**Assessment of PEM Life Test Data**

In determining the suitability of parts for long life electronic
systems, the key indicator for an electrical parameters is the range of
the statistical distribution for this parameter. Circuit design needs
(only) part critical electrical parameters to be within the circuit
application allowable values so that the circuit works to its prescribed
characteristics of functionality and performance. The relation of
circuit electrical parameters to piece part electrical parameters is a
function of circuit design and margins. In addition to margins necessary
to make the circuit work, additional margins are usually dictated either
by organizational processes (for example JPL D-8545) or common
industrial best practice.

Traditionally the only criterion for life test electrical parameter
variation has been that the part after life test meets specification
electrical parameters. For many MIL spec parts this criterion need only
be met at room temperature. The sample size in life test is very
insufficient to demonstrate long term reliability (the sample size would
have to be more than 10,000 to demonstrate less than 100 fits). In some
work (not clearly reported in the technical literature) a qualitative
review of the "parameter drift" is made by a parts engineer and parts
that drift more (according to a subjective criterion) are rejected for
flight parts.

I am interested here in what can be more objectively "squeezed" out of
PEM life test data and the implications of that data to circuit
applications. I will use some common statistical analysis techniques
(not unique to this activity) to get more information from the life test
data that will allow more assurance of the ability of PEM parts to meet
circuit requirements over a long duration space mission.

In the following, parameters during life test are measured and then
changes (delta's) are then computed. These changes are then described by
sample average and sample standard deviation. In this manner the changes
induced by an electrical test (the life test circuit) viewed as somewhat
similar to a circuit operation will be decoupled from the variations in
the electrical parameters before part usage. The task is then to
determine to reasonable approximation the expected range of these
changes so that the circuit may be checked to ensure it will work with
these changes. Most parameters are expected to have small changes (see
discussion later of operational amplifier critical parameters and
anticipated dependency on transistor characteristics and expected
statistical distributions). Range determination requires some assessment
of the statistical distribution observed in the experimental data.

If the parameter changes can be fitted reasonably to a **normal
distribution** there is a wealth of statistical inference that can be
drawn from the mean and standard deviation. Note that normal
distribution has a severely decreasing population at several standard
deviations from the average ("lean tail") so that experimental data
frequently cannot be fitted to a normal distribution. The range (defined
as maximum measurement in a sample minus the minimum measurement) of
samples drawn from a normal distribution follow the W distribution (Hald
II, pp.60-61 (Table VIII)). The W distribution describes the probability
of the range of a sample n taken from a normal distribution where the
average and standard deviation are known. As an approximation the
standard deviation may be taken as the value computed from a sample of
the sample population (e.g. the sample of the experiment). Selected
values of the W distribution are:

  ----------------------- ----------------------- -----------------------
  Sample Size (n)         W (90%)                 W(99%)

  5                       3.48 σ                  4.60 σ

  10                      4.13 σ                  5.16 σ

  20                      4.69 σ                  5.65 σ
  ----------------------- ----------------------- -----------------------

If the parameter can be fitted reasonably to a **student t
distribution** with degree of freedom ν the following inferences may be
drawn:

The student t distribution drawn on probability paper is shown in Hald
p.391 (Figure 15.2) and also in Figure 15.1 on p.392. In these figures
the student t distribution is shown in normalized parameters ((x-ξ)/s
where the standard notation is used with ξ being the average and s the
standard deviation *of the sample.*

As can be seen in the figures referenced, the student t distribution may
be used to curve fit experimental data that is not skewed (e.g. is
symmetrical around the origin). The central student t distribution is
used for data with zero average (in normalized parameters). Most
importantly, the student t distribution may be used to reasonably fit
data with "fat tails" e.g. with experimental distributions where the
data points do not fall off as rapidly from the average as the case of
the normal distribution.

In Bury pp. 124-125 the variance of the student t distribution where the
average is zero (central student t) is

$\frac{\nu}{\nu - 1}$$$

and the standard deviation is the square root.

Note that the student t has no standard deviation for one degree of
freedom (e.g. it is infinite).

The standard deviation for selected cases are:

  ----------------------------------- -----------------------------------
  Degrees of Freedom (υ)              Standard Deviation

  10                                  1.05

  5                                   1.12

  4                                   1.15

  2                                   1.41
  ----------------------------------- -----------------------------------

It is important to note that the use of the student t distribution is as
a curve fitting tool against experimental data expressed normalized
parameters. Frequently the student t distribution is used to compute
confidence levels for various parameters (average and standard deviation
and confidence levels thereof) of a normal distribution. No such
implication is appropriate here.

Once data is shown to reasonably fit a student t distribution,
inferences on the range can be made. Following Hald pp. 332-333 the
"studentized range" may be compared to the range of a samples from a
normal distribution (see W function above) with the good approximation:

$q = W(1 + \frac{3}{\nu}$)

Where q is the range (studentized range) of a student t distribution
with degree of freedom υ and W is the range of a normal distribution as
discussed above. This approximation is good provided the probability
desired for q is 0.95 to 0.99. It is seen that the studentized range is
approximately twice that of a normal distribution range for the same
sample probability and sample size.

A bimodal distribution may be used to curve fit experimental data if
that data (when plotted on probability paper) is not close to a straight
line (true for normal distribution) or to a student t with different
degrees of freedom. For a bimodal distribution to be a good fit to
experimental data, that data should be fit to two straight lines of
differing slopes. The data then must be segregated into the two sets
matching the two straight lines. The average and standard deviation of
these two different samples must then be computed. Since the sample size
of the resulting distributions is smaller than for the original
experimental data set, the standard deviation for each of the two
segregated samples is usually larger than for the entire original
dataset. This means that the range is wider if a bimodal fit is used.

Many other distributions are also available for curve fitting to
experimental data. Bury contains many commonly used distributions. These
distributions typically show skew (non-symmetrical behavior around the
average). Often these distributions can still be plotted on probability
paper, however, a function of the actual change in electrical parameter
is used. For example, a log normal distribution is plotted on
probability paper by using the log of the change in electrical
parameter. These other distributions have a different relation between
range estimates then the normal or student t distributions.

If the range is a stable parameter during a life test (e.g. if interim
measurements are taken throughout the life test and the range does not
change much) then one can reasonably assume (to engineering
approximation at least) that the change of the parameter during useful
life may be characterized by the range seen during life test. Range is
only one aspect of the parameter variation that must be considered in
assuring circuit operation is successful. To illustrate this
qualitatively consider a circuit operation where an allowance has been
made for "reliability or end of life variation" so that a parameter
range is known that will be acceptable (properly circuit operation is
error budgeted to include several factors besides reliability including
radiation degradations, thermal variations, etc.). I will call the
acceptable (or error budget) for reliability being LR. Then the observed
parameter range during life test as statistically computed using curve
fitting to various distributions is acceptable provided:

$$\text{LR} \leq \text{Observed}_{\text{Range}}$$

In the distributions described above the observed range is a number
multiplied by the standard deviation of the sample. Typically this
number is between 3 and 6. The number increases with the sample size and
also with the required probability (e.g. is greater for probability of
99% than for 90%). This matches the qualitative expectations we have for
proper allowance for reliability and end of life parameter variations.

[LT1028 (Operational Amplifier, High Speed, Ultra Low
Noise)]{.underline}

Critical parameters in application are voltage offset (Vos), current
offset (Ios) (only in high impedance input circuit), and input bias
current (Ib). To gain some appreciation for the changes seen during life
test the average, the standard deviation, the minimum and the maximum of
the entire sample (10 pieces) is compared against the range of the
parameter as specified by the part manufacturer.

The observations noted here are true through the entire range of life
test measurements including initial, post burn-in, after 500 hours of
life, and after 1000 hours of life.

The change in voltage offset average runs about 10-20% of the specified
range. Standard deviation runs about 10%-25% of the specified range.
Minimum and maximum values run as much as 50% of the specified values
except for one part which is below the minimum (about 90 versus 80
microvolts). Larger changes were observed at cold temperatures (-40°C).

Changes in input bias current runs about 10-20% of the specified range.
Standard deviation runs about one quarter of the specified range.
Minimum runs between 25% and 80% of the specified value and maximum runs
less than 50% of the maximum. Again largest minimums were seen at cold
temperature.

Input offset currents averages were usually below 10% of the maximum
range and standard deviations were even less. Minimum and maximum were
lower than about 15% of the specified value.

The discussion of statistical distributions will therefore be restricted
to Vos and Ib with the former receiving the most attention.

Voltage offset changes at 500 hours and room temperature when plotted by
MATLAB on probability paper shows a bimodal distribution with negative
values at a lesser slope (corresponds to greater standard deviation) and
a tighter distribution for positive values. Voltage offset changes at
1000 hours are much better fitted to a normal distribution. Sample size
standard deviations at both times are about 30 microvolts. Inferred
range with sample size taken as 10 and probability 99% is about 5 times
the standard deviation or 150 microvolts.

Input bias current changes at 500 hours and room temperature when
plotted by MATLAB on probability paper show a good fit to a normal
distribution. The worst case fit at large positive values would be to a
student t of 10 degrees of freedom. At 1000 hours the fit to normal
distribution is about the same. Since the sample standard deviation
varies between 30 and 45 nanoamps (does not increase with life test
duration) one can use either the normal distribution or student t with
10 degrees of freedom. Using the normal distribution for a sample size
as 10 and with probability of 99% the range is expected to be about 5
times the standard deviation or 220 nanoamps. Using the student t with
10 degrees of freedom the studentized range is 5% more for a result of
about 230 nanoamps.

Examining the data at hot temperature (+85°C), voltage offset change at
500 hours appears either bimodal or has one outlier at the highest
positive value. On the other hand voltage offset fits a normal
distribution well at 1000 hours. Standard deviations at both 500 and
1000 hours are close to 8. The data at 1000 hours supports an estimate
of the range as 40 microvolts.

Input bias current at high temperature fits well to a normal
distribution at 500 hours but is bimodal at 1000 hours (negative values
have different mean but similar slope as positive values). Standard
deviations of the changes are about 5.5. Therefore the range may be
estimated as 27.5 nanoamps.

[LT1013 (Operational Amplifier, Dual, Very High Slew Rate)]{.underline}

Critical parameters in application are voltage offset (Vos), current
offset (Ios) (only in high impedance input circuit), and input bias
current (Ib). To gain some appreciation for the changes seen the
average, the standard deviation, the minimum and the maximum of the
entire life test sample (22 pieces) is compared to the range of the part
manufacturer specified parameter range (symmetrical position and
negative value for these parameters).

The observations noted here are true through the entire range of life
test measurements including initial, post burn-in, and after 1000 hours
of life (I do not have interim life test measurements for this part).

There are six voltage offset measurement changes were taken at various
power supplies and common modes. Average values of all samples were
small compared to the standard deviation. At room temperature standard
deviation was less than 0.03 millivolts (part manufacturer specification
is 1.5 millivolts). The same is true of voltage offset changes at 85°C
except the standard deviation is closer to 0.055 millivolts. At cold
temperature (-40°C) the standard deviation is less than that observed at
room temperature but the average is about the same magnitude as the
standard deviation. Since all of these numbers are much less than the
part manufacturer specification limits I will do limited discussion of
the statistical shape of the change parameters. Change parameters may be
fitted to normal distribution at room temperature reasonably well, but
at high and low temperature there are 1-2 outliers (out of the 22 piece
test sample). However, since the standard deviation is so much less than
the prudent design range, I will not further analyze this data.

Input bias current changes showed smaller (or much smaller) averages
than standard deviations at all temperatures. Standard deviations also
were very small at all temperatures (typically less than 0.003 microamps
(part manufacturer over temperature specified limits is +/- 6
microamps).

Input offset current changes shows averages much below the standard
deviation at all temperatures. Standard deviations are also very low,
the highest being less than one nanoampere (at room temperature). Part
manufacturer specification limit is 400 nanoamperes at room temperature
and 600 nanoamperes over the temperature range.

[AD8028 (Operational Amplifier, Dual, Rail to Rail input/output, High
Gain Bandwidth)]{.underline}

Critical measured parameters in application are voltage offset (Vos),
current offset (Ios) (only in high impedance input circuit), input bias
current (Ib), output voltage swing (Vout). To gain some appreciation for
the *[changes]{.underline}* seen from before life test (after burn-in)
to after life test the average, the standard deviation, the minimum and
the maximum of the entire life test sample (10 pieces) is compared to
the part manufacturer specified parameter range.

The observations noted here are true through the entire range of life
test measurements including initial, post burn-in, and after 1000 hours
of life (I do not have interim life test measurements for this part).

There are six voltage offset measurement taken during the life tests at
various power supplies and output enable conditions (including 5 volt
and 3 volt device operation. Change from post-burn-in to post life test
were computed. Then, sample average and standard deviation was computed.
Data was taken and computed at room temperature (25°C), hot (+110°C) and
cold (-40°C). Normalized parameters (change minus average sample change
divided by standard deviation) were computed with the intent that
statistical analyses would be performed if the changes in any parameter
were significant or might affect circuit application. Both averages and
standard deviations were small for all voltage offsets. At room
temperature all averages and standard deviations were below 0.01
millivolts (part manufacturer spec limit is 0.8 millivolts). At hot,
both averages and standard deviations were below 0.02 millivolts (spec
limit is 1.6 millivolts). At cold, both averages and standard deviations
were below 0.04 millivolts (spec limit is 1.6 millivolts). A detailed
statistical analysis is not warranted for such small changes. However, a
sample plot was done on probability paper (scales) using MATLAB. The
curves could be fairly well fitted to a student t distribution with 3-5
degrees of freedom. Therefore, the expected range of these parameters
was approximately 5 times the standard deviation, which is still well
below any reasonable circuit application constraint.

Input bias current changes were very small. Standard deviations also
were very small at all temperatures (typically less than 0.03 microamps,
the largest was still less than 0.2 microamps) (part manufacturer spec
limit is 6 microamps over temperature).

Input offset current changes also were very small. Over temperature, the
largest average seen was about 45 nanoamps (part manufacturer spec limit
is 900 nanoamps). Largest standard deviation seen was about 23 nanoamps.

Output voltage swing was measured by swinging the output close to the
power supply at 5 volt nominal operation. Specification is 10
millivolts. The largest average value of the change or standard
deviation of the change in this measurement was 2 millivolts.

Open loop gain showed more variability. In all reasonable circuit
applications the open loop gain is heavily reduced by large amounts of
negative feedback and is not an important application parameter.

[LT1175IS8]{.underline}

This precision voltage regulator may be modeled (as all voltage
regulators) as a forward power loop (simplified as a transistor) whose
bias point is regulated by a feedback operational amplifier (with an
unknown closed loop gain driven by internal feedback). The gain stage
(negative feedback operational amplifier) measures the difference
between the power loop voltage output and a internally generated
reference voltage (may be thought of as a zener diode). Other features
such as low voltage lockout and over current are usually also
incorporated in the voltage regulator design.

Circuit application critical parameters are line regulation (difference
between programmed or desired output voltage and actual output voltage
(at nominal load). In this part (as is typical) this is expressed as a
percent with the lower number the better. Load regulation is the change
in output voltage between no load and maximum load (current). This is
also expressed as a percent.

The feedback gain circuit (typically achieved by an operational
amplifier used in a closed loop configuration with lots of negative
feedback) is worst case at cold temperature where the gain is lower. The
load and line regulation are expected to be worst at cold temperature.
Typical space operation is above zero degrees Celsius in a good
conservative design.

Most leakage currents (pin currents) are typically worst case at high
temperature and are driven by output transistor leakage current.

All parameters were measured at post burn-in, after 500 hours of life
test, and after 2000 hours of life test. Parameters were measured at
25°C, -40°C and +85°C (which are the part manufacturer characterized
limits) and at various intermediate points. Intermediate temperature
measurements did not show qualitatively different results.

Discussion here is limited to parameters that showed significant changes
(delta's) between initial, 500 hour and 2000 hour points. To make
discussion herein reasonably short, analysis and assessment is limited
to parameters where the standard deviation is greater than 10% of the
part manufacturer specification limit. Similar to parameters for the
other parts, the average of the changes was much smaller than the
standard deviation.

The following parameters showed significant changes according to the
criterion in the last paragraph at all temperatures: sense input
current, line regulation and load regulation. In addition, a few
parameters met this criterion only at cold temperature: output initial
accuracy and quiescent current. Since the latter two parameters are not
circuit critical parameters, they will not be further discussed here.

Line regulation at 25°C and 500 and 2000 hours life test duration were
plotted on probability scales by MATLAB and show reasonable fit to a
normal distribution. Line regulation at 85°C and 500 and 2000 hours life
test duration were plotted by MATLAB on probability scales. Reasonable
fit is achieved to a student t distribution of 2 degrees of freedom.
Line regulation at -40°C was also plotted by MATLAB at 500 and 2000
hours life test duration. Reasonable fit is achieves to a student t
distribution with 4 degrees of freedom.

At 25C the range may be estimated at 5 standard deviations. Standard
deviation is 0.0026, so the range is estimated at 0.0125%. Range
estimate is .0625%. (Specified maximum value of line regulation is
0.015% so that some allowance for this parameter must be made in the
circuit design error budget).

Load regulation at both 85°C and -40°C for 500 and 2000 hours life test
durations was plotted by MATLAB on probability scales and a reasonable
fit to normal distributions were found at both temperatures. At hot,
standard deviation is about 0.07, so that predicted range is about
0.35%. (Specified maximum value of load regulation is 0.35% so that some
allowance in circuit design error budget must be made for this
parameter).

Sense current was plotted by MATLAB on probability scales at all three
temperatures. Reasonable fit to a normal distribution was seen at room
and cold. At hot temperature, student t distribution fit was achieved
for 3 degrees of freedom. At room temperature the standard deviation of
the sense current changes is about 30 nanoamps so that an estimate of
the range is about 150 nanoamps. This is also (coincidently) the part
manufacturer's specified maximum parameters value, so significant
allowance must be made for changes in this parameter in the circuit
error budget.

[AD780BR]{.underline}

This device is an ultra precise voltage regulator (based on a band gap
reference) with tight parameters guaranteed by the part manufacturer
over the industrial temperature range (-40°C to +85°C). Characterization
parameters (typically met but not guaranteed) are specified over the
full military temperature range (-55°C to +125°C). A large life test
sample was used in this testing (49 parts). This may make it easier to
fit the data to a statistical distribution. Interim measurements were
also taken during the life test. For this analysis the delta parameters
were computed between the post burn-in measurements and data taken at
250, 500 and 1000 hours.

To restrict the discussion here I have evaluated the parameters at three
temperatures: 25°C, -40°C, +85°C and +125°C. To make the discussion here
more tractable, I first evaluate which parameters have significant
changes during the life test. I examine the changes in parameters and
note that where the changes are larger in magnitude, the average is
always smaller or much smaller than the standard deviation. Therefore, I
use the criterion previously used and examine parameters where the
change standard deviation is greater than one tenth (0.1) of the part
manufacturer specified range.

No parameter had this (still pretty small) change at all three
temperatures. Parameters now identified met this criterion (or almost
met this criterion at certain life test durations and temperatures as
follows:

1)  Output voltage (nominal 3.0V) change standard deviation met this
    criterion at 250 hours life test duration and -40°C

2)  Load regulation, sink mode B, change met this criterion at 500 hours
    and 25°C

3)  Output voltage (nominal 3.0V) change met this criterion at 500 hours
    and 85°C

4)  Load regulation, sink mode B, change (almost) met this criterion at
    1000 hours and 25°C

5)  Load regulation, sink mode B, change met this criterion at 1000
    hours and -40°C

6)  Output voltage (nominal 3.0V) change met this criterion at 1000
    hours and -40°C

7)  Output voltage (nominal 3.0V) change met this criterion at 1000
    hours and +85°C

8)  Load regulation, sink modes A and B, change met this criterion at
    1000 hours and +85°C

9)  Load regulation (shunt mode) change met this criterion at 1000 hours
    and +85C

10) Output voltage (nominal 3.0V) change met this criterion at 1000
    hours and +125°C

11) Load regulation, sink modes A and B, change met this criterion at
    1000 hours and +125°C

12) Load regulation (shunt mode) change met this criterion at 1000 hours
    and +125C

13) Output voltage (nominal 2.5V) change met this criterion at 1000
    hours and +125°C

I have therefore decided to perform a statistical analysis on Output
voltage (3.0V nominal) change at -40°C at all life test durations,
Output voltage (nominal 3.0V) change at 1000 hours and +125°C, Load
regulation (sink mode B) at 25°C and all life test durations, Load
regulation (sink mode B) at all temperatures at 1000 hours life test
duration, and Load regulation (shunt mode) at 1000 hours and +125°C.

Cold temperature output voltage change fits a student t distribution
with 2 degrees of freedom. Standard deviation is at most about 0.0003.
An estimate of the range would be 7 standard deviations or 0.0021 volts.
The part manufacturer specification for this parameter is 0.002 so that
error budget should be allocated (if the circuit needs precision
regulation at this cold temperature).

Load regulation (sink mode B) at 25°C at all life test durations can be
made to roughly fit a student t distribution with 2 degrees of freedom.
Standard deviation varies with test duration but is roughly
0.07millivolts. Estimating the range as 7 standard deviations yield a
range of 0/.49 millivolts. The part manufacturer specification for this
parameter is 0.75 millivolts so allowance should be made in circuit
error budget.

Load regulation (sink mode B) at 1000 hour durations for various
temperatures may be fitted to student t distribution with 2-4 degrees of
freedom. However, there is one anomalous data point at 85°C (one device
only). Raw data was reviewed and also data at all 5 temperatures was
reviewed. Only data for this one part at 85°C was anomalous. Probably an
electrical retest would be the appropriate step if such a situation
occurs in future testing. 125°C change can be fit to a student t
distribution with 4 degrees of freedom. Comparing the predicted range to
the part manufacturer specification limit would yield a more optimistic
situation than at room temperature. However, the part manufacturer does
not guarantee the 125°C limit.

Load regulation (shunt mode) change at 1000 hours and 125°C reasonably
fits a normal distribution. Standard deviation is about 0.1 millivolts.
Estimating the range as 5 standard deviations yield 0.5 millivolts. Part
manufacturer does not guarantee a limit at this temperature but reports
a typical number of .75 millivolts.