PB2017.05 ESD Spark Behavior and Modeling for Geometries Having Spark Lengths Greater Than the Value Predicted by Paschens Law

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Abstract—The insulation of plastic enclosures provides protection against direct electrostatic discharge (ESD) discharges to the system inside. However, seams between plastic parts are often unavoidable. To increase the voltage at which an ESD will penetrate the structure of the seam can be modified. Four plastic arrangements are constructed to investigate the spark length and current derivatives and to understand the ESD spark behavior for geometries having spark lengths longer than the values predicted by Paschen’s law. A two to threefold increase of spark lengths was found for sparks guided by plastic surfaces compared to spark length expected from the Paschen value at the same voltage level. In spite of the longer path, a faster spark development is observed for sparks along the plastic surface. Plastic arrangements that provide detour and fold-back paths hardly reduced the total spark length. No significant effects of the plastic materials or the polarity were observed. The spark length increased as the (absolute humidity) Absolute humidity (AH) increased, and the current derivative decreased by about 20% as the spark length increased with (relative humidity) Relative humidity (RH) changing from 9% to 65% at 29 °C. The spark resistance is modeled by a modified Rompe and Weizel’s law, which distinguishes the spark development in the air and along the plastic surface.

Index Terms—Electrostatic discharge, modeling, spark.


One method to achieve ESD robust electronic system design is to prevent sparking into the system. This can be achieved by a sufficiently insulating barrier. While even 0.3-mm plastic is usually not penetrated by ESD up to 25 kV difficulties are introduced by seams between plastic parts and openings that expose the inner circuits. The design choice can influence the tightness of adjacent plastic parts, the plastic materials, the wall thickness, and the distance to the electronics. Further the designer can shape the seams of adjacent plastic parts such that detours lengthen the spark path, or introduce fold-back structures that force the spark to develop against the electrostatic field direction. This article analyzes the effect of such design choices and provides an improved simulation model for the spark resistance which also includes situations in which the spark develops parallel to plastic surfaces.

A good starting point is to remind ourselves of Paschen’s law. It describes the relation between the static breakdown voltage as a function of the gap distance for homogeneous fields,

where U is the voltage in kilovolt (kV) and d is the distance in centimeter (cm).

However, a few practical electrode arrangements offer a homogeneous field. They differ in the following aspects:

1) Electrodes, such as Printed circuit board (PCBs) may have sharp tips or edges.

2) The spark path maybe guided along plastic surfaces as the spark penetrates between adjacent plastic parts of the enclosure.

3) The IEC 61000-4-2 test standard asks to approach the (device under test) Device under test (DUT) while attempting to discharge to it, thus, this leads to a change of the electric field strength.

It is generally known that sharp electrodes and paths parallel to insulating surfaces increase the length the spark can bridge. Our study has shown that for the voltage range relevant to ESD these geometric factors can increase the length by twofold or more relative to the value predicted by Paschen’s law. For approaching electrodes the opposite can happen: the discharge occurs at distances less than the value predicted by the Paschen’s law. This discrepancy is explained by a delay of the onset of the spark due to the statistical time lag while the gap is closing due to the approach velocity. This phenomenon leads to spark lengths below the Paschen value, reduced rise times and larger peak values. For approaching electrodes, the spark resistance behavior and its simulation is well documented in the literature [1], [2]. However, the lack of information on the spark behavior for spark lengths longer than the values predicted by Paschen’s law requires more investigation. Spark lengths longer than Paschen’s value are certainly more likely for voltages exceeding 8 kV as the effect of sharp edges is more pronounced at higher voltages. Further plastic surfaces, metallic edges, sur face contaminations, and humidity will reduce the statistical time lag. With reduced statistical time lag it becomes unlikely to experience shortening of the spark length due to the interplay of the approaching speed and statistical time lag. To our knowledge there is no comprehensive study of the spark behavior for plastic enclosures in a voltage range relevant to ESD.

The main questions addressed in this contribution are: What is the distance that a spark will bridge if it is guided by insulating surfaces of different shapes? How large are the current derivatives? And how can the current rise be simulated for this type of geometries?

This knowledge can guide the designer in selecting gap shapes and geometries that maximize the voltage needed to breakdown through a gap into an enclosure. If the breakdown cannot be prevented, the simulation models will allow us estimating the peak current derivative, which has been shown to be strongly related to soft-failures in products [5].


A. Experimental Setup

The experimental setup allows us controlling and capturing the following parameters.

1) The charge voltage is set by the operator. The setup is limited to 25 kV. The ground of the high voltage supply is connected to the ground plane, which forms the transmission line’s return path. The ESD current target is also grounded, thus the voltage is applied between the tip and the ESD current target.

2) The electrode shape and the geometry of the plastic arrangement is set to allow us straight, detour, and reversing spark paths, see Fig. 3.

3) Four different types of plastic materials have been used.

4) Testing is performed inside a climate chamber. The climate chamber does not allow us changing the air pressure. However, it has been shown that the value predicted by Paschen’s law is proportional to the air pressure for typical values on earth’s surface, i.e., it is reasonable that the results for spark lengths longer than the Paschen’s value will scale proportionally with air pressure. The testing has been done at a height of 300 m above sea level at about 40% RH and 23 °C.

5) The setup measures the discharge current and the electrode distance in the moment of the spark. The discharge current is measured via a current sensor as described in IEC 61000-4-2.

6) For measuring the spark length the moving electrode is attached to a slider via insulating fiberglass, see Figs. 1 and 2. The slider’s position is captured using a resistive position sensor. The discharge current triggers an S/H circuit that records the position of the moving electrode in the moment of the discharge.

A 7.5 m, 165 Ω transmission line was selected as discharging structure to provide a long enough square pulse. A value of 165 Ω is used as a compromise between the impedance of 266 Ω derived from the peak current definition of the IEC 61000-4- 2 standard (3.75 A/kV), charged cable discharges and lower impedance seen for the discharge of body worn equipment [3].

If no plastic arrangement or a straight path arrangement [see Fig. 3(a) and (b)] is used then the spark length equals the distance between the moving electrode and the current sensor. If arrangements (c) or (d) are used then the extra length within the plastic arrangement needs to be added to the electrode distance to obtain the total length of the spark path.

In plastic enclosures different types of interfaces are used. Some are folded back to increase the length of the gap and to force the spark to propagate against the electrostatic field. This approach increases the voltage needed for breakdown. To reproduce a set of different interfaces, the plastic surface has been machined to be smooth and pressed tightly to reproduce the situation encountered on products. One might expect that a spark cannot penetrate the gap between two plastic parts that are pressed together. However, it is known that holes as small as 10 µm will allow us a spark to penetrate through a seam, thus machining the surfaces on a milling machine does not prevent sparking. The four different plastic arrangements used for this investigation are shown in Fig. 3. They differ in the path style and path length. Arrangement (a) and (b) offer a straight spark paths parallel to the electrostatic field. Arrangement (a) used 3.2 mm thick plastic parts, while arrangement (b) uses 6.4 mm thick plastic parts. Arrangement (c) is created by offsetting two stacked 3.2 mm plastic parts. This leads to a detour for the spark, in which it partially travels perpendicular to the electrostatic field. Arrangement (d) forces the spark to travel against the electrostatic field. The total path that the spark travels along the plastic surface is 7.3 mm for arrangement (d). To obtain the total spark length, the section the spark bridges between the electrode and the plastic arrangement needs to be added.

If the voltage is above the minimal breakdown voltage the spark will partially travel guided by the plastic, and then bridge the section from the plastic surface to the rounded electrode in air. The length ratio between these two sections of the total spark length, depends on the plastic arrangement and the voltage. This experiment used a flat electrode (from the ESD current target) directly behind the plastic arrangement and the air discharge ESD generator tip as moving electrode. During the experiment the voltage was set and the electrode was approached towards the plastic arrangement. The approach speed was less than 10 mm/sec. While this speed is much less than the typical speed during ESD testing, it was observed that increasing the approach speed did not affect the results strongly for experiments that included a plastic arrangement. This indicates that the plastic arrangements lead to short statistical time lags. A short statistical time lag will lead to a breakdown at the moment the gap distance is reduced to a length that allows us a breakdown [1], [6].

The current sensor captures the current waveform leading to a system bandwidth (scope + cables + target) of about 3 GHz. Besides the spark length and the waveform, the peak current derivative is analyzed. This parameter has been selected as it has been shown that the peak current derivative often correlates to soft-failure thresholds on electronic systems [5].

B. Spark Length

The experiments involved setting the plastic arrangement, the charge voltage, and then approaching the electrodes. Sparking occurred above a voltage determined by the plastic arrangement. The current and the electrode distance at the moment of the sparking were measured and analyzed. Fig. 4 shows the spark lengths for different arrangements. Additionally, spark lengths according to Paschen’s law are included as a reference. A second reference shown in Fig. 4, is the discharge distance between two razor blades, arranged at 90o. The spark lengths obtained without plastic arrangement are very close to the Paschen’s values. This behavior is expected, as the arrangement is similar to the requirement for Paschen’s law, thus this result can be seen as indication for the correctness of the voltage setting and spark length measurement.

On the other extreme, Fig. 4 shows the measured breakdown distances for the razor blade setup. Here, additional measures were needed to reduce corona at the corners of the blades. The blades have been encapsulated in rounded electrodes for all regions except the region in which the sparking occurs. In spite of these measures, corona occurring at the sparking location (the center of the blades front edge) prevented the measurement for voltages exceeding 10 kV in the razor blade arrangement.

Using 3.2 mm thick plastic [see Fig. 3(a)] the spark values increased by a factor of about 2 compared to the Paschen value. For example for 10 kV, the values increased from 2.8 to 6–6.3 mm. Results using the 6.4-mm thick plastic [see Fig. 3(b)] indicate a further increase of the distance the spark can bridge. For example the value at 10 kV increased to 6.5–7 mm.

Intuitively one may expect that forcing the spark to propagate perpendicular or even against the electrostatic field would significantly decrease the distance a spark can bridge. However, the data does not support this hypothesis. The reverse arrangement having a plastic guided path of 7.3 mm, the spark length is 9.8–10.5 mm. This falls into a similar range as the arrangement (b) having a straight plastic guided path of 6.4 mm. The distance between the ESD current target and the air discharge tip is reduced because of the detour and the fold-back.

The seemingly counterintuitive observation that even a spark path against the electrostatic field will only marginally reduce the bridged distance, can be resolved if one considers that the developing spark modifies the local field as the streamer advances as an electrode [6].

While the underlying physical processes may not be fully understood the results clearly show a strong increase of the spark length if the spark is guided by plastic surfaces. This indicates that thicker plastic walls, or detour and reverse arrangements may not achieve the expected result of preventing sparking.

C. Effect of the Plastic Material

Table I presents data on the effect of selecting different plastic materials investigated for 15 kV using arrangement (b). The data are typical for other voltages and arrangements and give evidence that the selection of the plastic material does not strongly influence the sparking behavior. However, an important effect common to all plastic materials, is that the plastic arrangements significantly reduce the rise time and increase the peak current derivative of the discharge current. The results repeated well, see Fig. 5. This indicates that effects of possible surface changes or charge accumulation over repeated testing do not influence the results significantly. This is probably a result of having short pulses that transfer charges of less than 5 µC.

Further evidence to this effect is given by a direct comparison of the discharge currents presented in Fig. 5. (The waveforms are obtained using an RC discharge network instead of the transmission line structure.) Similar discharge currents have been observed for different plastic materials indicating that the discharge currents are independent of the plastic material. This might be explained by the fact that all plastic materials investigated have a somewhat similar relative permittivity in the range of 2.4–3.7. The data shown in Fig. 5 are supported by measurements at from –18 to 15 kV. The second, more pronounced effect is a faster spark development for the cases in which the spark is along a plastic surface. While the reason for the faster spark development has not been clarified in this study, the data clearly give evidence that the current derivative is increased, thus, the likelihood of damage or upset by ESD may be increased by spark paths along plastic surfaces. However, it was observed that the total distance the spark needed to bridge, has been increased by at least twofold.

We did not observe a strong effect of the polarity. This can be explained by the rather symmetric setup used in this investigation. One electrode is formed by the flat ESD current sensor while the other electrode is formed by the rounded ESD simulator air discharge tip.

D. Current Derivative

As discussed in the introduction section it is known that approaching electrodes can lead to spark lengths much shorter than the value predicted by Paschen’s law. During the delayed onset of the spark, the electrodes continue to approach, which increases the field strength in the gap. Once the discharge is initiated, the spark resistance will drop faster due to the increased field strength. This will also result into large peak current derivatives that may reach values of 1000 A/ns or more [1]. However, a long statistical time lag is required for voltages exceeding 8 kV to reach such high current derivative values. This requires a quasi-homogeneous field, clean electrodes and dry air. Most spark gap topologies encountered during air discharge mode ESD testing on products do not fulfill these conditions, i.e., the statistical time lag will be short. Throughout all our measurements, the peak current derivative remained in the range of 3–10 A/ns. Care must be taken to generalize these numbers as they will certainly be influenced by the source impedance of the discharge arrangement which was set to 165 Ω. However, the values will remain much lower than the values published for spark lengths shorter than the Paschen’s value [1]. Thus, one can use this value range to estimate the current derivatives and consequently induced voltages during ESD testing.

After having shown that neither polarity nor the plastic material selection strongly influenced the spark behavior the attention is moved to the effect of the plastic arrangement on the current derivative. The scatter plot presented in Fig. 6 details the effect of the spark length on the current derivative for different voltages and plastic arrangements. Data points on the left side of Fig. 6 shows spark lengths equal to the Paschen’s length (no plastic). The peak current derivatives slightly reduced as the voltage was increased from 10 to 15 kV. The middle section from 6–8 mm spark length of Fig. 6 shows results for plastic arrangement (a) that allows us a straight spark [see Fig. 3(a)] guided by 3.2-mm plastic. Although the spark length has increased from about 2.8 to 6.2 mm for 10 kV, the peak current derivative increased on average by 2 A/ns. This again indicates that the plastic surface not only allows us the spark to bridge larger distances, but also confirms that the spark develops faster in spite of its longer length. The right section of the plot presents the data for the detour and the counter field arrangements [see Fig. 3(c) and (d)]. The data is only shown for 15 kV, as not all arrangement showed a breakdown at lower voltages. In spite of spark lengths of more than 10 mm (about 3× the Paschen’s value for 15 kV) the spark development led to peak current derivatives in the range of 5.5–8.5 A/ns.

In summary, we conclude that the peak current derivatives for spark lengths longer than the values predicted by Paschen’s are in the range of 3–10 A/ns. The plastic guided spark can bridge up to 3 times the distance predicted by the Paschen’s value. In spite of these longer distances, the spark develops faster, leading to moderately increased peak current derivatives.

E. Influence of the Humidity

An ESD is affected by humidity by three mechanisms: in high humidity, the tribo-chargining is reduced [8]–[10], the conductivity of many materials is increased, leading to a faster charge decay, and the statistical time lag is strongly reduced [1]. The measurements, shown in Fig. 7, were conducted in a climate chamber where the environmental conditions can be varied between a relative humidity of 9% to 65% in a temperature range of 24 °C to 29 °C. The 6.4-mm plastic [see Fig. 3(b)] is used for the presented data at the right of the figure. For sparks guided by the plastic surface, the overall effect of the humidity on the current derivative is not strong. The spark length increases from 9.1–9.3 to 10.5–11.1 mm at 15 kV as the AH increases from 0.003 to 0.019 kg/m3. The data at 12 and 15 kV both show that by increasing AH, the peak current derivative decreases as the spark length increases. Overall, humidity has a minor effect on the experimental results.


It has been shown that the spark resistance law from Rompe and Weizel predicts the maximal current derivative over a large range of voltages and spark lengths in the range of 1.5–25 kV [1], [2], [11],

Based on the measured discharges in air while trying to achieve a good match between simulation and measurement, the spark constant KR is typically selected in the range of 0.5–1e–4 m2/(V2 s). However, the shorter rise times observed for sparks guided by plastic, (shown in Figs. 5 and 6), indicate that the spark develops faster if it is guided by plastic surfaces. To maintain a good match between simulation and measurements the value of KR needs to be modified if the spark is guided by a plastic surface. As seen in Fig. 8, a value of about 4e–4 m2/(V2 s) is most suitable for predicting the spark resistance drop for plastic guided sparks.

A second aspect that needs to be considered is that the initial spark originating from an ESD simulator tip, may be partially in air, until it reaches the plastic surface. Thus, one needs to consider both the spark distance in air and the spark distance along the plastic surface. Rompe and Weizel’s spark resistance law needs to be modified to simulate these cases. This is achieved by introducing a weighting function:

where p is the portion of the spark along plastic, Kplastic is the constant for the section parallel to the plastic and Kair is the constant needed to describe the spark development for the section in the air. The best match was achieved when Kair = 0.7e − 4 m2 /(V2 s) is used to model the section not guided by plastic, and Kplastic = 5e − 4 m2 /(V2 s) is used for the plastic guided distance.

To apply the modified law one should take the following steps:

1) Know the maximal voltage one wants to protect for.

2) Know the length of the section that is parallel to the plastic. This can be obtained from the mechanical drawings.

3) Estimate the total length of the spark, based, e.g., on information shown in Fig. 3.

4) Subtract the plastic guided path length from the total spark length. This provides the length of the section of the spark which is in air.

Using this information the both sections of the spark path are calculated and the modified law can be applied.

Using this modified spark resistance law, discharge waveforms have been simulated and compared to measurements, (for a constant path length along the plastic, but varying path lengths in the air [see Fig. 9]). This is achieved by increasing the voltage beyond the minimal value required to spark along the plastic. If the voltage exceeds this minimum limit, the distance of the section in air will increase. In the experiment the spark length section in air was measured using the setup shown in Fig. 1.

The peak current derivative is an important parameter as it often determines the peak induced voltage. The peak current derivatives and associated spark lengths are shown in Table II.

The results shown in Fig. 9 and Table II indicate that the modified law can predict the current waveform and especially the peak current derivative for sparks that are partially guided along a plastic surface.


This experimental investigation into spark distances, current rise times, and the modeling of the spark resistance showed that sparks guided along plastic surfaces can bridge distances two to three times longer than the distances predicted by Paschen’s law. The introduction of detour paths or paths against the electrostatic field did not increase the voltage needed to bridge a spark.

In spite of the longer spark path, a strongly reduced rise time was observed for sparks along plastic surfaces comparing discharges at the same voltage. This accelerated spark development requires a modification in Rompe and Weizel’s spark resistance law to allow us predicting the peak current derivative for arrangements in which the spark partially is guided by a plastic surface. Further the results indicate that using different types of plastic materials hardly affected the measured currents or sparking distance.

Increasing RH resulted in a longer spark and a slightly smaller current derivative. However, humidity was not a strongly influencing factor.

It is suggested that the seam structure of the plastic enclosure for the electronic system be carefully designed to increase the voltage at which ESD will penetrate into the system by providing detour paths or paths against the electrostatic field to allow us placing the electronics closer to the enclosure.