Tag Archives: Weibull distribution

Weibull Part III

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Folks,

Our discussion of Weibull Analysis continues…. Let’s say you have worked hard and assembled some SMT lead-free PCBs for thermal cycle testing. You used the best lead-free solder paste and some lead-free solder preforms as you assembled several through-hole components with the pin-in paste process.  You were a little concerned with the assembly process as the board was thermally and physically massive and the reflow process needed to be a bit above the recommended temperature and time.

The results of the thermal cycle testing are shown in Figure 1 below. You dutifully report the characteristic life (or scale) as 2,387 cycles and the first fail at 300 cycles. You were quite disappointed, as in the past similar, but slightly smaller boards, had a slightly higher scale, but more importantly, the first fail was about 1,000 cycles. Anyway, you write up your report and file it away.

Figure 1. A Weibull plot of the thermal cycle data.

Hold on! The data are screaming at you the something is going on. Look at the same data in Figure 2. Note two distinct lines shown in green. These two separate lines suggest very strongly that there are multiple failure modes. The line furthest to the right is likely the typical failure mode observed in the past. The line to the left is a new early failure mode. It could be due to something like oxidized pads or some other phenomena not seen when testing similar but smaller boards. Root cause failure analysis should be performed to try and understand to new failure mode.

Figure 2. A Weibull plot of the thermal cycle data with multiple failure modes noted.

Now for a human interest note: One of the rewarding aspects of being a professor at Dartmouth is the outstanding nature of many of the students. They are not just good academically, but often are talented artistically, athletically, etc. This point was brought home to me recently.  In a class I teach, ENGS 1: The Technology of Everyday Things, we were recently discussing the conservation of angular momentum (CoAM). One of the most striking ways to demonstrate CoAM is an ice skater’s spin. I went on the internet and could not find a good video of a spin. I then remembered that one of my former students, Julia Zaskorski, was on Dartmouth’s figure skating team. I asked her if she had a video she could share. It appears here. She is a materials science and physics major. Who knows, maybe we will see her at APEX or SMTAI in a few years.

Here is a little bio in her own words:

My name is Julia Zaskorski, and I’m a junior from Wellesley College taking part in the 12 College Exchange Program at Dartmouth. At Wellesley I am majoring in physics with the intent to pursue mechanical engineering. Despite Wellesley’s relationship with nearby MIT, Wellesley does not have its own engineering program, so I sought out the more self-contained curriculum and atmosphere at the Thayer School of Engineering.  In addition to the draw of the Thayer School, the Dartmouth Figure Skating team was also a hugely motivating factor for my exchange, as Wellesley does not have a team, let alone a rink.  I have known the coach of the Dartmouth team for several years now, and to finally see my name on the roster for the team is a dream come true.  The engineers, as well as the winter activities here in Hanover, pulled my heart to Dartmouth long before I’d ever set foot on campus. 

Cheers,

Dr .Ron

 

Weibull Analysis II: The Curse of the Early First Failure

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Folks,

In continuing our discussion on Weibull Analysis, let’s assume we assembled some SMT and through-hole PCBs with lead-free solder paste. On this board are also some bottom-side terminated (BTC) components (often called QFNs), that are also assembled with solder preforms.  A stress test is performed to test the BTCs. In such a test, the first fail in Weibull analysis is the most important data point. No matter the results of remainder of the data, these later fails cannot undo the effect of a very early first fail.

To understand this concept, let’s look at the Weibull chart below. In many high reliability applications, there may be a requirement that some small percentage of the components under test have at least some minimum reliability.

 

Figure 1.  Weibull Analysis with an Early Fail.

As an example, let’s say that 1% of the components cannot have less than 500 cycles of life.  By looking at Figure 1, we see that 1% have less than 150 cycles of life (see arrow.)  This one early outlier dramatically affects the Weibull Analysis.

However, if that outlier was removed, as seen in Figure 2, the data suggest that 1% of the components will have a life of 900 cycles. We can see the dramatic effect the first fail has on this result. Note that the first fail does not affect the “scale” or characteristic life much (2647 vs 2682). Hence, the characteristic life, is not a robust metric to use in a high reliability environment. However, the shape or slope is dramatically affected by the early fail as it changes from 2.22 to 4.23 when the early fail is “censored.”

Figure 2. Weibull analysis with the early fail removed (censored).

Why might an outlier like this exist? Almost certainly there is something unusual about the early fail. It might be something like an oxidized pad preventing good wetting of the solder. Perhaps something like this failure mode might be discovered in root cause failure analysis. However, I am typically opposed to censoring data, even with supportive failure analysis. I think the test should be done over. It is often too easy to talk yourself into accepting inconclusive failure analysis.

What is your opinion?

Cheers,

Dr. Ron

Interpreting Weibull Plots: I

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Folks,

A while ago I discussed the Weibull Distribution and its importance in electronics reliability analysis. This distribution has been used to evaluate the life of solder joints whether formed in SMT, wave, or even using solder preforms. In the next few posts, I would like to discuss how to interpret Weibull plots.

Let’s consider two Weibull plots from thermal cycle testing of lead-free solder joints as seen in Figure 1.

Figure 1. A Weibull plot of thermal cycle data for Alloy 2 and Alloy 4.

Both alloys have almost exactly the same scale, or characteristic life. You will remember that characteristic life is the number of cycles at which 63% of the test subjects fail. For Alloy 2 it is 2,593 cycles and for Alloy 4 it is slightly better at 2,629 cycles. However, these two alloys performed dramatically differently. The most striking difference is in their “spread.” We see this much greater spread for Alloy 4, when we plot a fit to the data as a normal distribution, as in Figure 2 below.

Figure 2. The best fit normal distribution plot for Alloy 2 and Alloy 4.

In the Weibull plot, the data for Alloy 2 has a very steep slope or shape factor, this indicates a tight distribution. A tight distribution is desirable as it facilitates more accurate prediction of thermal cycle life. Alloy 2 is clearly superior. So, in a Weibull distribution, not only is a large scale factor or characteristic life desired, but so is a steep slope or larger shape factor.

Next time we will talk about outliers.

Cheers,
Dr. Ron

Can Your Mortality be Modeled with Weibull Distribution?

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Folks,

In the last posting we saw how Weibull analysis helped us to determine that SACM lead-free solder (SAC 105 with about 0.1% manganese) has comparable (actually better) thermal cycle performance versus SAC 305 solder.  Software like Minitab will give us even more detailed information about the performance of the solder joints in stress testing as we see in Figure 1.

In addition to the Weibull plot, we also have the Probability Density Function (PDF), the Survival Function and the Hazard Function. The PDF tells us when it is most likely that a test board will fail in a test population, as shown by the inserted red line. We see that it is a little less than 2,000 cycles. The Survival Function shows the percent of surviving test boards. We observe that the expected life (the 50% point) is quite close to the maximum of the PDF. The Hazard Function tells us the rate at which the test boards are dropping out.  It increases with time, but there are few boars left so the PDF drops down at the end of the test, even though the fallout rate is the highest.

It is interesting (and perhaps appropriate in the wake of Halloween) to consider if human mortality follows a Weibull distribution. I used some data for the Centers for Disease Control that are a little over 10 years old for males in the US.  So, the mean life expectancy is a little low at 72 years. (I was a little lazy: the old data were a little easier to work with than new data, some conversions are needed to make it work.) The data appear in Figure 2.

As you can see, just like a solder joint, your life expectancy can be modeled quite well by the Weibull distribution.

Cheers,

Dr. Ron

Weibull Analysis of Solder Joint Failure Data II

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Folks,

Last time we introduced Weibull analysis. Let’s derive the relationships needed to calculate the slope, beta, and characteristic life, eta.

 

F(t) is the cumulative fraction of fails, from 0 to 1. By choosing Ln(t) as x and LnLn 1/(1-F(t) as y, we would expect a straight line. See the derivation above. It can be shown graphically that this fact is so. So if we plot F(t) versus t on logarithmic graph paper, the slope of the line will be beta. To determine eta, let t=eta, in the first equation below. The result is F(t) = 1-e-1 = 0.632. So the time at which 63.2% of the parts have failed, is eta, the characteristic life.

Let’s consider some data comparing SAC305 and SACM (SAC105 with about 0.1% manganese) BGA solder balls in thermal cycle testing. The primary test vehicle employed was a TFBGA with NiAu finish mounted on PCB with OSP finish. SACM is a new breakthrough soldering alloy that has better drop shock resistance than SAC105 and comparable thermal cycle performance to SAC305. The data follow. The first column is the sample number, the third and fifth columns are the number to thermal cycles to fail for SAC305 and SACM. The second and forth columns are rank of the sample number. One would think that the first number in the second column would be 100*(1/15) =6.67%, as it represents the cumulative percent of samples failed, but a slight correct factor is needed. By plotting the log log of rank as shown above (LnLn1/(1-F(t)) vs log of cycles at failure, we get the Weibull plot. The slopes of the best fit line is equal to beta and the number of cycles at rank = 63.2% is eta.

Fortunately software like Minitab 16 does the plotting and calculating of beta and eta automatically. The results are below:

We see that the shape (beta) for SAC305 is 1.76 and that of SACM is 6.09, the scale or characteristic life (eta) is 1736.8 and 2016.8 respectively.

These results are a strong vote of confidence for SACM. Its steep slope (high beta) suggests a tighter distribution, with more consistent solder joints and its characteristic life (eta) is also slightly greater.

I plan on teaching detailed workshops on this topic. I will keep you posted.

Cheers, Dr. Ron