1

Quantifying & Propagation of Uncertainty

Module 2

Lecture THREE (4-4)

3/14/05

2

What have you learned so far?

Determine Random Uncertainty in the Measurement of the Measurand Using Single measurement Using ONE sample Using M samples

Determine Overall Random uncertainty caused by Elemental Errors

Determine Total Uncertainty caused by Bias and Random uncertainties

Determine Total Uncertainty caused by more than ONE variable

3

Random & Bias ErrorsSingle Measurement Bx =0

4

Random & Bias Errors in Multiple Measurements

5

Determination of Total (Systematic and Random ) Uncertainties

21

1 1

22

k

i

m

jjix PBW

Total Systematic and Random Uncertainty Wx (RSS)

Bi’s are the systematic uncertainties caused by k elemental error sources and Pi’s are the random uncertainties caused by m elemental error sources

6

Examples

Lecture Slides (check values?) Readings (course web)

7

Mathematical Approach for Determining Uncertainties

o It allows us to study the impact of uncertainties caused by MORE THAN ONE INDEPENDENT variable on the TOTAL uncertainty on the DEPENDENT variable

o The mathematical mechanism to do this is “Partial Derivative”.

8

Calculate Total uncertainty

21

1

2

n

ii

i

dxx

RdR

dR is the Total uncertainty in the measurement of R (result-dependent variable) caused by Elemental uncertainties dxi in the variables xi (independent variables)

Using the RSS method

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Partial Derivative - Notation

The “total” change in the area is represented by the derivative dA as

dWW

AdL

L

AdA

Total Change in

area

Partial Change in area due to d W

Partial Change in area due to d L

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In Terms of Uncertainty in Measurements

dWW

AdL

L

AdA

Total Uncertain

ty

Sensitivity of A with respect to

L

Uncertainty in L

Uncertainty in W

Sensitivity of A with respect to W

11

TOTAL Uncertainty of Dependent Variable in Terms of Random and Bias Uncertainties of Independent Variables

dWW

AdL

L

AdA

Uncertainty in L

Uncertainty in W

21

1 1

22

L Lk

i

m

jLLL PBWdL

21

1 1

22

W Wk

i

m

jWWW PBWdW

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Determine Total Uncertainty

Using the RSS Formula

?)(30.28

0.4603.050

LdWdLdA2

1

21

22

22

units

W

Assume the dimensions of the rectangular (LxW= 60x50) and uncertainties in the measurements of L and W are + 0.4 mm and + 0.3 mm, respectively.

Meaning?

We are 95% Confident that True Value of Area = 3000.00 + 28.30 m2

Assuming that THERE is No Bias Errors

Other wise We are 95% Confident that

Mean Value of Area Measurements (population) = 3000.00 + 28.30 m2

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How to perform PD?

Transform the Multi-variable function to ONE variable function replace all variables with constants, except

the ONE variable that is differentiated Perform ordinary differentiation Replace back the constants with the

equivalent variables

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Example: Area A = L x W

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1 i.e ,constant assume

calculateTo

cLdL

dcL

L

cWW

LWLL

A

W

c

11

WLWLL

A

isThat

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Example: Area A = L x W

WcdW

dWc

W

cLL

LWWW

A

22

2 i.e ,constant assume

calculateTo

L

c

12

LLWWW

A

isThat

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Example: Area A = L x W

dWW

AdL

L

AdA

WLWLL

A

LLWWW

A

LdWdLdA W

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Example: Resistance

A

LR

Resistance

dAA

RdL

L

Rd

RdR

A

L

A

L

c

c

c

c

d

d

c

c

d

d

c

c

A

LR

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2

1

2

1

2

1

2

1

AAc

c

Lc

c

dL

d

c

Lc

L

A

L

LL

R

L

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for Similarly,

2

3

2

3

2

3

22

13

213

113

13

13

1

A

L

A

cc

Acc

AdA

dcc

dA

d

A

cc

dA

d

A

cc

A

A

L

AA

R

Another Differentiation

Rule?

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Example: Resistance

dAA

RdL

L

Rd

RdR

dAA

LdL

Ad

A

LdR

2

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Example: Resistance

Determine the Total uncertainty in measuring the

resistance R, for the nominal values of L = 2m,

A= 1 mm2 , and resistivity = 0.025x10-6 Ω.m. The

uncertainties in the measurement of L, A, and

resistivity are + 0.01m, + 0.1mm2, and +

0.001x10-6 Ω.m, respectively?

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Example: Electrical Resistance

dAA

RdL

L

Rd

RdR

dAA

LdL

Ad

A

LdR

2

General Formula

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Example: Electrical Resistance

21

222

dAA

RdL

L

Rd

RdR

21

22

22

22

2

dA

A

LdL

Ad

A

LdR

The RSS formula

21

22

22

22

2

dA

A

LdL

Ad

A

LdR

21

6

2

26

62

2

6

626

2

6101.0

101

210025.001.0

101

10025.010001.0

101

2

dR

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Determine Total Fractional Uncertainty Using Fractional Uncertainties of Variables

If the dependent variable R is a product of the measured variables, i.e

Nn

cba xxxCxR ....321Then, the fractional uncertainty in R is directly related to the fractional uncertainty of the variables

21

22

2

2

2

1

1 ....

n

n

x

dxN

x

dxb

x

dxa

R

dR

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The Two Forms are Equivalent

21

22

2

2

2

1

1 ....

n

n

x

dxN

x

dxb

x

dxa

R

dR

Which is equivalent to

21

22

22

2

11

.......dR

nn

dxx

Rdx

x

Rdx

x

R

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Back to Area Example Formula for Area A = L x W

WLWLL

A

LLWWW

A

LdWdLdA W

W

dW

L

dL

A

dALW

LdW

LW

WdL

LW

dA

LWBy Formula Divide

LdWWdLdA

2

22

22

mm30.28

50

3.0

60

4.0

5060

30,405060For

RSS theUsing

21

21

dA

dA

mm. dW mm. , dL, WL

W

dW

L

dL

A

dA

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Any Question?????

Any Question????Any Question????

Danke SchonTh

ank

you

Good Luck with Exam