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\begin{document}

\title{Design of Experiments: application to the reliability assessment of MEMS
devices - (Revision 2)~~}



\author[1]{Matteo Macchini}%
\author[1]{Maxime Auchlin}%
\author[1]{Alessio Mancinelli}%
\author[2]{Ivan Marozau}%
\affil[1]{EPFL}%
\affil[2]{CSEM (advisor)}%


\vspace{-1em}



  
  \date{May 08, 2024}


\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup





\selectlanguage{english}
\begin{abstract}
This report presents the application of the Plackett-Burman design of
experiment to a reliability study of a 3-axis commercial accelerometer
based on MEMS (MicroElectro-Mechanical System). The aim is to observe
the lifetime of the devices undergoing shock testing at 4,500g, after
having been through different environmental conditions supposed to be
representative of the space environment. In the study, 7 different
parameters are analyzed: the first step is environmental, with
temperature and humidity being varied. The second step consists in the
application of a mechanical solicitation, in the form of a vibration
test with varying frequencies and peak accelerations. The considered
response is the number of cycles of a sequence of shocks along all axes
of the device, necessary to produce~total failure. The experimental
results are used in order to compute the relative half-effects and
evaluate the corresponding normal plots for the error analysis.%
\end{abstract}%



\sloppy


\section*{Introduction}

{\label{966536}}

Nowadays reliability qualification tests for space applications are
based on MIL, NASA or ESA standards, the purpose being to ensure that a
device will perform nominally for a specified lifetime. Those tests can
help to understand which are the root-causes of the failure, in order to
mitigate them. However almost every testing procedure relies on the
survival rate while considering a single external constraint.

\par\null

It is useful to point out, in a first discussion, that in their standard
form they might fail to accurately represent real operation conditions,
where loads of different nature can simultaneously stress the device.
For example, the storage and then preparation on the launch pad in
French Guiana (the European spaceport) consists in a heterogeneous set
of tests, from a sequence of temperature and humidity oscillations, to
intense vibrations and shock in order to simulate~the launch and the~
separations of the different rocket's stages .~

\par\null

This work therefore provides a first view of the 8-runs 7-factors
application of the Plackett-Burman design. Two models are considered to
represent the experience: the first takes into account the main effects
and the second takes into account the interactions. The aim is to
observe any possible effect of the succession of thermal and mechanical
stress tests.~ A matrix of alias is computed in order~to reveal any
hidden interactions. Finally, a selection of effects and secondary
interactions is performed as well as an evaluation of their possible
random behavior. Figure 1 depicts the strategy used:

\par\null\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/MinMap/Reliability-of-a-MEMS-accelerometer}
\caption{{Mind map of the strategy.
{\label{533301}}%
}}
\end{center}
\end{figure}

\par\null

\section*{Experimental part}

{\label{986445}}

\subsection*{Choice of experience and
design}

{\label{861166}}

The testing method is based on a previous study -- named MEMS-REAL --
from CSEM on the reliability assessment of MEMS-based components for
space applications. A new method was developed, based on ESA and US
military standards, for components that are to be dedicated for space
applications.~ The method from MEMS-REAL is explained in the following
paragraph.

\par\null

Initially, a restricted set of five samples are tested over cycles with
increasing load (\emph{i.e.} 20\% greater temperature range, vibrations
intensity, etc.) starting from the manufacturer's nominal range of
operation, until complete failure occurs. This first step is necessary
to define the limit to which the devices can be pushed to. The load can
be provided by thermal cycling, thermal shocks, mechanical cycling
(\emph{i.e.} vibrations), mechanical shocks or pressure cycling. These
stress tests are chosen in order to represent, as well as possible, the
conditions that the device will undergo during spaceborne
operations.~The second step consists in the application of a load that
represents 25\% of the maximum value obtained from the first test. This
time, 20 samples are necessary and one evaluates of the number of cycles
necessary to reach 100\% of failure of the devices under a given unique
load.~ A Weibull statistics is then drawn and the characteristic
lifetime of the chosen device is defined.

\par\null

MEMS-REAL's method features three drawbacks:~

\begin{itemize}
\tightlist
\item
  a great number of samples is needed for each company willing to
  evaluate and then qualify a product for space,
\item
  performing of time consuming experiments on this large number of
  samples,
\item
  no consideration of possible interactions.~
\end{itemize}

These points can supposedly be mitigated by a good design, hence the
choice of a Plackett-Burman design as exploratory test plan. This report
therefore aims at verifying the applicability of the design of
experiments to MEMS reliability assessment.

\par\null

\subsection*{Test vehicle~}

{\label{684160}}

Japanese manufacturer Murata is producing the SCA-3100 with capabilities
summarized in Figure 2:

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.49\columnwidth]{figures/Mu-specs/Mu-specs-small}
\caption{{Murata's 3-axis MEMS accelerometer's characteristics.
{\label{805232}}%
}}
\end{center}
\end{figure}

The polymeric package is filled with a gel, designed to damp excessive
vibration and shocks, hence protecting the silicon subsystems and
metallic wire-bondings from fracture. This feature made the sample
particularly tough to brake during the vibration test campaign run in
MEMS-REAL. A property also found in the present study, even with the
devices having been through a first step of thermal testing.

~

\subsection*{Equipment~}

{\label{541257}}

The experimental part has been done at CSEM in Neuch\selectlanguage{ngerman}âtel, in
the~\emph{Advanced Manufacturing and Components Reliability} group.~The
focus has been given to environmental testing (namely thermal cycling
and humidity ingress combined), vibrations testing and mechanical
shocks.

\par\null

The environmental chamber ESPEC SH-662 permits to cycles between -60°C
to +180°C at maximum ramp-up rates of 2.5°C/min and ramp-down 1.7°C/min
. Humidity can be controlled over the range 15°C to 85°C for 85\%RH.
Outside of these bounds the humidity cannot be controlled.

For the vibrations testing, a low-force shaker for payloads up to 25kg
is used. The Brüel \& Kjær LDS V555 enables vibrations of frequencies up
to 2400 Hz and accelerations up to 100g when empty. With the test jig
and sample attached, the maximum load achievable is limited to 55g. A
reference accelerometer is screwed on the test jig, enabling active
control of the frequency and the acceleration.

\par\null

Finally, the response of the experiment is set to be the number of
mechanical shocks at 4,500g necessary to lead to a complete failure of
the sensor. The shock testing machine, a Shinyei PST-300, is made of a
pendulum, on the backplate of which the sample is fixed . The pendulum
can be lifted at a certain angle and then released, leading it to fall
and come hit a bumper placed at 0°, with a given kinetic energy. The
bumper can be chosen of different materials in order to cover a range of
5 to 10,000g, with a 2\% variation over reproducibility and a reference
accelerometer acquires the magnitude of the shocks for control.

\par\null

\subsection*{Test plan}

{\label{823830}}

A Hadamard matrix for 2\textsuperscript{3} runs is obtained from Matlab
by using the appropriate function\texttt{X=hadamard(8)}. The names of
the corresponding samples are indicated next to each line:

\par\null

\begin{equation}X=
\begin{pmatrix}
\begin{array}{rrrrrrrr}
  1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
  1 & -1 & 1 & -1 & 1 & -1 & 1 &-1\\
  1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\
  1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\
  1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\
  1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\
  1 & -1 & -1 & 1 & -1 & 1 & 1 & -1
\end{array}
\end{pmatrix}
\begin{array}{c}
\text{T08}\\
\text{T06}\\
\text{T02}\\
\text{T10}\\
\text{T07}\\
\text{T05}\\
\text{T01}\\
\text{T09}
\end{array}
\end{equation}


The factors are summarised hereafter:

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.49\columnwidth]{figures/Testplan/Testplan}
\caption{{Experimental factors tested following the Hadamard matrix. Due to
swapped humidity conditions, factor C (humidity initially) has been
redefined as the level of dryness, being equivalent to 1-(Relative
Humidity). ``atm'' defines ambient (atmospheric) humidity of the
laboratory.
{\label{912647}}%
}}
\end{center}
\end{figure}

The experimental procedure is separated over 3 phases: the
thermal-environmental phase with the combination of factors A, B, C and
D. Following is the mechanical step with factors E, F and G over the
three axes of the device (X-Y-Z sequentially). Finally, the samples are
fixed on the back-plate of the pendulum of the shocks machine and
undergo series of 6x5 shocks in all of the 6 main directions of the
device. After this sequence, the sample is removed from the jig and its
capability to measure the value of gravity is checked. A device is
considered as failed either when gravity cannot be measured accurately,
either if complete failure occurs.

\par\null

\subsubsection*{First step: Thermal-environmental
testing}

{\label{421475}}

Normally, thermal cycling procedure is either dictated by a standard
(MIL-STD-883K, Method 2002), either limited by the apparatus. In the
case of this study, the limits of the environmental chamber (ESPEC
SH-662) are taken as lower and upper bound. As for the temporal
conditions of the thermal cycling procedure, it was proposed to deviate
from the standard conditions significantly. Figure 4 shows the thermal
cycles built from the matrix of test. The factors has been set on the
legacy obtained from MEMS-REAL in order not to overstress the samples in
this first step.

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/ThermalCycles/ThermalCycles}
\caption{{Depiction of the thermal cycles following the factors defined in Figure
3 and the test matrix X. A cycle consists in the solid line curves,
while the dashed part indicate the initial and end sequences.
{\label{268482}}%
}}
\end{center}
\end{figure}

Special care has been taken to guarantee the same soak time in the
chamber, meaning that different heating and cooling duration have been
adapted for each run since the mean temperatures and ranges vary. As a
consequence, the time spent in the chamber for a single cycle is
rigorously the same for any condition: 5 hours and 7 minutes. Starting
conditions are also set accordingly, so that the time needed to reach
the first set point, from ambient conditions, remains constant. It is to
be noted that if the sample has to be removed from the chamber before
the end of the maximum number of cycles,~ the chamber is opened only
when the inside temperature matches the ambient, in order not to open it
when the inside temperature is smaller than the ambient (risk of
condensation and icing).

\par\null

\subsubsection*{Second step: Vibration
testing}

{\label{425318}}

The starting frequency is set at 5 Hz and increases at a rate of 2
octaves per minute until reaching the set point. This test is inspired
from MIL-STD-883J (Method 2005.2). The software automatically tunes the
increase of~\emph{g}, since the physical limitation of the apparatus
does not allow displacements big enough to reach the acceleration set
point at low frequencies.~

\par\null

The definition of a sweep is a logarithmical round trip from 5 Hz to
either 2,000 or 2,400 Hz -- and back again to 5 Hz following the same
path. The duration of a single sweep is 8 minutes and 30 seconds. It is
important to point out that vibration testing in the MEMS-REAL study
never managed to show any failure in the accelerometers produced by
Murata. During the present study, no failure after vibrations testing
was also recorded.

\par\null

\subsubsection*{Third step: Mechanical shocks
(response)}

{\label{363148}}

The first and second steps have been scaled so that 100\% of the samples
survived until this third step. The number of shocks necessary to reach
total failure were known from the previous MEMS-REAL tests and used as
comparison (Figure 5).~

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.49\columnwidth]{figures/mems-real-murat/mems-real-murat}
\caption{{Histogram of the number of failed Murata devices from MEMS-REAL. In this
case, shock testing was applied on pristine samples. {[}1{]}
{\label{617260}}%
}}
\end{center}
\end{figure}

Once the sample is observed to give inconsistent values of the
acceleration (g-value) or no signal at all, it is considered as broken.
Another round of shocks (6x5 in all main directions) is then done in
order to confirm the non-reversibility of the failure, meaning that some
element in the component is physically out-of-order.

\par\null

\section*{Results and analysis}

{\label{486715}}

The following table shows the experimental conditions and the response:\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Results/Output}

\end{center}
\end{figure}

\subsection*{Linear model}

{\label{877860}}

Since the test plan has been built following a Hadamard matrix, the
resulting orthogonal matrix comes in handy when dealing with the
computation of the matrix of dispersion, which translate into for the
Least Square Fit algorithm:

\begin{equation}
    \widehat{a}= (X^TX)^{-1}X^TY \equiv \frac{1}{N}I_NX^TY
\end{equation}

With \emph{N=8}, \emph{X} being given in Equation 1 and \emph{Y} from
Table 2. Computing this equation gives the half-effects and their
relative half-effects of the 7 factors:

~

\begin{equation}
\begin{pmatrix}
\begin{array}{rrrrrrrr}\alpha_0\\\alpha_1\\\alpha_2\\\alpha_3\\\alpha_4\\\alpha_5\\\alpha_6\\\alpha_7\end{array}
\end{pmatrix}
=
\begin{pmatrix}
\begin{array}{rrrrrrrr}197.5\\-45.0\\-122.5\\60.0\\52.5\\-115.0\\-97.5\\100.0\end{array}
\end{pmatrix}
\Rightarrow \widehat{Y}=X_{ij} \alpha_j = \begin{pmatrix}
\begin{array}{rrrrrrrr}30\\30\\ 150\\790\\150\\90\\280\\60\end{array}
\end{pmatrix}=Y \end{equation}

By taking these effects and computing the estimator vector, one gets the
same values of the response,~\emph{i.e.} the residues are zero. These
values can be graphically represented in terms of relative half-effects
(Figure 6):\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/rhe-1x1/corr-cropped}
\caption{{Graphical depiction of the Helf Effects with standard error and Relative
Half Effects without interactions.~ ``Cst'' is the constant, ``T\_m''
the mean temperature, "\textbf{\selectlanguage{greek}Δ}\selectlanguage{english}T" stands for the range, ``RH'' for
dryness, ``N\_T'' the number of thermal cycles, ``g'' the peak
acceleration and ``f'' the frequency of the vibrations and ``N\_V'' the
number of sweeps.
{\label{248578}}%
}}
\end{center}
\end{figure}

Since there are not enough degrees of freedom left due to the choice of
design, an ANOVA table cannot be used as analysis of errors. Hence, a
normal plot is made (Figure 7).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.49\columnwidth]{figures/normal-1x1/normal-1x1-NARROW-noLine}
\caption{{Normal plot of the seven factors half effects.
{\label{700029}}%
}}
\end{center}
\end{figure}

No particular grouping of the effects occur in the plot, meaning that
mostly none of the parameters relates to a random phenomenon. The model
-- a first degree linear without interaction, would therefore look
simply like:

\begin{equation}
\widehat{Y} = a_0 + a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5 + a_6x_6 + a_7x_7
\end{equation}

\subsection*{Linear model with
interactions}

{\label{165376}}

Let be an extension of the reasoning by considering the 2-by-2
interactions in this 7-factors model:

\begin{equation}
\widehat{Y}_{2x2} = \alpha_0 + \sum_{i=1}^7{\alpha_i}x_i+\sum_{i<j}^7{\alpha_{ij}x_ix_j} + \epsilon_{2x2}
\end{equation}

The second part of the equation only takes into account until the second
order. The matrix of test is then built by multiplying the columns of
the core matrix and the resulting model matrix is named
\(X_{2x2}\) :

\par\null\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/matrix-2x2/matrix-interactions-cropped}
\caption{{Matrix of the model \(X_{2x2}\) with interactions, half effects
(HE) and relative half effects (RHE).
{\label{665963}}%
}}
\end{center}
\end{figure}

\par\null

Using the least square fit algorithm, one finds the values for the half
effects and relative half effect, as highlighted in the two bottom lines
of the table shown in Figure 8. The normal plot of the comprehensive 2x2
interactions model has been built but provided no valuable information.
In order to emphasize any ``hidden'' interactions, the matrix of aliases
(Figure 9) for these effects is computed using the equation below:

\begin{equation}
A=\frac{1}{N}X^TX_{interactions}
\end{equation}

With~\(X\) being the model matrix without interaction
and~\(X_{interactions}\) the extended part of the global matrix with
interactions (\(X_{2x2}\)). Equation 6 results in the matrix
shown in Figure 9.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=1.00\columnwidth]{figures/alias-cropped/alias-cropped}
\caption{{Matrix of aliases~\(A\) and highlighting of the non-zero
values.
{\label{871250}}%
}}
\end{center}
\end{figure}

The corresponding table of contrasts is then built by combining the
non-zero values of~\(A\) (Equation 7).

\begin{equation}
\begin{cases}
  l_0 = a_0\\      
  l_1 = a_1+a_{23}+a_{45}+a_{67}\quad\quad \small\text{(1 = Mean temperature)}\\
  l_2=a_2+a_{13}+a_{46}+a_{57}\quad\quad \small \text{(2 = Temperature range)}\\
  l_3=a_3+a_{12}+a_{47}+a_{56}\quad\quad \small \text{(3 = Dryness)}\\
  l_4=a_4+a_{15}+a_{26}+a_{37}\quad\quad \small \text{(4 = Thermal cycles)}\\
  l_5=a_5+a_{14}+a_{27}+a_{36}\quad\quad \small \text{(5 = Peak acceleration)}\\
  l_6=a_6+a_{17}+a_{24}+a_{35}\quad\quad \small \text{(6 = Max. frequency)}\\
  l_7=a_7+a_{16}+a_{25}+a_{34}\quad\quad \small \text{(7 = Vibration sweeps)}
\end{cases}
\end{equation}

One can see from Equation 7 that the interactions effects are aliased
with the main effects. Expected interactions are:

\begin{itemize}
\tightlist
\item
  It is likely that a given thermal factor (\(a_{1-4}\)) shows
  interactions with the other thermal factors. Therefore are relevant
  the thermal
  cross-factors:~\(a_{12}\),~\(a_{13}\),~\(a_{14}\),~\(a_{23}\),~\(a_{24}\),~\(a_{34}\).
\item
  It is likely that a given mechanical factor (\(a_{5-7}\)) shows
  interactions with the other mechanical factors, hence for the
  mechanical counterpart:~\(a_{56}\), \(a_{57}\),
  \(a_{67}\).
\item
  The relation between the thermal and the mechanical parameters is not
  obvious. Imagining that it may depend on combined effect of a
  thermomechanical fatigue-related phenomenon, it would concern the rest
  of the 2x2 coefficents that appear in Equation 7.~
\item
  It could be possible that the contrasts~\(l_{1-4}\) only
  comprise the thermal 2x2 cross-factors, while~\(l_{5-7}\) only
  are aliased by the mechanical 2x2 cross-factors.
\end{itemize}

Since the interactions effects are pure alias, a Pareto chart can also
be used to represent the main effects' importance. By considering only
the magnitude (absolute value) of the half effects (since a Pareto chart
by definition can only have positive values), one gets (Figure 10).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/pareto-cropped/pareto2-cropped}
\caption{{Pareto chart of the main effects of the model with interactions.
{\label{195657}}%
}}
\end{center}
\end{figure}

The values obtained for the half effects in Figure 8 permit to compute
the estimated response for the number of shocks following a model with
interactions (Equation 8).

\par\null

\begin{equation}Y_i-\widehat{Y}_i=\epsilon_{2x2}=
\begin{pmatrix}
\begin{array}{rrrrrrrr}502.5\\502.5\\142.5\\-1778\\142.5\\322.5\\-247.5\\412.5\end{array}
\end{pmatrix}
\end{equation}

These residues are greater than the response, which indicates that the
full model with interactions is not an relevant way to model the
experiment. As an extension to the experiment, de-aliasing of the
effects would be imaginable and aleviate the doubts, provided 8 more
samples through mean of a full-foldover design. This method would enable
to separate the main effects from the interactions.

\par\null

\section*{Discussion and conclusion}

{\label{751796}}

On the thermal/humidity side, the plot of relative half effects (Figure
5) shows that the range of cycling has a more detrimental effect on the
device's~ lifetime compared to the mean temperature. This is an
interesting finding, since that could imply that failure could be more
likely linked to thermal fatigue phenomena, rather than a purely
Arrhenius-based accelerated aging. Humidity (through ``Dryness'') is the
second most important factor in this part, which shows a positive impact
on the lifetime (as expected for MEMS):~ lower is the humidity, longer
is the live of the device. Dryness, Mean temperature and Range of
cycling are most likely correlated.~ Interestingly, the number of cycles
-- within the frame of this experiment --~ does not appear as a
predominant factor. Surprisingly, the greater the number of cycles, the
greater the lifetime. This outcome is consistent with the fact that a
polymeric packaging filled with a damping gel should degrade faster
compared to the ceramic/metallic counterparts. Has~ to be noted that a
full scale reliability study (50 cycles) would maybe bring to a
different result. Finally, the normal plot indicated that, most likely,
none of these factors is random.

\par\null

On the mechanical side, the global view shows that all of the three
factors (peak acceleration, maximum frequency and vibration sweeps) have
big impact on the lifetime. The acceleration peak of vibration is
slightly more detrimental than the maximum frequency. On the other side,
the increasing number of cycles seems here to also increase the lifetime
of the device.~ A second run of experiments would maybe contradict this
statement.~

\par\null

Despite the normal plot indicating that most of the points are not
normal, the model with interactions cannot represent correctly~ the
experiment, as shown by the large values of the residues. Nevertheless,
a better knowledge of the outcome of a multi-parameter reliability
assessment has been obtained. It would now be interesting, as future
development, to extend the test plan with a focus on the most severe
effects, increase the number of cycles and possibly apply the fractional
factorial design.

\par\null

\section*{References}

{\label{941418}}

{[}1{]} Swiss Center for Electronics and Microtechnology
(CSEM),~\emph{Technical Memorandum: Test programme for reliability
assessment of MEMS products, 2016.}

{[}2{]} J.-M. Fuerbringer,~\emph{Design of Experiments (Doctoral
Program~ in Robotics, Control and Intelligent Systems)}, 2016.

{[}3{]} NIST/SEMATECH e-Handbook of Statistical
Methods,~\url{http://www.itl.nist.gov/div898/handbook/}, consulted in
January 2018. ~

\selectlanguage{english}
\FloatBarrier
\end{document}

