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SEMI E35-0305 © SEMI 1995, 2005 14 x x f x LSL LSL M d 1 (5) A1-4.4.3 For a characteristic with both upper and lower sp ecifications limits, use the following equa…
SEMI E35-0305 © SEMI 1995, 2005 13
A1-3 Limitations
A1-3.1 Application of this appendix requires that all measurement variances, biases, and process characteristic
distributions be accurately characterized. If this is not done, the resulting calculation may be in error.
A1-3.2 Application of this appendix also requires that measurement variances and biases be constant over the
measurement range interval of interest. If this is not the case, the resulting calculation may be in error.
A1-3.3 This appendix considers only the case of two convolved distributions, process characteristic variability and
measurement variability; consideration of additional distributions (such as would be required if the measurement
bias or variance were assumed to be a function of the measured value) is beyond the scope of this appendix.
A1-3.4 Extension of the model to the cases of decisions based on (1) measurements on multiple gauges, (2)
multiple inspections with the same gauge, or (3) on multiple characteristics is provided in §A1-5.
A1-4 Procedure
A1-4.1 Estimate the PDF for the process characteristic to be studied. This can be done using empirical data that
represents the process. Although actual data may be used to create a discrete PDF, it is sometimes convenient to use
the data to parametrically fit a PDF model (e.g., log normal).
A1-4.2 Unless already known, establish the bias and standard deviation for each measurement gauge to be
compared in accordance with SEMI E89.
A1-4.3 In all cases take the measurement influence into account so that it does not broaden the PDF. This may be
done by taking repeated measurements at each point in the measurement range and calculating the mean, or by
deconvolving the process characteristic PDF, f(x), and the measurement variability CDF, G(u), generally assumed to
be a Gaussian (or normal) distribution with arithmetic mean equal to the bias, so that:
x
x
uuG
M
u
M
d
2
exp
2
1
2
2
(1)
where:
= bias,
M
= standard deviation of the measurement distribution.
NOTE 1: The quantity σ
M
includes the effects of the change in bias over the time interval in which σ
M
has been established.
A1-4.4 Calculate
and
as follows. Note that the symbols in the equations for
and
have the following
meanings:
f(x) = PDF of process characteristic x,
USL = upper specification limit,
LSL = lower specification limit, and
(u) = Gaussian CDF (see Equation (1) in ¶A1-4.3).
A1-4.4.1 For a characteristic with only a USL, use the following equations to calculate
and
:
xxf
xUSL
USL
M
d1
(2)
xxf
xUSL
USL
M
d
(3)
A1-4.4.2 For a characteristic with only an LSL, use the following equations to calculate
and
:
xxf
xLSL
LSL
M
d
(4)
SEMI E35-0305 © SEMI 1995, 2005 14
xxf
xLSL
LSL
M
d1
(5)
A1-4.4.3 For a characteristic with both upper and lower specifications limits, use the following equations to
calculate
and
:
USL
LSL
M
USL
LSL
M
xxf
xLSL
xxf
xUSL
d
d1
(6)
xxf
xLSLxUSL
xxf
xLSLxUSL
USL
MM
LSL
MM
d
d
(7)
A1-5 Extensions of the Methodology to More Complex Situations
A1-5.1 In general, when a single characteristic on a unit is measured once on a single gauge and 100% sampling is
employed, the model will take the form described earlier. If the situation is more complex, the nature of the model
will be different. Factors that can affect the nature of the model include the following:
number of units examined (lot acceptance sampling vs. 100% sampling),
number of times a unit is inspected (single vs. multiple),
effect of the inspection process on the unit (destructive vs. non-destructive),
number of item characteristics examined for a single decision (one vs. many), and
cost functions associated with the business decisions (fixed vs. variable).
A1-5.2 Extension to Multiple Gauges
A1-5.2.1 To extend the model to multiple gauges, one must make the additional assumption that all measuring
gauges are measuring the same characteristic.
A1-5.2.1.1 In addition, define a conforming item as one that all gauges show the measured characteristic to be in
specification.
A1-5.2.1.2 A nonconforming item is taken to be one in which at least one gauge shows the measured characteristic
to be outside of specification.
A1-5.2.2 It is then possible to define a set of
and
error rates for each gauge. Let
1
,
2
, …,
n
be the
values
and
1
,
2
, …,
n
be the
values associated with the n different gauges.
A1-5.2.3 The overall
value,
T
, is calculated from the equation:
T
=
n
i
i
1
1
(9)
where:
xxf
USL
d
, and
f(x) = PDF of the characteristic being measured.
NOTE 2: This equation is based on the probability P that the item is conforming but that one or more gauges give a conforming
result:
T
= P[Item is conforming, 1 gauges show nonconforming]
= P[Item is conforming]
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– P[Item is conforming, All gauges show conforming]
A1-5.2.4 The overall
value,
T
, is calculated from the equation:
T
=
n
i
i
1
(10)
NOTE 3: This equation is based on the probability P that the item is nonconforming but that all gauges give a conforming result:
T
= P[Item is nonconforming, All gauges show pass]
A1-5.3 Extension to Multiple Inspections with the Same Metrology System
A1-5.3.1 Multiple inspection with the same metrology system is a special case of inspection with multiple gauges.
If the same measurement system is used to measure the item characteristic repeatedly, one merely lets
i
=
and
i
=
for all i, as the
and
error rates will not change for the same gauge.
A1-5.4 Extension to Decisions Based on Multiple Characteristics
A1-5.4.1 It is also possible to develop a model where a decision is based on more than one characteristic. As with
the case of multiple gauges, several assumptions must be made.
A1-5.4.1.1 The characteristics being measured are independent or an independent combination of their values is
used.
A1-5.4.1.2 All tests are performed before a decision to reject or pass is made.
A1-5.4.1.3 F(x) has been deconvolved from F·G.
A1-5.4.1.4 A conforming unit is defined as one in which all measured characteristics are shown to be within
specification.
A1-5.4.1.5 A nonconforming unit is taken to be one in which at least one characteristic is outside of specification.
A1-5.4.2 It is then possible to define a set of
and
errors for each gauge. Let
1
,
2
, …,
n
be the
values and
1
,
2
, …,
n
be the
values associated with the n different characteristics.
A1-5.4.3 The overall
value,
T
, is calculated from the equation:
T
=
n
i
n
i
ii
11
1
(11)
where:
USL
ii
xxf d
, and
f
i
(x) = PDF of the characteristic i.
This equation is based on the probability P that all characteristics conform, but that at least one test failed:
T
= P[All characteristics conform, At least one test failed]
= P[All characteristics conform]
– P[All characteristics conform, All tests passed]
A1-5.4.4 The overall
value,
T
, is calculated from the equation:
T
=
n
i
n
i
ii
P
11
1
(12)
where:
P
i
= proportion of observations within specification for characteristic i.
This equation is based on the probability P that one or more characteristics are nonconforming but that all tests
passed:
T
= P[1 characteristic nonconforming, All tests passed]