suggests that the large-area water-insulated system considered above could be operated at a much higher electric field than predicted by either Eq. ( 1) or (2).We discuss limitations of the design criterion, and present suggestions for future work, in Sec.IV.
In Appendix A, we demonstrate how the scaling suggested by Eq. (1) might be obtained from more fundamental considerations.In Appendix B, we evaluate one of the assumptions made in Sec.II.

II. WATER-BREAKDOWN RELATION FOR LARGE-AREA SYSTEMS
In this section we develop a dielectric-breakdown relation for large-area water-insulated pulsed-power systems.We begin by making the following simplifying assumptions.
(i) We assume that the characteristic time delay delay between the application of a voltage to a water-insulated anode-cathode gap, and the completion of dielectric failure of that gap (assuming such a failure can occur), can be approximated as follows: delay stat form : (3) In this expression stat is the statistical component of the delay time; i.e., the characteristic time between the application of the voltage and the appearance of the free electrons and ions that initiate the formation of streamers in the water.We define form to be the formative component: the time required for the streamers to propagate across the gap and evolve sufficiently to produce complete dielectric failure.Equation ( 3) is commonly assumed for modeling pulsed electrical breakdown of gas-filled spark gaps [35][36][37][38][39][40]; it is also assumed for millisecond-pulse water breakdown [28,29] and vacuum-insulator flashover [41].
(ii) We assume that the area of a water-insulated system of interest is sufficiently large that the appearance of free electrons and ions necessary to initiate a breakdown occurs somewhere in the system very early in the voltage pulse.Under this condition, the statistical time delay stat can be neglected, and the breakdown time delay is dominated by its formative component: (iii) We assume that breakdown dominated by the formative component can be studied experimentally with a point-plane electrode geometry [31].
(iv) We assume that when the point in a point-plane geometry is the anode, form is less than it is when the point is the cathode.This assumption is motivated by measurements performed by VanDevender and Martin [27] and Woodworth and colleagues [42], who observe that streamers that initiate from the positive electrode travel significantly faster than negative streamers.Hence we limit the analysis in this article to point-plane measurements made with a positive enhancement.
(v) We assume that voltage pulses of interest have normalized time histories that, to a reasonable approximation, are mathematically similar.We also assume that the water is homogeneous and isotropic, and has similar dielectric properties for all systems of interest.In addition, we assume that statistical fluctuations in the formative time can be neglected.Under these conditions, positive-enhanced point-plane breakdown in water is described by at most three independent variables.We can choose these to be form , E p , and d.
{When delay is dominated by its statistical component, i.e., when stat form and delay stat , then as shown in Refs.[29,41], the resulting water-breakdown relation, when expressed in a form similar to that given by Eq. ( 1), has identical exponents for the variables eff and A. Since the time exponent of Eq. ( 1) is much larger than the area exponent, this suggests that for the parameter regime over which Eq. ( 1) is valid, form can no longer be neglected.We elaborate on this point in Appendix A. As discussed in the last paragraph of Sec.II A 1 of Ref. [41], when the formative component dominates, i.e., when stat form and delay form , the resulting waterbreakdown relation is, under a certain set of conditions, independent of A. This is the parameter regime we consider in this article; i.e., we assume that the water-streamer transit-time relation is independent of A.g (vi) We assume that the dependence of the waterstreamer relation on d is weak and can be neglected; i.e., we assume the relation depends only on form and E p .This assumption is also made by VanDevender and Martin in Ref. [27].We evaluate this assumption in Appendix B.
When the above assumptions are valid, the peak field E p required to achieve complete dielectric failure is a function only of the time over which the voltage pulse is applied to the gap.The effective width of the voltage pulse eff in water-breakdown studies is usually quoted as the width at 63% of peak voltage [24 -27,30,31,43-49]; we adopt this convention herein to be consistent with the previous work.When voltage pulses of interest have normalized time histories that are mathematically similar (as assumed above), then eff / delay form .
Measurements performed with an ideal point-plane geometry, i.e., between an infinitely field-enhanced anode point and a flat cathode with infinite extent, are of course not possible.However, a number of measurements between a significantly enhanced anode electrode and a lessenhanced cathode have been described in the literature [27,42 -46,50]; these are summarized in Table I.
Assuming that for these experiments the normalized shapes of the voltage pulses are sufficiently similar, we plot E p as a function of eff in Fig. 1.Assuming E p is a power-law function of eff , we obtain from a regression analysis the following relation: E p 0:3300:026 eff 0:135 0:009: (5 This relation is plotted in Fig. 1.The uncertainties given in Eq. ( 5) are 1 values.Hence the eff exponent of Eq. ( 5) is within 3 of the exponent of the preliminary relation developed by VanDevender and Martin [27] for positive-streamer breakdown; this relation can be expressed as Sandia National Laboratories has successfully used Eq. ( 6) for the design of several of its accelerators, including PROTO II, PBFA I, and PBFA II.
Since 1977, the pulsed-power community has also used J. C. Martin's preliminary relation for positive-streamer breakdown [31]: This relation is often used even though Martin warned of the lack of data supporting Eq. ( 7) for voltages >1 MV 070401-3 [30,31].This relation is also plotted in Fig. 1.As suggested by the figure, Eq. ( 5) is more consistent with experiment than is Eq.(7).Equation ( 7) can be obtained from Fig. 7c-1 of Ref. [31].However, according to this figure, when V p > 1 MV, positive and negative streamers have the same average velocity, which is not consistent with the measurements reported in Refs.[27,42].We also note that Chapter 7c of Ref. [31] proposes a refinement of Eq. ( 7): This expression is less consistent with the data in Table I than is Eq.(7).
In Table II we present, for each of the relations given by Eqs. ( 5)-( 8), the normalized standard deviation of the differences between the data in Table I and the associated predictions of the relation.We define the normalized standard deviation n for the first relation [Eq.( 5)] as follows: where the sum is over the 25 measurements listed in Table I.The normalized standard deviation for each of the other three relations is similarly defined.

III. DESIGN CRITERION FOR LARGE-AREA SYSTEMS
Comparing Eqs. ( 1) and ( 5), and assuming that the eff exponents of these two relations are essentially the same, we find that Eq. ( 5) should be used instead of Eq. ( 1) whenever A * 10 4 cm 2 : ( As noted previously, the largest area of the data used by Eilbert and Lupton to develop Eq. ( 1) is 5520 cm 2 [24 -26,31,34].However, we caution that Eq. ( 10) may not be entirely meaningful.The water-streamer relation given in Chapter 7c of Ref. [31] for voltages <0:5 MV differs significantly from the relation given for voltages >1 MV, which suggests that water breakdown exhibits different behavior in these two voltage regimes.Equation ( 5) is based on voltages between 1 and 4.10 MV; however, much of the data used to develop Eq. ( 1) is less than 0.5 MV [24 -26,31], so it is uncertain whether Eqs. ( 1) and ( 5) can be combined to obtain Eq. (10).
In the absence of additional measurements, we make the tentative assumption that Eqs.(1) and ( 5) are both valid for voltages in excess of 1 MV.Assuming also that a 20% safety factor should be applied to Eq. ( 5) when used to design a system with A 10 4 cm 2 , we obtain the following design criterion: Equation ( 11) is consistent with measurements conducted by Maxwell Labs on the transfer capacitor of the BLACKJACK-3 pulse generator, which demonstrate that breakdown does not occur in a water-insulated system when E p 0:330 eff 0:119 and A 5:5 10 4 cm 2 [31,47].Equation ( 11) is also consistent with indirect measurements (i.e., direct measurements supplemented with circuit modeling) conducted on the Z accelerator [48], which show that breakdown does not occur when TABLE II.For each of the water-breakdown relations given as Eqs.( 5)-( 8), we present here the normalized standard deviation n of the differences between the measurements listed in Table I and the associated predictions of the relation.For the first relation, we define n as indicated by Eq. ( 9).We define n for each of the other relations in a similar manner.

Water-breakdown relation
Normalized standard deviation n of the differences between the measurements presented in Table I and the predictions of each relation The peak electric field required to achieve complete dielectric failure E p as a function of the effective pulse width eff .Each of the 25 measurements plotted here was obtained with a significantly field-enhanced anode and a lessenhanced cathode, as described in Refs.[27,42 -46,50].This data is summarized in Table I.We define E p as V p =d, where V p is the peak voltage in time across the anode-cathode gap, and d is the length of the gap.We define eff to be the width of the voltage pulse at 63% of peak.It appears that the data are more consistent with Eq. ( 5) than Eq. ( 7).
Presently, effective pulse widths for water-insulated systems of most interest range between 0:05 and 1 s.Hence, it is unfortunate that the literature only describes one point-plane measurement with eff > 0:3 s, as indicated by Table I and Fig. 1.However, Table III suggests Eq. ( 5) is reasonably accurate when eff 0:5 s.According to assumption (ii) of Sec.II, and also Eq. ( 11), dielectric breakdown of a sufficiently large area follows the same relation as a point-plane system.If assumption (ii) and Eq. ( 11) are in fact valid, the BLACKJACK-3 data given in Table III suggests that when eff 0:5 s, point-plane breakdown occurs when E p 0:330 eff > 0:119, which is consistent with Eq. ( 5).We note that this data is less consistent with Eq. ( 7), which predicts that the BLACKJACK-3 transfer capacitor should fail at a field that is 6% below the capacitor's normal operating field of 0:150 MV=cm [31,47].
An additional measurement with eff > 0:3 s is presented in Table 7c-II of Ref. [31].This measurement was performed at A 3000 cm 2 and eff 0:75 s.Under these conditions, breakdown was observed to occur when E p 0:160 MV=cm.(The peak voltage was on the order of 1.85 MV [31].)Since the area of this measurement is less than 10 4 cm 2 , we use Eq. ( 1) to estimate that had this measurement been made at 10 4 cm 2 , the peak electric field E p at breakdown would have been 0:149 MV=cm.Hence, this suggests that when eff 0:75 s, point-plane breakdown occurs when E p 0:330 eff 0:136, in reasonable agreement with Eq. ( 5).This measurement is less consistent with Eq. ( 7), which predicts that under these conditions, the breakdown field E p would be 0:115 MV=cm.
We revisit here the water-insulated system (with A 5 10 7 cm 2 ) considered in Sec.I.According to Eq. ( 11), when eff 0:5 s, the peak electric field of such a system should be limited to 0:136 MV=cm.This is 64% higher than the 0:083 MV=cm limit suggested by Eq. (1), and 164% higher than the 0:051 MV=cm limit suggested by Eq. ( 2), assuming 20% safety factors are applied to both Eqs. ( 1) and ( 2).[As discussed above, a 20% safety factor is applied to Eq. ( 5) to arrive at Eq. ( 11).]Consequently, if Eq. ( 11) is valid, large-area water-insulated systems of interest can be operated at significantly higher electric fields than suggested by either Eq. ( 1) or (2).

A. Limitations of the design criterion
Equation ( 5) can be rewritten as where v ave is the average streamer velocity across the anode-cathode gap.Assuming streamer propagation is driven by the electric field at the streamer tips, and that for frequencies of interest the dielectric constant of water is 80, then Eq. ( 5) [and hence Eq. ( 11)] are meaningful only when where c is the speed of light in vacuum.When Eq. ( 13) is not valid, the design criterion given by Eq. ( 11) should be replaced by one of the following form: where k is a suitably defined constant.This constant would be defined so that Eq. ( 14) guarantees that streamers cannot physically cross the gap during the duration of the voltage pulse.The velocity on the right-hand-side of Eq. ( 14) is 0:11c; we note that peak (i.e., not average) water-streamerpropagation velocities as high as 0:01c have been observed by Woodworth and colleagues [42].Even when Eq. ( 13) is valid, we caution that Eq. ( 11) is only applicable for the first pulse applied to a waterinsulated system during an accelerator shot, since Eq.(11) does not account for effects due to subsequent pulses (such as might be caused by reflections) on a system's dielectric strength.We also caution that Eq. ( 11) is not necessarily TABLE III.Conditions under which dielectric breakdown of water is observed not to occur.Each of these two observations was made on a large-area ( A * 10 4 cm 2 ) water-insulated system with a nominally uniform electric field.The quantity V p is the peak voltage in time across the anode-cathode gap, d is the length of the gap, E p V p =d, and eff is the temporal width of the voltage pulse at 63% of peak.The last column assumes E p is expressed in MV=cm, and eff in s.(The Maxwell-Lab data was taken on a transfer capacitor, with coaxial electrodes that have an outer radius of 60 cm and an inner radius of 48 cm [47].The peak field E p given here for this data is that at the outer conductor, which is the anode, and has been corrected for the coaxial geometry.)The observations summarized here are consistent with the design criterion given by Eq. (11) applicable at an interface between water and a solid insulator.

B. Suggestions for future work
For the measurements listed in Table I, 1 V p 4:10 MV, 1:25 d 22 cm, and 0:011 eff 0:6 s.However, these measurements do not include all physically reasonable combinations of the variables V p , d, and eff within these ranges, but only a small subset.In addition, it is not clear how far Eq.( 11) can be extrapolated beyond these ranges.Hence Eq. ( 11) is being proposed here only as a tentative design criterion; we suggest that additional experiments be conducted over a wider parameter regime to develop a definitive criterion.Such experiments would be similar to those described in Refs.[27,31,42 -46,49,50].
In addition, we note that Eq. ( 11) is valid only when the shape of the voltage pulse in question is, to a reasonable approximation, mathematically similar to those used for the measurements presented in Table I.To generalize Eq. ( 11) for use with arbitrary pulse shapes, we propose that a relation of the following form be developed [41,49]: where is the full width of the voltage pulse at its base, and and are constants.
Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

APPENDIX A: DEVELOPING A DIELECTRIC-BREAKDOWN RELATION FROM EXPRESSIONS FOR stat AND form
In this appendix we discuss how a dielectric-breakdown relation of the form given by Eq. ( 1) can be obtained from expressions for stat and form .
According to Refs.[28,29,41], when the dielectricbreakdown time delay delay of a system is dominated by stat , the exponents of eff and A in the corresponding breakdown relation, when expressed in the form given by Eq. ( 1), are identical.(This statement assumes that the relevant size variable of the system in question is the area A. For vacuum-insulator flashover, the relevant size variable is C, the insulator circumference [41].) For water breakdown dominated by stat , under the conditions studied in Ref. [ where 5 is a constant.The case 0 corresponds to when stat dominates; the case 1 when form dominates.When 0:95 we obtain E p 0:29 eff A 0:014 5 : (A8) We have performed a multiple-regression analysis on the water-breakdown data presented in [24 -26,31], and instead of Eq. ( 1), we obtain the following relation: The uncertainties presented in Eq. (A9) are 1 values.Hence, to within 2 (the usual standard for determining whether a discrepancy is significant [51]), Eqs.(A8) and (A9) are consistent.[However, we caution that such a comparison may not be meaningful.The data upon which Eq. (A1) is based was taken with voltages <0:2 MV, and much of the data upon which Eq. (A9) is based was taken with voltages <0:5 MV.The data upon which Eq. (A2) is based was taken with voltages between 1 and 4.10 MV.
According to Chapter 7c of Ref. [31], the water-streamer relation for voltages <0:5 MV differs significantly from the relation obtained for voltages >1 MV, which suggests that water breakdown exhibits different behavior in these two voltage regimes.Hence, it is not clear we can combine Eqs.(A1) and (A2) to obtain Eq. (A8), as described above, nor that we can compare Eq. (A8) to Eq. (A9).]When neither stat nor form dominates, a dielectricbreakdown relation may be more accurately expressed in a form similar to that given by Eq. (A4) than Eq.(A8), since an equation such as Eq.(A4) more accurately accounts for contributions from both stat and form to the delay time delay .

APPENDIX B: DEPENDENCE OF THE WATER-BREAKDOWN RELATION ON THE ANODE-CATHODE GAP d
In Sec.II, we make the simplifying assumption that the water-streamer transit-time relation developed in this article is independent of the anode-cathode gap d.We evaluate this assumption below.
Table I lists 25 point-plane measurements; the gap d is available for 17 of these.When we assume E p is a function of both eff and d, and perform a multiple-regression analysis on these 17 measurements, we obtain instead of Eq. ( 5) the following relation: E p 0:3600:053 eff d 0:0300:077 0:137 0:038: (B1) At the 95% confidence level, the exponent of d is between ÿ0:136 and 0.196.Hence, we presently do not have sufficient evidence to determine whether the exponent of d differs significantly from 0. Consequently, it appears that, for the available data, we are justified in assuming that the dependence of the water-streamer transit-time relation on d can be neglected.Of course, as more data become available, this assumption should be reexamined.

5 FIG. 1 .
FIG. 1. (Color)The peak electric field required to achieve complete dielectric failure E p as a function of the effective pulse width eff .Each of the 25 measurements plotted here was obtained with a significantly field-enhanced anode and a lessenhanced cathode, as described in Refs.[27,42 -46,50].This data is summarized in TableI.We define E p as V p =d, where V p is the peak voltage in time across the anode-cathode gap, and d is the length of the gap.We define eff to be the width of the voltage pulse at 63% of peak.It appears that the data are more consistent with Eq. (5) than Eq.(7). .
28], it appears that eff / delay ), we find that in general