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How do you know how long your design is going to last? Is there any way we can predict how long it will work? Why do Reliability Engineers get paid so much? 2 3 Definitions Random Experiment Event or Outcomes Event Space 4 What is an axiom? 2 of the 3 axioms 5 What is a random variable (RV)? What is a PDF? Math Definition 6 Normal Density pX(x) 1 fX (x ) e 2 1 x 2 2 x 7 Uniform Density 1 fX (x ) , b a a x b pX(x) 1/(b-a) x a b 8 Data for lifelengths of batteries (in hundreds of hours) 0.406 0.685 4.778 1.725 8.223 2.343 1.401 1.507 0.294 2.230 0.538 0.234 4.025 3.323 2.920 5.088 1.458 1.064 0.774 0.761 5.587 0.517 3.246 2.330 1.064 2.563 0.511 2.782 6.426 0.836 0.023 0.225 1.514 3.214 3.810 3.334 2.325 0.333 7.514 0.968 3.491 2.921 1.624 0.334 4.490 1.267 1.702 2.634 1.849 0.186 9 Exponential Density fX (x ) e x , x, 0 10 Example: Based on the data for lifelengths of batteries previously given, the random variable X representing the lifelength has associated with it an exponential density function with = 0.5. (a) Find the probability that the lifelength of a particular battery is less than 200 or greater than 400 hours. (b) Find the probability that a battery lasts more than 300 hours given that it has already been in use for more than 200 hours. 11 Square Roots of the lifelengths of batteries 0.637 1.531 0.733 2.256 2.364 1.601 0.152 1.826 1.868 1.126 0.828 1.184 0.484 1.207 0.719 0.715 0.474 1.525 1.709 1.305 2.186 1.228 2.006 1.032 1.802 1.668 1.230 0.577 1.274 1.623 1.313 0.542 1.823 0.880 1.526 2.535 1.793 2.741 0.578 1.360 2.868 1.493 1.709 0.872 1.032 0.914 1.952 0.984 2.119 0.431 WeiBull Distribution fX (x ) x e 1 x , x 0 12 Example: The length of service time during which a certain type of thermistors produces resistances within its specifications has been observed to follow a Weibull Distribution with = 1/50 and = 2 (measurements in thousand of hours). (a) Find the probability that one of these thermistors, to be installed in a system today, will function properly for over 10,000 hours. (b) Find the expected lifelength for the thermistor of this type. 13 Reliability (defn) Failure Rate 14 (t) burn-in service lifetime / constant failure rate wear-out mechanical devices electrical devices t 15 See the book for derivation of R(t). If the failure rate is constant, then R(t) = ? 16 Example: Consider a transistor with a constant failure rate of = 1/106 hours. (a) What is the probability that the resistor will be operable in 5 years? (b) Determine the MTTF and the reliability at the MTTF. 17 Def’n (Series System) = We model this as S1 S2 ... Sn 18 Definition: Redundancy Definition: Parallel System 19 S1 S2 . . . Sn 20 Example: Redundant Array of Independent disks (RAID) In a RAID, multiple hard drives are used to store the same data, thus achieving redundancy and increased reliability. One or more of the disks in the system can fail and the data can still be recovered. However, if all disks fail, then the data is lost. Assume that each disk drive has a failure rate of = 10 failures/106 hrs. How many disks must the system have to achieve a reliability of 98% in 10 years? 21 S2 S1 S3 S4 22 Quiz: Redundant Array of Independent disks (RAID) Your company intends to design, manufacture, and market a new RAID for network servers. The system must be able to store a total of 500 GB of user data and must have a reliability of at least 95% in 10 years. In order to develop the RAID system, 20-GB drives will be designed and utilized. To meet the requirement, you have decided to use a bank of 25 disks (25x20 GB = 500 GB) and utilize a system redundancy of 4 (each of the 25 disks has a redundancy of 4). What must the reliability of the 20 GB drive be in 10 years in order to meet the overall system reliability requirement? 23 What factors influence the failure rate? 24 Low Frequency FET, Appendix C. b T A Q E failures 10 hours 6 How would you find each of these? 25 Vcc = 5V IC RC = 100 VO = 5V or 0V = RB = 10k VI VO 2N3904 VI = 0 or 5V IB 26 CA Ambient, TA Junction, TJ JC Case, TC 27 TJ JC PD TC CA TA 28 SA Ambient, TA Heat Sink, TS CS thermal conductive paste Junction, TJ JC Case, TC 29 TJ JC TC PD CS TS SA TA 30 PD,max, mW with heat sink without heat sink slopes = -1JA mW/ oC TA, oC 31 So far, we have only looked at a single device. We are interested in collection of devices into a system! For example Vcc = 5V IC RC = 100 VO = 5V or 0V = RB = 10k VI VO 2N3904 VI = 0 or 5V IB 32