Comparison of the GUM and Monte Carlo Methods for the Uncertainty Estimation In Electromagnetic Compatibility Testing

Progress In Electromagnetics Research B, Vol. 34, 125–144, 2011
COMPARISON OF THE GUM AND MONTE CARLO
METHODS FOR THE UNCERTAINTY ESTIMATION IN
ELECTROMAGNETIC COMPATIBILITY TESTING
M. Azpu´rua*, C. Tremola, and E. Pa´ez
Instituto de Ingenier´ıa, Laboratorio de Electromagnetismo Aplicado,
Centro de Ingenier´ıa Ele´ctrica y Sistemas, Caracas, Venezuela
Abstract—The rigorous uncertainty estimation in Electromagnetic
Compatibility (EMC) testing is a complex task that has been addressed
through a simplified approach that typically assumes that all the
contributions are uncorrelated and symmetric, and combine them in
a linear or linearized model using the error propagation law. These
assumptions may affect the reliability of test results, and therefore,
it is advisable to use alternative methods, such as Monte Carlo
Method (MCM), for the calculation and validation of measurement
uncertainty. This paper presents the results of the estimation of
uncertainty for some of the most common EMC tests, such as: the
measurement of radiated and conducted emissions according to CISPR
22 and radiated (IEC 61000-4-3) and conducted (IEC 61000-4-6)
immunity, using both the conventional techniques of the Guide to
the Expression of Uncertainty in Measurement (GUM) and the Monte
Carlo Method. The results show no significant differences between the
uncertainty estimated using the aforementioned methods, and it was
observed that the GUM uncertainty framework slightly overestimates
the overall uncertainty for the cases evaluated here. Although the
GUM Uncertainty Framework proves to be adequate for the particular
EMC tests that were considered, generally the Monte Carlo Method
has features that avoid the assumptions and the limitations of the
GUM Uncertainty Framework.
1. INTRODUCTION
Accredited test laboratories are required to estimate the uncertainty
and to report it, according to the ISO/IEC 17025 — “General require-
ments for the competence of testing and calibration laboratories” [1].
Received 18 August 2011, Accepted 7 September 2011, Scheduled 8 September 2011
* Corresponding author: Marco A. Azpurua (bazpurua@fii.gob.ve).

126 Azpu´rua, Tremola, and Pa´ez
The aforementioned standard also recognizes that the complexity of
the measurement systems may in some cases preclude a rigorous eval-
uation of uncertainty, as happens in Electromagnetic Compatibility
(EMC) Testing.
The uncertainty estimation in EMC test is complex because
commonly: the equipment that is being tested was not designed
specifically for the test; the methods usually include setup factors
that affect the measurement; the test system is itself complex and
includes several separate but interconnected components (cables,
antennas, spectrum analyzers, filters, amplifiers and preamplifiers,
attenuators, RF switches, and others) and the quantities involved
may be electromagnetic fields, varying in space, and may be transient
or continuous [2]. Such complexity has been addressed through
a conventional approach that attempts to simplify the calculation
process while maintaining practical veracity in the uncertainties
produced [3].
The conventional uncertainty analysis, stated in the Guide to
the Expression of Uncertainty in Measurement (GUM) [4], is often
difficult to implement in complex systems, such as EMC test systems,
and requires several approximations and assumptions at each stage
of processing which may cast doubt on the results of the tests.
In that sense, the analysis of uncertainty in EMC testing using
Monte Carlo Methods is an alternative to resolve many of the
problems associated to the GUM uncertainty framework, including
non-symmetrical measurement uncertainty distributions, non-linearity
within the measurement system, input dependency and systematic
bias [5]. Nowadays, the Monte Carlo Method (MCM) is recognized
as a practical alternative by the Joint Committee for Guides in
Metrology (JCGM) of the Bureau International des Poids et Mesureson
(BIPM) and it has been included in the GUM as a supplement, since
2008 [6], and it has been widely used within many scientific disciplines,
such as metrology, geodesy, optics, hydrology, electronics, structural
mechanics, and various others [7–11].
The paper is organized as follows: on Section 2 it will be described
the general methodology followed by the conventional uncertainty
estimation and then, on Section 3, it will be explained the MCM
approach used to describe measurement uncertainties in the context
of GUM, next, on Sections 4 and 5 the conventional, and the MCM
techniques will be applied in the estimation of the uncertainty of
some of the most important EMC tests, respectively, and finally, a
comparison between the results was performed to reach the conclusions
of this study.

Progress In Electromagnetics Research B, Vol. 34, 2011 127
2. GUM UNCERTAINTY FRAMEWORK IN EMC
TESTING
The functional relationship (i.e., measurement model or equation)
between the measured (quantity intended for measurement) Y and the
set of input quantities {X1, X2, . . . , XN} in a EMC test measurement
process is given by,
Y = f(X1, X2, . . . , XN ). (1)
The measurement model f includes both corrections for
systematic effects and accounts for sources of variability, such as those
due to different observers, instruments, EUT (Equipment Under Test),
laboratories and times at which observations are made. Therefore, the
general functional relationship expresses not only a physical law but
also a measurement process. Some of these variables can be controlled
directly or indirectly, others can be observed but not controlled and
some others cannot even be observed.
An estimate of the measured Y , denoted by y, is obtained from (1)
using input estimates {x1, x2, . . . , xN} for the values of theN quantities
{X1, X2, . . . , XN}. Thus the output estimate y, which is the result of
the measurement, is given by,
y = f(x1, x2, . . . , xN ). (2)
The estimated standard deviation associated with the output
estimate or measurement result y, is defined as the combined standard
uncertainty and denoted by uc(y), and is determined from the
estimated standard deviation associated with each input estimate xi,
termed the standard uncertainty and denoted by u(xi) [4].
Each input estimate xi and its associated standard uncertainty
u(xi) are obtained from a distribution of possible values of the input
quantity Xi. This probability distribution may be experimentally
determined, that is, based on a series of observations Xi,j of Xi, or
it may be an a-priori distribution. Type A evaluations of standard
uncertainty components are founded on frequency distributions while
Type B evaluations are founded on a-priori distributions. It must be
recognized that in both cases, the distributions are models that are
used to represent the state of our knowledge [4].
2.1. Contributions from Type A Evaluations
The estimation of the contribution from Type A evaluations of
standard uncertainty of an input parameter is done by calculation
from a series of repeated observations, using statistical methods, and
resulting in an experimental probability distribution that is assumed

128 Azpu´rua, Tremola, and Pa´ez
to be normal. For any measurement method, this kind of contributions
should be evaluated following the procedure and configuration that are
typically involved in the test, using, if necessary, a standardized EUT.
All the observations must be independent and obtained under the same
conditions of measurement. The result will give a measure of the likely
contribution due to random fluctuations, for instance, uncontrolled
variations in the antenna position, the test environment, or losses
through cable reconnection [2]. Generally, the diversity of EUT makes
impractical to perform many repeated measurements on each type,
make and model of EUT. A predetermination of the uncertainty due to
random contributions is given by the experimental standard deviation
s(xi) of a series of m such measurements Xi,j ,
u(xi) = s(Xi,j) =
√√√√ 1
m− 1
m∑
j=1
(Xi,j −Xi
)2, (3)
where, Xi is the experimental average of themmeasurements, as shown
in (4).
Xi = 1m
m∑
j=1
Xi,j . (4)
Hence, u(xi) is used directly for the uncertainty due to random
contributions, excluding the effects of the EUT, when only one
measurement (m = 1) is made on the EUT and its result is the best
estimate of the input parameter, that is, xi = Xi,1. But if the result of
the measurement is close to a non compliance respect to the standard
that is being evaluated, it is advisable to perform several measurements
on the EUT itself, at least at those frequencies that are critical. In
this case (xi = Xi), and if the measurements are independent, the
uncertainty is reduced to the standard deviation of the mean, as shown
in (5).
u(xi) = s(Xi,j)√m . (5)
2.2. Contributions from Type B Evaluations
For an estimate xi of an input quantity Xi that has not been obtained
from repeated observations, the associated estimated standard
uncertainty u(xi) is evaluated by scientific judgment based on all
the available information on the possible variability of Xi [4]. The
pool of information may include: previous measurements, experience,
understanding of instrument behavior, manufacturer’s specifications,
data provided in calibration certificates (even if the reported

Progress In Electromagnetics Research B, Vol. 34, 2011 129
uncertainty is calculated from results of replicate measurements) and
uncertainties assigned to reference data taken from handbooks. This
kind of contributions may be used to model systematic effects, that
is, those that remain constant during the measurement but which
may change if the measurement conditions, methods or equipments
are altered [3]. If there is any doubt about whether a contribution is
significant, or not, it should be included in the uncertainty budget in
order to demonstrate that it has been considered. Some examples of
uncertainty contributions from Type B evaluations in EMC test are:
equipment calibration, mismatch errors, coupling effects, errors due to
constant deviations in the physical setup (antenna height, separation
form EUT and alignment),and instrument accuracy. If it is possible
and practical, corrections for systematic effects should be applied.
In the EMC testing uncertainty estimations of contributions form
Type B evaluations, it is required to choose a probability distribution
that models each source of variability. The probability distributions
that have been demonstrated to be more relevant, in EMC testing, are:
normal for uncertainties derived from multiple contributions, such as
calibration uncertainties with a statement of confidence; rectangular
distribution for contributions taken from manufacturers’ specifications
(a) (b)
(c) (d)
Figure 1. Probability density functions commonly used for Type B
evaluations of uncertainty in EMC testing. (a) Normal. (b) Uniform.
(c) Triangular and (d) U-shaped.

130 Azpu´rua, Tremola, and Pa´ez
and reading from digital instruments, triangular distribution when the
contribution has defined limits and where the majority of the values
between the limits lie around the central point like the measurements
taken with analog instruments and the U-shaped distribution is
applicable to mismatch uncertainty, where the probability of the true
value being close to the measured value is low [2, 3]. The standard
uncertainty associated with the probability distributions quoted above,
are shown in Figure 1.
It should be recognized that a Type B evaluation of standard
uncertainty can be as reliable as a Type A evaluation, especially
in a measurement situation where a Type A evaluation is based
on a comparatively small number of statistically independent
observations [3].
It is important not to “double-count” uncertainty components. If
a component of uncertainty arising from a particular effect is obtained
from a Type B evaluation, it should be included as an independent
component of uncertainty in the calculation of the combined standard
uncertainty of the measurement result only to the extent that the effect
does not contribute to the observed variability of the observations.
This is because the uncertainty due to that portion of the effect
that contributes to the observed variability is already included in the
component of uncertainty obtained from the statistical analysis of the
observations [4].
2.3. The Combined Standard Uncertainty
In order to calculate the combined standard uncertainty uc(y), the
conventional approach uses the law of propagation of uncertainty based
on a first-order Taylor series approximation of Y . This law, for the case
of correlated input quantities, is given by [4],
uc(y) =
√√√√
N∑
i=1
ci2u2(xi) + 2
N−1∑
i=1
N∑
j=i+1
cicju(xi)u(xj)r(xi, xj), (6)
where, ci is the estimated sensitivity coefficients (7) and r(xi, xj) is
the estimated correlation coefficient (8) calculated using the covariance
associated with xi and xj , u(xi, xj).
ci = ∂f∂Xi
∣∣∣∣
Xi=xi
, (7)
r(xi, xj) = u(xi, xj)u(xi)u(xj) . (8)

Progress In Electromagnetics Research B, Vol. 34, 2011 131
Usually in the EMC tests, the uncertainty budgets are calculated
assuming that the measured variables are uncorrelated and that the
value of the sensitivity coefficients is one (1) [3]. It is appreciated that
this approach is mathematically imprecise. However, it is assumed
that the approximations involved, do not cause an overall change in
the expanded uncertainty of any significance and well within the five-
percent criteria [2]. Consequently,the Equation (6) comes down to,
uc(y) =
√√√√
N∑
i=1
u2(xi). (9)
2.4. The Expanded Uncertainty
The expanded uncertainty, U , defines an interval about the
measurement result that will include the true value with a specified
level of confidence. The expanded uncertainty is obtained by
multiplying the combined standard uncertainty by a coverage factor,
k, which is set to approximately 2 (1.96) for a level of confidence of
95% assuming that the results are normally distributed. The result is
then expressed as follows,
Y = y ± U = y ± kuc(y). (10)
2.5. Disadvantages in Using the GUM Method of
Estimating Uncertainties in EMC Testing
Summarizing, the greatest disadvantages presented in the conventional
mathematical treatment of uncertainties in EMC testing are:
• Generally, the model of the measurement system is not linear.
Hence, the expected value of the system output is not equal to
the system output to the expected values of the inputs, that is,
E(y) 6= f(E(x1), E(x2), . . . , E(xN )) [6]. However, some of the
measurement models used in the EMC tests can be linearized by
the decibel’s logarithmic transformation of the measurements.
• Input variables may be correlated with each other, that is,
r(xi, xj) 6= 0 [3].
• The sensitivity coefficients, generally, differs from the assumed
value of 1, that is, ci 6= 1 [6].
• The result is not necessarily normally distributed given the
asymmetries of some contributions, such as, mismatch effects.
• The rigorous mathematical treatment of uncertainties using the
conventional approach, in complex systems like those involved in
EMC testing is undoubtedly impractical [2].

132 Azpu´rua, Tremola, and Pa´ez
In this regard, an alternative is to use the Monte Carlo Method,
which will be explained in general terms below.
3. UNCERTAINTY ESTIMATION USING MONTE
CARLO METHOD
In Monte Carlo techniques, both, the random and the systematic
components of the uncertainty are treated as having a random nature.
It is important to notice that the systematic component is not modeled
as random, it is the knowledge about it for which a probability
distribution is introduced [6].
This method basically consists in randomly generate a number M
of Monte Carlo trials (i.e., the number of model evaluations made)
in where the distribution function of the value of the output quantity
Y will be numerically approximated. It is further assumed that the
probability densities of the considered input quantities are a priori
known. Then, a sample vector of the input quantities can be drawn
repeatedly using pseudo random number generators. For each input
sample vector, the corresponding values of the output quantities are
calculated by using the corresponding functional relation. The set of
output sample vectors yields an empirical distribution which can be
used to approximate the correct random distribution of the output
quantities. All required measures (expectation value, variance and
covariance) as well as higher-order central moments such as skewness
and kurtosis can then be derived [7]. Before applying the MCM,
the conditions for valid application should be verified [6]. It is
recommended to use M ≥ 106 to estimate a 95% coverage interval
for the output quantity such that this length is correct to one or
two significant decimal digits [6]. It is also recommended to validate
the quality of the pseudo-random number generator to be used in the
calculations.
The MCM is implemented using an algorithm that can be
summarized as follows:
(i) There must be generated a set of N input parameters
{x1, x2, . . . , xN}, which are random variables distributed accord-
ing to a probability density function assigned to each input pa-
rameter. This process should be repeated M times for every input
quantity.
(ii) The functional relationship that model the measurement system
is then evaluated to obtain the output,
yj = f(x1,j , x2,j , . . . , xN,j), (11)

Progress In Electromagnetics Research B, Vol. 34, 2011 133
for j = 1, 2, . . . ,M . From this sample, it is possible to estimate
the probability density function of y.
(iii) The relevant estimates of any statistical quantity can be calculated
(average, variance, skewness and kurtosis of the output, among
others).
(iv) The output vector {y1, y2, . . . , yN} is sorted in ascending order to
obtain a vector y˜ = {y˜1, y˜2, . . . , y˜N}.
(v) The confidence interval [y˜r, y˜s] is found approximately through (12)
and (13) [12]:
r = round((M + 1)γ), (12)
s = round((M + 1)(1− γ)), (13)
where, γ is the significance level (γ = 0.025 for 95% of confidence)
and the function round(x) is used to represent the nearest integer
to x.
4. TYPICAL UNCERTAINTY BUDGETS IN EMC
TESTING
This section presents four examples of typical uncertainty budgets
relating to four different tests, two of which correspond to
electromagnetic interference (EMI) and the other two referred to
electromagnetic susceptibility (EMS). The examples were taken from
the document “The Expression of Uncertainty in EMC Testing”
of the United Kingdom Accreditation Service (UKAS) [3], with
minor changes. The approach adopted for emission measurements,
including sensitivity coefficients generally is to follow the methodology
currently being implemented in the Comite´ International Spe´cial des
Perturbations Radioe´lectriques, CISPR [13]. The contributions and
values in the following examples are used for reference only for the later
comparison with the results obtained applying the MCM. Laboratories
shall determine the uncertainty contributions for the tests they are
performing based on their own derived and available data.
4.1. Conducted Emissions According CISPR 22
The conducted disturbance level, CDL, is modeled through the
mathematical relationship shown in (14).
CDL = RI + LC + LAMN + dVSW + dVPA + dVPR
+dVNF + dZ + FST +MM +RS , (14)
where, RI , LC , LAMN , dVSW , dVPA, dVPR, dVNF , dZ, FST , MM
and RS represent, Receiver reading, Attenuation AMN (Artificial

134 Azpu´rua, Tremola, and Pa´ez
Table 1. Uncertainty budget for conducted emissions according
CISPR 22.
Source of
Uncertainty Type
Value
B1
Value
B2
Probability
Distribution ci
ui
B1
ui
B2
RI B 0.05 0.05 Rectangular 1 0.03 0.03
LC B 0.40 0.40 Normal 1 0.20 0.20
LAMN B 0.20 0.20 Normal 1 0.10 0.10
dVSW B 1.00 1.00 Rectangular 1 0.58 0.58
dVPA B 1.50 1.50 Rectangular 1 0.87 0.87
dVPR B 1.50 1.50 Rectangular 1 0.87 0.87
dVNF B 0.00 0.00 Rectangular 1 0.00 0.00
dZ B 3.60 2.70 Triangular 1 1.47 1.10
FST B 0.00 0.00 Rectangular 1 0.00 0.00
MM A 0.89 0.89 U-Shaped 1 0.63 0.63
RS A 0.50 0.50 Normal 1 0.50 0.50
uc = 2.17 1.94
U = 4.3 3.9
mains network) — Receiver, AMN voltage division factor, Receiver
error due to sine wave voltage, Receiver error due to Pulse amplitude
response, Receiver error due to Pulse repetition response, Error due
to finite signal to noise ratio, AMN Impedance, Frequency step error,
Mismatch and Measurement system repeatability, respectively [3].
The conventional uncertainty estimation is calculated using the
budget shown in Table 1. The budget is divided into two bands: 9 kHz
to 150 kHz (B1) and 150 kHZ to 30MHz (B2), due to changes in the
uncertainty contribution of the AMN impedance.
Therefore, the expanded uncertainty of the conducted disturbance
level measurement estimated through the conventional approach, for
a coverage factor, k = 2 (95% confidence level), is ±4.3 dB between
9 kHz and 150 kHz and, ±3.9 dB between 150 kHz and 30MHz.
4.2. Radiated Field Strength Measurement According
CISPR 22
The radiated filed strength, FS , is modeled through the mathematical
relationship shown in (15).
FS = RI + dVSW + dVPA + dVPR +AF + CL +AD +AH
+AP +AI + SI +DV +DB +DC + FST +MM +RS , (15)
where, RI , dVSW , dVPA, dVPR, AF , CL, AD, AH , AP , AI , SI , DV ,
DB, DC , FST , MM and RS represent, Receiver indication, Receiver

Progress In Electromagnetics Research B, Vol. 34, 2011 135
error due to sine-wave voltage, Receiver error due to Pulse amplitude
response, Receiver error due to Pulse repetition response, Error due
to finite signal to noise ratio, Antenna factor calibration uncertainty,
Cable loss uncertainty, Antenna directivity uncertainty, Antenna factor
height dependence, Mismatch and Measurement system repeatability,
respectively [3].
The conventional uncertainty estimation is calculated using the
budget shown in Table 2. The budget is divided into two bands:
30MHz to 300MHz (B1) and 300MHz to 1GHz (B2), due to changes
between different antennas. Both bands were measured at vertical
polarization and at 3m measurement distance from EUT.
Hence, the expanded uncertainty of the Radiated Field Strength
measurement estimated through the conventional approach, for a
coverage factor, k = 2 (95% confidence level), is ±5.4 dB between
Table 2. Uncertainty budget for radiated field strength measurement
according CISPR 22.
Source of
Uncertainty Type
Value
B1
Value
B2
Probability
Distribution ci
ui
B1
ui
B2
RI B 0.05 0.05 Rectangular 1 0.03 0.03
dVSW B 1.00 1.00 Rectangular 1 0.50 0.50
dVPA B 1.50 1.50 Rectangular 1 0.87 0.87
dVPR B 1.50 1.50 Rectangular 1 0.87 0.87
dVNF B 0.50 0.50 Normal 1 0.25 0.25
AF B 1.00 1.00 Normal 1 0.50 0.50
CL B 0.50 0.50 Normal 1 0.25 0.25
AD B 0.00 3.00 Triangular 1 0.00 1.73
AH B 2.00 0.50 Rectangular 1 1.15 0.29
AP B 0.00 1.00 Rectangular 1 0.00 0.58
AI B 0.25 0.25 Rectangular 1 0.14 0.14
SI B 4.00 4.00 Triangular 1 1.63 1.63
DV B 0.60 0.60 Rectangular 1 0.35 0.35
DB B 0.30 0.30 Rectangular 1 0.17 0.17
DC B 0.00 0.90 Rectangular 1 0.00 0.52
FST B 0.00 0.00 Rectangular 1 0.00 0.00
MM A −1.25 −0.54 U-Shaped 1 −0.88 −0.38
RS A 0.50 0.50 Normal 1 0.50 0.50
uc = 2.71 3.00
U = 5.4 6.0

136 Azpu´rua, Tremola, and Pa´ez
30MHz and 300MHz and, ±6.0 dB between 300MHz and 1GHz.
4.3. Radiated Immunity According IEC 61000-4-3
The mathematical model for the measurement process of the Radiated
Filed Strength, FS , is shown in (16).
FS = FSM + FSAW + PD + PAH + FD + FEUT +RS , (16)
where, DSM , FSAW , PD, PAH , FD and RS represent, the uncertainty
of the field probe used for monitoring the field strength, the
field strength acceptability window, the drift in the forward power
measurement, errors due to harmonics of the power amplifier, the
effect of field disturbance and the measurement system repeatability,
respectively [3]. The uncertainty contribution associated to the
antenna-EUT coupling and reflections due to EUT presence, FEUT ,
was added to the budget presented in [3] because it represents a
significant source of uncertainty [14].
It is remarkable that in order to perform EMC immu-
nity/susceptibility testing against a specified interference level it is
recommended that, in the absence of other guidance, the test shall
be performed at the specified immunity level increased by the stan-
dard uncertainty multiplied by a coverage factor k of 1.64 which under
normal circumstances would give a confidence of approximately 90%,
however, because the level is increased by this amount, there are only a
5% of probability that the immunity level remains below the required,
hence it is achieved a confidence of 95% that the immunity level has
been applied.
Table 3. Uncertainty budget for radiated immunity according IEC
61000-4-3.
Source of
Uncertainty Type Value
Probability
Distribution ci ui
FSM B 1.20 Normal 1 0.60
FSAW B 0.50 Rectangular 1 0.29
PD B 0.20 Rectangular 1 0.12
PAH B 0.35 Rectangular 1 0.20
FD B 0.35 Rectangular 1 0.29
FEUT B 0.32 U-Shaped 1 0.23
RS A 0.50 Normal 1 0.50
uc = 0.94
U = 1.5

Progress In Electromagnetics Research B, Vol. 34, 2011 137
The conventional uncertainty estimation is calculated using the
budget shown in Table 3. The budget is valid only for the “Re-
establishment of precalibrated field level” method. The measurement
uncertainty budget below is based on the assumption that it has been
demonstrated during calibration that the 6 dB field uniformity has been
achieved.
Hence, the expanded uncertainty of the Radiated Field Strength
Immunity Level estimated through the conventional approach, for a
coverage factor, k = 1.64 (90% confidence level), is ±1.5 dB.
4.4. Conducted Immunity According IEC 61000-4-6
The mathematical model for the conducted induced voltage level, CV L,
is given by (17),
CV L = VRMS + VLAW + PD + PAH +MV C +MAC +RS , (17)
where VRMS , VLAW , PD, PAH , MV C , MAC and RS represent, the
maximum induced current, the error specified for the RMS Voltmeter,
the Voltage level acceptability window, the signal generator drift, the
power amplifier harmonics, the contribution of the current coil, error
due to spectrum analyzer, mismatch between voltmeter and CDN,
mismatch between amplifiers and CDN and the measurement system
repeatability, respectively [3]. This model only applies when the test
is performed according to the CDN method described in IEC 61000-
4-6. For this type of test, the conventional uncertainty estimation is
calculated using the budget shown in Table 4.
Table 4. Uncertainty budget for conducted immunity according IEC
61000-4-6.
Source of
Uncertainty Type Value
Probability
Distribution ci ui
VRMS B 0.70 Rectangular 1 0.40
VLAW B 0.50 Rectangular 1 0.29
PD B 0.2 Rectangular 1 0.12
PAH B 0.70 Rectangular 1 0.40
MV C B −0.54 U-Shaped 1 −0.38
MAC B −0.16 U-Shaped 1 −0.82
RS A 0.50 Normal 1 0.50
uc = 1.22
U = 2.0

138 Azpu´rua, Tremola, and Pa´ez
Therefore, the expanded uncertainty of the Conducted Induced
Voltage Level estimated through the conventional approach, for a
coverage factor, k = 1.64 (90% confidence level), is ±2.0 dB.
5. UNCERTAINTY ESTIMATION IN EMC TESTING
USING MONTE CARLO SIMULATION METHOD
The expanded uncertainty of the aforementioned typical EMC test
was estimated applying the MCM for M = 106 and taking the same
probability density functions as proposed in [3] for the pseudo random
number generators. The results are given in the following subsections.
5.1. Conducted Emissions According CISPR 22
The conducted disturbance level, CDL, is modeled through the
mathematical relationship shown in (14). The expanded uncertainty
of CDL estimated using the MCM is ±4.2 dB for Band 1 and ±3.8 dB
for Band 2.
The histograms of relative frequency of the error in the conducted
disturbance level measurement are shown in Figures 2 and 3.
It is observed that the error in the conducted disturbance level
measurement, follows an approximately normal distribution.
-8 -6 -4 -2 0 2 4 6 80
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Error in the conducted disturbance level (dB)
De
nsi
ty

CISPR22_B1_MCMCISPR22_B1_GUM
Figure 2. Histogram of relative frequency of the error in the
conducted disturbance level measurements between 9 kHz and 150 kHz.

Progress In Electromagnetics Research B, Vol. 34, 2011 139
-6 -4 -2 0 2 4 60
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Error in the conducted disturbance level (dB)
De
nsi
ty

CISPR22_B2_MCMCISPR22_B2_GUM
Figure 3. Histogram of relative frequency of the error in the
conducted disturbance level measurements between 150 kHz and
30MHz.
-10 -5 0 5 100
0.05
0.1
0.15
Error in the Radiated Field Strength (dB)
De
nsi
ty

CISPR22_B1_MCMCISPR22_B1_GUM
Figure 4. Histogram of relative frequency of the error in the radiated
field strength measurements between 30MHz and 300MHz.
5.2. Radiated Field Strength Measurement According
CISPR 22
The Radiated Field Strength, FS , is modeled through the mathemat-
ical relationship shown in (15). The expanded uncertainty of FS esti-
mated using the MCM is ±5.3 dB for Band 1 and ±5.9 dB for Band 2.