Dynamic Variation of Contact Resistance
in Test Interfaces
Todd
Todd Sargent,
Sargent, MS
MS
Southwest
Southwest Test
Test Workshop
Workshop
June
June 10,
10, 2002
2002
You
You Can
Can Depend
Depend on
on inTEST
inTEST
Statistical Analysis doesn’t have to be limited to
standard control chart methods
• Isolate the random variable (contact resistance)
• Look at the distribution function: is it recognizable? Repeatable?
• Gaussian control charts are ubiquitously useful, but not necessarily the
best choice for all distributions.
• Gaussian control charts converge only in the limit of infinite number of
samples (the Central Limit theorem)
• Some distributions are difficult to manage, even with invocation of the
central limit theorem. Examine the distribution of the mean values of
your samples as a function of the number of samples: how quickly does
it converge to a Gaussian?
• If your variable fits a known distribution function, make use of that fact
for immediate assessment of reliability
• Control limits can be set, and judgments made, early, from smaller
sample sizes by considering the goodness of fit.
June 10, 2002
2
Force and Contact Resistance
Brush-Wellman,
http://www.brushwellman.com/www/Technical/DesignGuide/F
igure4
June 10, 2002
3
Test Fixture
June 10, 2002
4
Isolation of Dynamic Resistance Variation
••Physical
Physical mechanism
mechanism is
is spring
spring probe
probe tip
tip into
into aa via
via hole
hole
••Total
Total resistance
resistance is
is measured,
measured, and
and Rcontact
Rcontact is
is opened
opened and
and closed
closed
N
N times.
times. Resolution
Resolution == 11 milliohm.
milliohm. Rstatic
Rstatic is
is different
different for
for every
every
channel,
Rstatic) >>
channel, and
and variance(
variance(Rstatic)
>> Rcontact
Rcontact.. Contact
Contact Resistance
Resistance
<<
<< Total
Total resistance
resistance
••For
For each
each channel,
channel, the
the minimum
minimum value
value of
of N
N total
total resistance
resistance
measurements
measurements is
is subtracted
subtracted from
from all
all N
N values.
values.
••Remaining
Remaining N
-1 values
N-1
values are
are taken
taken to
to be
be the
the “contact
“contact resistance”.
resistance”.
••If
If Rcon
Rcon fails
fails our
our test,
test, we
we replace
replace the
the spring
spring probe
probe responsible.
responsible.
••Using
Using this
e are
this system,
system, w
we
are able
able to
to detect
detect cable
cable faults
faults that
that vary
vary by
by
only
’t anything
only aa few
few milliohms.
milliohms. There
There isn
isn’t
anything else
else that
that contributes.
contributes.
June 10, 2002
5
Data Example
Channel
A0
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
A11
R1
R2
0.012
0.016
0.007
0.017
0.004
0.008
0.001
0.013
0.007
0
0.017
0.013
R3
0
0.009
0.03
0
0
0.003
0.006
0
0
0.028
0
0.031
R4
0.021
0.007
0.005
0.011
0.011
0.004
0
0.014
0.013
0.01
0.015
0.01
R5
0.005
0.005
0.012
0.009
0.004
0.006
0.009
0.014
0.009
0.006
0.009
0.014
R6
0.009
0.009
0.016
0.012
0.004
0.006
0.004
0.015
0.002
0.009
0.015
0.002
R7
0.007
0.004
0
0.011
0.011
0
0.007
0.011
0.005
0.007
0.01
0.01
R8
0.001
0.009
0.008
0.008
0.013
0.002
0.006
0.019
0.005
0.006
0.013
0.009
A sample of Rcon data. Columns are repetition number, Rows
are channel. Typical # of channels > 900.
R9
0.008
0.005
0.013
0.004
0.009
0.008
0.01
0.017
0.016
0.002
0.012
0
R10
0.012
0
0.015
0.016
0.006
0.003
0.008
0.017
0.015
0.003
0.007
0.005
June 10, 2002
0.012
0.003
0.012
0.009
0.008
0.002
0.003
0.021
0.008
0.002
0.006
0.003
6
Plot the Distribution Function
Measurement Distribution
0.100
0.090
Measurement Distribution
0.080
Probability
0.070
0.060
0.050
0.040
0.030
0.020
0.010
0.000
0
5
10
15
20
25
30
35
40
45
50
Resistance (mohms)
June 10, 2002
7
Equations of some common
Probability Distributions
The Gaussian or
"Normal" distribution:
(z
G( z)
1
.e
µ)
2
The Chi-squared distribution
of degree ν:
2
2
χ ν ( z) χ 1 ( z)
2
χ 2 ( z)
....
2
χ ν ( z)
2
2 .σ
2 .π .σ
(ν
P χ2 ( z, ν )
2)
2
z
z
.e 2
µ ν
ν
ν
2
2 .Γ
2
σ
2
2 .ν
where z =random variable, µ =mean value, σ =std deviation, ν =degree
June 10, 2002
8
Scaling and Normalizing
α is a scale factor that is to be determined such that αz corresponds to
milliohms
(ν
f( α , z, ν )
2
( α .z)
α .z
2)
.e
µ α .ν
2
σ α . 2 .ν
ν
ν
2
2 .Γ
2
χ ( α , x, ν )
if x> 0 ,
f( α , x, ν )
∞
,0
The function is renormalized to
account for the scaling by α
f( α , x, ν ) d x
0
June 10, 2002
9
Differences Between Gaussian and
Chi-squared Distributions
• Gaussian (bell-shaped)
curve deviates
symmetrically about any
mean value
• Mean and variance are
independent
.5
• Chi-squared distribution
deviates asymmetrically
and is always positive
valued.
• Variance = 2 * Mean
0.154
0.4
0.2
0.15
dchisq( 5 .x , 5 ) 0.1
dnorm( x , 0 , 1 )
0.2
0.05
0
0
4
4
2
0
x
2
4
4
0
0
0
1
2
x
4
4
June 10, 2002
10
Distribution of Sample
means & std deviations
Sample Statistics Distributions
Probability
0.20
0.18
Channel Std Devn
0.16
Channel Mean
0.14
Contact Resistance Distribution
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0
5
10
15
20
25
Resistance (mohms)
30
35
40
June 10, 2002
11
Poisson (doesn’t) Fit
Poisson Comparison
0.250
Measurement Distribution
Poisson1
Probability
0.200
Poisson2
Poisson3
0.150
0.100
0.050
0.000
0
5
10
15
20
25
30
35
40
45
50
Resistance (mohms)
June 10, 2002
12
Data compared to Chi-Squared Degrees 1,2,3
Comparison of Various Degrees
0.120
0.100
Measured Data
X2 Deg 1
X2 Deg 2
Probability
0.080
X2 Deg 3
0.060
0.040
0.020
0.000
0
5
10
15
20
25
30
35
40
45
50
Resistance (mohms)
June 10, 2002
13
Goodness of Fit to Degree 5
Scale (α) varies with force
TLCI
R,0
χ α ,R,ν
1
0.1
α =
Iface
R
0.8
ν 5
χ α , R , ν 0.05
0
0
0.525
0
10
20
30
40
R
2
χ degree 5 fits best to the first 40 milliohms. Sporadic points out at >40 mohms tend
to skew the fit; it is decided to unweigh those points for the purpose of fitting.
M
Goodfit2 ( x, y )
(x
x
n=0
pchisq
Goodfit2 χv2 , TLCI
y)
<0 >
M
2
Goodfit2 ( χv , Iface ) = 0.022
n
, 100 = 1.475 .10
pchisq ( Goodfit2 ( χv , Iface ) , 100 ) = 3.147 10
40
163
51
10
163
=probability that these 2 functions would
occur at random over M data points.
BLCI FIT
June 10, 2002
14
Limits and Simultaneous Events
••Gaussian
Gaussian control
control charts
charts use
use 33 sigma
sigma limit
limit
••Compare
Compare the
the number
number of
of χχ22 deg
deg 55 std
std devns
devns that
that
correspond
correspond to
to the
the same
same probability:
probability:
pnorm( 3 , 0 , 1 ) = 0.999
qchisq ( pnorm( 3 , 0 , 1 ) , 5 )
= 6.268
2 .5
• Consider the probability that multiple events occur during one
test: Let NC=number of channels, NR=number of repetitions
• Probability of NC events occurring simultaneously, for NR
repetitions, where P(1) is probability of 1 contact:
• P(NC*NR) = P(1)^(1/(NC*NR)
June 10, 2002
15
Sample Deviation and Chart Limits
••Compute
Compute moving
moving average
average deviation
deviation from
from data
data (ex:
(ex: TLCI)
TLCI)
••Compute
Compute NSD,
NSD, the
the number
number of
of χχ22 deg
deg 55 std
std devns
devns that
that
correspond
correspond to
to P(NC*NR)
P(NC*NR) simultaneous
simultaneous events,
events, and
and scale
scale to
to
milliohms
milliohms using
using α
α fit
fit to
to product.
product.
••Compute
Compute control
control chart
chart limits
limits in
in usual
usual manner:
manner: limit
limit == mean
mean ++
NSD
NSD deviations
deviations (moving
(moving average)
average)
σ TLCI
T
α 1 . 2 .mean submatrix µ TLCI, 0 , T , 0 , 0
1
NSD
.qchisq pnorm( 3 , 0 , 1 )
1
NR .NC
σ TLCI = 2.78
16
,ν
α . 10 =
1.66
2.53
NSD = 15.051
α 1 . 2 .ν
LimitT
Limit3T
mean submatrix µ TLCI, 0 , T , 0 , 0
mean submatrix µ TLCI, 0 , T , 0 , 0
NSD.σ TLCI
T
3 .σ TLCI
T
June 10, 2002
16
Control Chart for TLCI Data
50
TLCI Chi-squared Control Chart
50
Resistance, milliohms
40
µ TLCI
T
Limit3
Limit
30
T
T
20
10
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
T
Sample, N reps, all channels
0
16
16
Mean
3 sigma
Control Limit
June 10, 2002
17
3D Chart of TLCI
TLCI Contact Resistance Distributions, 17 units
TLCI2
June 10, 2002
18
3D Charts of One BLCI Being Worked In
BLCI Contact Resistance Distributions, 6 units
BLCI2
BLCI Contact Resistance Distributions, 6 units
BLCI3
Magnified
Magnified view
view of
of 60
60 to
to 100
100 milliohm
milliohm
range.
range. Last
Last point
point == sum
sum of
of all
all values
values
greater
greater than
than 100
100 mohms
mohms
June 10, 2002
19
Series Connection
of multiple contact resistances
Placing
Placing two
two contacts
contacts in
in series
series gives
gives aa χχ22 distribution
distribution of
of degree
degree 10:
10:
0.154
0.2
0.15
dchisq α .R , 5
1
dchisq α .R , 10
1
0.1
0.05
0
0
0
1
10
20
R
30
40
40
June 10, 2002
20
Hypothetical Inferred Mechanism
•
•
Radial tolerance, r2 = x2 + y2 of x- & y- position random
variables (tolerances) is χ2 degree 2
Distance between spring probe and via centers is
described by the mating of two (χ2 degree 2) distributions:
1.
2.
•
•
The Spring probe position
The via hole position in the PCB
The sum or difference of two χ2 degree 2 distributions is χ2
degree 4
Variation of the z- coordinate by depth of spring probe may
be contributor of the fifth degree.
June 10, 2002
21
Summary
• Assumption that contact resistance distribution is described
by a Gaussian function is incorrect and seriously
underestimates the variance
• Control limits are very different when a non-Gaussian
function is used as a basis for the control chart
• Contact force affects the contact resistance distribution by
scaling the resistance variable
• Contact force does not affect the form or degree of the
contact resistance distribution.
• Two spring-probe interface contact resistances in series
are described by a χ2 distribution of degree ten.
• The mechanism of variation may be related to the pattern
tolerances in the connection planes
June 10, 2002
22