Electronic Product Design Teuvo Suntio - University of Ouluhakki/kurssit/LS/R_A_1.pdf · Electronic Product Design 52405S Slide 6/37 Definitions: Reliability R Reliability is the

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Teuvo SuntioTeuvo Suntio©

Professor of PowerProfessor of PowerElectronicsElectronics

at University of Ouluat University of Oulu

Reliability Engineering: System& Component Reliability

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Bibliography

G. H. Ebel, A. J. Lang, ’Reliability Approach to Spare Parts Problem’, NationalSymposium on Relaibility and Quality Record, 1963, pp. 421-430.

E. A. Elsayed,´Reliability Engineering’, Addison Wesley Longman, Inc.,1996, 737pp.

P. Jääskeläinen, ’Elektroniikan luotettavuus (405)’, Otakustantamo, 1978, Espoo

P. D. T. O’Connor, ’Practical Reliability Engineering’, John Wiley & Sons, 1995,431 pp.

T. Suntio, ’Evaluation of Power Systems Availability Using Simplified MethodsBased on Markov Chains’, IEEE International Telecommunications EnergyConference (INTELEC) Record, 1987, pp. 663-670.

T. Suntio, K. White, ’Reliability and Availability Performance as a Tool to OptimizePower Systems’, IEEE INTELEC Record, 1988, 49-56.

T. Suntio, S. Suur-Askola, ’DC-UPS System’s Reliability Performance: Facts andFiction’, Telecommunications Energy Special Conference (TELESCON) Record,1997, pp. 237-243.

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Dependability Concepts

Reliability is a word used to mean well and predictably behaving or workingproducts or systems. This constellation is, however, somewhat more complicatedincluding several concepts which have been used many times interchangeably.The different concepts are reliability performance, availability performance,maintainability performance, maintenance support performance as well asstorability and life. All these concepts together form a concept tree known asdependability concepts shown in Fig. 1. The overall reliability – especially of asystem - results from the combined effect of all the different concept areas.

AVAILABILITYPERFORMANCE

RELIABILITYPERFORMANCE

MAINTENANCESUPPORT

PERFORMANCE

MAINTAINABILITYPERFORMANCE

STORABILITY& LIFE

Fig. 1. Reliability or dependability concept hierarchy

We can conclude from Fig. 1 that availability performance covers all the aspectsrelated to ‘reliability’. It is also quite obvious that the overall availabilityperformance will be the result of team work including the technical system itself aswell as the organization maintaining it.

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A proper combination of high reliability equipment and a suitable maintenancephilosophy can result in high availability performance. Weaknesses in one aspectof availability, such as equipment reliability, may be compensated for byimprovement in another are such as more intensive maintenance and vice versa.

The quantitative measure of availability performance is availability defined as theability of item to perform its required function at a stated instant of time or overstated period of time. Availability is a function of time but will asymptoticallyapproach its steady-state or time invariant value which is normally used as anumerical value of availability.

The quantitative measure of reliability performance is reliability defined as theability of an item to perform a required function under stated conditions for astated period of time. The definitions of availability and reliability performances arequite similar but the following can be observed.

• availability and reliability are equal in non-repairable systems

• when repair and maintenance is taken into account the reliability of a device ornon-redundant system does not change but availability can be much higher but inthe case of a system having redundancy both the reliability and availability aregreatly affected by repair, maintenance and fault detection.

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Reliability engineering is one of the most useful tools to recognize, detect andrectify the problems which can affect adversely the operation of a system. Therelative figures or ratios of different implementations are an important source ofinformation. The absolute figures for availability and reliability indices are onlypredicted estimates of the future behavior and therefore, prone to errors.

The concepts described here will be defined more in detail later but the indicesrelated to these concepts are as follows.

Availability Performance: Availability A, unavailability U, mean-time-betweenfailures MTBF, mean-down time MDT.

Reliability Performance: Failure rate λλλλ, reliability R, mean-time-to failure MTTF

Maintainability Performance: Repair rate µµµµ, mean-time-to-repair MTTR

Maintenance Support Performance: spare parts shortage probability

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Definitions: Reliability R

Reliability is the probability that a product or service will operate properly for aspecified period of time (i.e., design life) under the design operating conditions(such as temperature or voltage) without failure.

In other words, reliability can be used as a measure of the system’s success inproviding its function properly. Reliability is denoted usually as R. According to itsdefinition, it is obvious, that reliability is a function of time, i.e, R(t).

Suppose that no identical components are subjected to a design operatingconditions test. During the interval of time (t -∆t, t), we observe nf(t) failedcomponents and ns(t) surviving components ( no = nf(t) + ns(t)). Since reliability isdefined as the cumulative probability function of success, then at time t, thereliability R(t) is

( )( ) s

o

n tR t

n=

In other words, if t is a random variable denoting the time to failure, then thereliability function at time t can be expressed as

t( ) ( )R t P t= >

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Reliability, Hazard Rate

The cumulative distribution function of failure F(t) is the complement of R(t), i.e.,R(t) + F(t) = 1. If the time to failure t has a probability density function (p.d.f) f(t),then we have

0

( ) 1 ( )t

R t f dτ τ= −∫Taking the derivative of the presented equation we obtain

( )( )

dR tf t

dt= −

If we consider the difference of the reliability values R(t) – R(t+ ∆t) within a shortperiod of time, i.e., t, t+ ∆t, and define the failure rate in this time interval as the theprobability that a failure per unit time occurs in this interval given that no failureshas occurred prior to t, the beginning of the interval. Thus the failure rate can beexpressed as

( ) ( )

( )

R t R t t

t R t

− + ∆∆ ⋅

The hazard function or rate h(t) is defined as the limit of the failure rate as ∆tapproaches zero.

( ) ( ) 1 ( ) ( )( ) lim ( )

( ) ( ) ( )t o

R t R t t dR t f th t

t R t R t dt R t∆ →

− + ∆= = − =∆ ⋅

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F(t), f(t), R(t), h(t)

Thus the keye equations that relate f(t), F(t), R(t) and h(t) are as follows

0( )

( ) ( ) 1 ( ) ( )

( )( ) ( )

( )( )

( )

t

t

o

h d

R t F t F t f d

dR tf t R t e

dtf t

h tR t

τ τ

τ τ

−

+ = =

∫= − =

=

∫

As an example we will estimatethe reliability data from a fictitiousdata source defined as follows. Amanufacture of light bulbs isinterested in estimating the meanlife of the bulbs. Two hundredbulbs are subjected to a reliabilitytest. The bulbs are observed, andthe failures in 1000-hour intervalsare recorded as shown in Table I.

Time Interval/hrs Failures in the interval

0 - 1000 100

1001 - 2000 40

2001 - 3000 20

3001 - 4000 15

4001 - 5000 10

5001 - 6000 8

6001 - 7000 7

Total 200

Table I.

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Example

We estimate fe(t), he(t), Re(t), and Fe(t) by using the following equations (thesubscript e stands for estimated). This results as shown in Table II.

( ) ( ) ( )( ) ( ) ( ) ( ) 1 ( )

( )f f e

e e e e eo s e

n t n t f tf t h t R t F t R t

n t n t h t= = = = −

∆ ∆

-4 4

e e ee

0 - 1000 5.0 5.0 1.0 0.0

1001 - 2000 2.0

Time interval/hrs f (t)/10 h ( ) /10 R ( ) F ( )t t t−

4.0 0.5 0.5

2001 - 3000 1.0 3.33 0.3 0.7

3001 - 4000 0.75 3.75 0.2 0.8

4001 - 5000 0.5 4.0 0.125 0.875

5001 - 6000 0.4 5.3 0.075 9.25

6001 - 7000 0.35 10.0 0.035 0.965

Table II.

The results show that the hazard rate of these bulbs are quite constant up to sixthtime interval, and then start to increase rapidly. Thus this period can be consideredas useful life period.

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Bathtub

The hazard rate h(t) is the conditional probability of failure in the interval t to (t+∆t),given that there was no failure at t. The hazard rate is also referred to asinstantaneous failure rate. The hazard rate is of greatest importance for systemdesigners, engineers, and repair and maintenance groups. It is useful in estimatingthe time to failure or between failure, spare parts policy, etc.

The shape of the hazard rate is typically referred to as a bathtub curve as shown inFig. 1.

Fig. 1. Bathtub-like behavior of hazard rate during the device life time

Ha

zard

Ra

teH

(t)

Early Life Region'Infant Mortality'

Constant Failure RateRegion 'Useful Life'

Wear Out Region

Time

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Constant Failure Rate

The early life is characterized by high failure rate but it is decreasing. The highinitial failure rate is usually caused by design or manufacturing flaws and will becorrected or reduced quite soon. The early failures can be minimized by increasingthe burn-in period of product or components before shipments re made, byimproving design and manufacturing process, and by improving the quality controlof the products. Some components such as electrolytic capacitors, storagebatteries, etc., tend to dry out and thereby, the component failure rate will startincreasing rapidly, i.e., wear-out.

The electronic and electrical components exhibit typically constant hazard or failurerate during the life time. The failure rate is typically denoted as λλλλ. This means thatthe random time to failure t is exponentially distributed having f(t), F(t) and R(t) asfollows.

( ) ( ) ( ) 1 ( )t t th t f t e F t e R t eλ λ λλ λ − − −= → = = − =

This means also that the time to failure t is exponentially distributed and number offailures will obey Poisson process.

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Mean Time To Failure MTTF

One of the measures of system’s reliability is the mean time to failure denoted asMTTF. It should not be confused with the mean time between failures denoted asMTBF´, which is related to repairable systems. MTTF can be computed as theexpected value of the reliability function R(t) resulting in the case of constant failurerate as follows.

0

1( )MTTF R t dt

λ

∞

= =∫

The unit of λ is usually denoted as 1 Fit that means 1 failure/ 109 hours. Inreliability engineering, it is customary to consider that one year equals to 8760hours. This means that a failure rate of 1 Fit equals to MTTF of 1.1416•105 a.

A system or product is a collection of components arranged according to a specificdesign in order to achieve desired functions with acceptable performance andreliability measures. A system configuration may be as simple as a series systemwhere all components are connected in series; a parallel system where allcomponents are connected in parallel; a series-parallel or a parallel-series wheresome of the components are connected in series and others in parallel resulting ina more complex network. Once a system is configured, its reliability must beevaluated and compared to acceptable reliability level.

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System’s ReliabilityNon-Repairable and Repairable

Systems

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System’s Reliability

We consider first non-repairable systems with time dependent reliability functionsR(t). The first step in evaluating a system’s reliability is to construct its reliabilitystructure. This can be done using e.g. a reliability block diagram (Fig. 2) or a faulttree (Fig. 3). A reliability block diagram shows which events or combination ofevents will lead to a system failure with associated reliabilities. A fault tree isconstructed using logic gates describing the process leading to a system failure;OR gate equals to a series connection, AND gate to a parallel connection and k/ngate to a special parallel connection of identical units, i.e., k-out-of-n, where n-kdefines the number of events leading to a fault.

Fig. 2. Reliability Block Diagram Fig. 3. Fault Tree

OR

AND

AND

k/n

OR

E1

E2

E3

E4

E5

E7

E6

Failure

E8

R1 R2

R4

R3

R5

R6

R7

R8

E1-4 E5-7E8

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Reliability; Non-repairable system

The system reliability will be evaluated by solving the combined reliabilities of thestructure. Next we defined the solutions to three basic structures - seriesconnection, parallel connection and k/n or k-out-of-n connections. We will consideronly events having constant failure rate.

R1 R2Series

Parallel

R1

R2

k-out-of-n

1 2 1 2

11

s s

n n

s i s iii

R R R

R R

λ λ λ

λ λ==

= ⋅ → = +

= → =∑∏

1 2 1 2 1 2

1

1 (1 )(1 )

1 (1 )

s

n

s ii

R R R R R R R

R R=

= − − − = + −

= − −∏

R

R

R

k-out-of-n

(1 )

!;0! 1

!( )!

ni n i

si k

nR R R

i

n n

i i n i

−

=

= ⋅ −

= = −

∑

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Reliability; Non-repairable system

We compute as a an example the combined reliability of the system shown in Fig.4.

R1 R2

R4

R3

R5

R6

R7

R8

Fig. 4. Reliability Block Diagram

1 2 3 4 5 6 7 8(1 (1 )(1 )(1 ))(1 (1 )(1 )(1 ))sR R R R R R R R R= − − − − − − − −

Many of the systems comprises of subassemblies or units, which can beconsidered as non-repairable systems. We must be able to evaluate their reliabilityas an equivalent failure rate λλλλeq. This can be done by computing MTTF of thesystem, and recalling that MTTR = 1/λeq. Most often the subsystems are a seriessystem resulting as λeq = λ1 + λ2 + …. + λn.

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Repairable System

The systems comprises of subassemblies, which can be replaced with a new unitin the case of failure. If a failure of one subassembly does not lead to a systemfailure, the system contains redundancy. Repairable systems are characterized byavailability defined as a probability that a system is functioning properly at astated instant of time. Availability is a function of time having a time dependentlimiting value A (i.e., steady-state availability) in the case of repairable systems.This steady-state value A and its complement unavailability U can be expressed bymeans of the mean time to failure MTTF and the mean down time MDT as follows

MTTF MDT

A UMTTF MDT MTTF MDT

= =+ +

It is quite common that the mean time to failure MTTF and the mean time betweenfailures MTBF are used as synonyms, but MTBF = MTTF + MDT. The repairablesystems are, therefore, characterized by means of MTBF and the non-repairableby means of MTTF. The value of MDT is usually in hours when the values of MTTFand MTBF are in years. This means that MTTF ≈ MTBF as time values but as aconcept they are different.

It is usual that the repair process of these equipment is assumed to be alsoaccording to the exponential p.d.f. having repair rate µµµµ as in the case of reliability.This results that the mean time to repair MTTR = 1/µ. For an individual item MDT =1/µ but not necessarily for a more complicated system.

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Repairable System

The availability A and unavailability U of an item having failure rate λλλλ and repairrate µµµµ is, therefore, as follows

MTTF MDT

A UMTTF MDT MTTF MDT

µ λλ µ λ µ

= = = =+ + + +

The availability A (i.e., MTTF and MDT) of a more complicated system has to beevaluated using specific methods. We will consider here only a method based onMarkov chains that is also known as state-space method. It is assumed that thestates are independent and the transition probabilities from any state to any otherstate are independent on the system state and will stay constant. If the time tofailure has exponential p.d.f., this will hold and Markov models can be used.

In a systems where we have redundancy, i.e., one failure of a special item doesnot lead to a system failure, the faults can be dormant. Thus we will classify thestates as follows:

* the failure of a component is detected immediately either by its internalmonitoring or some external monitoring of the system components, and thereby,the repair actions can be undertaken at once.

* the failure of a component is dormant, and will be detected only by means ofscheduled preventive maintenance actions, etc.

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Repairable System

The fault detection of a component can be taken into account by means of its faultdetection probability α. It can be estimated as a function of the steady-stateavailability Am of the monitoring system and its fault coverage cm. The faultcoverage of an item can be estimated as cm = λλλλd/λλλλT where λT is the total failurerate of the item to be monitored, and λd is the failure rate of the detectable faults. Ingeneral cm ≤ 1, but should be as close as possible to 1.

The Markov models for a series connection and a parallel connection of two itemsare presented in Fig. 5. The transitions between states are composed of the failurerates and repair rates of corresponding items. The state having highest number isusually the state of system failure, and the state denote 0 is the state where the allthe system components are functioning properly.

0 11 2λ λ+

µ

0

1

2

3

1λ

2λ

1µ2λ

1λ

µ

2µ

Fig. 5. Markov state-space diagrams

a) Series connection b) Parallel connection

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Repairable System

A Markov model for a system having n identical unit in parallel configured as (n-1) -out-of-n system with detected and dormant states is shown in Fig. 6. The state 1 isdetected state with one faulty unit and (n-1) good units and state 2 is dormant statewith one faulty unit (n-1) good units. The preventive maintenance rate is denotedby m.

0

1

2

3

nα λ⋅

(1 ) nα λ− ⋅

µ

m

( 1)n λ−

( 1)n λ−

µ

Fig. 6. Markov state-space diagram for (n-1)-out-of-n identical units

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Repairable System

The reliability and availability indices can be solved by means of the state-spacediagrams. The state transition matrix - denoted here as ASTM - contains the sum ofthe transition rates leaving each state (diagonal elements, minus sign) and the sumof the transition rates entering into each state from the other states (off-diagonalelements, plus sign) in such a way that the first row contains the transitions fromand into the system OK state, the last row the transitions from the system failurestate, respectively. Thus the transition matrix is a square matrix having as manyrows and columns as there are states. The transition matrices e.g. for the diagramsof Fig. 5 and 6 are as follows

(( 1) ) 0 0

(1 ) 0 (( 1) ) 0

0 ( 1) ( 1)

STM

n m

n nA

n n m

n n

λ µ µα λ λ µ

α λ λλ λ µ

− − − + = − − − + − − −

1 2

1 2

( )

( )STMAλ λ µ

λ λ µ− +

= + −

1 2 1 2

1 2 1

2 1 2

2 1

( )

( ) 0 0

0 ( ) 0

0

STMA

λ λ µ µ µλ λ µλ λ µ

λ λ µ

− + − + = − + −

Fig. 5 a) Fig. 5 b)

Fig. 6

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Repairable System

The system’s MTTF can be computed from the state-space diagram assuming thesystem as non-repairable, i.e., there is no transition from the final state or thesystem failure state. This means that we will use only (n-1) rows and columns fromthe state transition matrix. If we denote the average time the system will spend ineach up state as Ti, and the reduced state transition matrix as ASTM-1, we get amatrix equation as follows from which the average time values of each state can besolved.

1 1 11 1

1 1

1 1

0 0

. .. .

0 0

o o

STM STM

n n

T T

T TA A

T T

−− −

− −

− − ⋅ = → = ⋅

The mean time to failure MTTFs can be calculated as follows.

1

0

n

s ii

MTTF T−

=

= ∑

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To solve the system availability A, the last row of the state transition matrix has tobe manipulated in such a way that the last row will replace by the fact that the sumof the state availabilities Ai = 1 otherwise the matrix is not invertible because ofzero determinant. If we denote the manipulated state transition matrix as A’STM, thestate availabilities Ai can be solved as follows.

1 1 1

0 0

0 0

. .. .

1 1

o o

STM STM

n n

A A

A AA A

A A

−

′ ′⋅ = → = ⋅

The system availability As and unavailability Us can be solved then as follows.

1

0

n

s i s ni

A A U A−

== =∑

Repairable System

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Repairable System

The system mean time between failures MTBFs and mean down time MDTs can becomputed as follows using the system availability As and MTTFs.

1 s s

s s ss s

MTTF AMTBF MDT MTTF

A A

−= = ⋅

We calculated as an example the reliability and availability performance indices forthe system of Fig. 5a). The modified matrices and required indices for the seriesconnection are as follows

Example 1.

0 11 2λ λ+

µ[ ]1 1 2

1 21 2

( )

1( ) 1

STM

o o

A

T MTTF T

λ λ

λ λλ λ

− = − +

− + ⋅ = − → = =+

1 2

1 2

( )

( )STMAλ λ µ

λ λ µ− +

= + −

1 2

1 2 11 2

1 11 2

( )

1 1

( ) 0

1

STM

o s o

o s

A

A A A A

A A U A

λ λ µ

µλ λ µλ λ µ

αλ λ µ

− + ′ =

− + ⋅ + ⋅ = → = =+ +

+ = = =+ +

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0

1

2

3

1λ

2λ

1µ2λ

1λ

µ

2µ

1 2 1 2

1 2 1

2 1 2

2 1

( )

( ) 0 0

0 ( ) 0

0

STMA


λ λ µ

− + − + = − + −

1 2 1 2

1 1 2 1

2 1 2

1 2 1 1 2 2

1 2 1 1

2 1 2 2

21 2

1 1 2 20 2 1 1 21 2

2 1 1 2

( )

( ) 0

0 ( )

( ) 1

( ) 0

( ) 0

1(1 )

STM

o

o

o

ii

A

T T T

T T

T T

MTTF T

λ λ µ µλ λ µλ λ µ

λ λ µ µλ λ µλ λ µ

λ λλ µ λ µ λ µ λ µλ λ

λ µ λ µ

−

=

− + = − + − +

− + + + = −− + =− + =

= = + ++ ++ − −

+ +

∑

1 2 1 2

1 2 1

2 1 2

1 2 1 1 2 2 3

1 2 1 1

2 1 2 2

1 2

( )

( ) 0 0

0 ( ) 0

1 1 1 1

( ) 0

( ) 0

( ) 0

STM

o

o

o

o

A

A A A A

A A

A A

A A A



− + − + ′ = − +

− + + + + =− + =− + =+ + 3

21 2

1 1 2 20 2 1 1 21 2

2 1 1 2

1

(1 )( ) ( )s i

i

A

A Aµ λ λ

µ µ λ µ µ λ λ µ λ µλ λ µλ µ λ µ

=

+ =

= = ⋅ + +− − + ++ + − −+ +

∑

We calculated as an example the reliability and availability performance indices forthe system of Fig. 5b). The modified matrices and required indices for the seriesconnection are as follows

Example 2.

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0

1

2

3

nα λ⋅

(1 ) nα λ− ⋅

µ

m

( 1)n λ−

( 1)n λ−

µ

1

1 2

1

2

2

0

(( 1) ) 0

(1 ) 0 (( 1) )

1

(( 1) ) 0

(1 ) (( 1) ) 0

1(1+

(1 ) ( 1)(1 )( 1) ( 1)

STM

o

o

o

s ii

n m

A n n

n n m

n T T mT

n T n T

n T n m T

nMTTF T

m nnn n m

λ µα λ λ µ

α λ λλ µ

α λ λ µα λ λ

α λαµ α λλ

λ µ λ

−

=

− = − − + − − − +

− + + = −− − + =

− − − + =

= = − − +− −− + − +

∑ (1 ) + )

( 1)

n

n m

α λµ λ

−− +

We calculated as an example the reliability and availability performance indices forthe system of Fig.6. The modified matrices and required indices for the seriesconnection are as follows

(( 1) ) 0 0

(1 ) 0 (( 1) ) 0

0 ( 1) ( 1)

STM

n m

n nA

n n m

n n

λ µ µα λ λ µ

α λ λλ λ µ

− − − + = − − − + − − −

Example 3.

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1 2 3

1

2

1 2 3

2

0

(( 1) ) 0 0

(1 ) 0 (( 1) ) 0

1 1 1 1

0

(( 1) ) 0

(1 ) (( 1) ) 0

1

(1 )(

STM

o

o

o

o

s ii

n m

n nA

n n m

n A A mA A

n A n A

n A n m A

A A A A

A Am

n

λ µ µα λ λ µ

α λ λ

λ µ µα λ λ µ

α λ λ

µαλ µ=

− − − + ′ = − − − +

− + + + =− − + =

− − − + =+ + + =

= = −+ −∑ (1 )

(1 )) ( 1) ( 1)

( 1)

n nn n n m

n m

α λ α λµ λ λ µ λ

λ

−+ +− − + − +− +

We assume that the system of Fig. 6 comprises of three units having failure rate λ= 1/ 20 a = 5.7•10-6 1/h (5700 Fit). The repair rate µ is defined as follows: A failedunit or system will be corrected within 24 h at probability of 0.95. By this definitionthe repair rate can be computed as follows.

Example 4.

{ } 24 1 1 124 1 0.95 ln / 24 0.125 ; 8

0.05hP t h e h MTTR h

hµ µ

µ− ⋅> = − = → = ≈ = ≈

Example 3.

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The preventive maintenance rate m can be defined and computed in a similar way.If we assume that the preventive maintenance occur at probability of 0.95 withinone year, we get m ≈ 3.42•10-4 1/h.

The symbolic calculations are usually very tedious and therefore, the computing isnormally made using e.g. Matlab or Madcad software packages or some othersimilar tools. The resulting system MTTFs, availability As and MDTs for αααα = 0, 0.9and 0.99 are as follows.

0 217 0.99999579 8

0.9 2028 0.99999955 8

0.99 16180 0.9999999436 8

s s sMTTF A MDT

a h

a h

a h

α

It is quite obvious that the fault detection can have significant influence on thesystem availability and reliability performance. It will actually realize the potentialsof redundancy. If the failure of a unit will stay dormant and we do not have anypreventive maintenance, the system MTTFs is only about 17 a !

Example 4.

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Failure Groups

Many of the systems to be evaluated can be quite complicated or containnumerous states, which may result in difficulties to solve the system or even tocreate the necessary Markov diagrams. It is also quite useful to be able to trace theinfluence of different failure combinations in order to detect the significance orinsignificance of specific events. If we have independent failure groups orcombination of events which leads to system failure, i.e., the failure combinationsare in series in reliability sense, we can also evaluate the system reliability andavailability performance separately for each group and then combine the results asfollows.

1

1

11

1

ns s

s i s s s sni s s

i i

MTTF AA A MTTF MTBF MDT MTTF

A AMTTF

=

=

−= = = = ⋅∏∑

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Spare Parts Prediction

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The corrective maintenance will be normally carried out using replacement units orspare parts. The number of spare parts needed to satisfy specific requirements canbe predicted using the estimated failure rate of the replacement unit, shortageprobability and the mean time to repair or to deliver new spare parts. The numberof failures in unit time follows the Poisson distribution if the number of units n issufficiently large. If n is small, the distribution is binomial, but Poisson distributionwill give also good results, because the minimum number of spare units is alwaysequal to one.

According to Poisson p.d.f., the propability that exactly r failures will occur withintime interval T is

( )( )

!

rn T

T

n TP r e

rλλ −= ⋅

where n = total number of units

λ = equivalent failure rate of one unit

The probability that at least N units will fail during the time interval T is

0

( ) ( )N

T Tr

P r N P r=

≥ = ∑

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When we assume that T is the repair or delivery time of new spare parts and1 - Pr(r ≥≥≥≥ N) the shortage probability, the number of spare parts during T can beestimated assuming that one spare part is under repair. This means that thenumber of spare parts is N+1.

An Example:

Lets assume that we have a specific units in use (n) 50 pieces and the equivalentfailure rate of one unit (λ) is 5 kFit. The time to repair or deliver new spare parts (T)is 100 h. The required shortage propability is 0.001.

0.025100

(0.025)( )

!

( ) 1 ( )

0 0.097530991 0.0246901

1 0.02438275 0.00003074

r

hP r er

N P r P r

−= ⋅

−∑

The number of spare parts needed is 1+1.

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Component Reliability

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The system reliability evaluation is actually based on the predicted failure rates ofthe components used to implement the system. The actual failure rate of acomponent is dependent on its complexity (resistor vs. microcomputer) and on theenvironment where it is used as well as its design quality. This environmentincludes its operating temperature and other operating conditions such as appliedvoltage or current compared to rated values. All these factor will drastically affectthe resulting failure rate. In many applications, we could also have secondaryevents such as e.g. a fuse and high inrush currents, which actually may define theresulting failure rate of the associated component. These are actually called ascomponent stresses which has to be analyzed and a proper stress level or deratinglevel should selected. Very often the design rules of a company ( Quality SystemManual ) states these derating factors for specific components.

The component failure rate is initially predicted using different sources, whichdefines either a basic failure rate and different stress factors (e.g. MIL-HDBK 217;Reliability Prediction of Electronic Equipment) or general failure rate data inspecific application area (e.g. HRD4; Handbook of Reliability Data for Componentsused in Telecommunications Area, British Telecom), etc. When the systems orunits have been in use several years, the failure rate of the used components canbe predicted using the observed failure data. If MIL-HDBK 217 is use, the resultingfailure rates are usually quite pessimistic compared to the observed failure ratesfrom the field data.

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An indicative list of basic component failure rates for reliability predictions is asfollows.

Component Type Fit Component Type Fit

Resistors Integrated CircuitsComposition 1 Digital, CMOS, SSI 10Metal film 0.2 Digital, CMOS, MSI 15Resistor network 2 Op. Amps, single 20

Op. Amps, daul 30Capacitors Op. Amp, quad 50

Polystyrene, Polypropylene 2Ceramic 2 ElectromechanicalPolyester 0.5 Switch 30Aliminium electrolytic, I < 1A 5 Relay 100Tantalum electrolytic 5 Connector, n < 10 5

Connector, n > 10 20Inductive componnets

Coil or choke, signal 2 MiscellaneousTransformer, signal 5 Crystal 50Transformer, power 50 Oscillator 100

Printed Circuit Board 10Diodes Solder connection 0.03

Diode, P < 1W 1Diode, P > 1W 10Zener diode, P < 1 W 5LED, indicator 10

TransistorsTransistor, P < 1 W 5Transistor, P > 1 W 30

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Temperature is one of the stress factors having drastic effect on the componentreliability. Therefore, thermal design of an equipment is very important. Theindividual components shall also be selected in such a way that they can withstandthe environment they are supposed to be exposed. The effect of temperature canbe taken into account using Arrhenius acceleration factor. We assume that thebasic failure rate λλλλo is given at the temperature of To which is expressed in Kelvin(K) degrees ( 0oC = 273 oK). The failure at the new temperature level T1 can becalculated as follows

1

1 1( )

1o

E

T T ko eλ λ

∆− ⋅= ⋅

where

k = Boltzman’s constant 8.61 10-5 eV/oK

∆E = Activation energy in eV.

Typical values of ∆E for electronic devices are in the range of 0.5 eV ∼ 1.5 eV.Typical value is thus 0.75 eV corresponding to a doubling of failure rate for every10 K rise in temperature.

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An Example.

Lets assume that the failure rate of a specific component at 25oC is 5 kFit, but theactual operating temperature is 55oC. What is the actual failure rate. Assumetypical activation energy.

5

1 1 0.75( )

273 25 273 55 8.611055 25

14.49 5 72.46 o oC Ce kFit kFitλ λ −− ⋅

+ + ⋅= ⋅ = ⋅ =

This means that the mean time to failure MTTF is reduced from 570 a to 39 a.

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Electronic Product Design Teuvo Suntio - University of Ouluhakki/kurssit/LS/R_A_1.pdf · Electronic Product Design 52405S Slide 6/37 Definitions: Reliability R Reliability is the