Stochastic Modeling of Air Electrostatic Discharge Parameters

Yang Xiu, Samuel Sagan*, Advika Battini, Xiao Ma, Maxim Raginsky, and Elyse Rosenbaum University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA

*Now with IBM, Poughkeepsie, NY 12601 USA

Abstract—An automated tester is built for IEC 61000-4-2 air discharges. The relations between the parameters of the resulting waveforms are studied using stochastic modeling. The precharge voltage, peak current and rise time are interrelated, with a strong dependence on the humidity. However, there is no clear dependence of the peak current and rise time on the approach speed. Naïve Bayes method is used to predict the peak current and rise time from the precharge voltage and humidity. The likelihood that a tablet experiences a soft failure is predicted via logistic regression.

Index Terms--System-level ESD, air discharge, stochastic modeling.

I. INTRODUCTION

Prior works demonstrate that ESD-induced soft failures require a probabilistic description [1]. In this work, 960 air discharges with precharge voltage ranging from 8 kV to 15 kV were applied near the home button of a tablet that was streaming a YouTube video. Some of the discharges caused a soft failure. Two types of soft failures were observed: (1) YouTube is exited and the application has to be restarted; (2) different parts of the touch screen are activated (e.g., Share button, ‘Up next’ video, scroll bar). The incidence of soft failure is summarized in Fig. 1.

Figure 1. 960 air discharges to a tablet at precharge voltages ranging from

8 kV to 15 kV. Plot shows the fraction of ESD events that caused a soft failure at a specified precharge voltage. No failures were observed at ESD levels below 8 kV.

In [1], it is argued that soft failures are stochastic because the response of the equipment-under-test (EUT) depends on the instruction being executed at the moment the discharge occurs. For the case of air discharge, the ESD current waveform is highly variable [2, 3], as shown in Fig. 2, and this may be another reason why the system response appears to be stochastic.

There is expected to be some variability between the current pulses generated by consecutive air discharges, due to

the physics of spark formation [2, 3]. However, the variation will be heightened if the ESD gun position or, perhaps, its approach speed is not held constant between trials. The position of the gun relative to the EUT is very important; when testing the tablet, it was observed that moving the gun away from the home button and towards the screen decreased the incidence of failure. Furthermore, in some prior works, the peak discharge current is reported to be a weakly increasing function of the approach speed [4, 5, 6]. This work seeks to minimize variability by using an automated tester that will precisely control the position and approach speed of the ESD gun.

Figure 2. Twenty air discharges at 8 kV and a constant appoach speed are

applied to a tablet. The gun position relative to the home button may vary slightly between trials; humidity varies between trials performed on different days.

Thousands of waveforms are obtained using the automated tester. Using the acquired data, this work demonstrates a stochastic modeling technique to represent the waveform features, e.g. peak current and rise-time , as functions of the experimental conditions, such as the approach speed and the size of the EUT. Finally, this work investigates whether there is a deterministic relationship between features of the discharge waveform and soft failure occurrence.

II. AUTOMATED TESTER DESIGN

To improve the reproducibility of air discharge testing, an electromechanical system was developed to move the ESD gun towards an EUT at a precise speed. The system, diagramed in Fig. 3, includes features to automate air discharge testing. The ESD generator used in this work is the Noiseken ESS-B3011 with a Noiseken GT-30R ESD gun. Unlike previous automated testers [4, 7, 8], this one is built around an IEC 61000-4-2 standard testbed. Ref. [9] presents a tester that is built inside a temperature and humidity controlled chamber; although that tester is vertically oriented, similar to the one in this work, the EUT is placed on the ground plane rather than the horizontal coupling plane (HCP), which is a

significant deviation from the IEC 61000-4-2 setup. In this work, during ESD testing, the temperature and relative humidity values are monitored by a thermometer and a hydrometer, then recorded. It is observed that the temperature in the laboratory is nearly constant and the humidity is not.

Figure 3. A motor drives an adjustable gun-mounting arm up and down over

an EUT. Position sensors track the location of the arm, and a solenoid pushes the gun-trigger button on the ESD generator. An oscilloscope measures the discharge waveforms, and the system is controlled by a computer.

To perform air discharge testing, the user first calibrates the system. To do so, he or she manually guides the gun tip to the surface of the EUT using controls in the GUI running on the laptop. Next, the system raises the gun to a user-defined calibration height. The user also defines the trigger height and approach speed. The system will guide the gun downward towards the EUT at the set approach speed and will trigger the ESD generator when the gun tip is a distance equal to the trigger height above the EUT (these quantities are defined in Fig. 4). The discharge current is measured at the ESD gun tip by an F-65A current probe connected to a 3 GHz oscilloscope, and the pulse captured by the scope is automatically stored to the computer. The system can perform a user-defined array of air discharge testing without human interaction during runtime.

Figure 4. At the beginning of an experiment, the gun begins moving

downward from the calibration height (left). As it moves downward at a specified approach speed, the ESD generator is triggered at the specified trigger height (right).

The gun is mounted on a mechanical arm, and the zap point is adjustable in the plane of the tabletop. Two lockable slider joints allow the user to manually adjust this zap point. The arm extends over the ESD testbed from a low-friction slider on a vertical, t-slotted bar. The tower containing the t-slotted bar has a motor mounted on top. This motor drives the sprocket and chain system diagramed in Fig. 5 which pulls the arm and gun up and down.

Figure 5. The mechanical drive system includes a closed chain loop around a

motor-driven sprocket (red) and two idler sprockets (blue).

The system was designed to maintain high accuracy in vertical positioning while approaching the EUT at speeds up to 50 mm/s. To make precise steps, the motor’s torque must impart enough linear force on the arm to overcome friction and gravity. Notably, motor torque decreases when the motor coil currents cannot rise and fall fast enough to achieve the desired step rate, so positional accuracy is limited at high speed. Even in the worst case, 50 mm/s approach speed, this design theoretically maintains 33 µm positional accuracy during one zap. Position sensors ensure that this high accuracy will be maintained throughout all testing by eliminating drift over many zaps.

There are two photointerrupter position sensors in the control system—one above and one below the arm. The one below is manually adjusted to trip just below the point of contact between the gun and EUT. During operation, this sensor protects the EUT by stopping the motor if an EMI-induced upset at the microcontroller causes the system to move too far down over the EUT. These sorts of failures occur once in every 20 to 50 zaps at precharge levels of 12 kV to 15 kV and have been combatted with improved grounding, shielding, and elimination of ground loops.

The photointerrupter above the arm is set to the desired calibration height above the EUT surface by means of a semiautomatic procedure; the height is accurate to one full step, or 11 µm. After every zap, the system leverages this upper sensor to recalibrate itself and eliminate drift.

Since the ESD generator does not support GPIB or other communication protocols, a solenoid actuator is employed to trigger it. The control system triggers the solenoid at the point that the gun arrives at the trigger height during descent over the EUT.

Finally, scope waveforms are stored to the computer through GPIB immediately following gun contact with the EUT. To continue experimentation, the system will return to the calibration height, recalibrate to the upper position sensor, and automatically begin the next test.

Recall that the trigger height is selected by the user; measurements suggest that its value does not affect the experimental results. This claim is based on the results of an experiment in which the ESD gun was discharged to a rectangular metal sheet, positioned above and isolated from

the HCP. Multiple trials of the experiment were conducted and three parameters were varied: trigger height h, precharge voltage , and approach speed . In particular, values of 30 mm and 50 mm were used. Table 1 lists the measured values of peak current and rise time ; neither or

has an observable dependence on . It is concluded that trigger height is not a significant factor that affects the discharge waveforms, provided that its value is greater than the spark length. It is most convenient to select a value not much larger than the expected maximum spark length, in order to minimize the time needed for each experimental trial; a value of 30 mm is recommended.

Table 1. Measured values of the discharge waveforms’ and are not sensitive to changes in trigger height h. Each pair of and values at a given condition is averaged over five measurements. The measurements were taken on a single day and the relative humidity did not vary. (kV) Speed (mm/s) Height (mm) (A) (ns) 4 30 30 15.2 0.666 4 30 50 14.9 0.827 4 50 30 15.2 0.895 4 50 50 13.1 0.885 8 30 30 29.9 0.871 8 30 50 30.6 0.574 8 50 30 30.0 0.987 8 50 50 28.3 0.799

III. RESULTS

A simple EUT is used for the initial study of air discharge stochastic modeling. It consists of a 117 mm × 54 mm laminate; the top and bottom surfaces are coated with copper and those two surfaces are shorted together by conductive tape. The board is elevated 15 mm above the horizontal coupling plane (HCP) by non-conductive standoffs. The EUT is essentially floating; it is connected to the HCP by roughly 1 MΩ of resistance to bleed-off the charge between zaps.

The experimental variables are the precharge voltage , the trigger height h, the approach speed , and the relative humidity . The pre-charge voltage values, , ranged from 2 kV to 16 kV. For each set of ( , , , ), a minimum of 5 discharges were performed. For each discharge, Ipeak, , and the charge, , are extracted, as shown in Fig. 6.

A. Relationship between , and

Fig. 7 shows the relationship between Ipeak, and extracted from 3665 waveforms. The data are well

represented by the equation given in (1), where and α are fitting parameters.

= (1)

In [10], eq. (1) was proposed as an empirical equation, but it actually represents the expected response of the discharge circuit, which is modeled as shown in Fig. 8.

The spark can be represented by the Rompe-Weizel model, which describes the resistance based on the spark length [11]. The model has the form

Figure 6. Waveform metrics. is the charge delivered to the EUT (the red

area under the curve). is the peak current. Rise-time ( ) is measured from the time at which the current reaches 10% of its peak value to the time at which it rises to 90%.

Figure 7. 3665 discharges to the simple EUT. The relationship between ,

and is fit to (1).

HCP

Ltip

Rfast

Cfast_1 CEUT

Rspark

EUT

CHCP

Cslow

Rslow

Lstrap

Rfast

Cfast_2

Figure 8. Schematic used for circuit simulation of air discharge [12]. _ , _ and are precharged to the ESD voltage level. _ =

11 pF, _ = 1 pF, = 200 Ω, = 140 nH, = 150 pF, = 330 Ω, = 3 μH, and = 120 pF. For the simple EUT,

= 6.3 pF.

( ) = ( ) , (2)

where is the length of the spark, and is the discharge current through the spark. is an empirical parameter whose value lies between 0.5 m2/V2 and 1 m2/V2, and in the simulation it is set to 1 m2/V2. The simulation assumes that

once the spark forms, the discharge will not stop until the potentials across _ and are equal.

As indicated by (2), the spark resistance increases with the spark length, and a larger spark resistance decreases the peak value and increases the rise time of the discharge current pulse. In the measurements, it is observed that the peak current and rise time vary widely even when the precharge voltage is held constant. This implies that the spark length has a wide variation between air discharges. This is not unexpected; once the critical field is reached, there is a stochastic delay before the spark forms, and such delay can be on the order of milliseconds [3, 13]. Although the non-deterministic spark resistance causes the peak current and rise time measured at a fixed precharge voltage to vary, circuit analysis reveals that there remains a fixed relationship between the two parameters. In Fig. 9, the simulated peak current per kilovolt of precharge voltage is plotted versus the rise time. In the simulations, the precharge voltage is set to 15 kV and the spark length is varied between 1 mm and 10 mm. The log plot shows that the simulated results fall approximately on a straight line, similar to the measurement results in Fig. 7.

Figure 9. Simulated relationship between , and for the simple

EUT. Sim 1 assumes one precharge voltage (15 kV) and varies the spark length. Sim 2 calculates the precharge voltage given a spark length according to (3). The trend line of Sim 1 is shown as the solid blue line, and the trend line of Sim 2 is shown as the red dotted line.

A second set of simulation results are included in Fig. 9; here, it is assumed that there is one unique value of the spark length at a given precharge level, i.e., there is no variation of the spark length when the precharge level is held constant. The precharge level is derived from the breakdown field between two parallel plates, as shown below [14].

= ( + ln( )) (3)

In (3), and are parameters dependent on the composition of the gas, is the gas pressure, and is the distance between the plates. For air, = 365V/(cm ∙ Torr) and = 1.18. In this work, is taken to be one atmosphere (760 Torr). The log plot of the normalized peak current with respect to the rise time indicates that the simulated data approximately fall on a straight line, similar to the previous result. The data obtained from the two simulations can be fit to straight lines that have only slightly different slopes and y-intercepts. This implies that for air discharge, all measurements of the normalized peak current and the rise time, if plotted on a log scale, will fall close to one straight line.

It should be noted that when the spark length has a wider range of variation, there may be a deviation from the linear relation. The simulations of Fig. 9 are repeated with the spark length being allowed to vary over a wider range, from 1 mm to 50 mm; Fig. 10 shows the simulation results. The and

pairs in the log-scale plot do not strictly fall on a straight line. As increases, the curve starts to bend down, diverging from the line shown in Fig. 9. A similar downward trend can be discerned in the measurement data of Fig. 7.

Figure 10. Simulated relationship between , and for the simple

EUT. Sim 1 assumes one precharge voltage (15 kV) and varies the spark length. Sim 2 calculates the precharge voltage given a spark length according to (3). The simulations here have a wider range of the spark length than those shown in Fig. 9.

The slope and y-intercept of the line are affected by the EUT. In particular, in Fig. 8 is dependent on the size of the EUT as well as its vertical separation from the HCP. Any change to the equivalent capacitance or resistance of the discharge path alters the relationship between the peak current and rise time.

Fig. 11 illustrates the relationship between Ipeak, and for two relatively simple EUTs—two copper laminates of

different sizes. The smaller one is the simple EUT used in the previous experiments, and the larger one has dimensions 152 mm × 230 mm and is elevated 15 mm above the HCP. Fig. 12 shows the simulated waveform parameters for the larger EUT. From an examination of Fig. 9, Fig. 11 and Fig. 12, it is found that in both measurement and simulation, the magnitude of increases and the magnitude of α decreases when becomes larger.

(a) (b) Figure 11. Measured relationship between , and for (a) the simple

EUT and (b) the larger EUT. The measurements were performed on the same day with identical temperature and humidity. The best fit line for the data of 11(a) has a smaller slope than that in Fig. 7 because these data do not cover as large a range of and, notably, do not contain large values of .

Figure 12. Simulated relationship between , and for the larger

EUT. Sim 1 assumes one precharge voltage (15 kV) and varies the spark length. Sim 2 calculates the precharge voltage given a spark length according to (3). The trend line of Sim 1 is shown as the solid blue line, and the trend line of Sim 2 is shown as the red dotted line.

B. Effect on humidity

The measured and vary widely for a given set of , . The variability is related to the stochastic distribution of the delay, where delay refers to the time interval between when the critical breakdown field is reached and

when the spark is formed; the delay varies with the humidity [13]. Therefore, the air discharge waveforms depend on the humidity during the test. Fig. 13 and Fig. 14 show the measured and at two humidity levels; the ambient temperature was near constant during the experiments. It is observed that as the relative humidity increases, decreases and increases. This implies that when the humidity is higher, there is a longer spark length, or, equivalently, a shorter delay.

In the dataset of Fig. 13, is reduced significantly when the approach speed is below 10 mm/s and is 8 kV or larger; in the dataset of Fig. 14, is reduced significantly when is below 10 mm/s and is 8 kV or larger. These anomalies arise because the discharge waveform captured at a small approach speed, e.g. = 1mm/s, can be very different from one captured at a higher approach speed, as shown in Fig. 15. That observation suggests that the ESD gun and the EUT are not achieving an equal potential during the timeinterval in which ( ) is measured. This subject is addressed further in the following section.

(a) (b) Figure 13. Measured (a) and (b) of 275 discharges at various approach speeds and . The measurements were performed on a single day when the relative

humidity was 46%. The temperature during the test was 22.8 °C.

(a) (b) Figure 14. Measured (a) and (b) of 150 discharges at various approach speeds and . The measurements were performed on a single day when the relative

humidity was 21%. The temperature during the test was 22.4 °C.

(a)

(b)

Figure 15. Overlaid waveforms of five measurements at 15 kV when (a) =1mm/s and (b) = 20mm/s. The measurements are a subset of the ones used to create Fig. 13.

C. Multiple discharges

Multiple air discharges were observed to occur in some tests performed at precharge levels of 4 kV and above. The incidence of multiple discharges increases with the precharge voltage. The time interval between the discharges can be several seconds, and therefore it is not feasible to capture them all with a single trigger event at the scope. Fig. 16 shows example first and second discharges measured at 15 kV; it is observed that the two discharges have similar peak currents and rise times. Extracted parameter is the charge transferred to the EUT during the first current pulse only. It may not be a good measure of the total charge delivered. For a fixed EUT and gun position, the total charge transferred to the EUT should be a linearly increasing function of ; Fig. 17 shows that scales sub-linearly, especially at the higher precharge voltages, confirming that the charge transfer can be far from complete following the first discharge. Furthermore, it is observed that the low-speed experimental conditions that cause unusually small in Fig. 13 (or in Fig. 14) also produced small values of (see Fig. 17). However, it is not clear why the characteristics of the first discharge are dependent on the approach speed.

D. Effect of approach speed

Previous works on air discharge testing [4, 5, 6] have demonstrated an increase in as the approach speed

Figure 16. Overlaid primary and secondary discharges to the “simple EUT” at

15 kV. The approach speed was 25 mm/s.

Figure 17. Measured at various approach speeds and . The measured

and are shown in Fig. 13.

increases. In this work, measurements were performed to verify this dependency at four humidity levels. In each case, measurements were performed at five levels and with approach speeds of 10, 20, 30, 40 and 50 mm/s. The trigger height was held constant at 50 cm. For each set of , and

, five measurements were recorded. The measured and values at a given humidity and are used to fit a first-

order linear function of , such that

= + (4)

and = + . (5)

Table 2 and Table 3 show the p-values for the null hypothesis. A very small p-value supports the hypothesis that ≠ 0 or ≠ 0 and that or is linearly dependent on

. At low precharge voltages (2 kV and 4 kV), it is plausible that and are functions of . However, the overall results do not support a claim that there is a deterministic dependence of or on the approach speed at tens of millimeters per second. This conclusion is different from that in [4, 5, 6]; however, in this work, the measured and vary more widely between pulses, possibly due to the geometry of the EUT. The rectangular EUT contains sharp corners. Less irregular EUT were used in the prior works; for example, the EUT used in [4] is a stainless steel disk. In the prior works, and were found to be only weakly dependent on , and the large pulse variability obtained for the EUT used in this work could obscure a small deterministic dependency. It is anticipated that EUTs with more complex geometry will experience even greater pulse-to-pulse

variability, and the effect of approach speed will hardly be observable.

Table 2. P-values of the null hypothesis that = 0 in the linear regression of = + at different relative humidities and . is between 10 mm/s and 50 mm/s. The temperature was within the range of 22.4 °C to 22.8 °C. At each humidity level, V and v, five measurements were performed.

Humidity 2 kV 4 kV 8 kV 12 kV 15 kV 46% <0.001 0.040 0.487 0.250 0.116 32% 0.004 <0.001 0.016 0.068 0.203 21% 0.124 <0.001 0.546 0.787 0.261 17% 0.292 0.003 0.020 0.101 0.014

Table 3. P-values of the null hypothesis that = 0 in the linear regression of = + . Test conditions are the same as for Table 2. Humidity 2 kV 4 kV 8 kV 12 kV 15 kV

46% 0.034 0.014 <0.001 0.139 0.114 32% 0.168 <0.001 0.038 0.824 0.446 21% 0.151 0.003 0.043 <0.001 0.524 17% 0.132 0.085 0.190 0.801 0.131

IV. STOCHASTIC MODEL OF DISCHARGE

To model the distribution of and for arbitrary and , a generative modeling technique called naïve Bayes is used. In the naïve Bayes method, Bayes’ theorem is applied with a strong independence assumption on the input parameters. It is widely used for classification tasks, but may be modified to model continuous distributions [15]. Thanks to the independence assumption, naïve Bayes is computationally favorable and highly scalable. It is reported that naïve Bayes may generate meaningful results even if the independent assumption is violated [16].

Taking the control parameters ( , ) as features (denoted as ), and the waveform characteristics ( , ) as variables (denoted as ), Bayes’ theorem provides a way to calculate the posterior probability density,

| ( | ) = ( ) | ( | )( ) . (6)

In (6), ( ) is the prior density, which represents the marginal distribution of the variables without taking into consideration any information about the features. | ( | ) is the posterior density, which represents the conditional distribution of the variables given a set of features. | ( | ) is the likelihood function, and ( ) is the probability density of the features. The goal of our stochastic modeling endeavor is to obtain the posterior distribution of the variables given a specific set of features.

A key assumption of naïve Bayes method is the conditional independence of features given the variables, i.e.,

| ( | ) = ∏ | ( | ). (7)

This assumption simplifies the posterior calculation. Given a set of data with entries, ( , ) , kernel density estimation (KDE) [17] is employed to infer the pairwise joint densities ( , ) and the marginal ( ). To make sure that the model gives physically meaningful results, special care is needed during the KDE process to constrain the inferred distribution. For example, Ipeak, tr, and all should

be positive quantities, and must stay within the range 0, 100 . The reflection-based boundary correction method presented in [18] is used to enforce those constraints. Once ( , ) and ( ) are obtained, the feature-wise likelihood can be computed via Bayes’ theorem:

| ( | ) = ( , )( ) (8)

Finally, the overall likelihood is calculated using (6).

As illustration, the naïve Bayes method is applied to the stochastic modeling of a dataset containing extracted waveform characteristics and corresponding to among 2 kV, 4 kV, 6 kV, 8 kV, 10 kV, 12 kV, 14 kV, 16 kV and among 17%, 21%, 32%, and 46%. Although the trigger height and approach speed were experimental variables, they are not included as features in the modeling procedure based on the evidence that they are not significant factors affecting the discharge waveforms.

Figures 18-21 show the modeling process for the case of 8 kV precharge voltage and 40% relative humidity. Specifically, Fig. 18 overlays the prior density ( ) and

two conditional densities, | ( |8) and | ( |40) , where is a dummy variable representing

Ipeak, and is marginalized out by integrating the joint density of and over all . Fig. 19 presents the same distributions, except that represents tr, and is marginalized out. Fig. 20 and Fig. 21 show the marginal probability density of the resulting posterior distributions, | , ( |8,40) and | , ( |8,40).

Figure 18. Peak current marginal distribution of the prior distribution and the

conditional distribution w.r.t. each feature for the 8 kV and 40% case.

Figure 19. Rise time marginal distribution of the prior distribution and the

conditional distribution w.r.t. each feature for the 8 kV and 40 % case.

Figure 20. Peak current marginal distribution of the posterior distribution,

given an 8 kV precharge voltage and 40% relative humidity, is plotted in blue. The posterior is sampled and a normalized histogram of the peak currents of those samples is overlaid.

Figure 21. Rise time marginal distribution of the posterior distribution, given

an 8 kV precharge voltage and 40% relative humidity, is plotted in blue. The posterior is sampled and a normalized histogram of the peak currents of those samples is overlaid.

The model is validated against independently acquired measurement data with different sets of features. The posterior distribution is sampled using the Metropolis-Hastings algorithm [19], which is a Monte Carlo technique. Specifically, a Markov chain whose equilibrium distribution equals the posterior distribution is constructed. States of the Markov chain are used as samples from the posterior distribution, and the normalized histogram is compared to that derived from the validation data, with the same binning. The root-mean-square error (RMSE) between the normalized values in each histogram over every bin, given by

= ∑ ( ) − ( ) , (9)

provides a metric for prediction error. Here, is the total

number of bins, ( ) is the normalized value in bin of the

histogram for the predicted samples, and ( ) is the normalized value in bin of the histogram for the measured samples. A lower RMSE corresponds to a better prediction. The validation results for 4 different sets of features are collected in Table 4.

The predicted and measured samples for the 15 kV precharge voltage and 32% relative humidity case are shown in Fig. 22. For this specific case, 10,000 predicted datapoints and 55 measured datapoints are plotted. Most of the measured datapoints are clustered in the high-density region of the predicted datapoints. Although the predicted datapoints are

scattered over a larger region, this is expected since the number of predicted datapoints is significantly larger than the number of measured datapoints. Table 4. RMSE of the validation result calculated based on normalized

histogram with 100 bins along each dimension. (kV) (%) RMSE

15 17 0.0022 15 21 0.0022 15 32 0.0015 15 46 0.0017

Figure 22. The predicted and measured and given features of 15 kV

precharge voltage and 32 % relative humidity.

The modeling methodology described in this section can be expanded to include more features. The additional features might include the size of the EUT, the length of cable connected to the EUT, and the angle between the gun tip and the EUT.

V. TABLET SOFT FAILURES

A tablet was placed above the HCP and 960 air discharges from 8 kV to 15 kV were performed around the home button. The experimental outcomes are summarized in Fig. 23. Fig. 24 shows normalized plotted versus tr. The

correlation coefficient between log / and log ( ) is 0.289, whereas the coefficient for the simple EUT was -0.919. The discharges to the tablet do not follow the relationship in (1), as evidenced by the correlation coefficient being far from -1 (recall,α is negative). It is suspected that in many experiments there exist multiple discharges due to the complex structure of the tablet.

The data of Fig. 23 are fit to a logistic regression function

ln = , (10)

where is the probability that a soft failure occurs, is the feature vector padded with a leading 1, and represents the coefficients. The p-values for the null hypothesis with different feature sets are shown in Table 5. It is observed that for the tablet EUT, the likelihood of soft failure is insensitive to , but it is affected by and . The fitted probability

function with = , is shown in (11) and plotted in Fig. 25.

ln 1 − 1 = 4.417 − 0.179 + 0.056 (11)

It is noted that increases with increasing but decreasing . It had not been anticipated that the soft failure likelihood would be a decreasing function of the peak current or that it would be insensitive to the rise time. Further study is needed to determine if the failures are correlated with any properties of the secondary discharge pulses. Since the failure likelihood increases with , it is surmised that the total charge transferred to the EUT is an important parameter.

Figure 23. Soft failure occurrences—the application halts or the system hangs.

Failure likelihood varies across the feature space ( , ).

Figure 24. The relationship between , and for 960 discharges to

the tablet.

Table 5. (a) P-values of the null hypothesis that = 0 in the logistic regression ln 1 − 1⁄ = + .

= 0 0.029 0.323

0.659 (b) P-values of the null hypothesis that = 0 or = 0 in the logistic

regression ln 1 − 1⁄ = + + . ( , ) = 0 = 0 , 0.283 0.517 , 0.006 0.060 , 0.031 0.861 (c) P-values of the null hypothesis that = 0 , = 0 or = 0 in the

logistic regression ln 1 − 1⁄ = + + + . ( , , ) = 0 = 0 = 0 , , 0.008 0.062 1.000

Figure 25. Soft failure occurrence from 960 air discharges to the tablet and the

logistic regression model. The dots at the bottom represent experiments in which soft failures did not occur; the dots at the top are the experiments in which soft failures did occur. The plotted surface is the model.

Fig. 26 shows that the incidence of soft failure increases as the humidity becomes lower. For the simple EUT, increases as the humidity decreases, but there is no such trend in the tablet data. In addition, the tablet data do not show a clear linear dependency of or on the approach speed, similar to the finding for the simple EUT. The tablet data were collected over multiple days; the tablet is placed on the test bed at the start of a session and its placement may not be precisely the same as for the previous session. It is observed that the parameters of the discharge waveform are very sensitive to any minor repositioning of the zap point, and it is conjectured that this sensitivity results from the complex structure of the tablet.

Figure 26. 600 air discharges to a tablet. Plot shows the fraction of ESD events

that caused a soft failure at different humidity levels. The temperature remained constant.

VI. CONCLUSIONS

Well-designed commercial products have a low probability of soft failure from air electrostatic discharge. Nevertheless, the soft failures must be characterized to ascertain whether they can be tolerated, for example, to evaluate if they pose a safety hazard. An automated air discharge tester is needed for that purpose and a suitable design is presented in this work. The ESD source is a commercial, IEC 61000-4-2 ESD gun.

If the equipment under test, EUT, can be modeled simply as a capacitor, both circuit analysis and measurement data reveal that ⁄ is inversely proportional to , where

is the peak value of the first (i.e., primary) current pulse arising from the discharge event, is the precharge voltage, and is the rise time of the first current pulse. For more complicated EUT, such as a computer tablet, the high correlation between the parameters of the primary discharge is lost.

Although a deterministic relation between the values of and is found for relatively simple EUT, the values themselves are nondeterministic, even if the gun approach speed and position relative to the EUT are well controlled. This finding is attributed to the stochastic processes underlying spark formation. Relative humidity affects , in a deterministic manner for the simple EUT, with the average decreasing as humidity increases. The discharge parameters are insensitive to the gun approach speed for the EUTs used in this work.

It is not unusual for there to be multiple air discharges in a single experimental trial, especially at precharge levels above 4 kV. The time interval between the ESD current pulses is many orders of magnitude larger than the duration of a single pulse, making it infeasible to record the multiple pulses using an oscilloscope. As a result, it is difficult to measure the total amount of charge delivered to the EUT.

For a particular commercial tablet, the soft failure incidence is observed to increase as the relative humidity decreases. Since consumer electronics are not used in humidity-controlled environments, it is advisable to conduct ESD testing at low humidity levels to ensure robustness of the product. By using logistic regression, it is found that the occurrence of soft failures is independent of , and it increases as decreases and increases. These findings suggest that total charge may be a controlling parameter and/or that the upsets may be triggered by a secondary discharge.

The naïve Bayes algorithm may be used to predict characteristics of the discharge waveform based on the features of the EUT and the experimental setup. In this work, a generative model for , was formulated as a function of , .

ACKNOWLEDGEMENT

Funding for this research was provided in part by the National Science Foundation under the CAEML I/UCRC award no. CNS 16-24811.

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