---
title: Rompe–Weizel Spark Resistance Model
url: https://www.emergentmind.com/topics/rompe-weizel-spark-resistance-model
type: topic
---

# Rompe–Weizel Spark Resistance Model

Searching arXiv for recent and foundational papers on the Rompe–Weizel spark resistance model and related ESD arc modeling.
The Rompe–Weizel spark resistance model is a classical, energy-based description of arc-channel resistance in which the instantaneous spark resistance is determined by the time integral of discharge current squared. In the quasi-static air-gap electrostatic discharge (ESD) regime, the model provides a direct link between spark resistance, series load resistance, and energy partitioning. For discharges in ambient air across a fixed gap, recent measurements over \(R_L = 0.1\)–\(10{,}000\,\Omega\), \(C = 100\), \(700\), or \(1300\,\mathrm{pF}\), and \(l = 1.27\)–\(6.35\,\mathrm{mm}\) show that the fraction of stored energy delivered to a series victim load is largely independent of gap length within experimental uncertainty, while remaining strongly dependent on capacitance and load resistance [2509.05791].

## 1. Physical setting and circuit formulation

In the reported quasi-static ESD configuration, discharge occurs in ambient air across a fixed gap between two \(3.75\,\mathrm{cm}\) diameter graphite spheres mounted in a modified Ross Engineering SG-40-H spark gap. The gap length \(l\) is set and held constant during breakdown, with tested values \(l = 1.27\), \(2.54\), \(3.81\), \(5.08\), and \(6.35\,\mathrm{mm}\). Air pressure is approximately \(630\,\mathrm{torr}\) (\(0.83\,\mathrm{atm}\)) in Golden, Colorado; temperature and humidity are not reported. An external capacitor \(C\) is charged to the breakdown voltage \(V_b\) through a large charging resistor and then discharges through a series path comprising the spark gap and a “victim” load resistance \(R_L\), where \(R_L = R_x + R_c\) is the sum of a low-inductance series resistor and a current viewing resistor (CVR). The discharge loop has non-negligible stray inductance, with \(L_{\mathrm{OAS}} \approx 1.1 \pm 0.2\,\mu\mathrm{H}\) measured for the open-air system and \(L \approx 0.33\)–\(1.25\,\mu\mathrm{H}\) used during resistance extraction to align voltage and current zero-crossings [2509.05791].

The initial condition is a capacitor charged to \(V_b\), followed by discharge onset at \(t=0\). The stored energy is

$$
E_{\mathrm{cap}}=\tfrac{1}{2} C V_b^2.
$$

Under a uniform-field approximation with breakdown field \(E_{\mathrm{th}}\), one may write \(V_b = E_{\mathrm{th}} l\), giving

$$
E_{\mathrm{cap}}=\tfrac{1}{2} C E_{\mathrm{th}}^2 l^2.
$$

Neglecting inductance momentarily, the lumped discharge equations are

$$
V_{\mathrm{cap}}(t)=i(t)\,[R_L+R_s(t)],
$$

and

$$
\frac{dV_{\mathrm{cap}}}{dt}=-\frac{i(t)}{C}.
$$

Including loop inductance yields the corrected resistive-branch relation

$$
R_T(t)=\frac{V(t)-L\,\frac{di(t)}{dt}}{i(t)}=R_s(t)+R_L.
$$

This formulation separates the measurable total branch resistance from the time-varying spark resistance \(R_s(t)\). The load energy is

$$
E_{\mathrm{load}}=\int_0^T i(t)^2 R_L\,dt,
$$

the spark-channel energy is

$$
E_{\mathrm{spark}}=E_{\mathrm{cap}}-E_{\mathrm{load}},
$$

and the load-energy fraction is

$$
\eta=\frac{E_{\mathrm{load}}}{E_{\mathrm{cap}}}.
$$

The complementary fraction \(1-\eta\) is the spark-energy fraction.

## 2. Rompe–Weizel constitutive law

The Rompe–Weizel model is applied in this context with two core assumptions during arc formation: constant effective energy cost per electron \(\varepsilon\), treated as an empirical parameter aggregating ionization, dissociation, excitation, and heating channels in low-temperature air plasma, and constant electron mobility \(\mu\). The discharge channel is regarded as a uniform cylinder of length \(l\), cross-section \(A\), and conductivity

$$
\sigma = e\,\mu\,n_e,
$$

where \(e\) is the elementary charge and \(n_e\) is the electron density. Within this formulation, the resulting spark resistance is independent of the channel cross-sectional area and depends only on the history of the discharge current [2509.05791].

The central Rompe–Weizel relation is

$$
R_s(t)=\left[\frac{2a_R}{l^2}\int_0^t I(t')^2\,dt'\right]^{-1/2},
$$

with

$$
a_R\equiv \frac{e\,\mu}{\varepsilon}.
$$

As used in the study, \(a_R\) has units of \(\mathrm{cm}^2/(\mathrm{s}\cdot \mathrm{V}^2)\). The reported literature range in air is \(a_R \approx 0.5\)–\(2\,\mathrm{cm}^2/(\mathrm{s}\cdot \mathrm{V}^2)\), while the modern ESD study treats \(a_R\) as an empirical fit parameter [2509.05791].

Combining the same assumptions with Joule heating gives an energy-based form for spark-channel energy accumulated up to \(t_f\):

$$
E_{\mathrm{spark}}=
\sqrt{
\frac{2l^2}{a_R}
\int_0^{t_f} I^2(t)\,dt
}.
$$

The final, post-breakdown spark resistance is then

$$
R_{S\!F}=\frac{l^2}{a_R E_{\mathrm{spark}}}.
$$

If all stored energy were delivered to the spark, the minimum possible final spark resistance would be

$$
R_{S\!F,\min}=\frac{2}{a_R C E_{\mathrm{th}}^2},
\qquad
E_{\mathrm{th}}=\frac{V_b}{l}.
$$

A notable consequence is that \(R_{S\!F,\min}\) is independent of \(l\), because \(E_{\mathrm{th}} = V_b/l\) and \(E_{\mathrm{cap}} \propto l^2\). This gap-length cancellation is central to the subsequent energy-partition result.

## 3. Energy partitioning with a series load resistor

When the Rompe–Weizel resistance law is combined with the capacitor discharge equations and energy conservation, the series-loaded discharge yields a closed scaling relation between the load-energy fraction \(\eta\), the load resistance \(R_L\), and the minimum final spark resistance \(R_{S\!F,\min}\) [2509.05791]:

$$
(1-\eta)^2 = 2\,\eta\,\frac{R_{S\!F,\min}}{R_L}.
$$

The physically relevant root is

$$
\eta=
1+\frac{R_{S\!F,\min}}{R_L}
-\sqrt{
2\,\frac{R_{S\!F,\min}}{R_L}
+\left(\frac{R_{S\!F,\min}}{R_L}\right)^2
}.
$$

This relation predicts monotonic growth of \(\eta\) with \(R_L\). In the small-load limit \(R_L \ll R_{S\!F,\min}\),

$$
\eta \approx \tfrac{1}{2}\,\frac{R_L}{R_{S\!F,\min}}
\left(1-\frac{R_L}{R_{S\!F,\min}}\right),
$$

and to first order,

$$
\eta \approx \frac{R_L}{2R_{S\!F,\min}}.
$$

In the large-load limit, the reported asymptote is

$$
\eta \approx 1-\sqrt{\frac{2R_{S\!F,\min}}{R_L}}.
$$

For small loads, the model further yields a scaling collapse

$$
\frac{\eta}{C R_L}=\frac{a_R E_{\mathrm{th}}^2}{4},
$$

which demonstrates linear scaling of \(\eta\) with both \(C\) and \(R_L\). In the measured regime, this scaling is accompanied by empirically weak dependence on gap length for \(l \gtrsim 1\,\mathrm{mm}\) [2509.05791].

Two equivalent expressions connect the final spark resistance to the energy fraction:

$$
R_{S\!F}=\frac{1}{a_R C E_{\mathrm{th}}^2}
\left[
1+\sqrt{1+a_R C E_{\mathrm{th}}^2 \eta}
\right],
$$

and

$$
R_{S\!F}=\frac{R_L}{\eta}\left(\frac{1}{\eta}-1\right).
$$

The second form is independent of \(l\), which aligns with the experimental finding that \(\eta\) is largely gap-length invariant across \(1.27\)–\(6.35\,\mathrm{mm}\). A common misconception is therefore that fixed-gap ESD energy delivery to a series resistor must vary strongly with gap length; in the tested quasi-static regime, the dominant dependence is instead on \(C\), \(R_L\), and the fitted \(a_R\), not on \(l\) itself [2509.05791].

## 4. Experimental calibration and observed regimes

The experimental validation spans \(R_L = 0.1\)–\(10{,}000\,\Omega\), \(C = 100\), \(700\), and \(1300\,\mathrm{pF}\), and \(l = 1.27\)–\(6.35\,\mathrm{mm}\), with loop inductance of order \(1\,\mu\mathrm{H}\). Within experimental uncertainty, \(\eta(C,R_L)\) is effectively independent of \(l\) over this range. For small \(R_L\), \(\eta\) scales linearly with \(C\) and \(R_L\); as \(R_L\) increases, \(\eta\) rises monotonically toward unity. The crossover near \(\eta \approx 0.5\) occurs at smaller \(R_L\) for larger \(C\), consistent with the inverse dependence of \(R_{S\!F,\min}\) on \(C\). The fitted final spark resistance \(R_{S\!F}\) increases with \(R_L\) because more of the initial energy is absorbed by the load. In waveform analysis, measured \(R_s(t)\) exhibits an initial rapid drop at breakdown and then remains nearly constant during the bulk of the current pulse in an example with \(C = 700\,\mathrm{pF}\), \(R_L = 0.0983\,\Omega\), and \(l = 1.27\,\mathrm{mm}\) [2509.05791].

For discharges at \(l = 3.81\,\mathrm{mm}\), the empirically fitted Rompe–Weizel constant takes the following values:

| \(C\) | \(a_R\) |
|---|---|
| \(100\,\mathrm{pF}\) | \(0.4169\,\mathrm{cm}^2/(\mathrm{s}\cdot\mathrm{V}^2)\) |
| \(700\,\mathrm{pF}\) | \(1.8140\,\mathrm{cm}^2/(\mathrm{s}\cdot\mathrm{V}^2)\) |
| \(1300\,\mathrm{pF}\) | \(1.7730\,\mathrm{cm}^2/(\mathrm{s}\cdot\mathrm{V}^2)\) |

The study reports that \(a_R\) increases with \(l\) for the two larger capacitances, while the \(100\,\mathrm{pF}\) dataset shows a general upward trend with a slight depression at larger \(l\). All fitted values remain within the literature range cited in the study. Example traces show \(V_b \approx 11\,\mathrm{kV}\) for \(l = 3.81\,\mathrm{mm}\) and \(C = 700\,\mathrm{pF}\), while some theoretical plots use \(V_b = 4.57\,\mathrm{kV}\). The paper does not report statistical goodness-of-fit metrics or uncertainty bounds for \(a_R\) beyond plotted trends, and the weak gap-length dependence of \(\eta\) is shown through min–max bars rather than formal confidence intervals [2509.05791].

The same study includes a worked numerical example for \(l = 3.81\,\mathrm{mm}\), \(C = 700\,\mathrm{pF}\), \(V_b = 11\,\mathrm{kV}\), and \(a_R = 1.814\,\mathrm{cm}^2/(\mathrm{s}\cdot\mathrm{V}^2)\), yielding \(E_{\mathrm{cap}} \approx 0.042\,\mathrm{J}\), \(E_{\mathrm{th}} \approx 28.9\,\mathrm{kV/cm}\), and \(R_{S\!F,\min} \approx 1.9\,\mathrm{k}\Omega\). Using the closed-form \(\eta\) relation, the reported load-energy fractions are approximately \(0.02\) at \(R_L = 50\,\Omega\), \(0.178\) at \(R_L = 1{,}000\,\Omega\), and \(0.546\) at \(R_L = 10{,}000\,\Omega\), illustrating the monotonic increase of load-coupled energy with series resistance [2509.05791].

## 5. Relation to effective-resistance methods and related discharge studies

A separate spark-discharge study, "The energy distribution structure and dynamic characteristics of energy release in electrostatic discharge process" [1509.06106], investigates capacitor-stored energies at nominal \(10\,\mathrm{J}\), \(100\,\mathrm{J}\), and \(1000\,\mathrm{J}\) levels using tungsten electrodes, a Tektronix DPO7254C oscilloscope, a Tektronix P6015A high-voltage probe, and a Pearson 110A current monitor. That study evaluates wire, switch, and plasma resistances by an energy-conservation method rather than by a direct Rompe–Weizel law. The released capacitor energy is

$$
E_{\mathrm{release}}=\tfrac{1}{2} C (U_0^2-U_1^2),
$$

and the effective total resistance is

$$
R_{\mathrm{total}}=\frac{E_{\mathrm{release}}}{\int_0^{t_f} I(t)^2\,dt}.
$$

Using separate short-circuit tests,

$$
R_{\mathrm{short}}=\frac{E_{\mathrm{release,short}}}{\int_0^{t_f'} I'(t)^2\,dt},
$$

and therefore

$$
R_{\mathrm{plasma}}=R_{\mathrm{total}}-R_{\mathrm{short}}.
$$

The same framework yields

$$
E_{\mathrm{plasma}} = R_{\mathrm{plasma}} \int_0^{t_f} I(t)^2\,dt,
$$

with corresponding expressions for wire and switch energy. In that study, discharge durations increase from approximately \(0.06\,\mathrm{ms}\) at the \(10\,\mathrm{J}\) level to approximately \(0.3\,\mathrm{ms}\) at the \(1000\,\mathrm{J}\) level; peak spark power increases from approximately \(700\,\mathrm{kW}\) to approximately \(1600\,\mathrm{kW}\); \(R_{\mathrm{total}}\) decreases from approximately \(0.7\,\Omega\) to approximately \(0.082\,\Omega\); and \(R_{\mathrm{plasma}}\) decreases from approximately \(0.10\,\Omega\) to approximately \(0.012\,\Omega\) [1509.06106].

The same source reports spark energy fractions of approximately \(0.0830\), \(0.1297\), and \(0.1305\) for the \(10\,\mathrm{J}\), \(100\,\mathrm{J}\), and \(1000\,\mathrm{J}\) levels, respectively, with wire and switch fractions changing correspondingly. It argues that energy-conservation-based dynamic resistance is more precise than resistance inferred from current oscillation parameters. A related synthesis in that source maps these data to a simplified Rompe–Weizel-type resistance law,

$$
R_{\mathrm{RW}}(t)=\frac{A}{\int_0^t I(\tau)^2\,d\tau}+B,
$$

where \(A\) and \(B\) are phenomenological coefficients rather than the canonical \(a_R\) of the classical Rompe–Weizel formulation. This suggests continuity between the classical inverse-\(\int I^2 dt\) dependence and broader effective-resistance approaches in spark-discharge diagnostics [1509.06106].

By contrast, the quasi-static air-gap ESD study does not present a direct comparison to Toepler, Cassie–Mayr, or other arc models. It instead focuses on Rompe–Weizel because the model provides a simple, energy-based resistance law that links the current history directly to energy partitioning [2509.05791].

## 6. Assumptions, limitations, and engineering use

The Rompe–Weizel model, as deployed in quasi-static ESD analysis, is not a complete microphysical plasma simulation. Its constant-\(\mu\) and constant-\(\varepsilon\) assumptions are empirical closures, and prebreakdown processes, leader formation, channel expansion, and detailed thermal and kinetic evolution are not modeled explicitly. The channel is treated as a uniform cylinder, and the uniform-field relation \(V_b = E_{\mathrm{th}} l\) is used. The reported interpretation is that increasing \(l\) increases stored energy and, by implication, increases discharge channel radius so as to keep \(R_{S\!F,\min}\) independent of \(l\). This suggests that the model is most appropriate for the arc phase of quasi-static ESD rather than for highly transient, fast-rising fields, strongly nonuniform geometries, or cases with rapid parameter evolution driven by thermal and kinetic processes [2509.05791].

Inductance occupies a secondary but non-negligible role. It affects voltage–current phase and therefore instantaneous resistance extraction, but the energy-partition expressions depend on \(\int i^2 dt\) rather than explicitly on \(L\). For resistance extraction from simultaneous \(V(t)\) and \(I(t)\), the reported procedure is to use a constant \(L\) chosen so that voltage and current zero-crossings match, after which

$$
R_s(t)=R_T(t)-R_L.
$$

In time-domain circuit simulation, the reported implementation is a behavioral resistor obeying

$$
R_s(t)=\left[\frac{2a_R}{l^2}\int_0^t i(t')^2\,dt'\right]^{-1/2},
$$

with \(R_L\) in series and loop inductance retained when the geometry produces significant loop area. The initial condition is a capacitor set to \(V_b\), with the spark branch closed at \(t=0\) or triggered at \(V_b\), and the spark resistance then evolved according to the Rompe–Weizel law [2509.05791].

For design and safety analysis, the model provides direct guidance. In the small-load regime, \(\eta \approx (a_R E_{\mathrm{th}}^2/4)\, C R_L\), so reducing \(C\) and \(R_L\) reduces energy delivered to the load. In the large-load regime, \(\eta \to 1\), meaning most of the stored energy is delivered to the series element rather than to the spark channel. The reported use case is therefore the setting of design limits for sensitive electronics and energetic materials by constraining \(E_{\mathrm{load}} = \eta E_{\mathrm{cap}}\) below damage thresholds [2509.05791].

The principal empirical conclusion is narrow but important: in fixed-gap, quasi-static air ESD over the tested ranges, the Rompe–Weizel model links current-history-dependent spark resistance to energy partitioning with a closed-form law, and the resulting load-energy fraction is largely independent of gap length for \(l \ge 1\,\mathrm{mm}\). A plausible implication is that, within this regime, accurate characterization of \(a_R\), \(C\), \(V_b\), and \(R_L\) is more consequential for predicting victim-load energy than further refinement of the fixed gap length itself [2509.05791].

Source: https://www.emergentmind.com/topics/rompe-weizel-spark-resistance-model