Formative times in electrical discharges in overvoltaged gaps are analyzed with a model having no spatial dependence and with simple assumptions about discharge channel temperature T and discharge voltage. The model treats the early temporal evolution of the discharge. Specifically, the dissipative voltage drop, V*, across the discharge is taken to be a step function of T. Thermal quasi-equilibrium is assumed in the discharge medium, and it is shown that d(In t/sub d/)//sub d/(In theta )=-1, i.e., theta t/sub d/=constant, where theta is the fractional overvoltage and t/sub d/ is the formative time lag, in agreement with measured values of t/sub d/ for much of the experimentally explored range of theta . Highly-time-resolved ( approximately 92 ps) experimental data are presented for the first 10 ns of electrical discharge initiation; these data suggest that the authors' model should provide a reasonable representation of t/sub d/ when t/sub d/>10-100 ns. >

A simple method is proposed for improving the accuracy of WKB eigenvalues by using the WKB eigenfunctions as trial functions in a variational-principle expression for the eigenvalues. The first-order eigenvalues obtained from this estimate are shown to differ from the exact values by terms of order ε6 (where ε is an appropriately defined small parameter that specifies the accuracy of the WKB approximation). For comparison, the fifth-order WKB eigenvalues also differ from the exact values by terms of order ε6. Higher-order variational eigenvalues are also defined, and the third-order estimate is shown to differ from the exact by terms of order ε10. The accuracy of these variational and WKB estimates are illustrated with a numerical example.

We compare experimental data on overdamped arc discharges with a lumped-circuit discharge model employing the Arc Welder's Ansatz (AWA). The AWA prescribes that the arc resistance varies in time as R-alpha (t) = V*/\I(t)\, where V* is a positive constant and I (t) is the discharge current. In the circuit, in addition to the time-dependent arc resistance R-alpha and a small arc inductance L-alpha, there is an external time-dependent series resistance R(o), inductance L(o), and source capacitance C; the values of R(o), L(o), and C are given. We compare the AWA theory with an observed arc current pulse I (t) by using the given values of R(o), L(o), and C and normalizing the theory to one point on the data curve; an adequate fit to the data is obtained. We also compare data for the dynamic arc resistance R-alpha (t) with the AWA prescription and with other available theories for R-alpha, which also use a one-point normalization. The AWA form for R-alpha compares favorably with the other theories. Next, we show that many theories for the arc resistance near threshold (where the current I --> 0+ as t --> 0+) can be reduced to the form R-alpha is-proportional-to 1/I(P), p > 0, and we argue that for proper threshold behavior, one must have p = 1. The AWA theory meets this criterion; most other theories do not. Although all theories of R-alpha are subject to questions regarding the onset of the arc, we conclude that our AWA theory-in its present form-accounts for overdamped arc discharge data after onset at least as well as other existing theories.

Electromagnetic radiation from narrow-bore electrical discharges is analyzed with emphasis on the asymptotic behavior at high frequencies, f. This part of the radiated spectrum is dominated by discontinuities in the discharge current and its derivatives. When a discontinuity in the current is present, the envelope of the radiated field falls off with large f as f -1 for almost all observation angles. If the current is continuous but has discontinuities in its first time derivative, the envelope falls off like f - (1 + 1/n) (where n is an integer ≥ 2) for a range of observation angles. If the current and its derivatives are continuous, then the fields fall off at least as fast as f -2 . To some extent, the presence of discontinuities is model-dependent. However, the above frequency dependencies persist over a large range of high frequency even when current variations are rapid but not precisely abrupt.

The existence of thresholds for electrical discharge onset suggests a functional relation between macroscopic resistivity and current. At low current, the resistivity should be inversely proportional to the magnitude of the current. Macroscopic models which employ this scaling predict many empirically observed properties of transient electrical discharges such as: (i) thresholds for the onset of current, (ii) the abrupt termination of current in active regions of a current channel, (iii) current restart in passive regions of current channels, (iv) leaders, and (v) residual charge, both in channels and at sources when current terminates. An overview of research with these models is presented and examples are used to illustrate the results that have been obtained. These models are shown to predict current channel formation and describe results of efforts to benchmark theory with experimental data. >

Experimental data for three 1.7-m-long transient discharges are compared with an AWA lumped-circuit discharge model developed earlier by the authors (R.T. Robiscoe, A. Kadish, and W.B. Maier, III, J. Appl. Phys. vol.64, 1988) in which the arc resistance is taken from the 'arc welder's ansatz', R/sub a/=V*/ mod I mod , where V* is a positive constant and I is the discharge current. In addition to the arc resistance, a small series resistance R is present in the external circuit. A single value for each R and V* is deduced from the data, and these values are used to characterize all three discharges. Adequate agreement of the experimental data with the model is obtained; for example, it is possible to predict the proper number of current reversals for each discharge and abrupt termination of current flow after a finite time. It is suggested that the AWA lumped-circuit model provides a better representation of the data than a standard lumped-circuit RLC model and hence is more useful as a tool for prediction and interpretation of discharge.< >

A model for freely propagating transient electrical discharges, such as lightning and punch-through arcs, is developed in this paper. We describe the electromagnetic fields by Maxwell’s equations and we represent the interaction of electric fields with the medium to produce current by ∂J/∂t=ω2(E−E*Ĵ)/4π, where ω and E* are parameters characteristic of the medium, J≡current density, and Ĵ≡J/‖J‖. We illustrate the properties of this model for small-diameter, guided, cylindrically symmetric discharges. Analytic, numerical, and approximate solutions are given for special cases. The model describes, in a new and comprehensive fashion, certain macroscopic discharge properties, such as threshold behavior, quenching and reignition, path tortuosity, discharge termination with nonzero charge density remaining along the discharge path, and other experimentally observed discharge phenomena. Fields, current densities, and charge densities are quantitatively determined from given boundary and initial conditions. We suggest that many macroscopic discharge properties are properly explained by the model as electromagnetic phenomena, and we discuss extensions of the model to include chemistry, principally ionization and recombination.

The nonlinear evolution of transient electrical discharges initiated from a small charge spot on dielectric surfaces is analyzed with a transmission line model. The relation between the resistance per unit length, R̂, and the current, I, is assumed to be given by a local Arc Welder’s Ansatz: R̂‖I‖=E*, where E* is a positive constant. Comparison is made with a similar study of discharges initiated from a large charge spot. While both studies predict conditions under which charge is, or is not, transported to a dielectric edge, significant differences in the two cases are revealed. For example, if charge is not transported to an edge, current reversal is possible if the charge spot is small, but can only be unidirectional if the spot is large.

The nonlinear dynamics of charge transport due to an electric discharge on a dielectric surface is analyzed using a transmission line model. The relation between the resistance per unit length, R̂, and the current, I, is assumed to be given by the local arc-welder’s ansatz, R̂‖I‖=E*, where E* is a positive constant. The model predicts that a discharge initiated in the vicinity of a charge spot can propagate partway down a current channel and abruptly terminate before transporting charge to the dielectric edge. This behavior is similar to leader phenomena observed in lightning and other electrical discharges. We show that the direction of the current along the current channel is constant throughout such a discharge. The minimum voltage at the charge spot that allows charge to be transported to the dielectric edge is determined. This critical voltage Vl depends on the length l of the current channel. We show that the ‘‘average field,’’ Vl/l, decreases as l increases. When the charge spot voltage is less than the critical voltage, we obtain upper and lower bounds for both the arc duration time and amount of charge removed from the charge spot.

The space-time dynamics of surface flashover discharges is studied using a nonlinear one-dimensional transmission line model. When the current I is not zero, the relation between the resistance per unit length, R̂, and I is assumed to be given by a local arc welder’s ansatz, R̂‖I‖=E*, where E* is a constant. The model predicts a threshold for discharge, and abrupt local termination and spontaneous restart of the discharge current. If at a place on the discharge path it happens that the charge gradient fails to exceed the threshold condition when the current vanishes, then the current will abruptly terminate there. However, if a discharge current flows in a region adjacent to one where the current has terminated, the edge of the current-free region can be ‘‘ignited,’’ resulting in the ‘‘active’’ region encroaching on the ‘‘quiet’’ one. A formula for the speed of encroachment is derived. Formulas are also derived for current pulse waveforms and the charge transported during the discharge.

Summary Form only given, as follows. Charge transport in a surface flashover arc has been analyzed using a transmission line model. The generator is a charge spot whose capacitance is taken to be large compared to the capacitance of the charge path. Resistance per unit length, R, is assumed to satisfy a local arc welder's ansatz, R mod I mod =E.*, where I is the local current and E* is a constant with the dimensions volts per meter. If charge flow is in the interval 0or=E(1+x/sub 1//L)/2, charge will not arrive at the far end of the line. The current will first abruptly terminate everywhere on the line. For configurations of different lengths but fixed x/sub 1/, the inequality shows that, for a flashover to occur, the minimum value of V/sub 0//L is a decreasing function of the path length, L. The inequality can also be used to show that if the initial charge on the line is localized near the generator, then, for a flashover to occur, the initial potential gradients must be very large compared to E*.<>

Summary Form only given, as follows. The dynamics of flashover arc discharges is studied using a transmission line model in which the resistance per unit length, R, and the current, I, are assumed to satisfy the local arc welder's ansatz R mod I mod =E*, where E* is a constant that has the dimensions of an electronic field. The model predicts a local threshold condition for onset of current flow: mod delta V/ delta x mod >E*, where V(x,t) is the electric potential across the transmission line at position x and time t. This condition can be interpreted as a condition for stripping of charge by electric fields parallel to the arc path. The model also predicts abrupt termination of arc current in some parts of the transmission line, while other parts remain active. In general, passive regions where current flow has terminated remain electrically charged. Current in the active regions can either quench or generate restart of current in a neighboring passive region.<>

An electrical breakdown on a highly charged dielectric surface can result in a discharge along the surface, i.e., a flashover arc. We construct a simple circuit model for such an arc: the discharge of a capacitor C (related to the initial charged area) through a series inductor L and resistor R (related to the arc considered as a plasma). The arc current assumes a very simple form over most of its dynamic range, and such measured arc quantities as total charge transport, pulse width, peak current, and rise time are easily calculated. Moreover, straightforward a priori estimates of C, L, and R values give calculated arc quantities in good agreement with observation, for both typical magnitudes and areal scaling. We also analyze the effect on areal scaling of allowing the arc resistance R to ‘‘switch’’ during the evolution of the arc, from a small value characteristic of the arc plasma to a large value characteristic of the dielectric surface. Finally, we consider some aspects of the electromagnetic radiation generated by the arc.

Electrical breakdown of highly charged insulating systems can result in an arc discharge, i.e., a sudden, intense pulse of current. We model such arcs by a simple circuit: the discharge of a capacitor C (related to the initial charge reservoir) through a series inductor L and resistor R. For R=V*/‖Ia‖, where V* is a positive constant and Ia is the arc current, an essentially arbitrary dependence for L=L(Ia), a constant capacitance, and a circuit starting voltage V0, we establish four remarkable results for the subsequent arc discharge: (1) no discharge occurs at all unless ‖V0‖>V*; (2) if n is the largest non-negative integer for which ‖V0‖≥(2n+1)V*, then the arc current will reverse sign precisely n times and will decline in amplitude by 2V* at each extreme; (3) the discharge stops abruptly at a final voltage Vf=(−1)n+1[V0−(n+1)2V* sgn V0]; (4) maxima and minima in Ia occur at voltages V=±V*. Results (1) and (3) provide the threshold condition and finite final potential necessary for any realistic arc discharge theory, while result (2) suggests an experiment to look for a finite number of current oscillations in a highly driven arc. Result (4) suggests an experimental method for determining V*. Finally, the empirical areal scaling laws for arcs are reproduced with this model. The usual phenomenological treatments of arc start and stop voltages, current ringing, and areal scaling are thus modeled by a single parameter, V*. These results are generalized to voltage-dependent capacitance, C(V).

We analyze the nature of radiation signals detected from electrical arcing events originating on highly charged dielectric surfaces. Starting from a simple model of the arc current pulse, we calculate the electromagnetic radiation signal incident on a distant radio-frequency receiver. The receiver is tuned to some central frequency and operates in a given bandwidth. Because the receiver responds to the arc current derivative rather than the arc current itself, the detected radiation pulse profile can differ substantially from the current pulse profile, even when the receiver bandwidth is large compared to the reciprocal pulse width. This signal distortion significantly affects the interpretation of arc radiation data. To illustrate the point, we show that at high frequencies the detected pulse width carries very little information on the arc itself, and that even a scan of pulse width versus frequency may provide a misleading picture of possible arc dynamics. Finally, we argue that arc radiation data from radio-frequency measurements are most useful at low frequencies, when the receiver frequency and bandwidth are both comparable to the reciprocal arc duration.

We develop a theory describing the propagation of an electromagnetic pulse propagating in one dimension through a medium of essentially arbitrary conductivity and polarizability. An integral solution for the pulse amplitude is obtained by means of the slowly varying envelope approximation. Within the limits of this approximation (which requires linear coupling of the pulse to the medium, smooth variation of the pulse envelope, and pulse widths large enough to contain many cycles of the pulse carrier frequency), our solution is quite general, and it allows relatively easy comparison of the evolution of various pulse shapes as they travel through the medium. We apply the theory to an analysis of a pulse propagating in a lossy plasma, and show that among pulses with similar initial widths and energies, the initial pulse shape can have significant effects on the subsequent pulse distortion, broadening, and energy transport. Specifically, for a lossy plasma, we find that the pulse energy transport is enhanced for pulses which are initially well localized about a central maximum.

We develop a theory describing the propagation of a pulse of electromagnetic energy through a gas in which breakdown is occurring (by means of the pulse-induced electron cascade). The theory is based on solutions to a model wave equation for the pulse electric field, which incorporates the electron cascade in an approximate fashion, and which is derived from the appropriate Maxwell equations coupling the pulse fields to the cascading electron current. Nonlinear effects are ignored, but solutions to the model wave equation appear useful in identifying major trends in the pulse propagation, such as the pulse attenuation and its dependence on the input pulse nominal frequency, spectral breadth, width, rise time, and overall shape. Our results are of relatively simple analytical form, they semiquantitatively confirm available data, and they imply that the pulse energy transport through the electron cascade can be enhanced by choice of certain combinations of the input pulse parameters.

We calculate explicit corrections to the Weisskopf-Wigner laws governing the spontaneous radiative transition 2P → 1S in a non-relativistic two level hydrogenlike atom. Deviations from the usual exponential decay and lorentzian emission spectrum prove to be unobservably small, even at the sub-ppm detection levels characterizing experiments searching for atomic parity violations.