2.882 System Design and Analysis
February 28
Taesik Lee © 2004 What we’ll do today
• Information content for multi-FR
– Basic statistics/probability
• Allowable tolerance (linear tolerancing) vs. statistical
tolerancing
Taesik Lee © 2005
Review
Information content
dr u
P ( FR ) =
� f ( FR )dFR
dr l
dr u
I = - log2 P = - log2 P(FR) = - log2
� f (FR)dFR
dr l
|dr|
Design Range
Common Range,
AC
p.d.f.
f(FR)
System Range,
p.d.f. f(FR)
dru
drl
|sr|
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FR
Normal Distribution
We want to know this area
(probability)
p.d.f.
f(FR)
F(z), value from the standard
normal distribution function
-1s
X ~ N(m, s2)
0
1s
FR
Then, Z = (X-m)/s ~ N(0, 1)
z
1
2
3
4
5
6
phi(z)
0.84134474
0.977249938
0.998650033
0.999968314
0.999999713
0.999999999
1-phi(z)
prob between +z, -z
0.15865526
0.68268948
0.022750062
0.954499876
0.001349967
0.997300066
3.1686E-05
0.999936628
2.87105E-07
0.999999426
9.90122E-10
0.999999998
Taesik Lee © 2005
ppm
ppb
317310.5
45500.12
2699.934
63.37207
0.57421
574.21
0.00198 1.980244
HW1, #5
• System range, FR1 ~ N(m,s2)
• Design range drl =< FR1 =< dru
• l = (dru + drl)/2 – µ
Q: Information Content ?
l
p.d.f.
f(FR)
N(m,s2)
drl
(drl+dru)/2
l
u)/2 + l
drl - (dr+dr
In terms of s multiple: {drl - (drl+dru)/2 + l}/s
Taesik Lee © 2005
dru
FR
dru – (drl+dru)/2 + l
{dru – (drl+dru)/2 + l}/s
Review
Multiple FR system range
Example
� FR1
� Ø1
0
ø
� DP1
�
�
�
� =
Œ
�
œ
�
FR 2
�
º1
1ß
�
DP2
�
Design range
FR1: [-0.5 , 0.5]
FR2: [-2.0 , 2.0]
Design range
DP2
FR2
DP1
FR1
DP variation (joint pdf) is
mapped onto FR space;
only a projection is shown
here
Assuming statistical
independence between
DP1 and DP2, the joint pdf
of (DP1,DP2) is a product
of pdf(DP1) and pdf(DP2)
Taesik Lee © 2005
Detecting change in system range
“Monitoring marginal probability of each FR is not
only inaccurate but potentially misleading”
Example
Design range
� FR1
� Ø1
0
ø
� DP1
�
�
�
� =
Œ
�
œ
�
FR 2
�
º1
1ß
�
DP2
�
FR1: [-0.5,0.5]
FR2: [-2,2]
Design parameter variation
Initial
DP1: U[-1,1]
DP2: U[0,1.5]
After change
DP1: U[-1,1]
DP2: U[-1,1.6]
Taesik Lee © 2005
Information
Contents
Information
Contents
FR2
FR2
2.5
2
Joint p.d.f.
(FR1,FR2)
1.5
-1
FR2
2.6
2
2.5
1 pFR2
0.5
1
0.5
FR1
-1
FR2
Marginal
pdf (FR2)
Joint p.d.f.
(FR1,FR2)
.6
p.d.f
-1
1
Design range
Marginal
pdf (FR1)
0.5
pFR1
-1
1
-1
-2
(b)
-2
(a)
p.d.f
DP1: U[-1,1]
DP2: U[0,1.5]
Marginal
pdf (FR1)
0.5
pFR1
-1
FR1
(c)
1
FR1
DP1: U[-1,1]
DP2: U[-1,1.6]
(c)
Before DP2 change
After DP2 change
Wrong
Correct
pFR1
pFR2
pFR1× pFR2
pFR1,FR2
Before
0.5
0.9583
0.4792
0.5
After
0.5
0.9654
0.4827
0.499
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Marginal
pdf (FR2)
0.3846 p.d.f
FR1
Design range
(b)
p.d.f
0.6 p
FR2
-1
-1
-2
(a)
2.6
Review
Allowable tolerance
•
•
•
•
Defined for DP
Tolerances that DPs can take while FRs still remaining completely
inside design ranges
Unconditional tolerance
Conservative tolerancing
FR2
DP2
Ø
X
ŒX
º
Design range
O
ø
X
ϧ
FR1
DFR1
DDP1 =
A11
DFR2 - A21� DDP1
DDP2 =
A22
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Allowable tolerance area
DP1
Design range is mapped
onto DP space
Linear tolerancing vs. Statistical tolerancing
� FR1
� Ø 1 0
ø �DP1
�
�
� =
Œ
�
�
œ
�
FR2
� º0.4 1ß
�
DP2
�
Linear tolerancing
(-0.6,2.04)
Design range
FR1: [-0.6,0.6]
FR2: [-1.8,1.8]
Statistical tolerancing
DP2
(0.6,1.56)
-0.6
0.6
DP1
(-0.6,-1.56)
3sFR1 = 0.6 � sFR1 = 0.2
Therefore, sDP1 = 0.2
Allowable tolerance
Var(FR2) = 0.42Var(DP1) + 12 Var(DP2)
Thus, sDP2 = 0.5946
DP1: [-0.6,0.6]
DP2: [-1.56,1.56]
3 sDP1 = 0.6
3 sDP2 = 1.784
Taesik Lee © 2005