Abstract
Silicon micro-machined accelerometers are used in automobile air bag deployment systems to sense a crash event. Identifying failure rates and predicting the lifetimes at 100,000 km, which equates to a five year lifetime, are of prime importance. Data from Motorola’s customer returned product tracking system were used to identify accelerometer field failures from the years 1997 through 2000. The Weibull failure distribution was used because of its ability to model various hazard rates. It was initially suspected that the failure rates were not constant over the lifetime based on the failure mechanisms. Maximum likelihood estimation (MLE) was the preferred method for estimating the parameters for the Weibull distribution. A method of identifying service time for suspended devices was developed and is a key element of the model since the vast majority of the devices deployed are still functional in the field. A modification of the hazard rate equation for the Weibull distribution allowed the estimate of the characteristic life based on a ppm (parts per million) level improvement for some specific failures. ReliaSoft’s Weibull++ 5.0 software was used for analysis. [Editor's Note: Weibull++ 6 is now available.]

Introduction
Motorola’s Sensor Products Division (SPD) has been supplying solid state accelerometers for more than three years. These devices contain a two chip solution consisting of a surface micro-machined “g-cell” and a CMOS mixed signal control chip. The principle application of the device is in automobile air bag deployment systems. A thorough knowledge of field performance is imperative for this safety critical application. SPD has developed a database that captures information on customer returns including failure mechanisms, manufacturing date codes and kilometer readings at the failed point. Analysis of the data helped the product team to focus on the top failures and track their progress in reducing field failures. Analysis of the failures resulted in product improvements and continued tracking of field returns validated the effectiveness of the improvements.

Analysis Method
The analysis method is described next and the glossary at the end of this paper provides more information on the reliability terms used herein.

Suspended units: To be able to make a five year (100,000 km) prediction, a critical piece of information is the “time in service” (in kilometers) of the non-failed devices, which are treated as suspended items. The number of devices was obtained from monthly shipment information for the years of interest. The distance driven for the suspended items had to be estimated. This estimation was accomplished by establishing an average distance driven for each year of 20,000 km, which translates into 1,667 km per month [See Ref. 1]. Table 1 details the calculated monthly distances traveled for the years 1997, 1998 and 1999 based on data extracted in December of 1999. Note that zero distance is logged for the last four months of 1999. It was determined from the records of returned failures and customer information that it typically took about three months from device shipment for installation into an air bag module and subsequent placement in service to occur.

Failed units: The failed unit information was taken from the customer return tracking system and a second special database maintained by the accelerometer group, which included the final failure mechanism and the distance traveled in kilometers or miles. All distances with mileage were converted to kilometers. Also included in the data were the date codes that allowed us to identify the work week and year of manufacture.
It was decided that the two-parameter Weibull failure distribution would be used based on its ability to model various hazard rates. The MLE method was the recommended procedure used to determine the Weibull parameters ( = slope and = characteristic life) [See Ref. 2]. The following attributes of the MLE method are key and become evident as the sample size becomes large enough (where “large enough” indicates >20 and MLE estimates are considered accurate for >10 failures):

• Lack of Bias: Expected value equals or is centered on target to the true parameter.
• Minimum Variation: Lower dispersion.
• Sufficiency: Estimator considers all of the statistical information available in the data.
• Consistency: Estimate asymptotically approaches the true value with larger size samples.

Discussion of Results
Distributions of the different failure modes were analyzed, their parameters estimated and plots were made using ReliaSoft’s Weibull++ 5.0 software. All failures with recorded distance were included in the analysis. The results of this analysis were detailed by failure mode into the following categories: Slope, ; Characteristic Life, ; Reliability at 100,000 km, R(t); and ppm at 100,000 km.

The reliability, R(t), was calculated using the two-parameter Weibull equation:

A prediction for the 100,000 km ppm was made for each failure mode and for the three year total. A point estimate of the quality level can be obtained through the use of the instantaneous failure rate at time t or hazard rate equation. The hazard rate equation [See Ref. 3] is defined as follows, where f(t) is the density function:

The failure modes for each year were combined and Weibull plots were made of each year. The results of all three years were plotted on one Weibull graph (Figure 1), which clearly shows the reduction in the ppm level by year. These reductions were accomplished through detailed failure analysis and corrective actions for the failure mechanisms.

Conclusions
Establishing a method for determining the service time (kilometers driven) for the suspended units provided the first step in modeling field failure rates for the accelerometer products. The product return database developed by the accelerometer team provided the key details of the failure modes and failure times. By using the reliability software, ReliaSoft’s Weibull++, the distributions of each failure mode were analyzed and the Weibull parameters, and , were estimated. The use of MLE proved to be the appropriate estimation method. The ability to make predictions of future returns is foreseen as an additional use of this analysis. Finally, we were able to provide management with timely and easily understood graphical data that showed a large decrease in field returns.

Glossary
Some of the reliability terms used in this paper include:

• Reliability Function at Time t, R(t): One minus the cumulative distribution function (cdf).
• Characteristic Life, : Time where 63.2% {exp(-1)} of devices will fail. Scale parameter for the Weibull distribution.
• Slope Parameter, : The parameter that controls the shape of the Weibull probability density function (pdf).
• Hazard Rate, h(t): Instantaneous failure rate at time t. The importance of the hazard function is that it indicates the change in the failure rate over the life of a population of devices. The rate is constant for the exponential failure distribution. For the Weibull distribution, the hazard function is decreasing for < 1, increasing for > 1 and constant when is exactly 1 (exponential distribution).
• Maximum Likelihood Estimator: The probability of the sample is written by multiplying the density function evaluated at each point. The product, containing the data points and the unknown parameters, is called the likelihood function. By finding parameter values that maximize this expression, we make the set of data observed more likely.
• Suspensions: Devices removed from the test before they fail. In the case of warranty data, the number of units still operating in the field are treated as suspensions.

References:
[1] Estimation arrived at from discussion with customers and historical evidence.
[2] Tobias, Paul A. and David C. Trindade, Applied Reliability, Van Nostrand Reinhold, 1995.
[3] Lu, Ming-Wei, “Automotive Reliability Prediction Based on Early Field Failure Warranty Data,” Quality and Reliability Engineering International, March - April 1998, p.103-108.