nis 81_1994 uncertainty in emc

This publication contains policy, guidance and requirements

applicable to NAMAS testing laboratories

NIS81EDITION 1 • MAY 1994

The Treatment of Uncertainty

in EMC Measurements

Contents

Section Page

1 Introduction 2

2 Concept of uncertainty 2

3 Steps in establishing an uncertainty budget 3

4 Compliance with specification 8

5 References 10

Appendix I Summary of equations used 11

Appendix II Examples of uncertainty budgets 12

1 Radiated emissions 12

2 Conducted emissions 14

3 Radiated immunity 15

Appendix III Calculation of hp 17

Appendix IV Calculation of uncertainty in logarithmic orlinear quantities 18

© Crown Copyright 1994

NAMAS Executive, National Physical Laboratory, Teddlngton, Middlesex; TW11 OLWj England

Tel: 081';;9437140 Fax: 081-943 7134 Telex: 262344 N~L G

EDITION 1 • MAY 1994 PAGE 1 OF 18

NIS 81 • TREATMENT OF UNCERTAINTY IN EMC MEASUREMENTS

1 Introduction

1. 1 The general requirements for the estimation and reporting ofuncertainty aregiven in the NAMAS Accreditation Standard, M10. General gUidance on theestimation and reporting of uncertainties in testing is given in NAMASpublication, NISBO. This publication provides specific gUidance on theapplication of the principles set out in NISBO for laboratories seeking orholding NAMAS accreditation for EMC testing.

1.2 It is recommended that laboratories follow the methods for estimatinguncertainty described in NISBO and in this publication except where themethod is defined in the test specification.

1.3 It should be noted that the small differences in uncertainty estimatesobtained using the methods given in other, similar, documents [3], [4], [5]and [6] are not considered to be significant.

lA This publication does not attempt to define what the uncertaintycontributions are, or what they should be, since these are dependent on theequipment used and the method of test. However, examples of uncertaintybudgets are given in Appendix 11 for some common EMC measurements andhave been made as realistic as possible.

2 Concept of uncertainty

2.1 When a measurement is made the result will be different from the true ortheoretically correct value. This difference is the result of an error in themeasured value and it should be the aim of the measurement process tominimise this error. In practice the extent to which this can be achieved maybe limited and a statement of uncertainty is used to reflect thequality/ accuracy of the measured result as compared with the true value.A statement of uncertainty is incomplete without an accompanyingstatement of the confidence that can be placed in the value of theuncertainty.

2.2 Uncertainties arise from random effects and from imperfect correction forsystematic effects. The recommendations ofthe InteITlational Committee forWeights and Measures (CIPM)[l], which will be followed by NAMAS, are thatuncertainty components be grouped into two categories, based on theirmethod ofevaluation. These categories are referred to as Type A and Type B.

2.3 Type A evaluation is by calculation from a series of repeated observationsand therefore includes random effects. The statistically-estimated standarddeviation is sometimes called a Type A standard uncertainty for convenience.Type B evaluation is by means other than Type A. For example, byjudgement based on data in calibration certificates, previous measurementdata, experience with the behaviour of the instruments, manufacturers'specifications and all other relevant information. This category includesuncertainties arising from systematic effects. The components evaluated by

PAGE 2 OF 18 EDITION 1 • MAY 1994


Type A and Type B methods are combined together to produce an overallvalue of uncertainty.

2.4 NIS8l. in line with other gUidance documents on uncertainty, recommendsthat the reported uncertainty is calculated from the root sum of squares ofthe standard deviations of the individual components multiplied by acoverage factor, k, of 2, which approximates to a level of confidence of 95%.If a higher level of confidence is required then k = 3 (CL of 99.7%) can beused.

3 Steps in establishing an uncertainty budget

3.1 Decide on the range of measurement to which the budget will apply.

An uncertainty budget is a list of the probable sources of error with anestimation of their uncertainty limits and probability distribution. It is likelythat some uncertainty contributions will not be the same for the completerange of the measurement and a decision has to be made about thebreakdown that will be most appropriate. A single budget covering thecomplete range may mean that a larger uncertainty is assigned than isstrictly necessary. However, this may be preferable in some cases where itis not necessary to over complicate the calculation and reporting process.Priority should be given to calculating the uncertainty in the region of thetest specification limit, or limits.

3.2 Type A evaluation ofuncertainty components.

3.2.1 Random effects result in errors that vary in an unpredictable way while themeasurement is being made or is repeated under the same conditions. Theuncertainty associated with these contributions can be evaluated bystatistical techniques from repeated measurements. An estimate of thestandard deviation, s(qJ, of a series of n readings, qk' is obtained from:

1 ~ -2-(-1) Lt (qk-q)n k-)

where q is the mean value of n measurements.

(1)

3.2.2 The random component of uncertainty can be reduced by making repeatmeasurements in the process of testing the equipment under test (EUT).This yields the standard deviation of the mean, s( ij). given by:

EDITION 1 • MAY 1994

s(q) (2)

PAGE 3 OF 18


3.2.3 Practical considerations will normally mean that the number of repeatreadings will be very small and will often be limited to only a single reading.It is satisfactory to use a predetermination of S(qk) for the measurementsystem, based on a larger number of repeats, provided the system, method,configuration and conditions etc. are truly representative of the test.However, such a predetermination will not include the contributions of theparticular EUT. The value of n to be used to obtain s( q) under thesecircumstances is the number of measurements made in the process oftesting and not the number ofmeasurements made in the predetermination.Repeat measurements should be undertaken when the measured result isclose to the specification limit.

3.2.5 A value for the random contributions of the measurement system is in anycase an essential part of the uncertainty assessment and a type A evaluationshould be made on the 'typical' processes and configuration involved in thetest. For example, in the case of open site measurements, the type Aevaluation could include reconnecting the antenna and receiver andadjusting the antenna height to maximise the receiver reading.

3.2.6 The standard uncertainty, U(xt) , of an estimate XI of an input quantity q,based on a type A evaluation is therefore:

(3)

3.3 Type B evaluation: list all the other signtficant contributions touncertainty with an estimation of their limit value.

3.3.1 Contributions to uncertainty arising from systematic effects are those thatremain constant while the measurement is being made but can change ifthemeasurement conditions, method or equipment is altered. If there is anydoubt about whether a contribution is significant it should be included inthe uncertainty budget in order to demonstrate that it has been considered.

3.3.2 Normally, all corrections that can be applied to the measured result shouldbe applied. However, in some cases it may be impractical or unnecessary tocorrect for all known errors. For example. the calibration certificate for anEMC receiver may give actual measured input results at specific readings.with an associated uncertainty. It is possible to correct subsequent readingsby using this calibration to achieve the lowest possible uncertainty. However.it is more practical to use indicated values with no corrections applied, inwhich case the manufacturer's specified uncertainty should be used,provided it has been confirmed by an accredited calibration or, where thisis not obtainable, a route acceptable to NAMAS.

3.3.3 The individual uncertainty contributions should be in terms of the variationin the quantity being measured, rather than the influence quantity, and allin the same units. Most EMC measurements are derived from readings usinglogarithmic scales (eg dBllV) , corrections for the gain or loss of systemcomponents are in dB, specification limits are generally given in dB and

PAGE 4 OF 18 EDITION 1. MAY 1994


instrument specification limits are normally in dB. In these cases it isrecommended that the uncertainty calculations are made in terms of dBs.In some cases, for example, where the addition of signals is the dominantcontribution it may be more correct to calculate the uncertainty in terms ofabsolute values, eg Vfm. The use of dB, percentages or absolute values isdiscussed in Appendix IV

3.3.4 It is relatively straightforward to assign a value to the uncertaintycontribution when there is already evidence on which to base the value,such as a calibration certificate or manufacturer's specification. In othercases there may be little or no data available and an estimation has to bemade based on experience or on other relevant published material. In suchcases it is generally safer to overestimate the size of a contribution untilmore substantial evidence is available.

3.3.5 Most contributions to uncertainty can be adequately represented by asymmetrical distribution about the nominal or measured result, for instancethe uncertainty attributed to a receiver. However, some contributions are notsymmetrical and these are most simply dealt with by calculating separatepositive and negative values for the total uncertainty. The decision onwhether this is appropriate will depend on the difference between the twovalues and the need for rigour in the uncertainty estimation. An example ofan asymmetric uncertainty is the addition of two signals at the samefrequency where the resultant is dependent on their relative phase, asoccurs with multiple reflections in a screened room and mismatchuncertainty.

3.3.6 The basis of the RSS approach relies upon uncorrelated contributions. Thejudicious selection of test equipment and measurement method can ensurethat adverse correlation between individual contributions is avoided orminimised. If adverse correlation between any contributions is known orsuspected then the most straightforward approach is to sum the standarduncertainty of these contributions arithmetically. In some situations it isnecessary to use the same items of test equipment for different steps in themeasurement process. For example, in the pre-calibration for radiatedimmunity measurements it is essential that the same transmit antenna isused for the calibration and testing.

3.4 Assign a probability distribution and detennine the standarduncertainty ofeach contribution.

The probability distribution of an uncertainty describes the variation inprobability of the true value lying at any particular difference from themeasured or assigned result. The form ofthe probability distribution will notnecessarily be a regular geometric shape and an assumption has to bemade, based on prior knowledge or theory, that it approximates to one of thecommon forms. It is then possible to calculate the standard uncertainty,u(x;) , for the assigned form from simple equations. The three maindistributions of interest to EMC measurement are normal, rectangular andU shaped.



3.4. I Normal:

This distribution can be assigned to uncertainties derived from multiplecontributions. For example, when a NAMAS calibration laboratory providesa total uncertainty for an instrument this will have been calculated at aminimum level of confidence of 95% and can be assumed to be normal. Thestandard uncertainty ofa contribution to uncertainty with assumed normaldistribution is found by dividing the uncertainty by the coverage factor, k,appropriate to the stated level of confidence.

For Normal Distributions: ut uncertaintyXI) = ---:---..::..

k(4)

where k =2 if the reported level of confidence is 95%. (Strictly speaking fora level of confidence of 95%, k = 1.96, however, the difference this makes tothe combined uncertainty is not significant)

3.4.2 Rectangular:

This distribution means that there is equal probability ofthe true value lyinganywhere between the prescribed limits. A rectangular distribution shouldbe assigned where a manufacturer's specification limits are used as theuncertainty, unless there is a statement of confidence associated with thespecification, in which case a normal distribution can be assumed.

For Rectangular Distributions: u(xt) (5)

where al

is the semi-range limit value of the individual uncertaintycontribution.

3.4.3 V Shaped:

This distribution is applicable to mismatch uncertainty[7). The value of thelimit for the mismatch uncertainty, M, associated with the power transfer ata junction is obtained from 2010glO(l±lrallrLlldB, or

100(( 1± 1rail r L1)2 - 1) % where r a and r Lare the reflection coefficients forthe source and load. As stated in para 3.3.5, mismatch uncertainty isasymmetric about the measured result, however, the difference this makesto the total uncertainty is often insignificant and it is acceptable to use thelarger of the two limits ie 2010glO(l - Irail r LIl

For V-Shaped Distributions:

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.fi(6)

EDITION 1 • MAY 1994


3.5 Detennine the combined standard uncertainty.

The combined uncertainty, Ue(y), is obtained by taking the square root of thesum of squares of the individual standard uncertainties. If any of thestandard uncertainties are not already in tenns of the measured quantitythen they should be converted using the appropriate functional relationship.Cl' for example, the uncertainty in the measurement distance on an opensite should be converted to the uncertainty in the received signal strength,then:

(7)

Any contributions with known or suspected adverse correlation should beadded together, then for m contributions:

(8)

3.6 Detennine the expanded uncertainty.

The expanded uncertainty, U, defines an interval about the measured resultthat will encompass the true value with a specified level of confidence. p%.The expanded uncertainty is obtained by multiplying the combined standarduncertainty by a coverage factor, k, thus:

U = kuclY) (9)

The level of confidence recommended by NAMAS for EMC testing is 95%which can be obtained with k = 2. However, if random error in themeasurement process is a significant proportion of the total and S(qk) hasbeen determine from a relatively small number of repeat measurements thenthe value of k will need to be increased in order to maintain the specifiedlevel of confidence. This revis.ed value of k is le" and can be obtained usingthe procedure given in Appendix Ill. The need to use le" in place of k can bedetermined by applying the following criteria:

if uc(Y} / s( q} < 3 then a value for ~ should be obtained using theprocedure given in Appendix Ill.



3.7 Reporting ofresult.

The result of the measurement, after all approprtate corrections have beenmade, is y and may typically be reported as follows:

The measured result is: y dBJlV :t U dB

for a level of confidence of approximately 95%, (k = 2).

Alternatively an overall uncertainty may be given for results that are plottedor tabulated.

4 Compliance with specification

4.1 NAMAS requirements state that when a product is tested against a declaredspecification then the report must contain a statement indicating whetherthe results show compliance or non-compliance with the specification. Thisdecision can be made by the test laboratory if the value of the test result ishigher or lower than the specification limit by a margin greater than theestimated uncertainty. A problem arises when the margin between themeasured result and the specification limit is less than the measurementuncertainty. In these cases the laboratory cannot be sure that compliance,or non-compliance, has been demonstrated.

4.2 For some products and circumstances it may be appropriate for the user tomake a judgement of compliance based on whether the measured result iswithin the specified limits, with no account taken of the uncertainty. Thisis sometimes referred to as "shared rtsk" since the end user takes some ofthe risk of the product not meeting the specification. The implications ofsuch a rtsk will vary considerably. It may be acceptable to ignoremeasurement uncertainty for non-safety crttical performance, such as theEMC immunity characteristics of radio and television for example, but whentesting a heart pacemaker or the ADS system on a vehicle the user mayrequire that the risk of the product not complying is negligible. In whichcase the uncertainty must be taken into account.

4.3 EMC testing is carried out on a very wide range of products intended for avartety of applications. It is not therefore possible, or appropriate, forNAMAS to recommend standard rules for judging compliance. If a testspecification lays down the criteria then this should be followed, but this isa rare occurrence. If there is a recognised agreement, betweenregulatory/ certification bodies and manufacturers for instance, then thiscan be followed but again this is not common. If neither the testspecification nor regulatory bodies set down rules for compliance then thegUidance given in this publication should be followed.

4.4 Ifan agreement, code of practice or specification stipulates that uncertaintycan be ignored when judging compliance then all parties should know whatthat uncertainty is. The responsibility for calculating and declaring theuncertainty rests with the test laboratory.



4.5 In the absence of any specification criteria. guidance. or code of practiceNAMAS accredited EMC test laboratories should advise the client when theuncertainty involved in the measurement makes a Judgement ofcompliancedifficult. Examples of appropriate statements are given below:

Case A Case B CaseC Case D

TT

A

1A

upper ---+- 1limit

TA

1A

1The product complies The measured result Is The measured result Is The product does not

below the specification above the specification complylimit by a margin less limit by a margin lessthan the measurement than the measurementuncertainty: It Is not uncertainty; It Is nottherefore possible to therefore possible to

determine compliance at determine compliance ata level of confidence of a level of confidence of

95%. However, the 95%. However. themeasured result measured result

Indicates a higher Indicates a higherprobability that the probability that the

product tested complies product tested does notwtth the specification comply wtth the

limit. specification limit.



5 References(1) Guide to the Expression of Uncertainty in Measurement. BIPM, IEC. IFCC,

ISO. IDPAe, IDPAP, OIML. International Organisation for Standardization,Geneva. Switzerland, ISBN 92-67-10188-9, First Edition, 1993

(2) The Expression of Uncertainty in Testing. NIS80, NAMAS (to be published)

(3) The Expression of Uncertainty and Corifidence in Measurement, NAMASPublication NIS3003, Edition 7 May 1991.

(4) Guidelines for the expression of uncertainty of measurement in calibration.WECC Doe. 19-1990.

(5) Guide to the evaluation and expression oJuncertainties associated with theresults of electrical measurements. Def Stan 0026/lssue 2 Sept 1988.

(6) Uncertainties in the measurement ofmobile radio equipment characteristics.ETSI Technical Report, ETR028, March 1992. .

(7) Hams, LA. and Warner, F.L. Re-examination oJmismatch uncertainty whenmeasuring power and attenuation. lEE Proc. Vol 128 Pt H No.l February1981.


NIS 81. TREATMENT OF UNCERTAINTY IN EMC MEASUREMENTS

APPENDIX I

Summary of equations used1. estimated standard deviation from a sample of n readings:

1 ~ - 2-(-1) L (qk - q)n k-I

2. standard deviation of the mean of n readings:

s(q)

3. standard uncertainty resulting from type A evaluation.

u(xt) = s(q)

4. standard uncertainty for contributions with normal probability distribution:

uncertaintyu(xj) = k

5. standard uncertainty for contributions with rectangular probabilitydistribution:

6. standard uncertainty for contributions with U shaped probabilitydistribution:

M

fi7. standard uncertainties in terms of the measured quantity:

u/(y) = ct•u(xt)

8. combined standard uncertainty:

m

Uc(y) E u;(y)/-1

9. expanded uncertainty:



APPENDIXll

Examples of typical uncertainty budgetsThe following examples give the likely uncertainty contributions for the more·common EMC measurements. The contributions and values are not intended toimply mandatory requirements. Laboratories should determine theuncertainty contributions for the tests they are performing. Where theuncertainty contribution is considered insignificant a '0' has been used.

Example 1

Measurement of vertically polarised field strength between 30 dBrVIm and 60dBrVIm over the frequency range 30 MHz to 1 GHz on an open area test site at 3mand lOm

Uncertainty (dB)Probability

81conlcal Antenna Log periodicContribution Distribution

Antenna

3m lOm 3m lOm

Ambient signals - - - -

Antenna factor calibration normal (k = 2) ±1.0 ±1.0 ±1.0 ±1.0

Cable loss calibration normal (k = 2) ±0.5 ±0.5 ±0.5 ±0.5

Receiver specification rectangular ±1.5 ±1.5 ±1.5 ±1.5

Antenna dlrectlvlty rectangular +0.5 0 +3.0 +0.5-0 -0 -0

Antenna factor variation with height rectangular ±2.0 ±2.0 ±0.5 ±0.5

Antenna phase centrc variation rectangular 0 0 ±1.0 ±0.2

Antenna factor frequency Interpolation rectangular ±0.25 ±0.25 ±0.25 ±0.25

Measurement distance variation rectangular ±0.6 :0.4 ±0.6 ±OA

Site Imperfections rectangular ±2.0 ±2.0 ±2.0 ±2.0

Mismatch

Receiver VRC: r l= 0.2Antenna VRC: r e= 0.67 (81) 0.3 (Lp) U-shaped +1.1 +1.1 ±0.5 ±O.5Uncertainty limits 20Log(l±r1r e) -1.25 -1.25

System repeatablllty (previous assessment Std Deviation ±0.5 ±O.5 ±0.5 ±0.5of s(q,J from 5 repeats. 1 reading on EUT)

Repeatablllty of EUT • - - - -

Combined standard uncertainty u,,(y) normal +2.19 +2.16 +2.52 +1.74-2.21 -2.20 -1.82 -1.72

Expanded uncerialnly U normal (k =2) +4.38 +4.32 +5.04 +3.48-4.42 -4.40 -3.64 -3.44



Calculation for 3m biconical antenna, positive value:

In this example it is probable that uc(y) / S(qk) > 3, unless the repeatability of theEUT is particularly poor, and a coverage factor of k = 2 will ensure that the level ofconfidence will be approximately 95%, therefore:

U = 2 uc(y) = 2 x ±2.19 = ±4.38 dB

Notes concerning example 1

1. 1 Ambient signals have not been considered in this budget since theuncertainty will be very dependent on relative signal levels and will not affectall frequencies equally. The effect on a measurement result due to ambientsignals should be assessed at the time measurement is made and, ifnecessary, the uncertainty should be increased.

1.2 The antenna and cable will reqUire traceable calibrations for which anuncertainty would have been estimated using NAMAS recommendations 1•

based on a normal probability distribution with k = 2.

1.3 The receiver uncertainty would probably be obtained from themanufacturer's specification for which a rectangular distribution would haveto be assumed.

1.4 The antenna factor uncertainty does not take account ofantenna directivity.Unless a detailed analysis is made of all the variables and their effect on thereceived signal, an estimation has to be made of the limit values and arectangular distribution assumed. The angle ofincidence with respect to theantenna bore sight will generally be greater for a 3 m range. Since theantenna calibration is with respect to bore sight it is reasonable to assumethat the actual signal strength will not be less than the indicated reading,but could be higher by an unknown amount, resulting in an asymmetricuncertainty .

1.5 The antenna factor may vary with height and since the height will not alwaysbe the same in use as when the antenna was calibrated an additionaluncertainty is required. A calibration certificate from NPL will normally givegUidance on the value for this contribution.

1.6 The phase centre for log periodic antennas will vary with frequency and isnot accounted for in calibration.

1.7 The uncertainty in the measurement distance will be relatively small but willhave some effect on the received signal strength. The increase inmeasurement distance as the antenna height is increased is an inevitableconsequence of the method reqUired by most radiated emission specificationstandards and is therefore not considered to be a contribution touncertainty.



1.8 Site imperfections are difficult to quantify but may include the followingcontributions:

-unwanted reflections from adjacent objects.-ground plane imperfections: reflection coefficient, flatness and edge effects.-losses or reflections from "transparent" cabins for the EUT or site coverings.-earth currents in antenna cables (mainly effects biconical antennas).

The specified limits for the difference between measured site attenuation andthe theoretical value (±4 dB) need not be included in total since themeasurement ofsite attenuation includes uncertainty contributions alreadyallowed for in this budget, such as antenna factor.

1.9 The contribution from repeatability of the EUT needs to be assessed at thetime. It would be reasonable to base this on repeat measurements at oneor two frequencies. There is no need to make an accurate assessment of therepeatability of the EUT if the results are clearly well within the specificationlimit.

Example 2

Measurement of conducted emissions between 30 dBpV and 60 dBpV over thefrequency range 9 kHz to 30 MHz.

g

Probability Uncertainty (±dB)Contribution Distribution

9 kHz - 150 MHz 150 - 30 MHz

Receiver specification rectangular 1.5 1.5

LlSN coupling specification rectangular 1.5 1.5

Cable and Input attenuator calibration normal (k = 2) 0.3 0.5

Mismatch

Receiver VRC: f l= 0.03LlSN VRC: f a= 0.8 (9 kHz) 0.2 (30 MHz)

Uncertainty limits 20Log(l±f.fa) U-shaped 0.2 0.05

System repeatablllty (previous assessment of standard dev. 0.2 0.35S(qk) from 10 repeats. 1 reading on EUTl

Repeatabillty of EUT • - -

Combined standard uncertainty u.,(y) normal 1.26 1.30

Expanded uncertainty U normal (k = 2) 2.5 2.6

Calculation for 9 kHz to 150 kHz ran e:

'" ± + 0.22

+ 0.22 '" ± 1.26 dB-y

As with example 1 it is probable that uc(y) / S(qk) > 3 and k = 2 will suffice.therefore:



U = 2 x Uc(y) = 2 x ±1.26 = ±2.5 dB


2.1 It is probable that there will be an attenuator at the receiver input, providinga low VRC, so even though the LISN output is not close to son, particularly atlow frequencies, the mismatch uncertainty is relatively insignificant.

2.2 Since the two budgets produce almost the same total it would be sensible toquote a single figure covertng the whole frequency range, say ±2.6dB.

Example 3

Radiated ElectIic Field Immunity Measurements at 3 V/m and 30 MHz to 300 MHz.

Uncertainty (±VIm)Contribution Distribution

30 MHz - 300 MHz

Field strength monitor calibration (±1.0 dB) normal (k = 2) 0.37

System repeatablllty (previous assessment of s(qJ std. deviation. 0.3from 5 repeats. 1 reading on EUn

Repeatabtlily of EUT • -

Combined standard uncertainty normal 0.35

Expanded uncertainly normal (k = 2.4) 0.84

Calculation of combined standard uncertainty:

(0.37)2 + 0.32 = 0.35 V/m2

since uc(y) / S(qk) < 3 Appendix III was used to obtain a value for kp

0.354

veJf = -0-.-3-:-4 --0-.-1-8-5....,..4--:r- + --00-

= 7.4

where the number of readings to obtain S(qk) was 5, giving vt = 4

from the table in Appendix III le" = 2.4, therefore:

U = 2.4 x 0.35 = 0.84 V/m




3. 1 The measurement method used for this example is a substitution techniquewhere the field strength has been pre-calibrated without the EUT present asproposed in draft versions of a revision of IEC 801-3: 1984. The standardrequires the uniformity of field strength to be between 0 and +6 dB but it isnot considered necessary to include this as a contribution to the uncertainty.

3.2 In this case the field strength monitor reading and the specification limit willbe in terms ofV/m, while the monitor calibration uncertainty and the systemrepeatability will most probably be given in dBs. It is recommended that theuncertainty is calculated in terms of V/m if this how the specification limit isdefined, however, if the calculation is made in dBs the difference in this caseis insignificant.

3.3 In order to determine the probability of compliance at a level of confidence of95% it is necessary to test at a field strength of 3.84 V/m (specification level+ 0.84 V/m). If a failure is detected at this level but not at the specificationlimit this is equivalent to case B in section 4.5, case C occurs when theproduct fails at a level of 3.0 - 0.84 = 2.16 V/m and above.

3.4 The system repeatability should be based on repeat measurements of the fielduniformity calibration, including re-positioning the transmit antenna andresetting the input power level, at a number of field monitor locations andfrequencies.

• A value for the uncertainty attributable to the equipment under test is notincluded in these examples. It will need to be considered dUring the testingand included in the uncertainty of the test if it is a significant contribution.

PAGE 16 OF 18 EDITION 1. MAY 1994

NIS 81. TREATMENT OF UNCERTAINTY IN EMC MEASUREMENTS

APPENDIX "I

Calculation of kp

When random errors in a measurement system are comparable to the systematicerrors the expanded uncertainty calculated using equation (8) may be anunderestimation, unless a large number of repeat readings have been made.

In these circumstances a coverage factor ~ will need to be obtained from the tdistribution, based on the effective degrees offreedom, velf ' of ucfy) and the requiredlevel of confidence.

The effective degrees of freedom is calculated from:

U:(y)velf = -:-----..,..---:-------:--

u~(y) ui(y) r.4(y) ~(y)--+--+-_......+--

VI v2 v3 vm

The degrees of freedom, Vt' ofthe standard uncertainties based on type B evaluationcan be assumed to be infmite in most cases. For standard uncertainties obtainedfrom a type A evaluation Vt = n - 1, where n is the number of readings used tocalculate S(qk)'

The value of ~ is obtained from t-distribution tables for the appropriate level ofconfidence. The following table gives values of~ for various degrees of freedom vel[for a level of confidence of 95%, (actually 95.45%). Values of kp for other levels otconfidence are. given in reference [1]

Veil 1 2 3 4 5 6 7 8 10 20 50 -kp 13.97 4.53 3.31 2.87 2.65 2.52 2.43 2.37 2.28 2.13 2.05 2.0

The criteria given in para 6.3 to detennine the need to use the procedure given inthis Appendix is based on the conclusion that if uc(y) / u(CJk ) > 3 and all the othercontributions are assumed to have infinite degrees of freedom, then velf > 81 (34

),

giving a value for ~ of less than 2.05, which can be approximated by k = 2.



APPENDIX IV

Calculation of uncertainty in logarithmic or linearquantitiesA general expression that describes a measurement, y, with its uncertainty, U, inrelative values, based on the product of a series of input quantities, -Xi, and theiruncertainties, u(Xj) in relative values, is given by:

y(l ± U) = x\[l ± u(xt )) • X2[l ± u(X2)).-X3[l ± u(-X3)) ....xN[l ± u(xN)) IV(l)

the uncertainty terms are:

[l ± U] = [l ± u(x\)).[l ± u(x)].[l ± u(-X3)).[l ± u(xN))

the total uncertainty can be approximately expressed as:

U = u(xt ) + u(X2) + u(-X3) .... + u(xN)

IV(2)

IV(3)

which is in a form that can be treated by the RSS approach. However, by takinglogs of equation IV(2) then:

10g[1 ± V] = 10g[1 ± u(xt )] + 10g[1 ± u(X2)] + 10g[1 ± u(-X3)] + 10g[1 ± u(xJ] IV(4)

this is a more exact expression for the total uncertainty than equation IV(3) and isalso amenable to RSS treatment. However, if u(Xil in linear terms is relatively largethen:

log[ 1 + u(xl)) :I' log[l - u(xJ] IV(5)

Whether it is correct to combine uncertainties in linear form, eg %, or logarithmicform, eg dB, will depend upon whether their probability distributions can be betterdescribed in linear or logarithmic form. If the uncertainties for the majorcontributions are supplied in terms of dB it can only be assumed that theprobability distribution that is assigned to them should also be in dBs.

In practice the difference in the calculation of u.,ly) between dBs or % is relativelysmall. In example 1 the positive value for the biconical antenna at 3m is +2.19 dBwhen calculated in dB. If the contributions in dB are converted to percentagevoltage ratios and the calculated Ue(y) converted back to dB the result is +2.13 dB,a difference of 0.06 dB. However, the two expanded uncertainties are +4.38 dB(calculated in dB) and +3.83 dB (55.5%), a diffemece of 0.55 dB.

It is recommended that if the specification limit is given in dB terms, eg dBJ.lV, andthe contributing uncertainties are mostly stated in dB, then the uncertaintycalculations should be made in dBs. If the specification limit is given in absoluteterms, eg Vfm then the calculations should be made in absolute units.

PAGE 18 OF 18 EDITION I • MAY 1994

nis 81_1994 uncertainty in emc

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