TTC Uncertainty

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Introduction to Measurement Uncertainty

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- 2 -

Overview IntroductionStatistics Concept & Descriptive StatisticsDefinitions

Measurement Uncertainty

Type A EvaluationsType B Evaluations Putting It All Together RSS Reporting Uncertainty Special Cases

Example Uncertainty BudgetSummaryReferences


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- 3 -

1999ISO/IEC 17025

Introduction



- 4 -

Statistics Concept & Descriptive Statistics

(),

(Sample Statistic)(Mean)(Median)(Mode)(Standard Deviation)

(Trimmed Mean)

(Weighted Mean)


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- 5 -

Sample Concept

Population

Sample



- 6 -

Sampling and Statistics 3 kinds of sampling methods

Simple random sampling () Stratified random sampling () Cluster sampling ()

Any numerical summary from a sample is a statisticand eachone has a different sampling distributionthat describes itstheoretical behavior.

This theoretical behavior provides the guidelines for theinference process.

The most common type of sampling is simple random samplingwhere every item in the population is equally likely to be selected.


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- 7 -

TAF Policy

ISO95%

()



- 8 -

TAF Policy

/

///


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- 9 -

TAF Policy

4.3.1.2



- 1 0 -

TAF Policy

(c)

/,

,(b)

(a)


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- 1 1 -

TAF Policy4.3.1.2

/50%

(4.3.2)

(4.3.2)



- 1 2 -

Error

Random

Systematic

Definitions

Error


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- 1 3 -

Random Errors component of the uncertainty of measurement which, in the

course of a number of measurements of the same measurand,

varies in an unpredictable way.

Noise (random noise)

Careless measurements

Low resolution instruments Dropped digits

Definitions

Random Errors



- 1 4 -

Definitions

Systematic Errors

Systematic Errors() component of the uncertainty of measurement which, in the

course of a number of measurements of the same measurandremains constant or varies in a predictable way.

Mis-calibrated instrument

Unaccounted cable loss


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- 1 5 -

Definitions

Low P recisionLow Accuracy

Low PrecisionHigh Accuracy

High PrecisionLow Accuracy

High P recisionHigh Accuracy

Errors Data Sheet

(Precision) :Random Error (Accuracy) :System Error



- 1 6 -

Definitions

Correction / Correction Factor:The value / the numerical factor by which the

uncorrected result of a measurement is

multiplied to compensate for an assumedsystematic error.

Confidence level:The probability of the accumulated error of a

measurement being within the stated range ofuncertainty of measurement


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- 1 7 -

Repeatability

Closeness of the agreement between the results ofsuccessive measurements of the same measurand

carried out subject to all the following conditions.

the same method of measurement; the same observer; the same measuring instrument; the same location; the same conditions of use; repetition over a short period of time



- 1 8 -

Reproducibility

Closeness of agreement between the results ofmeasurements of the same measurand, where theindividual measurements are carried out changing

conditions such as:

method of measurement;observer;measuring instrument; location;conditions of use; time.


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- 1 9 -

Error

(Uncertainty) ?



- 2 0 -

For Random Errors(

)For Systematic Errors

error


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- 2 1 -

SS standardstandard WW

work piecework piece

II instrumentinstrument PP personperson EE environmentenvironment MM methodmethod



- 2 2 -

Type A

pdfISO GUM 2.3.2Type B

Confidence level

ISO GUM 2.3.3


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- 2 3 -

AB



- 2 4 -

- Type Axinxij

xiA

==

n

j ijx

nix

1

1

( ) 21

)(1

1i

n

j

iji xxn

x

= =

( )n

x ix

i

)( =

ANOVA


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- 2 5 -

- Type B The probability distribution of the measured quantity describes

the variation in probability of the true value lying at anyparticular difference from the measured or assigned result.

The form of the probability distribution will often not be known,and an assumption has to be made, based on prior knowledgeor theory.

Choose the expected statistical distribution and determine itsstandard uncertainty

Note: Contribution must be in terms of the variation in the measuredquantity, not the influence quantity.



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Type B Evaluations There are a number of common distributions for uncertainty

contributions:

Normal distribution:

Examples:

Results of Type A evaluations Expanded uncertainties of components

where Ui is the expandeduncertainty of thecontribution and k is thecoverage factor (k = 2 for95% confidence).

-4 -3 -2 -1 0 1 2 3 4

68%

99.7%

95%

k

Uu ii =


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- 2 7 -

Type B Evaluations Rectangular distribution measurement result has an equal

probability of being anywhere within the range of ai to ai.

Examples:

Equipment manufacturer accuracy values (not fromstandard uncertainty budget)

Equipment resolution limits.

Any term where only maximalrange or error is known.

-2ai

-ai

0 ai

2ai

100%

3

ii

au =

Standard Uncertainty:



- 2 8 -

Type B Evaluations U-shaped distribution measurement result has a higher

likelihood of being some value above or below the medianthan being at the median.

2

ii

au =

-2ai

-ai

0 ai

2ai

Examples:

Mismatch (VSWR)

Distribution of a sine wave

Resistors (Culling)



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Type B Evaluations Triangular distribution non-normal distribution with linear fall-

off from maximum to zero.

-2ai

-ai

0 ai

2ai

100%


6

ii

au =

Examples: Alternate to rectangular or

normal distribution whendistribution is known to peakat center and has a knownmaximum expected value.



- 3 0 -

Each Distribution Types

-2ai

-ai

0 ai

2ai

Normal Distribution U-Shaped Distribution Triangular Distribution

-2ai

-ai

0 ai

2ai

-2ai

-ai

0 ai

2ai

Rectangular Distribution


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ISO GUM 2.3.4

(Sensitivity Coefficients)

ii XfC = /

=

=N

i

ic uu1

2

)( iii xucu =

The combined standard uncertainty is assumed to have a normal distribution.



- 3 2 -

(coverage factor) k(yU)(yU)23(y-Y)/uc(y)k1.9695 k2.57699

)(yukU c=


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- 3 3 -

1.73100%399.73%

1.7199%2.57699%

1.6595%1.9695%

157.74%168.27%

()

()



- 3 4 -

/2/2/2 /2/2/2 1 - 1 -

xx

xx

xkx

k

x

xkx

k


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- 3 5 -

t

X1 , ... , Xnn .

nXXX +++ ...21 X



- 3 6 -

n(n-1)

(n-2)

ISO GUM

(Effective degree of freedom)Welch-Satterhwaite

formula

=

=n

i i

i

ceff

x

xv

1

4

4

)(u

)(u


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- 3 7 -

t t



- 3 8 -

t

t ( 20)0)

t ( 10)0)

00

zz,, tt


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- 3 9 -

t Degrees Area in Upper Tail

of Freedom .20 .10 .05 .025 .01 .005

. . . . . . .

50 .849 1.299 1.676 2.009 2.403 2.678

60 .848 1.296 1.671 2.000 2.390 2.660

80 .846 1.292 1.664 1.990 2.374 2.639

100 .845 1.290 1.660 1.984 2.364 2.626

.842 1.282 1.645 1.960 2.326 2.576



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Reporting Uncertainty The standard uncertainty is the common term used for

calculations. It represents a 1 span (~68%) of a normaldistribution.

Typically, measurement uncertainties are expressed as an

Expanded Uncertainty, U=k uc, where kis the coverage factor. A coverage factor of k=2 is typically used, representing a 95%

confidence that the measured value is within the specifiedmeasurement uncertainty.

Reporting of expanded uncertainties must include both theuncertainty value and either the coverage factor or confidenceinterval in order to assure proper use.


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- 4 1 -

eff

Step8

=Uc * K

KUC=

Ui2Uiciui

U

UAUB

A,B

Ui2

ui*ci

(

,T )

(A,B)

Step9

Step7

Step6

Step5

Step4

Step3

Step2

Step1



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For Type A analyses with only a small number ofsamples, the standard coverage factor is insufficientto ensure that the expanded uncertainty covers the

expected confidence interval. Must use variable kp.

RSS math works for values in dB! However,distribution of a linear value may change whenconverted to dB.

Uncertainties typically always determined inmeasurement output units.

Special Cases

2.002.052.132.282.372.432.522.652.873.314.5314.0kp

50201087654321N-1


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- 4 3 -

Uncertainty Calculations

0.111Convert to reflection coefficient

1.25Max SWR (Frequency dependent)

6. Mismatch

4.08%System Contribution, coverage factor of 2

4%5. Sensor Calibration Uncertainty

0.03%Zero set / Power = 300nW / 1mW

4. Zero Uncertainty (Mode and frequency dependent)

0.8%Meter Uncertainty

0.8%Instrumentation linearity

30nW/100uW = 0.03%Noise=Measurement noise x free

run multiplier = 50nW x 0.6 = 30nW

3. Calculate meter uncertainty

1GHz2. Frequency

1mW1. Power level



- 4 4 -

Uncertainty Calculations

4.46%8. Expanded Uncertainty (k=2)

2.23%

7. Combined measurement uncertainty for k=1

0.115Convert to reflection coefficient

1.26Max DUT SWR (Frequency dependent)

2

rss

2

SensorDUTc

2

sys

2

)(Max)(MaxU

+

=


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- 4 5 -

Example Uncertainty BudgetContribution Source Value Unit Distribution u_j (dB)

Mismatch: Transmit Side 0.00 dB U-Shaped 0.00

Analyzer Output Port Source Reflectivity Manufacturer -35.00 dB

Analyzer Output Port VSWR 1.04

Antenna Input Port VSWR (1775-2000) Measured 1.45

Antenna Input Port Reflectivity -14.72 dB

Cable Loss (S21 & S12) Measured 8.00 dB

Mismatch: Receive Side 0.01 dB U-Shaped 0.00

Analyzer Input Port Load Reflectivity Manufacturer -42.00 dB

Analyzer Input Port VSWR 1.02

Antenna Output Port VSWR Measured 1.35

Antenna Output Port Reflectivity -16.54 dB

Cable Loss (S21 & S12) Measured 3.00 dB

Network Analyzer Measurement Uncertainty Manufacturer 0.40 dB Rectangular 0.23

(Full Two-Port Calibration, 50 dB path loss, Wide Dynamic Range device)

Transmit Cable Loss Variation Measured 0.05 dB Rectangular 0.03

(Due to flexing, etc.)

Mounting Accuracy: Reference Antenna Calculated 0.00 Rectangular 0.00

Antenna Mounting (PLS Laser Aligned & Custom Mounts) 0.13 inches

Range length 14.50 feet

Reference Antenna Gain Uncertainty Manufacturer 0.22 dB Normal 0.11

Miscellaneous Uncertainty CTIA 2.1 G.13 0.20 dB Normal 0.10

Total Uncertainty, u_c Type B RSS 0.28

Expanded Uncertainty, U k = 2 0.55Validity Range: 1775-2000 MHz



- 4 6 -

Summary

Accuracy, PrecisionRepeatability

Microsoft Word


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- 4 7 -

References1. NIST Technical Note 1297-1994, Guidelines for Evaluating

and Expressing the Uncertainty of NIST MeasurementResults, Barry N. Taylor and Chris E. Kuyatt.

2. NIS-81, The Treatment of Uncertainty in EMCMeasurements, NAMAS

3. ISO/IEC Guide 17025, General requirements for thecompetence of testing and calibration laboratories.

4. Introduction to Measurement Uncertainty, Michael Foegelle,ETS-Lindgren

Thanks for Your Attention !

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