Energy Loss to Conductors Operated at Lineal Current Densities 10 Ma=cm: Semianalytic Model, Magnetohydrodynamic Simulations, and Experiment

We have developed a semianalytic expression for the total energy loss to a vacuum transmission-line electrode operated at high lineal current densities. (We define the lineal current density j ' B= 0 to be the current per unit electrode width, where B is the magnetic field at the electrode surface and 0 is the permeability of free space.) The expression accounts for energy loss due to Ohmic heating, magnetic diffusion, j Â B work, and the increase in the transmission line's vacuum inductance due to motion of the vacuum-electrode boundary. The sum of these four terms constitutes the Poynting fluence at the original location of the boundary. The expression assumes that (i) the current distribution in the electrode can be approximated as one-dimensional and planar; (ii) the current IðtÞ ¼ 0 for t < 0, and IðtÞ / t for t ! 0; (iii) j ' 10 MA=cm; and (iv) the current-pulse width is between 50 and 300 ns. Under these conditions we find that, to first order, the total energy lost per unit electrode-surface area is given by W t ðtÞ ¼ t B ðtÞ þ t B ðtÞ, where BðtÞ is the nominal magnetic field at the surface. The quantities , , , , , and are material constants that are determined by normalizing the expression for W t ðtÞ to the results of 1D magnetohydrodynamic MACH2 simulations. For stainless-steel electrodes operated at current densities between 0. An effective time-dependent resistance, appropriate for circuit simulations of pulsed-power accelerators, is derived from W t ðtÞ. Resistance-model predictions are compared to energy-loss measurements made with stainless-steel electrodes operated at peak lineal current densities as high as 12 MA=cm (and peak currents as high as 23 MA). The predictions are consistent with the measurements, to within experimental uncertainties. We also find that a previously used electrode-energy-loss model overpredicts the measurements by as much as an order of magnitude.

A MITL is a vacuum transmission line that operates at an electric field sufficiently high to cause electrons to be emitted from the MITL's cathode, and at a magnetic field sufficiently high to prevent most of the emitted electrons from striking the anode [12][13][14][15][16]. MITLs are commonly used in pulsed-power accelerators to transmit electromagnetic power and energy to a load [17][18][19][20].An idealized representation of a one-dimensional (1D) steady-state MITL is given by Fig. 1 [16].In the steady state, the free electron current (i.e., the electron current in the vacuum gap) flows approximately parallel to both the anode and cathode electrodes [12][13][14][15][16].The magnitude of the bound current in the anode I a is greater than the bound cathode current I k , since I a ¼ I k þ I f , where I f is the free electron current, which is commonly referred to as the electron-flow current.
Optimizing the design and performance of a pulsedpower accelerator requires an understanding of how the conductors of its MITL system perform when operated at high lineal current densities.More specifically, it is required to understand how much energy is lost to such conductors as a function of time.We define the lineal current density j ' to be the current per unit conductor width; hence j ' B= 0 where B is the magnetic field at the conductor surface and 0 is the permeability of free space.(SI units are used for equations throughout.)The lineal current density is to be distinguished from the areal PHYSICAL REVIEW SPECIAL TOPICS -ACCELERATORS AND BEAMS 11, 120401 (2008) 1098-4402=08=11(12)=120401 (14) 120401-1 Ó 2008 The American Physical Society current density; i.e., the current per unit conductor area.As suggested by Fig. 1, the lineal current densities of interest in a MITL are those at the MITL's anode and cathode surfaces, and depend on I a and I k , respectively.When a MITL is well insulated, I a $ I k [12][13][14][15][16].
The calculations presented in Refs.[17][18][19][20] suggest that a peak value of j ' on the order of 10 MA=cm, with a pulse width on the order of 100 ns, may be required to achieve z-pinch-driven thermonuclear fusion.Similar current densities may also be required for advanced equation-of-state, radiation-physics, radiation-effects, astrophysics, and other high-energy-density-physics experiments.
The total energy loss associated with the operation of a conductor at such lineal current densities has four principal components: (i) Ohmic heating of the conductor [21][22][23]; (ii) diffusion of magnetic field into the conductor [21][22][23]; (iii) j Â B work performed on the conductor [23]; and (iv) the energy loss associated with an increase in vacuum inductance due to motion of the vacuum-conductor boundary.The sum of these four terms constitutes the Poynting fluence at the original location of the boundary.
Analytic calculations of the Ohmic and magneticdiffusion losses under various conditions are presented by Knoepfel in Refs.[21,22].The Knoepfel results are applied by Singer and Hunter [23] to copper conductors operated at high lineal current densities.The Singer-Hunter calculations assume that the resistivity of copper is proportional to the temperature , which implies that the Ohmic and diffusive losses scale as B 3 .This approximation is valid at near-solid densities (9 g=cm 3 ) and temperatures 1080 C. Such temperatures correspond to peak lineal current densities 0:7 MA=cm [21,22,24].(In Appendix A, we give Knoepfel's relation for the increase in the conductor-surface temperature as a function of the peak lineal current density [21,22].) The approximation / is, however, not applicable at higher temperatures.In particular, at densities $9 g=cm 3 , the resistivity of copper is expected to be relatively constant for temperatures between 10 000 and 30 000 K, with a value on the order of 100 -cm [25,26].This is consistent with Fig. 10.23 of Ref. [21], which suggests that the resistivity of solid-density copper peaks at $100 -cm, over a similar temperature range.Hence, at lineal current densities significantly in excess of 0:7 MA=cm, the B 3 scaling assumed by the Singer-Hunter model of Ohmic and magnetic-diffusion losses is not quite applicable.
Singer and Hunter [23] also estimate the energy loss due to j Â B work performed on a copper conductor, using results presented by Knoepfel [21,22].Estimates are given in weak-and strong-magnetic-field limits, and assume that the magnetic field at the surface of the conductor has a step-function time history.Figure 1 of Ref. [23] plots the sum of Ohmic, magnetic-diffusion, and j Â B losses as a function of time.This figure is, however, not selfconsistent, since the Ohmic and diffusive components assume that the current increases as t 1=2 , whereas for the j Â B component, the current time history is assumed to be a step function.
Even though copper has a significantly lower roomtemperature resistivity than stainless steel, the conductors of MITLs in pulsed-power accelerators are often fabricated from stainless, which has high-voltage, vacuum, fabrication, and mechanical properties superior to those of copper and many other materials [27].At near-solid densities ( $ 8 g=cm 3 ) and temperatures between 4000 and 30 000 K, the resistivity of stainless steel is, like copper, expected to be relatively constant, with a value on the order of 100 -cm [26].
In this article, we develop a semianalytic model of the total energy loss to a conductor operated at peak lineal current densities as high as 10 MA=cm, for current pulse widths between 50 and 300 ns.The model does not assume that the resistivity is proportional to the temperature; but rather that for current densities and time scales at which most of the energy loss occurs, the resistivity can be approximated as a constant.The model accounts for the Ohmic, magnetic-diffusion, and j Â B energy losses in a self-consistent manner.The model also accounts for the loss associated with the vacuum-inductance increase due to motion of the vacuum-conductor boundary, a loss mechanism not considered in Refs.[21][22][23].As shown in this article, this loss can be a factor of 2 greater than the j Â B loss.[16].This figure assumes that the electromagnetic power flows in the positive-z direction.The electric-and magnetic-field vectors point in the negative-x and negative-y directions, respectively.The quantity g is the anode-cathode gap, is the thickness of the electron sheath, I a is the magnitude of the bound anode current, I f is the electron-flow current, I k is the bound cathode current, V a is the total MITL voltage, and V is the voltage at the edge of the sheath.
The semianalytic model is developed in Sec.II.In Sec.III we express the model in terms of an effective resistance that can be used in circuit simulations of the operation of pulsed-power accelerators.In Sec.IV, we normalize the expressions developed in Secs.II and III to results of numerical 1D magnetohydrodynamic (MHD) simulations of a stainless-steel conductor subjected to high lineal current densities.The simulations were performed using MACH2, a 2 1  2 D MHD simulation code [28].The simulations extend the work presented by Rosenthal and colleagues in Ref. [26], and complement simulations performed by Spielman, Chantrenne, and McDaniel [29].A description of the MACH2 model can be found in [26,28].The simulations use the electrical and thermal conductivities discussed in [30] and the equation-of-state tables presented in [31].In Sec.V we describe measurements, which were performed on the Z accelerator, of the energy loss to stainless-steel conductors [32].In Sec.VI we compare the measurements with predictions of the semianalytic model, and those of a model previously used by the pulsedpower community.Suggestions for future work are discussed in Sec.VII.
The expression we develop in Sec.III for the effective resistance of a conductor is most accurate at high lineal current densities; low current densities are discussed in Appendix A. The conductor-energy-loss model previously used by the pulsed-power community is outlined in Appendix B. The comparison in Sec.VI between measurements and predictions implicitly assumes that the currentdensity profiles of the experimental configurations described in Sec.V are 1D and planar; Appendix C examines this assumption.The comparison in Sec.VI also assumes that the time history of the current applied to the experimental hardware can be approximated as a linear ramp; this assumption is examined in Appendix D.

II. SEMIANALYTIC MODEL OF THE TOTAL ENERGY LOSS TO A CONDUCTOR
In this section we develop a semianalytic expression for the total energy loss to a conductor operated at high lineal current densities.The model makes a number of simplifying assumptions, and hence is accurate only to first order.
The total energy loss (per unit conductor-surface area) W t is assumed to be the sum of four components: where W r is the loss due to resistive (i.e., Ohmic) heating, W m is the energy of the magnetic field that has diffused into the conductor, W w is the j Â B work performed throughout the conductor, and W ÁL is the energy required by the change in vacuum inductance due to motion of the vacuum-conductor boundary.We consider the case where the total current IðtÞ applied to a conductor is a linear function of time: (2) Although there are an infinite number of possible current waveforms, we restrict our analysis in this article to such a linear ramp.This allows us to use results presented earlier by Knoepfel [21,22], and to simplify our semianalytic model.(It is straightforward to apply the approach presented herein to other power-law functions.)When Eqs. ( 2) and ( 3) are a reasonable approximation to the current time history, and the resistivity of the conductor can be approximated as a constant, then as shown by Knoepfel [21,22] where is the resistivity of the conductor and BðtÞ is the magnetic field at the conductor surface.
We do not develop analytic expressions for W w and W ÁL , which would be outside the scope of this article; we instead estimate upper bounds for these quantities.To obtain an upper bound for W w we assume a snowplow model: we assume that (i) the magnetic pressure at the conductor surface is significantly greater than that inside; (ii) material-strength effects can be neglected; (iii) the mass density of the conductor before it is compressed is approximately given by its room-temperature value; and (iv) the magnetic pressure snowplows the conductor mass as it is accelerated.Under these simplifying assumptions where In these expressions v is the characteristic inward velocity of the total accreted mass, m is the accreted mass per unit area, and 0 is the initial room-temperature mass density of the conductor.Combining Eqs. ( 2), (3), and ( 5)- (7) gives The energy loss (per unit conductor-surface area) due to the increase in vacuum inductance W ÁL is approximately given by Combining Eqs. ( 2), ( 3), ( 6), (7), and ( 9), one obtains an upper bound on W ÁL : ENERGY LOSS TO CONDUCTORS . . .Phys.Rev. ST Accel.Beams 11, 120401 (2008) 120401-3 We note that It follows from Eqs. ( 8) and ( 10) that Equations ( 4) and (12) suggest that, in general, where , , , , , and are material constants.In this article we determine the constants for a stainless-steel conductor by normalizing Eqs. ( 13) and ( 14) to the results of 1D MACH2 simulations.

III. EFFECTIVE RESISTANCE OF A CONDUCTOR SYSTEM
We develop below an expression for an effective resistance R eff of a system of conductors operated at high lineal current densities.The expression can be used to account for conductor-energy-loss effects in circuit simulations of the operation of a pulsed-power accelerator.As shown below, R eff is nonlinear since it is a function of IðtÞ.
Following Parks and Spence [33], we obtain an effective resistance by equating the time-rate-of-change of W t to an Ohmic power loss: where S is the total surface area of the system of conductors.Combining Eqs. ( 2), ( 3), (15), and ( 16), and assuming that the conductor system is cylindrically symmetric so that (where r is the distance from the symmetry axis), we obtain where Assuming that the conductor system consists of cylindrical and radial-disk conductors, we obtain from Eq. ( 18) the following: where The first term on the right-hand sides of Eqs. ( 22) and ( 23) The quantities ' i and a i are the length and radius of the ith cylindrical electrode, respectively.The quantities c j and b j are the outer and inner radii of the jth disk electrode, and # j is the angle made by the jth disk electrode with respect to the horizontal.
For the special case when ¼ 2, Eq. ( 23) becomes A similar result is obtained for the second term on the right-hand side of Eq. ( 23) when ¼ 2.

IV. 1D MHD MACH2 SIMULATIONS
We have calculated numerically the energy loss to a stainless-steel conductor operated at high-lineal-current densities by performing 1D MHD simulations using the MACH2 code [28].The simulations incorporate Lee-More-Desjarlais electrical and thermal conductivities [30] and SESAME equation-of-state tables [31].The simulations use Lagrangian coordinates, and apply a linear-ramp current pulse to a solid stainless-steel cylinder that has an initial radius of 1 cm.The initial grid spacing and time step used are 3 m and 1 ps, respectively.The simulations conserve energy to within 0.002%.When test simulations are performed with the resistivity held constant in MACH2, the simulation results agree with the predictions of the Knoepfel relation [Eq.( 4)] to within 1% (when j Â B W. A. STYGAR et al. Phys.Rev. ST Accel.Beams 11, 120401 (2008) 120401-4 work can be neglected).A more complete description of the MHD simulations will be presented in a companion article by Rosenthal and colleagues [34].Twenty MACH2 simulations were performed.These reach peak currents that range from 3.14 to 62.8 MA; hence the nominal peak lineal-current densities range from 0.5 to 10 MA=cm.(The actual peak current densities are somewhat higher than the nominal values because the cylinder is compressed slightly by the magnetic field.)The simulations reach peak current in times that range from 50 to 300 ns.Results of the simulations are summarized by Table I.
The results are used to estimate the values of the six material constants of Eqs. ( 13) and ( 14) in the following manner: The constants and are obtained from a linearized least-squares fit to Eq. ( 13) of the results listed in the first three columns of Table I.To within the uncertainties of the fit, ¼ 1=2 and ¼ 2. Assuming these values, we find that for the results listed, the average value of ¼ 3:36 Â 10 5 .Hence, Similarly, Eq. ( 14) and columns 1, 2, and 5 of Table I are used to find that Consequently, W t ¼ ð3:36 Â 10 5 Þt 1=2 B 2 ðtÞ þ ð4:47 Â 10 4 Þt 5=4 B 4 ðtÞ: Equations ( 25)-( 33) are, of course, most accurate for the rise times and peak lineal current densities considered by Table I, when the conductor is a 1-cm-radius solid stainless-steel cylinder.Table I compares the values at peak current of W r þ W m , W w þ W ÁL , and W t as predicted by the semianalytic model [Eqs.( 31)-( 33)] to those predicted by MACH2.For the 20 TABLE I. Summary of conductor-energy-loss calculations.These assume a linear-ramp current pulse is applied to a 1-cm-radius solid cylinder of stainless steel.The numerical results are obtained from 1D Lagrangian MHD simulations performed using the MACH2 code [28,30,31].The losses presented are those obtained at peak current.The expression for W w þ W ÁL given by Eq. ( 32) predicts a loss that disagrees with the numerical result for many of the conditions considered, but the disagreements occur when the magnitude of this component is significantly less than that of W r þ W m .As suggested by the last three columns, the total energy loss W t given by Eq. ( 33) agrees to first order with the simulation results.

Time to peak current (ns)
Peak nominal lineal current density (MA=cm)  4) and ( 31), we find that in contrast to its nominal room-temperature value of 72 -cm, the characteristic resistance of stainless steel at high current densities is 110 -cm.Comparing Eqs. ( 12) and ( 32), we find that for the conditions studied herein, Eq. ( 12) is always greater than Eq. ( 32), as expected.For a current pulse that rises in 300 ns and peaks at 10 MA=cm, the loss predicted by Eq. ( 12) is 41% greater than that predicted by Eq. (32).The difference between Eqs. ( 12) and (32) exceeds 41% for the other cases considered.
Table I lists only the energy losses at peak current.We caution that the semianalytic estimates of W r þ W m , W w þ W ÁL , and W t are significantly less accurate at times when the current is less than half its peak value; i.e., early in time.However, at such times the losses are much less than at peak current, and for many cases of practical interest, are significantly less important.

V. MEASUREMENTS OF ENERGY LOSS TO STAINLESS-STEEL CONDUCTORS
Measurements of the total energy loss to a system of stainless-steel conductors operated at high lineal current densities were performed on four Z-accelerator shots [32].The load for these shots was the conductor system itself, which was fielded at the center of the Z stack-MITL system.A cross-sectional view of the system is presented by Fig. 2. Two load configurations were used; these are outlined by Figs. 3 and 4 The conductor energy loss determined experimentally is estimated from the following expression: The quantity P stack is the total electrical power that flows into the stack-MITL system.The quantities L i , C i , and R i are the inductance, capacitance, and resistance of the ith component of the system.The quantities I i and V i are the current and voltage at the ith component.The last term of Eq. ( 34) accounts for MITL flow electrons that are lost to the anode in the vicinity of the system's vacuum doublepost-hole convolute [35][36][37][38][39][40][41][42].All the currents and voltages on the right-hand side of Eq. ( 34) were measured using the differential-output B-dot and D-dot monitors that are described in Refs.[43,44]; a few of the monitors are depicted by Fig. 2.
For the shots described above, the Z MITL system was magnetically insulated very early in the pulse [2,4,9].(The MITLs were designed to be insulated early, for loads of interest, to minimize the energy that is lost to electron-flow current [2,4,9].)Once insulation is established, most of the electron-flow current launched in the outer MITLs is lost in the vicinity of the convolute [37][38][39][40][41][42].The lost current is accounted for by the final term on the right-hand side of Eq. (34).For the experiments described herein, this term does not exceed 2% of the total energy delivered to the stack-MITL system.The electron-flow current that originates in the inner MITL (which is located downstream of the convolute) is negligible [2,4,9].
The term on the right-hand side of Eq. ( 34) that includes R i is summed over all the conductors located outside a 5.38-cm radius.We define this term to be that due to the energy loss to conductors operated at low lineal current densities.As discussed in Appendix A, we arbitrarily define the boundary between low and high current densities to be 0:63-0:72 MA=cm.We assume that for conductors outside a 5.38-cm radius, the sum of the Ohmic and diffusive losses is given by Eq. ( 4), and use the roomtemperature resistivity for .For conductors outside a 5.38-cm radius, the W w and W ÁL losses can be neglected.semianalytic model, when expressed in terms of the effective resistance developed in Sec.III.For the conductor system used in the experiments, the effective resistance R eff is given by the following expression: where X 1 and X 2 are given by Eqs. ( 19), (20), and ( 25)- (30).The plots of Figs.5-8 start at the extrapolated beginning of the load current, ignoring a small prepulse.The plots end 20 ns after peak current.
For each of the four shots, the total experimental uncertainty (due to random and systematic errors) in the measured value of the energy loss is estimated to be AE100 kJ; hence, the excellent agreement between theory and experiment suggested by Figs.35).The conductor system used as the load on this shot is that illustrated by Fig. 3.The plot includes a representative error bar.
useful only in demonstrating that Eq. ( 35) is consistent with experiment to zeroth order.Table II compares, for each shot, the peak value of the measured energy loss to that predicted theoretically by the semianalytic model [Eq.(35)].The table also compares the measurements to the predictions of an energy-loss model that was previously used by the pulsed-power community [45]; this model is outlined in Appendix B. Since the measurements have large uncertainties, they only validate Eq. ( 35) to zeroth order.However, the measurements exclude the possibility that the previous model [Eq.(B3)] is correct.In fact, the previous model predicts that for Z shots 507, 532, and 589, the energy loss is a factor of 1.7-2.0greater than the total energy delivered to the Z stack-MITL system; hence, it is clear that the previous model is significantly in error.
The comparisons made by Figs.5-8 and Table II implicitly assume that the 1D-planar approximation applies to the experiments.This assumption is examined in Appendix C. The comparisons also assume that the time history of the load current can be approximated by a linear ramp; this assumption is examined in Appendix D.

VII. SUGGESTIONS FOR FUTURE WORK
The analytic results developed in Secs.II and III assume that the current time history can be approximated as a linear ramp.As mentioned previously, analytic results for other power-law functions can be obtained using the procedure outlined herein.
The MHD simulations described in Sec.IV were performed using the MACH2 code [28,30,31].It would be of interest to repeat the simulations described in Sec.IV with other MHD codes, to determine whether they give comparable results.
As discussed in Sec.IV, Eqs. ( 13) and ( 14) were normalized to the results of MACH2 simulations that assume a linear-ramp current pulse is applied to a 1-cm-radius stainless-steel cylinder.If desired, one could assume a current time history that deviates slightly from a linear ramp, a different conductor geometry, and a different con-ductor material, to obtain results more relevant to a problem at hand.Doing so would, of course, result in a different set of material constants , , , , , and .
The energy-loss measurements discussed in Sec.V were performed by taking the difference of two quantities that are similar in magnitude; hence, the measured losses have a large uncertainty.Such uncertainties could be reduced by designing an experimental arrangement considerably simpler than that indicated by Fig. 2, without a double-posthole vacuum convolute, and with more current and voltage measurements performed closer to the high-lineal-currentdensity conductors under study.It would also be of interest to measure energy loss to conductors other than stainless steel, and to compare such measurements with predictions.(35).We also compare the measurements with predictions of a previously used conductor-loss model [Eq.(B3)].The measurements validate Eq. ( 35) to zeroth order.The measurements also suggest that the previous model is significantly in error, and can predict losses that are too large by as much as an order of magnitude.16)-( 23), (B1), and (B2) suggest that, for a system of cylindrical and disk-shaped electrodes, Eq. (B2) is generalized as follows:

Z-shot number
We use Eq.(B3) to predict the conductor energy loss for the experiments described in Sec.V; the predictions are listed in Table II.Equation (B3) assumes copper conductors, whereas the experiments of Sec.V used stainless steel.Since the hightemperature resistivity and room-temperature mass density of copper are comparable to those of stainless, Eqs. ( 4) and (12) suggest that if Eq. (B3) were correct, it would predict losses that are comparable to the observed values.Instead, as suggested by Table II, Eq. (B3) predicts losses that are as much as an order of magnitude greater.In fact, for Z shots 507, 532, and 589, Eq. (B3) predicts losses that are a factor of 1.7-2.0greater than the total energy delivered to the Zaccelerator stack-MITL system.Hence, it appears Eq. (B3) is significantly in error, and predicts losses that can be too large by as much as an order of magnitude.

APPENDIX C: VERIFICATION THAT THE 1D-PLANAR ASSUMPTION APPLIES TO THE EXPERIMENTS
The semianalytic model developed in Secs.II and III assumes that the current-density and magnetic-field profiles in the conductor can be approximated as onedimensional and planar.The MHD simulations discussed in Sec.IV assume that the current is carried by a 1-cmradius solid cylinder of stainless steel, since such a system is approximately 1D and planar (given the millimeter-scale resistive skin depth of our 50-300 ns ramped currents), and since such a geometry is likely to be that of most interest for future systems that drive a z-pinch load.
However, the stainless-steel electrodes of Fig. 3 include a 0.3-cm-radius conductor on axis.The radius was chosen to be this small to achieve on Z a current density on the order of 10 MA=cm.In Sec.VI, we compare measurements made with the electrodes of Fig. 3 to the predictions of the semianalytic model.To determine whether the 1Dplanar approximation is applicable when the minimum radius is 0.3 cm, we performed a 1D Lagrangian MACH2 simulation that assumes a current pulse with a peak nominal lineal current density of 12 MA=cm, and a 150-ns rise time, is applied to a 0.3-cm-radius stainless cylinder.
The results of the simulation are presented in Table III, along with the predictions of the semianalytic model.It appears that the model is, in fact, applicable to a conductor with such a small radius, under the conditions studied.However, we caution that the agreement indicated by Table III is due, in part, to compensating effects.TABLE III.Summary of conductor-energy-loss calculations performed to quantify the accuracy of the 1D-planar approximation.These assume a linear-ramp current pulse is applied to a 0.3-cm-radius solid cylinder of stainless steel.The numerical results are obtained from a 1D Lagrangian MHD simulation performed using the MACH2 code [28,30,31].The losses presented here are those obtained at peak current.As suggested below, the total energy loss W t given by Eq. ( 33) agrees to first order with the simulation result.

FIG. 1 .
FIG. 1. (Color) Idealized 1D magnetically insulated transmission line (MITL) in planar geometry[16].This figure assumes that the electromagnetic power flows in the positive-z direction.The electric-and magnetic-field vectors point in the negative-x and negative-y directions, respectively.The quantity g is the anode-cathode gap, is the thickness of the electron sheath, I a is the magnitude of the bound anode current, I f is the electron-flow current, I k is the bound cathode current, V a is the total MITL voltage, and V is the voltage at the edge of the sheath.

Figures 5 -
Figures 5-8 plot the measured energy loss as a function of time for each of the four Z-accelerator shots described in Sec.V. Also plotted is the energy loss predicted by the

FIG. 3 .
FIG. 3. (Color) Cross-sectional view of the system of stainlesssteel conductors fielded as the load on Z-accelerator-shots 507, 532, and 589.The illustration is to scale.Except for the on-axis post, the conductors are 0.3-cm thick.The outer radius of the conductors is 5.38 cm.

FIG. 7 .FIG. 8 .FIG. 6 .FIG. 5 .
FIG. 7. (Color)Comparison of the measured conductor energy loss on Z-shot 533 with the prediction of the semianalytic model, as given by Eq. (35).The conductor system used as the load on this shot is that illustrated by Fig.4.The plot includes a representative error bar.

TABLE II .
Comparison of the conductor energy loss measured on Z-shots 507, 532, 533, and 589 with predictions of the semianalytic model, as expressed by Eq.