Uncertainty evaluations in EMC measurements

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1 Uncertainty evaluations in EMC measurements Carlo Carobbi Dipartimento di Elettronica e Telecomunicazioni Università degli Studi di Firenze Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

2 Uncertainty issues with EMC measurements Large uncertainties Extensive use of log-units Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

3 Large uncertainties Budgeted uncertainties in emission measurements (CISPR ): Conducted emission: 4 db (2σ) Radiated emission: 5 db (2σ) Variability contributions due to test set-up and EUT not included Similar figures apply to the uncertainty of immunity test levels (i.e , ) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

4 Extensive use of log-units Large dynamic range (both of frequency and amplitude) implies use of log-units Measuring instruments scale, calibration factors, majority of uncertainty contributions, test limits in log-units Mathematical model: weighted sum of quantities in log-units Final result in log-units Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

5 Mathematical model: examples from CISPR Radiated emission E = V r + AF + L c + Σ n (δ n ) Conducted emission V = V r + L amn + L c + Σ n (δ n ) δ n represent a correction in log-units Corrections have: Zero expectation (in most cases) Non-negligible uncertainty Independent input quantities Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

6 Log-units and uncertainties Frequently asked questions Correctness of the use of log-units in uncertainty budgets (e.g. sum of db 2 ) Conversion of uncertainties from log-units to linear units (more frequent) and vice versa (less frequent) Mixing quantities expressed in log and in natural units Most appropriate unit for uncertainty: log or linear? How to deal with asymmetric uncertainty intervals Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

7 From log-units to linear units 9 3 Log-units: range = max min center = (max+min)/2 Linear units: range factor = max/min center = (max min) 1/2 Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

8 Transformation rules (I) Y = 10 log(x) is a quantity in log-units y is the best estimate of Y U y is the uncertainty of y: Y within y U y and y + U y with a stated coverage probability then x = 10 y/10 is the best estimate of X U x = 10 U y/10 is the uncertainty of x: X within x / U x and x U x with the same coverage probability Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

9 Transformation rules (II) y = 1/N (y 1 + y 2 + y y N ) If the arithmetic mean y is the best estimate of Y x = (x 1 x 2 x 3 x N ) 1/N then the geometric mean x is the best estimate of X Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

10 Transformation rules (III) N 1 = ( ) 2 uy ( ) yn y NN ( 1) n= 1 If the standard deviation u(y) is adopted as the uncertainty of y N 1 = ( ) 2 10log ug( x) 10log xn 10log( x) NN ( 1) then the geometric standard deviation u g (x) is the corresponding choice for uncertainty of x n= 1 Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

11 Correctness of db 2 N 1 = ( ) 2 uy ( ) yn y NN ( 1) The unit of the deviation y n y is db, thus the unit of (y n y) 2 is db 2 and that of u(y) is db db 2 may sound odd but it is correct n= 1 Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

12 Transformation rules (IV) The following intervals provide the same coverage probabilities Log-units Linear units y uy ( ) < Y< y+ uy ( ) x / u ( x) < X < x u ( x) 2 2 y 2 uy ( ) < Y< y+ 2 uy ( ) x / u ( x) < X < x u ( x) 3 3 y 3 uy ( ) < Y< y+ 3 uy ( ) x / u ( x) < X < x u ( x) g g g g g g Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

13 Log-normal pdf median/geometric std. dev. median = geometric mean median geometric std. dev. If Y follows a normal pdf then X follows a lognormal pdf Area below each tail nearly 0.16 σ y = 2 db in the case plotted in figure Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

14 Type A evaluation: positive linear quantities with large relative dispersion Transform to log-units (or use the geometric mean and standard deviation in linear units) because values lower than x δ less probable than values greater than x + δ (X must be positive, due to physical reasons), while values lower than x/k and greater than x k may be expected to occur with the same probability, then asymmetric pdf in linear units and symmetric pdf in log-units Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

15 Example You need to evaluate the best estimate and the dispersion of the electric field over the UFA ( ). You can proceed as follows: Calculate the arithmetic mean and the standard deviation of the field in log-units (transform the field meter readings from V/m to db(v/m)) Calculate the geometric mean and the geometric standard deviation of the field in linear units (V/m) You obtain the same results Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

16 Type B evaluation: conversion from linear units to log-units Quantities contributing to uncertainty and naturally expressed in linear units need to be converted to log-units Example: you assume (type B evaluation) that a correction factor should be applied to AMN impedance, which amounts to 1.0 ± 0.2 (1 std. dev). Need to convert to the corresponding log-unit additive correction. Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

17 Transformation rule Y = 20 log(x) if x is the best estimate of X and u(x) is the standard uncertainty of x, then the best estimate of Y is y = 20 log(x) [u(x)/x] 2 and the standard uncertainty is u(y) = [u(x)/x] y = ( 0.2 ± 1.7) db using the figures in the example Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

18 Asymmetric intervals in db The correction in db has asymmetric limits L( ) and L(+) Simply evaluate the best estimate of the correction as δ = ½ [L( ) + L(+)] and the standard uncertainty as u(δ) = ½ [L(+) L( )] / d d is the appropriate divisor (depends on the pdf) Formulas valid when asymmetry is low (e.g. low mismatch) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

19 Questions and answers Correctness of the use of log-units in uncertainty budgets (e.g. sum of db 2 ) Use permitted, see 11 Conversion of uncertainties from log-units to linear units (more frequent) and vice versa (less frequent) See 8, 9, 10, 12, 17 for transformation rules Mixing quantities expressed in log and in natural units Use the appropriate transformation rules and mix, need to resort to numerical techniques in some cases (not frequent) Most appropriate unit for uncertainty: log or linear? There is not a simple answer, it depends on the specific case, see however 14 and 15 for a general statement How to deal with asymmetric uncertainty intervals See 7, 12, 13, 14, 17, 18. It is intrinsic to the lin to log transformation and vice versa. Need to resort to numerical techniques in some cases (not frequent) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

20 Uncertainty evaluations in EMC measurements Carlo Carobbi Dipartimento di Elettronica e Telecomunicazioni Università degli Studi di Firenze Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

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