Dynamic Variation of Contact ResistanceDynamic Variation of Contact Resistancein Test Interfacesin Test Interfaces

Todd Sargent, MSTodd Sargent, MSTodd Sargent, MS

Southwest Test Workshop

June 10, 2002

Southwest Test WorkshopSouthwest Test Workshop

June 10, 2002June 10, 2002

You Can Depend on inTESTYou Can Depend on inTEST

June 10, 2002 2

Statistical Analysis doesn’t have to be limited to Statistical Analysis doesn’t have to be limited to standard control chart methodsstandard control chart methods

•• Isolate the random variable (contact resistance)Isolate the random variable (contact resistance)•• Look at the distribution function: is it recognizable? RepeataLook at the distribution function: is it recognizable? Repeatable?ble?•• Gaussian control charts are ubiquitously useful, but not necessGaussian control charts are ubiquitously useful, but not necessarily the arily the

best choice for all distributions.best choice for all distributions.•• Gaussian control charts converge only in the limit of infinite nGaussian control charts converge only in the limit of infinite number of umber of

samples (the Central Limit theorem)samples (the Central Limit theorem)•• Some distributions are difficult to manage, even with invocationSome distributions are difficult to manage, even with invocation of the of the

central limit theorem. Examine the distribution of the mean valcentral limit theorem. Examine the distribution of the mean values of ues of your samples as a function of the number of samples: how quicklyour samples as a function of the number of samples: how quickly does y does it converge to a Gaussian?it converge to a Gaussian?

•• If your variable fits a known distribution function, make use ofIf your variable fits a known distribution function, make use of that fact that fact for immediate assessment of reliability for immediate assessment of reliability

•• Control limits can be set, and judgments made, early, from smallControl limits can be set, and judgments made, early, from smaller er sample sizes by considering the goodness of fit.sample sizes by considering the goodness of fit.

June 10, 2002Brush-Wellman,

http://www.brushwellman.com/www/Technical/DesignGuide/Figure4

3

Force and Contact ResistanceForce and Contact Resistance

June 10, 2002 4

Test FixtureTest Fixture

June 10, 2002 5

Isolation of Dynamic Resistance VariationIsolation of Dynamic Resistance Variation

•Physical mechanism is spring probe tip into a via hole••Total resistance is measured, and Rcontact is opened and closed N times. Resolution = 1 milliohm. Rstatic is different for every channel, and variance(Rstatic) >> Rcontact. Contact Resistance << Total resistance•For each channel, the minimum value of N total resistance measurements is subtracted from all N values.•Remaining N-1 values are taken to be the “contact resistance”.•If Rcon fails our test, we replace the spring probe responsible.•Using this system, we are able to detect cable faults that vary by only a few milliohms. There isn’t anything else that contributes.

•Physical mechanism is spring probe tip into a via holePhysical mechanism is spring probe tip into a via hole••Total resistance is measured, andTotal resistance is measured, and Rcontact Rcontact is opened and closed is opened and closed N times. Resolution = 1 milliohm. N times. Resolution = 1 milliohm. Rstatic Rstatic is different for every is different for every channel, and variance(channel, and variance(RstaticRstatic) >> ) >> RcontactRcontact. Contact Resistance . Contact Resistance << Total resistance<< Total resistance••For each channel, the minimum value of N total resistance For each channel, the minimum value of N total resistance measurements is subtracted from all N values.measurements is subtracted from all N values.••Remaining NRemaining N--1 values are taken to be the “contact resistance”.1 values are taken to be the “contact resistance”.••If If RconRcon fails our test, we replace the spring probe responsible.fails our test, we replace the spring probe responsible.••Using this system,Using this system, wwe are able to detect cable faults that vary by e are able to detect cable faults that vary by only a few milliohms. There isnonly a few milliohms. There isn’’t anything else that contributes. t anything else that contributes.

June 10, 2002A sample of Rcon data. Columns are repetition number, Rows

are channel. Typical # of channels > 900. 6

Data ExampleData Example

Channel R1 R2 R3 R4 R5 R6 R7 R8 R9 R10A0 0.012 0 0.021 0.005 0.009 0.007 0.001 0.008 0.012 0.012A1 0.016 0.009 0.007 0.005 0.009 0.004 0.009 0.005 0 0.003A2 0.007 0.03 0.005 0.012 0.016 0 0.008 0.013 0.015 0.012A3 0.017 0 0.011 0.009 0.012 0.011 0.008 0.004 0.016 0.009A4 0.004 0 0.011 0.004 0.004 0.011 0.013 0.009 0.006 0.008A5 0.008 0.003 0.004 0.006 0.006 0 0.002 0.008 0.003 0.002A6 0.001 0.006 0 0.009 0.004 0.007 0.006 0.01 0.008 0.003A7 0.013 0 0.014 0.014 0.015 0.011 0.019 0.017 0.017 0.021A8 0.007 0 0.013 0.009 0.002 0.005 0.005 0.016 0.015 0.008A9 0 0.028 0.01 0.006 0.009 0.007 0.006 0.002 0.003 0.002A10 0.017 0 0.015 0.009 0.015 0.01 0.013 0.012 0.007 0.006A11 0.013 0.031 0.01 0.014 0.002 0.01 0.009 0 0.005 0.003

June 10, 2002 7

Plot the Distribution FunctionPlot the Distribution FunctionMeasurement Distribution

0.000

0.010

0.020

0.030

0.040

0.050

0.060

0.070

0.080

0.090

0.100

0 5 10 15 20 25 30 35 40 45 50

Resistance (mohms)

Prob

abili

ty

Measurement Distribution

June 10, 2002 8

Equations of some common Equations of some common Probability DistributionsProbability Distributions

The Gaussian or "Normal" distribution:

The Chi-squared distributionof degree ν:

χ ν2 z( ) χ 1

2 z( ) χ 22 z( ) .... χ ν

2 z( )

G z( ) 1

2 π. σ.e

z µ( )2

2 σ2..

µ ν

P χ2 z ν,( ) z

ν 2( )2 e

z2.

2

ν

2 Γν

2. σ

2 2 ν.

where z =random variable, µ =mean value, σ =std deviation, ν =degree

June 10, 2002 9

Scaling and NormalizingScaling and Normalizing

α is a scale factor that is to be determined such that αz corresponds to milliohms

µ α ν.

f α z, ν,( ) α z.( )

ν 2( )2 e

α z.

2.

2

ν

2 Γν

2.

σ α 2 ν..

The function is renormalized toaccount for the scaling by α

χ α x, ν,( ) if x 0> f α x, ν,( )

0

∞xf α x, ν,( ) d

, 0,

June 10, 2002 10

Differences Between Gaussian and Differences Between Gaussian and ChiChi--squared Distributionssquared Distributions

•• Gaussian (bellGaussian (bell--shaped) shaped) curve deviates curve deviates symmetrically about any symmetrically about any mean valuemean value

•• Mean and variance are Mean and variance are independent

•• ChiChi--squared distribution squared distribution deviates asymmetrically deviates asymmetrically and is always positive and is always positive valued. valued.

•• Variance = 2 * MeanVariance = 2 * Meanindependent

4 2 0 2 40

0.2

0.4

.5

0

dnorm x 0, 1,( )

44 x0 2 4

0

0.05

0.1

0.15

0.20.154

0

dchisq 5 x. 5,( )

41 x

June 10, 2002 11

Distribution of Sample Distribution of Sample means & std deviationsmeans & std deviations

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

0 5 10 15 20 25 30 35 40Resistance (mohms)

Prob

abili

ty

Channel Std Devn

Channel Mean

Contact Resistance Distribution

Sample Statistics Distributions

June 10, 2002 12

Poisson (doesn’t) FitPoisson (doesn’t) FitPoisson Comparison

0.000

0.050

0.100

0.150

0.200

0.250

0 5 10 15 20 25 30 35 40 45 50

Resistance (mohms)

Prob

abili

ty

Measurement DistributionPoisson1Poisson2Poisson3

June 10, 2002 13

Data compared to ChiData compared to Chi--Squared Degrees 1,2,3Squared Degrees 1,2,3Comparison of Various Degrees

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0 5 10 15 20 25 30 35 40 45 50

Resistance (mohms)

Prob

abili

ty

Measured Data

X2 Deg 1

X2 Deg 2

X2 Deg 3

June 10, 2002 14

Goodness of Fit to Degree 5Goodness of Fit to Degree 5Scale (Scale (αα)) varies with forcevaries with force

0 10 20 30 400

0.05

0.1TLCI R 0,

χ α 1 R, ν,

IfaceR

χ α 0 R, ν,

R

α0.525

0.8=

ν 5

χ2 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 40Goodfit2 x y,( )

0

M

n

x y( )2

x n=Goodfit2 χv Iface,( ) 0.022=

=probability that these 2 functions would occur at random over M data points.pchisq Goodfit2 χv2 TLCI 0< >, 100, 1.475 1051. 10 163=

pchisq Goodfit2 χv Iface,( ) 100,( ) 3.147 10 163= BLCI FIT

June 10, 2002 15

Limits and Simultaneous EventsLimits and Simultaneous Events•Gaussian control charts use 3 sigma limit••Compare the number of χ2 deg 5 std devns that correspond to the same probability:

•Gaussian control charts use 3 sigma limitGaussian control charts use 3 sigma limit••Compare the number of Compare the number of χχ22 deg 5 deg 5 stdstd devnsdevns that that correspond to the same probability:correspond to the same probability:

pnorm 3 0, 1,( ) 0.999=qchisq pnorm 3 0, 1,( ) 5,( )

2 5.6.268=

•• Consider the probability that multiple events occur during one Consider the probability that multiple events occur during one test: Let NC=number of channels, NR=number of repetitionstest: Let NC=number of channels, NR=number of repetitions

•• Probability of NC events occurring simultaneously, for NR Probability of NC events occurring simultaneously, for NR repetitions, where P(1) is probability of 1 contact:repetitions, where P(1) is probability of 1 contact:•• P(NC*NR) = P(1)^(1/(NC*NR)P(NC*NR) = P(1)^(1/(NC*NR)

June 10, 2002 16

Sample Deviation and Chart LimitsSample Deviation and Chart Limits•Compute moving average deviation from data (ex: TLCI) ••Compute NSD, the number of χ2 deg 5 std devns that correspond to P(NC*NR) simultaneous events, and scale to milliohms using α fit to product.•Compute control chart limits in usual manner: limit = mean + NSD deviations (moving average)

•Compute moving average deviation from data (ex: TLCI) Compute moving average deviation from data (ex: TLCI) ••Compute NSD, the number of Compute NSD, the number of χχ22 deg 5 deg 5 stdstd devnsdevns that that correspond to P(NC*NR) simultaneous events, and scale to correspond to P(NC*NR) simultaneous events, and scale to milliohms using milliohms using αα fit to product.fit to product.••Compute control chart limits in usual manner: limit = mean + Compute control chart limits in usual manner: limit = mean + NSD deviations (moving average)NSD deviations (moving average)

σ TLCITα1 2 mean submatrix µ TLCI 0, T, 0, 0,.. σ TLCI16

2.78=α 10.

1.66

2.53=

NSD 1

α1 2 ν..qchisq pnorm 3 0, 1,( )

1NR NC.

ν,. NSD 15.051=

LimitT mean submatrix µ TLCI 0, T, 0, 0, NSD σ TLCIT.

Limit3T mean submatrix µ TLCI 0, T, 0, 0, 3 σ TLCIT.

June 10, 2002 17

Control Chart for TLCI DataControl Chart for TLCI Data

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

10

20

30

40

50

Mean3 sigmaControl Limit

TLCI Chi-squared Control Chart

Sample, N reps, all channels

Res

ista

nce,

mill

iohm

s

50

0

µ TLCIT

Limit3T

LimitT

160 T

June 10, 2002 18

3D Chart of TLCI3D Chart of TLCI

TLCI Contact Resistance Distributions, 17 units

TLCI2

June 10, 2002 19

3D Charts of One BLCI Being Worked In3D Charts of One BLCI Being Worked In

BLCI Contact Resistance Distributions, 6 units

BLCI2

BLCI Contact Resistance Distributions, 6 units

BLCI3

Magnified view of 60 to 100 milliohm range. Last point = sum of all values greater than 100 mohms

Magnified view of 60 to 100 milliohm Magnified view of 60 to 100 milliohm range. Last point = sum of all values range. Last point = sum of all values greater than 100 greater than 100 mohmsmohms

June 10, 2002 20

Series ConnectionSeries Connectionof multiple contact resistancesof multiple contact resistances

Placing two contacts in series gives a χ2 distribution of degree 10:Placing two contacts in series gives a Placing two contacts in series gives a χχ22 distribution of degree 10:distribution of degree 10:

0 10 20 30 400

0.05

0.1

0.15

0.20.154

0

dchisq α1 R. 5,

dchisq α1 R. 10,

401 R

June 10, 2002 21

Hypothetical Inferred MechanismHypothetical Inferred Mechanism

•• Radial tolerance, Radial tolerance, rr22 = x= x22 + y+ y22 of xof x-- & y& y-- position random position random variables (tolerances) is variables (tolerances) is χχ22 degree 2degree 2

•• Distance between spring probe and via centers is Distance between spring probe and via centers is described by the mating of two (described by the mating of two (χχ22 degree 2) distributions:degree 2) distributions:

1.1. The Spring probe positionThe Spring probe position2.2. The via hole position in the PCBThe via hole position in the PCB

•• The sum or difference of two The sum or difference of two χχ22 degree 2 distributions is degree 2 distributions is χχ22

degree 4degree 4•• Variation of the zVariation of the z-- coordinate by depth of spring probe may coordinate by depth of spring probe may

be contributor of the fifth degree.be contributor of the fifth degree.

June 10, 2002 22

SummarySummary•• Assumption that contact resistance distribution is described Assumption that contact resistance distribution is described

by a Gaussian function is incorrect and seriously by a Gaussian function is incorrect and seriously underestimates the varianceunderestimates the variance

•• Control limits are very different when a nonControl limits are very different when a non--Gaussian Gaussian function is used as a basis for the control chartfunction is used as a basis for the control chart

•• Contact force affects the contact resistance distribution by Contact force affects the contact resistance distribution by scaling the resistance variablescaling the resistance variable

•• Contact force does not affect the form or degree of the Contact force does not affect the form or degree of the contact resistance distribution.contact resistance distribution.

•• Two springTwo spring--probe interface contact resistances in series probe interface contact resistances in series are described by a are described by a χχ22 distribution of degree ten.distribution of degree ten.

•• The mechanism of variation may be related to the pattern The mechanism of variation may be related to the pattern tolerances in the connection planestolerances in the connection planes