7.2.1. Miner’s approach

This approach is generally used in HALT/HASS tests to justify the lack of damage of the products without latent defects. The lack of damage rests on a theoretical perspective rooted in Miner’s rule (Miner 1945). According to this rule, the cumulative damage of the product is linear. Hence, if one or more products are subjected to “n” burn-in cycles without observing any failures, then the burn-in test cycle to be applied to all the products will damage the healthy products by no more than 1/n. Damage of 2% is generally admitted. In this case, the POS1 test involves the application of 50 burn-in cycles, as illustrated in Figure 7.3.

A second hypothesis concerns the estimation of damage of products submitted to a HASS cycle. From a reliability perspective, what determines the useful service life of the product (flat sector of the bathtub curve) is the minimum of random variables representing each aging failure mechanism (that can vary depending on the life profile).

This can be schematically represented by Figure 7.4.

If all the laws of reliability of failure mechanisms were known, it could be verified if the specified reliability objective has been met. Unfortunately, this is not the case, as the component manufacturers do not provide this information in the datasheet of the component. Though experience shows that, in general, aging failure mechanisms are not observed during product operation, the minimum of the aging mechanisms with respect to the product service life is not known. A conservative approach is therefore to take the case in which the minimum corresponds exactly to the end of the service life.

Following the failures observed during the “n” cycles, certain information can be extracted, as listed in Table 7.1.

If one or several failures are observed on the first burn-in cycle, this proves a certain effectiveness of burn-in. This scenario is however unlikely given the small number of products undergoing the POS1 test. Hence, POS2 is proposed.

7.2.2. Approach according to the physical laws of failure

There is obviously no physical law of failure taking into account the simultaneous effect of the thermal cycling, vibrations and energizing. Moreover, there is no recent model taking into account the influence of the slope in the thermal cycling, this being particularly important during the precipitation phase. However, from a conservative perspective, it is possible to take into account the aging occurring during the POS1 test by considering only the effect of thermal amplitude according to the Coffin–Manson law. The focus is on the acceleration factor between the POS1 conditions and those observed during operation by the end user. This is given by:

[7.1]

A small problem remains to be solved however, since the thermal amplitude is different between the precipitation and detection phases. However, based on the Sedyakin principle, an equivalent amplitude (Bayle 2019) can be found in terms of reliability, which is given by:

[7.2]

PROOF.–

According to the Sedyakin principle:

or

or

or

End

Consequently, the acceleration factor can be written as:

or even

[7.3]

Therefore, the 50 cycles of POS1 can be reflected by operational cycles in the following form:

[7.4]

EXAMPLE.–

Consider an operational application with the life profile given in Table 7.2.

Assume the precipitation and detection phases have the following characteristics:

• – ΔTprecipitation = 150°C;
• – ΔTdetection = 110°C;
• – N1 = N2 = 2.

Let us analyze the aging generated by POS1 for the products without defects. For thermal cycling, the following failure mechanisms can be considered:

• – Breakage of brazed joints with a coefficient m = 2.

An equivalence in operational cycles is therefore obtained for POS1 for brazed joints

or

This result is below the 7,300 cycles of the life profile of the previous table. But it is important to note that the acceleration factor model that has been taken into account is very conservative. On the other hand, the focus is on the aging for one burn-in cycle. In this case, it can be seen that it represents 2,163/50 ~ 43 cycles or barely 0.6% (43/7,500).

• – Bonding removal or breakage, for active components, with a coefficient m = 4.

or

This result shows that, while being conservative, the POS1 test covers the 7,300 cycles of the life profile considered in this example. If no failures are observed, it is reasonable to believe there will be none during operation.

7.2.3. Zero-failure reliability proof approach

It is interesting to see what this means in terms of reliability for the POS1 test. Similar hypotheses to those in section 7.2.2 lead us to obtain:

• – Products without maintenance

According to equation:

where:

NC is the level of confidence;

AF is the acceleration factor from equation [7.1];

N is the number of products undergoing POS1 test;

Npos1 is the number of thermal cycles of POS1;

Nm is the number of thermal cycles of the mission;

β is the shape parameter of Weibull law.

Resuming the data in the previous example and moreover considering that N = 4, NC = 90% and β = 5, the following results are obtained:

Brazed joints

Bonding

• – Products without maintenance

According to the equation:

Assuming the same hypotheses as previously leads to:

Brazed joints

Bonding 40,195 cycles

to be compared to 7,300 of life profile.

7.3.1. Influence of parameter Q

Let us fix p = 2% and observe the evolution of this probability as a function of Q and n. Figure 7.6 is obtained.

It can be seen that the number of products Q subjected to burn-in does not influence the probability of detecting a defect during POS2 testing. Furthermore, it is important to acknowledge that it is necessary to have a significant number of products for POS2 testing in order to achieve a reasonable probability of detecting one or more latent defects.

7.3.2. Influence of parameter p

Let us fix Q = 1,000 and observe the evolution of this probability as a function of p and n. Figure 7.7 is obtained.

It can be noted that the proportion of defects has a significant impact on the probability of detecting at least one defect during the POS2 test. Moreover, in order to have a reasonable probability of detecting a defect, the required number of products is all the larger as the defect proportion is larger, which is in fact logical.

Indeed, since generally p ≪ 1, equation [7.6] can be approximated by:

[7.7]

Hence, the value of n corresponds to a given probability

[7.8]

Based on equation [7.8], it is possible to set a probability to filter a defect and estimate the number of products “n” that must be subjected to POS2 testing. But how can this probability be fixed knowing that it is an ascending function of n? The economic aspect can then open up a possibility. Indeed, given:

• – Ce is the cost of POS2 test;
• – Cp is the cost of a product;
• – Cr is the cost of repair due to manufacturing defect;
• – Ct is the total cost,

it can be written that:

Indeed, the cost due to products having a defect depends on their number Q.p as well as on the fact that they were not filtered out by burn-in. Based on equation [7.8], it can be written that:

[7.9]

This equation is differentiated with respect to n:

This derivative is zero if:

or

or finally

[7.10]

Since “n” is a non-zero integer, the following inequality must be valid: .

NOTE.–

• – The cost of the POS2 test is not involved in the search for the optimum number of products undergoing POS2 test.
• – Due to the previous inequality, it is possible to have no optimum.
• – In case there is one, the optimum is a minimum since
  
• – The minimal cost obtained is given by:

EXAMPLE.–

Ce = 20,000

Cp = 200

Cr = 10,000

Q = 10,000

P = 2%

The resulting optimal number of products for POS2 test is n* = 150. The total corresponding cost is ~ 420,000.

7.3.3. Summary of the POS2 test

It should be recalled that the objective of the POS2 test is to estimate the effectiveness of burn-in. The number of products subjected to burn-in is set in advance, and therefore this parameter cannot be changed. Neither can the proportion “p” of defects; it simply remains the number of products at each POS2 stage to optimize the probability of having a failure in this test.

If no failure is observed after the third stage of POS2 testing, several hypotheses can be formulated to explain this fact:

• – The manufacturing process is excellent (p = 0.1%), and few defects are generated during this stage. The previous figure shows that with 100 products having undergone POS2 testing, there is only 10% chance of observing a failure. No conclusion can be drawn on burn-in, except that it may be useless. This situation must be confirmed by the absence of youth failures with the final user.
• – Burn-in is not at all effective and does not filter out any latent defects. Theoretically, there are 2 solutions:
  • - wait for feedback information that confirms whether or not there are youth failures observed by the end user;
  • - or, modify the levels of physical contributions implemented for burn-in, as they are not considered serious enough.

The general rule is to wait for the operational feedback information.

7.4.1. Precipitation stage

The precipitation stage is conducted with the equipment in operation and under continuous monitoring. However, there is no need to stop the test during the precipitation stage if there are functional drifts. As a general rule, the temperature limits, HOL and LOL, during the precipitation stage are defined using the results of HALT with a 10°C reduction applied to the HOT and LOT limits:

• – high operational limit: HOL = UTOL – 10°C;
• – low operational limit: LOL = LTOL + 10°C.

A slightly different definition can sometimes be found in which the temperature limits are defined based on HALT results with a temperature reduction applied to the UTOL and LTOL limits of the HALT test defined as follows:

• – high operational limit: HOL = HOT – 10% x HOT;
• – low operational limit: LOL = LOT + 10% x LOT.

In the case of thermal cycling (type RCT), the temperature ramps are fixed at the value found during the tests for the VRT limit search; this can go up to 60°C/min in the chamber per piece of equipment (10°C/min locally on the burnt-in material). These temperature ramps are instructions of the HALT/HASS chambers (QUALMARK type). It is important to instrument any equipment confined without forced ventilation. In general, the maximum value reached is 35 to 45°C/min at the product level.

The vibration limit HV during the precipitation stage is determined from the HALT test and is decreased by 50%: HV = HVOL/2. It is important to instrument the equipment when several pieces of equipment are in the chamber in order to verify the homogeneity of vibration levels after installations.

7.4.2. Detection stage

In the detection stage, the constraints are generally applied at the specification level, making it possible to evidence obvious defects. The detection stage is conducted with the equipment under operation and under monitoring. The high and low limits of temperature during the detection stage rely on the maximal operational specification levels, and there is no notion of margin.

The temperature ramps can be weak, for example, 3°C or 10°C per minute if the equipment is to be tested during these ramps. The higher limit of vibration is reduced by 60 to 80% of the value fixed in precipitation, with the definition of a minimum level generally of 5 Grms, below which there is no descent.

The applied vibration can have a sawtooth form, ascending and descending by 5 Grms to X% of the value fixed in precipitation or in a constant manner.

At the end of each level, there is an on/off stage that may involve the rated voltages and currents as well as, in certain cases, with the minimum, maximum levels of voltage and current.

Figure 7.9 proposes an example of the HASS cycle that combines temperature levels and random vibrations.

The overall HALT/HASS method is illustrated in Figure 7.10.

7.5.1. Constraints extrinsic to the equipment manufacturer

Several constraints extrinsic to the equipment manufacturer can be considered:

• – recommendation of the system;
• – operational context (life profile, production volume, etc.);
• – a diminished brand image in front of the client;
• – the seriousness of the operational life profile;
• – existence of strong competition.

7.5.2. Constraints intrinsic to the equipment manufacturer

Several constraints intrinsic to the equipment manufacturer can be considered:

• – recommendation by the company strategy;
• – cost constraints in the case of poor reliability of the product being operated by the client;
• – very demanding reliability objectives;
• – the quantity of products to be delivered.

Hence, the economic criteria and the manufacturing constraints guide the choice for or against a burn-in process.

7.5.3. Decision criteria

The overview diagram given in Figure 7.11 proposes a certain number of qualitative manufacturing criteria that may influence the decision of the equipment manufacturer to implement a burn-in process. Other criteria and constraints may be involved in the final decision (see below).

• – Final user specification:

The final user can, according to their own experience, require a burn-in test to be conducted and include it in the specifications they send to the manufacturer. In this case, it is a contractual demand that states the necessity for the manufacturer to conduct a burn-in test. This situation is typically encountered in the case of space equipment, in applying the ECSS standards required by the users. If not specified by the final user, the choice in conducting a burn-in test is made according to the other criteria given in Figure 7.11.

• – Safety constraints:

  If the manufactured equipment/product is such that:

  • - a high reliability level of the full product is required (typically civilian avionics);
  • - if it is defective due to a youth failure, it endangers the operation of the complete product, a burn-in process should be implemented, which will eliminate a maximum number of youth defects in order to reach the required reliability level.
• – Production volume:

  If the manufacturer has the elements described in section 7.5.3 available, it is possible to calculate if implementing a burn-in process is of interest, depending on the number of parts to be manufactured. Indeed, depending on the coverage rate, the effectiveness of the burn-in process and the proportion of products with latent defects, the minimal quantity to be produced before detecting a defect may be higher than the overall production. In this latter case, implementing a burn-in process presents no interest. If the manufacturer does not have sufficient elements, the choice of conducting a burn-in process is made according to the other criteria in Figure 7.2.

• – System equipment recommendation:

  If the equipment/manufactured product is integrated by a system manufacturer before delivery to the final client, it can, depending on its own experience, or in response to client demand, require the implementation of a burn-in and include it in the specifications it sends to the manufacturer.

• – Feedback from the manufacturer:

  In the absence of demands or external elements leading to the implementation of a burn-in process, the feedback from the manufacturer on similar products/equipment can be used to guide its choice. If for a product/equipment of similar complexity, involving identical manufacturing processes, burn-in implementation has previously revealed a level of latent defects that was unacceptable with respect to the reliability level expected for the new product, a burn-in process would logically be once again implemented.

• – Interest of burn-in implementation:

  This analysis can be conducted when parameters such as coverage rate, effectiveness and proportion of products with latent defects are known or estimated.

  The precondition for burn-in implementation is to have the possibility to observe products with youth defects during burn-in.

  Several parameters are involved in this case:

• – the volume N (quantity) of products to be delivered. This is a purely deterministic quantity known in advance;
• – the coverage rate τc of the monitoring implemented during burn-in. This is the capacity of burn-in to be able to observe a product failure. This quantity is estimated;
• – burn-in effectiveness τeff, meaning its capacity to transform a latent failure into an obvious one;
• – the proportion p of products having a latent failure induced by the manufacturing process. This quantity is difficult to estimate precisely, as it may vary throughout the manufacturing process (variability of means, operators, components, etc.).

The number of products Q for which a manufacturing defect can be revealed can then be estimated by:

[7.11]

The proportion of products Pr for which a manufacturing defect is observed is then:

[7.12]

For the probability of observing defects revealed by burn-in to be as high as possible, it is our proposal to estimate the lower bound of this proportion Pr for a given risk level α (Gau 2010).

This is given by:

[7.13]

where qF is the quantile of Fisher–Snedecor distribution (Gau 2010).

The quantile of a distribution law is a reciprocal function of the corresponding distribution function. The quantity of the Fisher–Snedecor law can be easily calculated, for example, using Excel with the function “INVERSE.LOI.F”.

Hence, if N = 100, α = 10% and Q = 2, the proportion of defects amounts to 2%, but the lower bound at risk α = 10% amounts to 0.36%. The lower bound of the number of products having a youth defect revealed by the burn-in can now be estimated by burn-in from the following relation:

[7.14]

EXAMPLE.–

Given the following data:

• – 20 < N < 100,000
• – τ_c = 90%
• – τ_eff = 80%
• – 1% < p < 10%

The curve shown in Figure 7.12 is obtained.

Hence, estimating that p = 5%, a burn-in presents interest provided that the number of defective products is above 100. Moreover, for a total of 100,000 products installed, even with a defect proportion of 1%, 677 defective products could be detected over the 1,000 products generated by the manufacturing process.

Obviously, while the amount of products to be delivered is known, the proportion of defective products cannot be known in advance. However, a reasonable estimation is possible due to the experience with products from previous generations. Setting a minimum of 10 potentially defective products, the curve shows that the least number to be delivered are:

• – 200 products if the proportion of defects is 10%;
• – 500 products if the proportion of defects is 5%;
• – more than 1,000 products if the proportion of defects is 2%;
• – more than 2,000 products if the proportion of defects is 1%.

7.7.1. No burn-in test is conducted

The overall cost induced by the products depends on:

• – the repair cost C1 during operation of products having youth defects or:
  
• – the repair cost C2 during operation of products having no youth defects or:
  

The overall cost is therefore given by:

or even:

[7.16]

The average number of breakdowns N2(t) during operation for the products without manufacturing defects can be calculated using the Poisson process theory. Indeed, the observed catastrophic breakdowns can be modeled by an exponential distribution of parameter λ. Considering the effect of maintenance, a homogeneous Poisson process is obtained, whose average number of breakdowns at instant “t” is given by:

The average number of breakdowns N1(t) during operation for the products with manufacturing defects can be calculated using the renewal process theory (Bayle 2019) and is given by:

This can be illustrated with the following diagram:

7.7.2. Burn-in test is conducted

As already noted, the set of products can be divided into two categories:

• – those that have youth defects, and their number is p.N;
• – those that have no youth defects (healthy products), and their number is (1 - p).N.

Among the products having a latent youth defect, a certain number will be detected, while others will not, because the coverage rate of the tests is generally not 100%. Figure 7.13 illustrates the various possible cases for a product.

The overall cost C(t) induced by the products depends on:

• – burn-in cost given by:
  
• – operation cost given by:
  

The overall cost with burn-in is: C_AD (t) = Cost_Burnin(t) + Cost_Operation(t) or still:

To compare these two costs, it is important to study the difference between them. Let us note this difference by ΔC(t) = C_AD(t) − C_SD(t) or:

The calculation of N1 is complex and requires knowledge on the parameters of the Weibull law modeling youth breakdowns. To simplify the theoretical approach, the following approximation can be made (as they are differentiated only by a youth failure):

Hence: