methodologies for the analysis of reliability of electronic devices
TRANSCRIPT
METHODOLOGIES
FOR THE ANALYSIS
OF RELIABILITY OF
ELECTRONIC
DEVICES
Tongji University Shanghai • Politecnico di Torino • Politecnico di Milano
“POLITONG” Sino-Italian Double Degree Project
Faculty of Information Technology Engineering
Electronic Engineering Degree
STUDENT INFO Paolo Vinella
ID: 163485
Introducing CRF FIAT Powertrain
CRF is the R&D Center founded to improve
competitiveness of the products of Fiat Group
Research activities in automotive engineering,
manufacturing, advanced materials, ICT, electronics a
Internship: introduction and goals
1. Analysis of state of the art techniques and development;
2. Research oriented to the expansion with new methods;
3. Software development (on NI CVI environment) to
improve FIAT LTA software’s capabilities;
4. Validation testing based on real data.
GOAL: derive in an analytic form the confidence boundaries of reliability expectation of electronic
devices in a time-varying multi-stresses scenario.
Electronic devices and Reliability
• In datasheets, electronic devices characterized with reliability parameters THE FACT
• Reliability information usually limited to very specific / fixed stress conditions LIMITATIONS
• “Accelerated life time testing”
allows to estimate the reliability also when stresses are different from the nominal ones and even time-variant
POSSIBLE SOLUTION
Reliability
Reliability information guarantees a specific device to operate properly before the first failure occurs.
A good reliability expectation:
1. Reduces economic and time costs;
2. Allows producer to establish more realistic / optimized
thresholds concerning:
Warranty
Maintenance
through product
lifetime
Repair
Stocks
Ingredients for an engineered
Lifetime Analysis
Most probable value – average (mean) expectation time
Confidence level – established percentage that expresses
the degree of reliability of our range
Confidence interval – range of credible values for that
parameter’s expectation
Example: Gaussian distribution
AVG VALUE ± CONFIDENCE_INTERVAL @ CONFIDENCE_LEVEL OF [ ]%
Accelerated Lifetime Testing guidelines
Collect electronic samples to be subjected to accelerated tests
Define stresses (voltage, temperature, humidity,…)
Collect results from stresses: for each sample it is possible to know if it is still alive or dead after a certain time
Parameter estimation of Weibull Distribution with GLL + Likelihood model
Confidence boundaries computation for Probability / Reliability / Failure Rate / Cumulative functions
Statistical Model Weibull distribution as probability density function: suitable to
represent reliability of mechanical/electronic components
For a considered time value:
Weibull Cumulative function
Weibull Reliability function
Weibull Failure Rate function
𝜂 : scale parameter represents the bond of the dependence of Weibull
distribution function on stress profiles that are applied to the samples;
𝛽 : shape parameter is expression of the slope of the Weibull function.
Weibull probability function 𝑓 𝑡 =𝛽
𝜂
𝑡
𝜂
𝛽−1𝑒−
𝑡
𝜂
𝛽
𝒇(𝒕, 𝜼)
𝒕
𝜼 = 𝟓𝟎
𝜼 = 𝟏𝟎𝟎
𝜼 = 𝟐𝟎𝟎
𝜷 = 𝟐
Mathematical model for the
applied stresses (T.V. scenario) GLL (General Log-Linear) model to link applied stresses to
the scale parameter η of Weibull function:
𝜂 = 𝜂 𝑥1, 𝑥2, … , 𝑥𝑛 = 𝑒𝑎0+ 𝑎𝑖∙𝑔𝑖 𝑓𝑖 𝑥𝑖𝑛𝑖=1
Furthermore, inside Weibull equations, replace the term 𝑡
𝜂 with:
𝐼 𝑡, 𝑓𝑖 = du
𝑒 𝑎0 + 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=1
𝑡
𝑡=0
• The GLL model can be “injected” into Weibull function.
• Unknown parameters to be determined, in order to
define the model: 𝛽, 𝑎0, 𝑎1, 𝑎2, … , 𝑎𝑛.
MLE: parameter estimation Maximum Likelihood Estimation is a method of estimating the
parameters of a statistical model.
Applied to a data set (stress tests) given a statistical model (Weibull).
Likelihood function: the vector of parameters (𝛽 , 𝑎0 ,𝑎1 ,𝑎2 ,… , 𝑎𝑛 ) is its absolute maximum point.
𝑙𝑛𝐿 = ln 𝑓 𝑡𝑗
𝐹𝑒
𝑗=1
+ ln𝑅 𝑡𝑧
𝑆
𝑧=1
( Fe → number of failures; S → number of successes; 𝑡𝑖 → sample extraction time )
FIAT LTA relies on a Genetic Algorithm for this goal.
Once parameters are known, we can analyze how to compute confidence boundaries…
(Local) Fisher Matrix: the road
toward boundaries computation
Each of the parameters (𝛽, 𝑎0, 𝑎1, 𝑎2, … , 𝑎𝑛) represented as normal distribution:
o the absolute maximum of the MLE function provides their mean value; a
o to compute their standard deviation, the Fisher matrix is what we need.
MLE function
(logarithmic form 𝑙𝑛𝐿)
solutions (𝛽 , 𝑎0 ,𝑎1 ,𝑎2 ,… , 𝑎𝑛 )
FISHER MATRIX: made of
second order derivatives of 𝒍𝒏𝑳 function
INVERSE (Local) Fisher Matrix
The inverse matrix [ℱ−1] of [ℱ] is really meaningful because it
contains the variance and the covariances among the parameters
and it can be interpreted as follow:
ℱ−1 =
Var 𝛽 Cov 𝛽, 𝑎0 Cov 𝛽, 𝑎1
Cov 𝑎0, 𝛽 Var 𝑎0 Cov 𝑎0, 𝑎1
Cov 𝑎1, 𝛽 Cov 𝑎1, 𝑎0 Var 𝑎1
… Cov 𝛽, 𝑎𝑛
… Cov 𝑎0, 𝑎𝑛
… Cov 𝑎1, 𝑎𝑛… … …Cov 𝑎𝑛, 𝛽 Cov 𝑎𝑛, 𝑎0 Cov 𝑎𝑛, 𝑎1
… … …… Var 𝑎𝑛
Just so you know: expression of
terms inside the [F] matrix
𝜕2𝑙𝑛𝐿
𝜕𝛽2 = −1
𝛽2 − ln2 𝐼 ∙ 𝐼𝛽𝐹𝑒
𝑗=1
− ln2 𝐼 ∙ 𝐼𝛽𝑆
𝑧=1
𝜕2𝑙𝑛𝐿
𝜕𝑎𝑟𝜕𝛽≡
𝜕2𝑙𝑛𝐿
𝜕𝛽𝜕𝑎𝑟= 𝐼−1 𝐼𝛽 − 1 + 𝛽𝐼𝛽 ln ( 𝐼) ∙
𝑔𝑟 (𝑓𝑟 𝑢 )
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
du
𝑡𝑗
0
𝐹𝑒
𝑗=1
+
+ 𝐼𝛽−1 + 𝛽𝐼𝛽−1 ln ( 𝐼) 𝑔𝑟 (𝑓𝑟 𝑢 )
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
du
𝑡𝑧
0
𝑆
𝑧=1
𝜕2𝑙𝑛𝐿
𝜕𝑎𝑟𝜕𝑎𝑠≡
𝜕2𝑙𝑛𝐿
𝜕𝑎𝑠𝜕𝑎𝑟= 1 − 𝛽 𝐼−2 1 + 𝛽𝐼𝛽
𝑔𝑠 (𝑓𝑠 𝑢 )du
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
𝑡𝑗
0
∙ 𝑔𝑟 (𝑓𝑟 𝑢 )du
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
𝑡𝑗
0
+
𝐹𝑒
𝑗=1
+𝐼−1 𝛽 1 − 𝐼𝛽 − 1 𝑔𝑠 (𝑓𝑠 𝑢 ) ∙ 𝑔𝑟 (𝑓𝑟 𝑢 )
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
du
𝑡𝑗
0
+
+ 𝛽 1 − 𝛽 𝐼𝛽−2 𝑔𝑠 (𝑓𝑠 𝑢 )du
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
𝑡𝑧
0
𝑔𝑟 (𝑓𝑟 𝑢 )
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
du
𝑡𝑧
0
− 𝐼𝛽−1 𝑔𝑠 (𝑓𝑠 𝑢 ) ∙ 𝑔𝑟 (𝑓𝑟 𝑢 )
𝑒 𝑎𝑖 ∙ 𝑔𝑖 (𝑓𝑖 𝑢 )𝑛𝑖=0
du
𝑡𝑧
0
𝑆
𝑧=1
Variance estimation of (𝛽, 𝑎0, 𝑎1, 𝑎2, … , 𝑎𝑛)-dependent distribution functions
For a function 𝜃 𝑡, 𝛽 , 𝑎0 ,𝑎1 ,… , 𝑎𝑛 :
The variance is a time-dependent function
NOTE: the parameter-dependent functions are Weibull Probability /
Cumulative / Reliability / Failure Rate functions!
Var𝜃 𝑡 =𝜕𝜃
𝜕𝛽
2
Var 𝛽 + 2𝜕𝜃
𝜕𝛽
𝜕𝜃
𝜕𝑎𝑖 Cov 𝛽, 𝑎𝑖
𝑛
𝑖=0
+ 𝜕𝜃
𝜕𝑎𝑖
𝜕𝜃
𝜕𝑎𝑗 Cov 𝑎𝑖 , 𝑎𝑗
𝑛
𝑗=0
𝑛
𝑖=0
Confidence boundaries (yes, finally
here we are ☺)
Expression of the function 𝜃 𝑡 with its confidence interval (once its
variance Var𝜃 𝑡 is known):
𝜃 𝑡 ± 𝑘𝑎 Var𝜃 𝑡
mean value variance
a particular constant
(index of the confidence level 𝛿)
Operative part of the Internship
FIAT LTA software updated in confidence boundaries computation with
analytical approach each time a derivative has to be determined:
1. Fisher matrix computation;
2. Variance estimation.
Previous version of LTA relies on numeric derivation approach.
“Analytical approach” means instead that all the derivatives must be
determined by hand, then fed to the software source code.
FIAT LTA in boundaries computation:
before VS now
Advantages over the previous version of LTA: A
1. More efficient/reliable way to determine confidence boundaries;
2. Required time now essentially independent from number of devices
under test, applied stresses and number of profiles;
3. Output results generated without noticeable delay (less than one sec).
Previous algorithm based on numeric derivation suffers of high latency-to-
output. Sometimes, it crashes without providing any output.
Let’s compare the new FIAT LTA with a competitor solution: Reliasoft® ALTA
EXAMPLE: confidence boundaries
computation - FIAT LTA vs Reliasoft ALTA INPUT SET: 11 electronic components under test, all Failures. A single Profile made of one
stress. The stress is an applied voltage increasing as step function over time, 2V to 7V.
EXAMPLE: confidence boundaries
computation - FIAT LTA vs Reliasoft ALTA
Fiat LTA and Reliasoft® ALTA use two different ways to estimate the parameters of the distribution – LTA relies on a genetic algorithm:
Here a comparison between Fisher matrixes computed with FIAT LTA (with the new method implemented during the internship) versus ALTA:
𝐹𝑖𝑠ℎ𝑒𝑟𝐴𝐿𝑇𝐴 =2.791302 −4.770014 −8.342983−4.770014 78.912452 95.879401−8.342983 95.879401 119.288033
𝐹𝑖𝑠ℎ𝑒𝑟𝐿𝑇𝐴 =2.789696 −4.749727 −8.319259−4.749727 78.840362 95.793655−8.319259 95.793655 119.183768
𝑎0 𝑎1 𝛽
Reliasoft® ALTA 9.8421 -3.9985 2.6783
FIAT LTA 9.8442 -4.0006 2.6750
EXAMPLE: confidence boundaries
computation – Reliasoft® ALTA PLOTS
EXAMPLE: confidence boundaries
computation – FIAT LTA PLOTS
USER CAN
SEE HOW
RELIABILITY BEHAVES
CHANGING
THE MISSION
PROFILE!
∀ time value:
% of broken devices
∀ time value:
% of survived devices
units of failure for unit of time
in respect to the ones that
survive (i.e. “1 failure/month”)
CONCLUSIONS
Internship experience as an extremely useful opportunity
Work on a concrete project inside a real workplace environment like R&D
department of FIAT
A taste of research into statistical/mathematical models
Opportunity to devise an algorithm taking into account optimization and
no “software overhead”
FINAL THANKS
Ing. Massimo Abrate – Company Tutor
Prof. Alessio Carullo – Academic Tutor