Predicted Reliability and Optimal Procurement
of an Integrated Circuit
ESE 499: Systems Design Project
Fall 2006
Pankaj Chhabra Student Washington University 
Gary L. Crawford
(advisor) The Boeing Company Washington University Adjunct Professor 
Introduction
My project has two stages:
The circuit I analyzed is meant for use in
an automobile. It alerts the driver to power supply failure.
This circuit schematic provides only partial information about the
electronic components, meaning that only a rough prediction can be made
regarding circuit reliability. These predictions are made using
the most common attributes held by these components. I analyzed
the circuit's reliability in the automobile environment (the circuit's
intended environment) and in commercial airplanes and space flight
(nonintended environments). MILHDBK217F contains models for predicting electronic component failure rates. Specific component failure rates are found my multiplying a base failure rate by a number of factors, called "pi factors." Base failure rates are obtained depending on component characteristics, while pi factors are dependent upon variables such as temperature, power rating, and environment. For example, the failure rate of a resistor can be determined from the following formula: 

where the Greek letter lambda stands for failure rate and pi stands for the aforementioned pi factors. λ_{p} is therefore the part failure rate, while λ_{b} is the base failure rate. π_{T} is the temperature factor, π_{P} is the power factor, π_{S} is the power stress factor, π_{Q} is the quality factor and π_{E} is the environment factor. The value for each of the pi factors can be looked up in a table after computing relevant parameters, such as part stress and junction temperature, for example. A great deal of the work for this project came from computing parameters in order to find the pi factors.
The failure rate is related to another important measure called Mean Time Between Failures (MTBF). MTBF is related to the failure rate by the following equation:
MTBF = 1/λ.
When determining the reliability of a circuit, one wishes for a low failure rate, and similarly a high MTBF. Both MTBF and failure rate allow for a direct determination of circuit reliability. Reliability fits the exponential model, and depends on failure rate in the following way.
R = e^{λt}
Similarly, reliability depends on MTBF in the following way:
R = e^{} ^{t/MTBF}
Failure rates (or MTBFs) of component parts are simply summed in order to find the failure rate (or MTBF) of the circuit as a whole. It is important to note that the failure rates, and MTBF values, determined are in terms of operating hours only, and do not included downtime. As mentioned previously, I analyzed the integrated circuit over three environmental conditions.
Reliability Prediction
Since the circuit is meant for use in a car, it should first be tested in such an environment. In MILHDBK217F terms, this environment is called "ground mobile." I found the failure rates for the circuit at a lowbound temperature (0°C) and a highbound temperature (75°C). The failure rates are given in the following table.
Component 
Failure Rate (0°C) 
Failure Rate (75°C) 
Average Value 
10k resistor 
0.0742 
0.139 
0.1066 
220k variable resistor 
0.0239 
0.0445 
0.0342 
22 μF, 16V capacitor 
0.2212 
3.24 
1.7306 
10k variable resistor 
0.0596 
0.1113 
0.08545 
2N3055 transistor 
0.0343 
0.0343 
0.0343 
12V lamp 
12.8304 
12.8304 
12.8304 
ua741 operational amplifier 
0.2 
5.6 
2.9 
pushbutton switch 
4.14 
4.14 
4.14 
Total failure rate (failures per 10^{6} hours) 
17.5836 
26.1395 
21.86155 
The average failure rate found is 21.9 failures per 10^{6} hours. This failure rate is extremely low. Therefore, the reliability can be seen graphically, using the exponential model of reliability, as an exponential decay curve that looks very much like a straight line.
The circuit may also be reliable if used in a commercial airplane. In MILHDBK217F terms, this environment is called "airborne, inhabited, cargo." The failure rates are found at the same lowerbound and upperbound temperatures as in the automobile case. The failure rates for this circuit in the commercial airplane environment are summarized in the following table.
Component 
Failure Rate (0°C) 
Failure Rate (75°C) 
Average Value 
10k resistor 
0.0835 
0.1748 
0.12915 
220k variable resistor 
0.0268 
0.05596 
0.04138 
22 μF, 16V capacitor 
0.1327 
3.28 
1.70635 
10k variable resistor 
0.06705 
0.13997 
0.10351 
2N3055 transistor 
0.0589 
0.0589 
0.0589 
12V lamp 
17.1068 
17.1068 
17.1068 
ua741 operational amplifier 
0.2 
14 
7.1 
pushbutton switch 
2.3 
2.3 
2.3 
Total failure rate (failures per 10^{6} hours) 
19.97575 
37.11643 
28.54609 
The average failure rate for the commercial airplane condition is 28.5. failures per 10^{6} hours. This failure rate is low, as in the case of an automobile, but since airplanes operate for a greater portion of the day than automobiles, the circuit is not reliable enough for use in such an environment. The reliability can be seen graphically, using the exponential model of reliability.
Space Flight
The space flight condition is known as the "space flight" environment, simply, in MILHDBK217F terms. The temperature lowbound in this case is still 0°C, but its high bound is now 150°C. The failure rates for this circuit can be seen in the following table.
Component 
Failure Rate (0°C) 
Failure Rate (150°C) 
Average Value 
10k resistor 
0.0023 
0.00776 
0.00503 
220k variable resistor 
0.0007438 
0.00248 
0.0016119 
22 μF, 16V capacitor 
0.00553 
1.08 
0.542765 
10k variable resistor 
0.00186 
0.00621 
0.004035 
2N3055 transistor 
0.0019 
0.0019 
0.0019 
12V lamp 
2.99 
2.99 
2.99 
ua741 operational amplifier 
0.2 
15120 
7560.1 
pushbutton switch 
0.115 
0.115 
0.115 
Total failure rate (failures per 10^{6} hours) 
3.3173338 
15124.20335 
7563.760342 
Notice that the high temperature failure rate for this circuit is extremely high. This high value suggests that the circuit would probably not be functional after takeoff of a space mission. Nevertheless, the reliability curve for this circuit under space flight conditions can be seen to follow the exponential model.
Optimal Procurement
Using vendor data sheets for specific models of the components needed to build the integrated circuit, I determined the failure rates for all component offerings. I also found pricing information for these components at Digikey. I then ran a binary integer program to determine which specific components I should buy if I wanted to build the circuit. The objective function for this linear program simply minimizes pricetoMTBF ratio, while the constraints make sure that I only procure one model of each component. I ran my linear program using the CPLEX 300 solver in MPL. The following table shows the information used in making the procurement decision, along with whether the specific component is a highreliability, lowcost part.
Part 
Manufacturer 
Unit Price 
Average MTBF 
Price/Average MTBF 
Buy? 
2n3055 transistor 
ON Semiconductor 
1.05 
28.0112 
0.037485 
Yes 

STMicroelectronics 
1.68 
28.0112 
0.059976 
No 
12V lamp 
JKL Components 
1.14 
0.114548 
9.9522 
Yes 

NKK Switches 
1.92 
0.114548 
16.7616 
No 
10k variable resistor 
Bourns 
27.34 
100.5025 
0.272033 
No 

Bourns 
2.61 
95.34706 
0.02737368 
No 

Panasonic 
0.73 
95.34706 
0.00765624 
No 

Panasonic 
0.41 
100.5025 
0.0040795 
Yes 
220k variable resistor 
Panasonic 
0.63 
9.534706 
0.0660744 
No 

Murata 
0.2 
19.19754 
0.010418 
Yes 
10k resistor 
Yageo 
0.05 
1072.731 
0.00004661 
Yes 

Yageo 
0.05 
28.26456 
0.001769 
No 
pushbutton switch 
Judco 
1.06 
0.241546 
4.3884 
Yes 

Lumex 
3.21 
0.187688 
17.10288 
No 
capacitor 
TDK 
1.55 
3.625159 
0.4275675 
No 

Panasonic 
0.86 
3.625159 
0.237231 
Yes 
operational amplifier 
Texas Instruments 
0.4 
6.904647 
0.057932 
Yes 

National Semiconductor 
0.84 
6.543218 
0.1283772 
No 

STMicroelectronics 
0.21 
6.217745 
0.1302723 
No 
Summary
This circuit is reliable only in the automobile environment, its intended application. Also, components may be purchased that balance cost considerations and meet high reliability standards. This project properly reflects the work of a reliability engineer, one of the subspecialties of systems engineering that is not often covered.
This project was chosen mainly due to my interest in reliability, which was thanks in great part to my project advisor, Gary L. Crawford. For those of you interested in reliability, I suggest taking Reliability and Quality Control (ESE 405), taught in the Fall 2006 semester by Mr. Crawford.