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Nisr United States Department of Commerce

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National Institute of Standards and Technology

NIST Technical Note 1389

Measurement Uncertainty of

Radiated Emissions

David A. Hill

Motohisa Kanda

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NIST Technical Note 1389

Measurement Uncertainty of

Radiated Emissions

David A. Hill

Motohisa Kanda

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

March 1997

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U.S. DEPARTMENT OF COMMERCE, William M Daley, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1389, 36 pages (March 1997)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1997

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
CONTENTS

Page

Abstract 1

1. INTRODUCTION 1

2. SEPARATE AND COMBINED UNCERTAINTIES 2

3. SETUP OF EQUIPMENT UNDER TEST 4

3.1 Theory for Radiated Emissions 4

3.2 Measurement Uncertainties 6

4. MEASUREMENT PROCEDURE 7

4.

1

Sampling Issues 8

4.2 Operator Issues 8

4.3 Type A Uncertainty 9

5. FACILITY 10

5.1 Open Area Test Site (OATS) 10

5.2 Anechoic and Semi-Anechoic Chambers 11 5.3 Shielded Enclosure 11

5.4 Reverberation Chamber 12

5.5 TEM and GTEM Cells 12

5.6 Three-Loop System 13

6. ANTENNA 13

6.

1

Standard-Antenna Method 13

6.2 Standard-Field Method 14

6.3 Standard-Site Method 14

6.4 Reference-Antenna Method 15

7. RECEIVER 15

8. CONCLUSIONS AND RECOMMENDATIONS 15

REFERENCES 17

APPENDIX A. PROPAGATION OF UNCERTAINTY 22

APPENDIX B. UNCERTAINTY OF A PRODUCT OF QUANTITIES 24

HI

MEASUREMENT UNCERTAINTY OF RADIATED EMISSIONS

David A. Hill and Motohisa Kanda

Electromagnetic Fields Division

National Institute of Standards and Technology

Boulder, Colorado 80303

We present a summary of current knowledge and techniques for evaluating

measurement uncertainty of radiated emissions. The important quantity for

compliance testing is the expanded uncertainty (typically for a coverage factor

k = 2), and we discuss the validity of obtaining the combined uncertainty from an RSS sum of separate uncertainties. A generic uncertainty model includes the

following separate sources of measurement uncertainty: setup of equipment under

test, measurement procedure, facility, antenna, and receiver. Measurement

uncertainties for radiated emissions are large and not well quantified, and

recommendations are made for further study.

Key words: anechoic chamber; antenna factor; combined uncertainty; compliance

testing; expanded uncertainty; measurement uncertainty; open area test site; propagation of uncertainty; radiated emissions; reverberation chamber; shielded

enclosure.

1. INTRODUCTION

Electromagnetic compatibility (EMC) tests and measurements typically have large

uncertainties of at least several decibels [1,2]. Large measurement uncertainty can lead to

overdesign and extra expense of a product to achieve EMC or to underdesign and failure to achieve EMC. Consequently, it is important to have a good knowledge of EMC measurement

uncertainty and to minimize it to the extent possible.

Most EMC tests and measurements fall into one of the following four categories: (1)

radiated emissions, (2) radiated immunity, (3) conducted emissions, or (4) conducted immunity.

This report will concentrate on radiated emissions measurements, but many of the individual

results will apply equally well to radiated imunity measurements since both types of measurements

deal with radiated fields. The purposes of this report are to identify the numerous sources of

uncertainty in radiated emissions measurements and to evaluate these uncertainties where

possible. In many cases this evaluation is not now possible, and further experimental or

theoretical study will be required. Some estimates and examples of radiated emissions

measurement uncertainties have been made in references [1] and [2].
A diagram for a generic radiated emissions measurement is shown in figure 1 . The

equipment under test (EUT) could be any electronic device, such as a digital computer, that is capable of emitting electromagnetic radiation. We include cables in figure 1 because they

typically affect radiated emissions. In a formal compliance test, the measurement setup and

procedure are normally specified in detail, but many other radiated emissions measurements are made without set procedures. This report covers both cases because they involve the same

radiated field measurement issues and uncertainties.

The measurement facility could be an outdoor facility, such as an open area test site (OATS) as specified by the Federal Communications Commission (FCC) for measuring radiated

emissions from computing devices [3], or an indoor shielded facility, such as an anechoic chamber

or a reverberation chamber. The receiving antenna could be any well characterized antenna. A

narrowband example is a half-wavelength, tuned dipole, and broadband examples are biconical

and log-periodic antennas. A typical receiver is a spectrum analyzer, and in formal compliance

tests a minimum receiver bandwidth is normally specified.

The organization of this report is as follows. Section 2 covers methods for reporting

separate measurement uncertainties and combined or total uncertainty. Sections 3 through 7 cover the separate uncertainties associated with EUT setup, measurement procedure, facility,

antenna, and receiver respectively. Section 8 includes conclusions and recommendations for

further study. Appendix A examines propagation of uncertainty as derived from a Taylor series

expansion. Appendix B examines propagation of uncertainty for the special case where an

estimated value is based on a product of measured values.

2. SEPARATE AND COMBINED UNCERTAINTIES

In a radiated emissions test, the measured quantity can be electric field strength (V/m),

magnetic field strength (A/m), or radiated power (W). All three are positive, real quantities.

However, they are frequently expressed in decibels (dB) relative to some convenient level. For

example, radiated electric field measurements or emissions limits [4] are frequently referred to

1 |iV/m and are expressed in dB(l )aV/m). When field strength or power is expressed in decibels

relative to some reference level, it can be positive or negative.

If the standard uncertainty of a measured emission is evaluated statistically, then it is classified as Type A [5,6]. Type B uncertainties are those which are evaluated by other means. It

is strongly recommended that corrections be applied for known systematic effects, but in practice

such corrections may not be made because of limited resources or because of the insignificance of

a particular systematic effect. Even when a correction is made, a type B uncertainty remains due

to an incomplete knowledge of the value of the correction. In keeping with the notation in reference [5], we use the symbol w, to represent standard uncertainty due to the /th effect. It is equal to the positive square root of the estimated variance w/.

The usual method for combining separate standard uncertainties to determine the

combined standard uncertainty u^ [5] is the root sum of squares (RSS):

«.' = E "A (1)
The justification of the RSS combination in eq (1) is discussed in Appendix A. For radiated

emissions measurements where the relative w, values are not necessarily small, the RSS

combination may not be valid. This issue requires further study. In addition to the combined standard uncertainty u^ of a measured emission, an expanded

uncertainty, U = ku^, is also a usefijl quantity. The coverage factor k can be chosen to achieve a desired confidence level [5]. NIST generally uses a value of A: = 2, and this value was also chosen

in a recent uncertainty analysis of radar cross-section measurements [7]. For a normal

distribution, the choice oik = 2 yields a 95 percent certainty that the error is bounded by ±U. A

probable error (equal to 0.675 u^ has been defined as the error that is exceeded 50 percent of the

time[l. Sec. 8.1.2]. In some cases probability distributions other than normal, such as rectangular, U shaped, or log normal, have been found useful in EMC measurement uncertainty studies [1,2]. The

rectangular distribution has been suggested for cases when the "worst case" limits are known

(perhaps from manufacturer specifications) but no additional information is available. This

distribution assigns an equal probability lying anywhere between the prescribed limits. If the

worst-case limits are ±a„ then the standard uncertainty w, can be derived: u. = aj\fl . Although

this is an appropriate value for the standard uncertainty, some modification is required in using it for calculating the expanded uncertainty. For a coverage factor k = 2, the expanded uncertainty

Uj is U^ = lu^ = 2ajyj3 . Since this value for 95 percent confidence exceeds the prescribed limit

a,, it is obviously an overestimate. Possible modifications are (1) to revisit the initial rectangular

distribution assumption, (2) to set f/, = a,, or (3) to set u=aJ2 before calculating u^ and U.

In the EMC community, relative errors or uncertainties have commonly been expressed in

decibels [8,9]. If the combined standard uncertainty is small, then the logarithmic standard

uncertainty is linearly related to the relative value. For example, if we are measuring an electric or magnetic field strength, the approximate relationship is

u^ (dB) - 8.686 u^. (2)

If we are dealing with radiated power rather than field strength, then the coefficient 8.686 in eq

(2) is changed to 4.343. Generally radiated-emissions measurements do not have small combined

standard uncertainties, and eq (2) is not valid. A convenient assumption that is commonly made

in EMC measurements is that the probability density flinction is log normal [1]. Then decibel

values have a normal distribution, and there is no need to convert to relative values in performing

the RSS in eq (1). Some theoretical and experimental justification for the log normal distribution

is given in [10]. The theoretical explanation is that the measured quantity can be represented by a product of a large number of random factors so that the log of the measured quantity becomes a sum to which the central-limit theorem [11] applies. For convenience we will also quote

uncertainties in decibels in most of this report, but uncertainties should generally be converted to

relative values before evaluating eq (1). However, for cases involving large uncertainties, the

convenient assumption of a log normal distribution allows the use of decibel values and yields

symmetric positive and negative uncertainties in decibels. It also avoids the nonphysical resuh of

negative radiated power in the negative uncertainty interval. The issue of how to handle large
negative uncertainties needs further study to put it on a rigorous basis.

The issue of how to use uncertainty estimates in compliance testing is currently unresolved

[12]. One approach [2] is that the measured emission level plus the expanded uncertainty U

(usually for k = 2) must be less than the specified emission limit for one to state that the EUT

complies. By the same approach, the EUT does not comply if the measured emission level minus

the expanded uncertainty exceeds the specified limit. If the measured emission level is within ±U

of the specified limit, then it is unknown whether the EUT complies. The other approach is based on the assumption that the specification limit already includes a margin for the expanded

uncertainty. Then compliance is determined only by the measured emission level, and the

measurement laboratory does not have to take its measurement uncertainty estimate into account

in determining whether the EUT complies. Regardless of which approach is ultimately agreed

upon, it is desirable to quantify and to reduce measurement uncertainty.

3. SETUP OF EQUIPMENT UNDER TEST

3 . 1 Theory for Radiated Emissions

Before considering measurement issues, we review some basic radiation theory. The

actual sources and currents within a real EUT are typically very complicated, but we can simplify

the radiation theory by surrounding the EUT by a surface S as shown in figure 2. Then the

radiated electric field Eif) outside S can be written as a surface integral of the tangential

components of the surface electric field Eif') and magnetic field Hif) :

Eif) = V^ n n'^Eif')gdS' * -^V^V^n n'^Hif')gdS\ (r\

where g = eiq?i-jk^)/i4nR), R = \r-f'\, k^ = a)7ji~e^, the vector r' designates the source

point on S, and « ' is the outward unit normal to S. The time dependence is exp(/a)/), so eq (3)

represents one spectral component of a general time-varying field. Tai has called eq (3) the Franz

formula [13], and there is a dual equation for the radiated magnetic field. We have assumed that

the region outside S is free space with permittivity Eq and permeability [Iq, but the effect of a

perfectly conducting ground plane (assumed for an OATS facility) could be included by image

theory.

Equation (3) could apply to any radiator enclosed within the surface S. The equivalence

principle [14] states that the fields in the integrands could be replaced by surface electric currents

KJf) and magnetic currents KJf) radiating in free space:

KJf) - ji'xHir') and KJr') - -n'^Eif). (4)

Since the currents in eq (4) are only equivalent currents, they radiate the same field as the EUT

outside S, but do not provide much insight to the radiation process.
Better insight can be obtained by shrinking the surface down to the boundaries of the

EUT. Figure 3 shows a simple example where the EUT comprises two metal boxes with

apertures and a connecting cable. By breaking the surface into three parts, we can write the

radiated field in eq (3) as a sum of three terms:

E(r) - EJj) . EJf) * EJir), (5)

where E^ is the electric field radiated by the metal surface S^, E^ is the electric field radiated by

the apertures S^, and E^ is the electric field radiated by the cable surface S^. If we assume that the metal surface S^ is perfectly conducting, then the tangential electric

field is zero, and the tangential magnetic current determines the actual surface current:

«'x£(F') = and n'^Hif") - KJj') on S^. (6)

From eqs (3) and (6), we can write E^ as

EJjr) - -J— V X V X II KJjr')gdS '. /yx

If Sn, is a real metal with a nonzero skin depth, then K^ is the current flowing within a thin layer

at the surface.

The field radiated by the total aperture surface cannot be simplified because both the

tangential electric and magnetic fields are nonzero. Thus the expression for E is similar to E in eq(3):

EJf) -^^

jl n'^E(f')gdS' . Vx Vx fj n' x H(r')gdS '. (8)

The aperture surface S^ is not necessarily free space. It could be any penetrable material such that

both the tangential electric and magnetic fields are nonzero.

If the cable is thin, then the field radiated from S^ depends only on the bulk cable current

4(1,'), where V is the linear position along the cable. The resultant expression for E^ is

^/r) = -!- V X V X Ie^I(L ')gdL, ^9)

where the integration is along the cable length L^ and e^ is a unit vector directed along the cable.

Even if the EUT is very complex with many apertures and cables, the total radiated field
can still be broken down as in eq (5). The integrals in eqs (7) through (9) need to extend over the

total conductor surface, aperture surface, or cable length.

The far-field approximations to eqs (7) through (9) are obtained by setting k^» 1, where

r = \r\, and retaining only r'^ terms:

EJf) ^ ^ ^"^ ^"^ // KJine^ ' dS\ (10)

47ir ;

-jk e'^

^^ u • -/

Anr i

and

EST) - ^-^ r - r X fe'j^^L ')J^^^'^' dL \ (12)

47tr ,

where r|o = (|ic/^o) ^"d r = f/r. When ^ is not large, the general eqs (7) through (9) need to be

used. Then the curl and curl-curl operations generate f^ (induction field) and f^ (quasi-static

field) terms in addition to r' terms.

3.2 Measurement Uncertainties

The general categories of uncertainty sources related to the EUT setup can be lumped

into:

(1) EUT mode of operation,

(2) EUT physical configuration, and

(3) cable configuration.

Cable configuration could be lumped under EUT physical configuration, but we choose to treat it as a separate category because it is important and sometimes dominates the radiation.

If an EUT has M modes of operation, then a complete test would include M emission

tests, one for each mode of operation. (Since an emission measurement for each of the M modes

of operation includes sampling in frequency, angle, and polarization, the number of required

individual measurements could be very large.) This is generally too time consuming to be

practical, and for some EUTs, such as computers, M is not even well defined. An emission test
typically requires that the test be performed for the mode of maximum radiation. Unfortunately,

the mode of maximum radiation might well be a function of frequency. Typically the

manufacturer will have some knowledge of the likely radiation mechanism and the mode of

maximum radiation. However, this issue requires further study, and we know of no estimates of

uncertainty due to multiple modes of operation.

The physical EUT configuration affects the geometry of the surfaces, S„ and S^, and the

cable layout L^ as shown in figure 3. Even if the surface and cable currents and the aperture fields

remain unchanged when components are moved, the radiated electric field as given by eqs (7)

through (9) will be affected by changes in r ' (and hence R). Estimates of the effects of

configuration changes could be made from eqs (7) through (9), but in general the currents and

aperture fields in the integrands are not known and would change with changes in the EUT

configuration. Modern computational techniques [15,16] and experimental studies would be

useful for estimating radiation changes (and uncertainties) due to EUT configuration. Some

analytical error estimates based on simple path length effects have been presented [1, Sec. 8.4.1],

but these estimates neglect phase effects. Cable configuration effects could be lumped with general EUT configuration effects, but

they are so prevalent [17,18] that we chose to separate them. In some radiated emissions tests, the lengths, terminations, and positions of power cables and interconnecting leads are prescribed

in detail [1, Sec. 9.2.2], but often they are to be configured to maximize emissions. Radiated

emissions from simple cable geometries can be described analytically [3, Sec. 8.1]. These simple

geometries are useful for determining the dependence of cable radiation on parameters such as cable length, loop area, current magnitude, and frequency, and they could be used to generate

useful numerical results for variations in emissions as a function of cable layout. Experimental

studies would be usefijl for more complex EUTs with multiple boxes and cables. Such studies

have not yet been performed with the goal of estimating measurement uncertainty of radiated

emissions. An expanded uncertainty of 2 dB has been suggested for cable layout effects [8], but

this general area, especially termination effects, needs further study.

Since we have no precise uncertainty estimates on any of the EUT categories (mode of

operation, physical configuration, or cable layout), this is clearly a weak area in overall

measurement uncertainty knowledge that merits extensive study. This is particularly true because

the uncertainties due to EUT setup are expected to be large [8]. In the following sections, we will see that some estimates of uncertainties due measurement techniques and facilities already exist,

but there are also large gaps.

4. MEASUREMENT PROCEDURE

Measurement procedures and facilities for radiated emissions measurements are summarized in reference [1, Ch. 8]. The details of the procedure differ for various facilities, but

sampling, operator, and repeat issues are common to all the methods.
4.

1

Sampling Issues

The total number A^, of measurements required in a test can be written

N^ -. M-S'A'H'P'F, (13)

where A/ is the number of modes of operation, S is the number of setup configurations (box

locations and cable layouts), A is the number of azimuthal angle locations for the receiving

antenna, H is the number of heights for the receiving antenna, P is the number of polarizations

(usually two) of the receiving antenna, and F is the number of frequency samples. For even modest values ofM, S,A,H, P, and F, the total number A^, of measurements required to minimize

sampling errors can be very large (on the order of 10^).

Since A', is constrained by time and other practical considerations, there will be errors caused by undersampling. Some considerations in choosing M, S, A, H, and F efficiently have

been discussed in [1, Sec. 9.4], but the measurement uncertainties resulting from undersampling

have apparently not been studied. Some analytical studies of field sampling effects based on the

electrical size of the EUT could be performed using modern antenna measurement theory [19],

and this covers A and H effects. However, the uncertainties caused by undersampling in M, S,

and F need to be studied experimentally.

4.2 Operator Issues

Operator errors can occur in setting up the EUT and the receiving equipment and in taking

instrument readings (particularly for analog readouts). Automation of measurements has

eliminated some operator errors (such as instrument misreadings), but it introduces possible

uncertainty due to software errors. The uncertainties due to the operator instrument readings or software have not been evaluated and are usually assumed to be negligible under normal

conditions.

However, we can evaluate the uncertainty in the measured electric (or magnetic) field

strength due to uncertainty in distance (caused by operator setup). For cases where the receiving

antenna is in the far field, the field falls off as the reciprocal of distance r. From the theory for the

propagation of uncertainty as covered in Appendix A, the standard uncertainty u^ in the measured

electric field strength E due the standard uncertainty u{r) is

u

r BE

dr

u(r)--u(r). (14)

r

Since the relative uncertainty u/E equals u{r)/r, the standard uncertainty in decibels is

«^(dB) = -20 log,, 1 - hOl ^ 8.686 ^^. (15)
The final expression in eq (15) assumes that u{r)lr is small. For example, the table of typical

uncertainties in reference [2] gives a rectangular distribution with u^ = 0A/\/3 dB ^ 0.23 dB for a distance of 10 m. From eq (15), this corresponds to u(r)/r - 0.027 or 2.7%. This relative

uncertainty seems large for a distance measurement, but might be based on the difficulty of

defining the distance from a complicated EUT. For cases where the receiving antenna is not in the far field of the EUT, the distance dependence is more complex and probably unknown, and

eqs (14) and (15) are not valid. However, some approximations for r'^ quasi-static, near-field

variation can be obtained [1].

4.3 Type A Uncertainty

If « repeat runs of an emission measurement >^i, are made, then the standard uncertainty

u^(y) equals the estimated standard deviation of the mean y which by type A evaluation is [5]

u,(y) = ^ n(n - 1) w

E (y^y)'

1/2 1 where y = — E >'f « fc.1 (16)

Equation (16) can be considered an evaluation of system repeatability based on n repeat

measurements. For example, reference [2] suggests a typical Type A standard uncertainty value

of 0.5 dB for five repeat runs. In the example table in reference [2], this Type A uncertainty is added to the Type B uncertainties by RSS.

In a typical radiated emissions measurement, time and efficiency dictate that no additional

repeat measurements are made (« = 1). In such cases, it is reasonable to base the Type A

uncertainty on previous system repeat measurements by setting the standard uncertainty equal to

the standard deviation s(y):

«.(y) = s(y) = n- 1

n \

E (y. y?

b.\ )

1/2 (17)

By comparing eqs (16) and (17), we see that the standard deviation in eq (17) is reduced by a

factor of «'^'^ when n repeat runs are performed on the actual emissions measurement and the

standard uncertainty is given by eq (16). If no previous system repeat runs are performed, then

the Type A uncertainty needs to be accounted for in the separate uncertainties as discussed in Appendix A. When a radiated emission >» is a fijnction of a number of quantities x„ the uncertainty

of>' can be determined from the uncertainties of x, by the propagation-of-uncertainty equation as

in Appendix A. If a Type A evaluation of the uncertainty of the input estimate w(x,) is performed,

the appropriate expression is the same as eq (16):

u(x)

\ n(n - 1) w

E (^a-^.)'

1/2 — 1

, where x, = — E ^,j « fc.1 (18)
and X, ;. is the Ath sample ofx,. In selecting the coverage factor k for the expanded uncertainty U of quantities, the

uncertainty in the estimated uncertainty must be accounted for in order to achieve the desired

confidence level (95%). A table of/: values is given in [5]. For large n, k approaches 2, but

smaller values oin require a somewhat larger value of A:.

5. FACILITY

A variety of facilities can be used for radiated emissions measurements, and they attempt

to simulate different radiation conditions, such as free space, half space, or guided wave.

Consequently, we will discuss the various facilities and their contribution to measurement

uncertainty separately.

5.1 Open Area Test Site (OATS)

An OATS attempts to simulate a half-space environment bounded by a perfectly

conducting plane. The ideal geometry is shown in figure 4 where the EUT and the receiving

antenna are located over an infinite ground plane and separated by a horizontal distance D (for

example 3 or 10 m). For this ideal geometry, the OATS makes no contribution to measurement

uncertainty, but a real-world OATS will have construction imperfection limitations and ambient

fields that contribute to measurement uncertainty. Practical construction limitations of OATS are: (1) finite extent of the ground plane, (2) irregularities in the ground plane surface, (3) imperfect

conductivity of the ground plane, and (4) reflecting objects in the vicinity.

Item 1 is unavoidable, but a typical requirement is that the ground plane be flat and

obstruction-free over an elliptical area with a major axis at least twice the horizontal measurement

distance and a minor axis at least ^3 times the horizontal measurement distance [1, Sec. 7.1.1.1].

An estimate of the magnitude of fields diffracted from the edges of the ground plane could be

obtained from the geometrical theory of diffraction [20]. An experimental evaluation of edge

reflections could be obtained on a large ground plane by performing site attenuation

measurements [21] at various distances from ground plane edges.

Ground plane surface irregularities can occur where ground plane sections join or where

wire-mesh screens buckle from weathering or extended use. The Rayleigh roughness criterion

[22] is a simple requirement for the maximum surface deviation h for which a rough surface will act like a smooth surface. It is based on a maximum phase variation of Till for reflected rays and

for arbitrary incidence angle is given by /? < A(/8, where Xq is the free-space wavelength. For a maximum frequency of 1 GHz, this criterion gives /? < 3.75 cm, but this will need to be reduced as the maximum frequency is raised to 2 GHz or higher. This is a fairly crude criterion, and

perturbation theory for rough surface scattering [23] could be used to provide estimates of

measurement uncertainty caused by ground plane roughness.

All ground planes are imperfect conductors, but solid metal sheets have conductivities on

the order of 10^ S/m and can be considered perfect conductors for practical purposes. Wire mesh

screens [24] can be used for ground planes if the mesh periodicity is sufficiently small (typically

10
less than Xq/\0). Reflection coefficients for nonmetal materials depend on incidence angle and

polarization [25] and can depart significantly from the ideal value of one. Reflecting objects should be well clear of the elliptical area described above, but even more distant reflectors can introduce significant reflections if they large enough (vehicles,

buildings, etc.). If sites are covered, the cover can introduce significant reflections, particularly if

it is wet or dirty.

Because of the all the factors that contribute to OATS uncertainty, measurement

uncertainty studies have not attempted to analyze the individual effects separately. Instead the

approach has been to perfiDrm site attenuation measurements [21] or radiated emission

measurements [26,27] with the same source at multiple sites. These results are very useful for

evaluating uncertainty for measurements made on OATS, but they do not allow isolation of the

component of uncertainty due to OATS from other uncertainty sources (antennas, mismatch,

receiver, etc.). Reference [2] shows a worst-case uncertainty (rectangular distribution) of 2 dB

for OATS imperfections, and that is consistent with the uncertainties determined in references

[21,26,27]. As expected, it is less than the maximum 4 dB deviation from the theoretical ideal site attenuation required for qualification of an OATS for radiation emission measurement from

computing devices [3].

The uncertainty due to ambient fields is variable and hard to assess. There are certain

times and frequencies when the ambient field levels at the National Institute of Standards and

Technology OATS are too high to perform antenna calibrations. Kolb [27] reports the same

problem with radiated emission measurements at a number of sites. Reference [2] lists ambient

signals as a possible source of measurement uncertainty, but does not give any typical values.

5.2 Anechoic and Semi-Anechoic Chambers

Anechoic (absorber-lined) chambers attempt to simulate a free-space environment with no

reflections. Semi-anechoic chambers have a conducting floor and attempt to simulate an OATS

with only a ground reflection. Semi-anechoic chambers can be used as an alternative to OATS if they can demonstrate equivalence by showing agreement of site attenuation measurements with

theory [28]. Both anechoic and semi-anechoic chambers have an advantage over OATS because

they are shielded from external ambient fields. The primary source of measurement uncertainty due to the anechoic chambers is wall

reflections (single or multiple). Absorber design and the ability to analyze [29] and to measure

[30,31] absorber reflectivity are improving, but it is still difficult to evaluate measurement

uncertainty in anechoic chambers. A recent paper [32] showed agreement between measured and

theoretical site attenuation within 4 dB at frequencies from 35 to 1000 MHz, but poorer

agreement at 30 MHz. Because of the push for indoor measurements and improvements in absorber, anechoic and semi-anechoic chamber measurement uncertainty should eventually

approach OATS measurement uncertainty.

5.3 Shielded Enclosure

Shielded enclosures are typically well-shielded rectangular cavities with metal walls [1,

11
Sec. 7.2]. They have the advantage of shielding from external ambient fields and are specified for

MIL-STD-462 radiated emissions measurements. However, they are known to produce large

uncertainties [33] in radiated emissions measurements.

Strong wall reflections and cavity resonances [1, Sec. 8.2.2.2] affect both the EUT

radiation and the receiving antenna factor [34]. Radiated emissions measurements with a wellcharacterized spherical dipole radiator have been performed at a number of different shielded

enclosures [35], and large variations have been reported from site to site and day to day. The

overall results in reference [35] imply an expanded uncertainty of approximately 10 dB with

values increasing to 20 dB at cavity resonances. These uncertainty values are so large that the RSS uncertainty combination with other

uncertainties based on a first-order Taylor series is clearly not valid. However, in such a case the

clear course of improvement would be to concentrate on reducing the large uncertainty due to shielded enclosure effects. A step in this direction is to introduce some absorber in shielded

enclosure measurements [36] with the intent of decreasing cavity Q. It remains to be determined

how effective this will be and how well this will work at low frequencies where absorber is less effective. The limit of this approach would be to go to an anechoic chamber.

5.4 Reverberation Chamber

The reverberation chamber (also called mode-stirred chamber) is a high-Q, muUimode

cavity which uses either mechanical [37] or frequency [38,39] stirring to obtain field uniformity.

Up to now only mechanical stirring has been used for emissions measurements. The quantity

measured is the total radiated power [40,41] rather than the maximum electric field strength at a

specified distance.

The measurement method is to compare the power received with a reference receiving

antenna when the EUT is radiating with that when a known power is fed to a well-characterized

transmitting antenna as in figure 5. The main source of measurement uncertainty is imperfect

spatial uniformity in the chamber fields. The expanded uncertainty due to this effect is approximately 3 dB [37,41], but this value can be reduced with improved stirring and data

processing [42]. Antenna patterns have little effect in reverberation chamber measurements, but

input impedance and efficiency affect the reference measurement and can introduce an additional

expanded uncertainty on the order of 1 dB [41,42].

5.5 TEM and GTEM Cells

The transverse electromagnetic (TEM) cell is a single-mode, guided-wave structure as shown in figure 6. The center region is a rectangular coaxial transmission line, and transition

sections taper down to the two ports. If the EUT is electrically small, the free-space radiation

pattern and the total radiated power can be determined from two-port measurements with the

EUT in several orientations [43]. Because the TEM cell needs to be operated in the single-mode

region below the cutoff frequencies of the higher-order modes [44], this is a low-frequency

method.

An uncertainty analysis of standard field generation with TEM cells [45] has shown an

12
expanded uncertainty of 0.8 dB. Measurement of an electric dipole moment [43] with the

geometry in figure 6 has shown a low expanded uncertainty of 0.2 dB, but a more complex EUT

which will have more electric dipole and magnetic dipole components can be expected to have a

larger measurement uncertainty. Consequently, the 0.8 dB measurement uncertainty for TEM cell measurements found in [45] is probably a realistic value for expanded uncertainty.

The gigahertz transverse electromagnetic (GTEM) cell [46] is a tapered version of figure

6 with a broadband termination in place of Port 2. It has the advantage of a broad bandwidth, but

has larger measurement uncertainty because it is a nonuniform structure with a complex

termination. It has been used to perform radiated emissions measurements from 30 to 1000 MHz,

but further work is needed on evaluating its measurement uncertainty for general EUTs.

5.6 Three-Loop System

The three-loop system [47] comprises three orthogonal loop antennas, each with two

oppositely located loads as shown in figure 7. If an electrically small EUT is located at the center,

the radiation characteristics can be determined in terms of three equivalent electric dipoles and

three equivalent magnetic dipoles in a manner analogous to TEM cell measurements [43]. Both

methods are restricted to low frequencies, but the three-loop system has the advantage that the

EUT does not have to be rotated.

Some experimental uncertainty analysis has been performed for the three-loop system [48]

by using a controlled spherical dipole radiator as the source. The expanded uncertainty is approximately 2.5 dB, but further uncertainty study is required.

6. ANTENNA

This section discusses methods for determining antenna factors of antennas used for

radiated emission measurements for the frequency range between 30 and 1000 MHz. Antennas

applicable for this measurement are linearly polarized antennas such as tuned dipoles, biconical

dipoles, and log periodic arrays. The typical calibration methods used for this task include

standard antenna, standard field, standard site, and reference antenna [49,50].

6.1 Standard-Antenna Method

The standard-antenna method used at NIST involves half-wave dipole receiving antennas

at the open area test site (OATS) typically for the frequency range between 30 and 1000 MHz

[51,52]. The NIST standard antennas are self-resonant, half-wave dipoles with a high-impedance,

balanced voltmeter built across the center gap of the dipole. The calibrating field strength is determined from the effective length and the open-circuit rf voltage induced in the standard

dipole. The effective length of the standard dipole having a length near self resonance is presently

calculated through a simple Schelkunoff formula [53] and is accurate within 0.1 dB (expanded

uncertainty). We are now in the process of recalculating this effective length by use of the

method of moments, and this will be reported later along with the uncertainty of the numerical

13
theoretical model.

Other sources of uncertainty in the standard-antenna method involve uncertainties in the

dc to rf voltage transfer and the various instruments used such as spectrum analyzers, rf

vohmeters and power monitors, uncertainties in the disturbance of the field caused by the ground

screen edges, interconnecting cables and nearby buildings, and uncertainties associated v/ith

antenna alignments, antenna heights and separation distances. Uncertainty analyses of these

effects are currently underway at NIST.

6.2 Standard-Field Method

The standard-field method used to establish reference cw field strengths in the NIST

anechoic chamber uses a series of standard-gain, open-ended waveguides from 200 to 500 MHz

and pyramidal horns from 500 to 1000 MHz [54-56]. The gain of an open-ended waveguide has

been studied experimentally and theoretically [54,55]. The expanded uncertainty of this gain is about 0.1 dB [54,55], but fijrther analysis by the method of moments is underway at NIST.

Standard fields above 500 MHz are produced by a series of standard-gain, pyramidal horns.

Currently we use a simple formula derived by aperture integration for the far-field and near-field

gains [56]. The expanded uncertainty of the near-field gain is estimated to be 0.5 dB [57], but

this area requires fiirther study. Other uncertainty sources involve uncertainty in the magnitude of

multipath reflections within the anechoic chamber, the various instruments such as directional

couplers, incident and reflected power monitors, and uncertainty associated with antenna

alignments and antenna separation distances.

For the calibration of vertically polarized monopole antennas over the frequency range of

30 kHz to 300 MHz, standard cw fields are established at specific field strengths for specified

distances fi^om the transmitting monopole on the OATS [58]. The field strength is calculated in terms of the applied voltage, the length and diameter of the transmitting monopole, the distance

between the transmitting monopole and the observation point, and its height above the ground

plane. The expanded uncertainty of the vertical electric field is estimated to be about 0. 1 dB. The

receiving antenna is usually far enough away from the transmitting monopole that the field

incident on the receiving antenna can be treated as a plane wave (constant amplitude and phase

along the receiving antenna). If the separation distance is not large enough, the incident field is not constant along the receiving antenna, and this requires the "non-planarity" correction [59].

There is also another proximity correction that results from the mutual impedance between two

monopoles. These non-planarity and proximity corrections can be applied properly to the

calculation of the antenna factor. That should eliminate most of the uncertainty associated with

these factors. The other sources of uncertainty listed in the standard-antenna method are also

applicable for the standard-field method.

6.3 Standard-Site Method

The standard-site method requires three sets of attenuation measurements between two

different antennas under indentical geometries using three different antennas taken in pairs.

Therefore, this method is also called the three-antenna method. If two identical antennas are to be

14
calibrated, their antenna factor can be obtained from a single attenuation measurement: thus, this

is called the two-antenna method. The standard-site method is based on an ideal (perfect) ground

plane, and thus the uncertainty of this calibration method is strongly dependent on the quality of

the ground plane. Otherwise, the measurements on which the standard-site method is based are attenuation measurements, i.e., ratio measurements; thus absolute measurements of the power

and/or voltage are not necessary. It is, thus, believed to be a relatively accurate measurement

provided that the ground plane is ideal. A detailed uncertainty analysis of the standard-site

method is currently under study at NIST.

6.4 Reference-Antenna Method

The reference-antenna method provides a method of antenna calibration based on the use of a (calibrated) standard antenna. The antenna factor is then derived simply by substitution

against the standard antenna. The ratio between the two measurements of received voltage

during the substitution is the difference (in decibels) between the antenna factors of the standard

antenna and the unknown antenna. The main source of uncertainty in the calibration using this

reference antenna method is therefore the uncertainty of antenna factor of the standard antenna

used as a reference. The other sources of uncertainty have been discussed previously.

7. RECEIVER

Impedance mismatch can exist at both the antenna and the receiver. Antenna impedance

mismatch could be accounted for either in the antenna factor calibration or together with receiver

impedance mismatch [60]. Some estimates of the combined mismatch contribution to uncertainty

have been proposed in references [1,2,9], and they tend to have an expanded uncertainty of

approximately 1 dB depending on the antenna type. Under some assumptions, the probability

distribution is U-shaped [2,60], but rectangular or normal would serve as well for setting the

expanded uncertainty. Further work is needed here, but it needs to be done in connection with

antenna factor calibration and uncertainty analysis.

Receivers for radiated emissions measurements are thoroughly discussed in reference [1,

Ch. 3] and uncertainty considerations are covered in reference [1, Ch. 8]. A rectangular

probability density with a worst-case uncertainty of ±1.5 dB is suggested in reference [2], and this

is consistent with reference [9]. Bandwidth issues need flirther study if this value is to be reduced.

8. CONCLUSIONS AND RECOMMENDATIONS

In this report we have summarized the major sources of uncertainty in radiated emissions

measurements. As indicated in Sections 3 through 7, many uncertainty sources are not well

understood and require further study. Consequently, the combined uncertainty and expanded

uncertainty f/are not well known. This causes a serious difficuhy in compliance testing if we

follow the approach [2] that the measured emission plus U must be below the emission limit for

15
one to state that the EUT complies. We need to know U (or an accurate estimate) to follow this

approach. Even if we follow the competing approach [12] that the measurement uncertainty is included in the emission limit, the expected expanded uncertainty is needed to set that limit with

confidence. The need for a good estimate of U cannot be avoided.

One difficulty in determining measurement uncertainty for radiated emissions is the large

number of uncertainty sources. For example, Table 1 taken from reference [2] lists 13 uncertainty

contributions for a measurement of vertically polarized radiated emission performed on an OATS.

Most of these items could be the subject of an uncertainty study in their own right, and most of

them can be further subdivided into separate uncertainty contributions. For example, the

contribution Site imperfections in Table 1 is subdivided into four subtopics in our Section 5.1, and

five other types of measurement facilities are discussed in Section 5. No uncertainty estimates for

the contributions Ambient signals and Repeatability o/EUThtq given in Table 1, and this reflects

the difficulty of their evaluation. Actually the contribution Repeatability ofEUT is only one of

many issues raised in our Section 3 on SETUP OF EQUIPMENT UNDER TEST. Also our

subtopics in Section 4 on MEASUREMENT PROCEDURE are not included in Table 1. The

overall point is that a complete list of uncertainty components is long and most of them require

further study.

A second difficulty in determining measurement uncertainty of radiated emissions is that

the separate and combined uncertainties are not necessarily small. Even though the expanded

uncertainties in Table 1 [2] are on the order of 4 dB, other estimates that include additional effects

give expanded uncertainty estimates on the order of 10 dB or greater [1,8]. When the

uncertainties are this large, the standard propagation of uncertainty formulas and RSS

combination that are based on first-order Taylor series may not be valid (see Appendix A). Also

the use of decibel specification rather than relative values may not be valid when the measured

quantity rather than the logarithm is normally distributed.

Even though the importance of determining EMC measurement uncertainty is well

recognized [12] and the general methods of quantifying and specifying measurement uncertainty

are now agreed upon in the international community [6], further work in radiated emissions

measurement uncertainty is clearly needed. Because of the large number of uncertainty

components, both experimental and theoretical approaches are needed.

Experimental work can contribute to improved knowledge and statistical data bases in the

following areas (in the order of their previous discussion):

• EUT mode of operation

• EUT configuration (including cable layout)

• Measurement procedure (including sampling issues)

• OATS imperfections

• Ambient fields

• Anechoic and semi-anechoic chambers

• Shielded enclosures

• Reverberation chambers

GTEM cells

• Three-loop system

• Antenna factor

16
• Impedance mismatch

• Receivers (including bandwidth issues)

Similarly, theoretical and computational work can contribute to improved knowledge and

statistical data bases in the following areas: Propagation and combination of uncertainties when uncertainties are not small

Use of decibel versus relative uncertainty

EUT configuration (including cable layout)

Measurement procedure (including sampling issues)

OATS imperfections

Anechoic and semi-anechoic chambers

Reverberation chambers

Three-loop system

Antenna factor

Impedance mismatch

Although this report has addressed only radiated emissions measurement uncertainty, most

of the same issues appear for radiated immunity measurements. Maxwell's equations are

reciprocal, so many of the same uncertainty quantities can be obtained directly by reciprocity [61].

However, some of the differences in the monitoring of the EUT, the facility use, and the

measurement procedure in radiated immunity measurement [1] require separate uncertainty

treatment.

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materials from 30 to 1000 MHz. IEEE Trans. Electromag. Compat., 34: 304-307; 1992.

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[34] Camell, D.; Koepke, G.; Smith, R.; Rakoski, B. A standard source method for reducing

antenna factor errors in shielded room measurements. Natl. Inst. Stand. Technol. Tech.

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[35] Koepke, G.; Randa, J. Screened-room measurements on the NIST spherical-dipole

standard radiator. J. Res. Natl. Inst. Stand. Technol, 99: 737-749; 1994.

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[38] Hill, D.A. Electronic mode stirring for reverberation chambers. IEEE Trans. Electromag.

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Gaussian noise excitation for reverberation chambers and applications to radiated

susceptibility testing. Natl. Inst. Stand. Technol. Tech. Note 1375; 1996.

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7; 1993.

[41] Hill, D.A.; Camell, D.G.; Cavcey, K.H.; Koepke, G.H. Radiated emissions and immunity

of microstrip transmission lines: theory and reverberation chamber measurements. IEEE

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[42] Ladbury, J.M.; Koepke, G.H.; Camell, D.G. Evaluation of the NASA Langley Research

Center mode-stirred chamber facility. Natl. Inst. Stand. Technol. Tech. Note; in

preparation.

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radiating sources from tests inside a TEM cell. IEEE Trans. Electromag. Compat., 23:

113-121; 1981.

[44] Hill, D. Bandwidth limitations of TEM cells due to resonances. J. Microwave Power, 18:

181-195; 1983.

[45] Kanda, M.; Orr, R.D. Generation of standard electromagnetic fields in a TEM cell. Nat.

Bur. Stand. (U.S.) Technical Note 1319; 1988.

[46] Wilson, P. On correlating TEM cell and OATS emission measurements. IEEE Trans.

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Electromag. Compat., 37: 1-16; 1995.

[47] Kanda, M.; Hill, D.A. A three-loop system for determining the radiation characteristics of

an electrically small source. IEEE Trans. Electromag. Compat., 34. 1-3; 1992.

[48] Novotny, D.R.; Masterson, K.D.; Kanda, M. An optically linked three-loop antenna

system for determining the radiation characteristics of an electrically small source. IEEE

Int. Symp. Electromag. Compat.: 300-305; 1993.

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Measurements in Electromagnetic Interference (EMI) Control, ANSI C63.5, 1988.

[50] Kanda, M. Standard antennas for electromagnetic interference measurements and

methods to calibrate them. IEEE Trans. Electromag. Compat, 36: 261-273; 1994.

[51] Kanda, M. Standard probes for electromagnetic field measurements. IEEE Trans.

Antennas Propagat., 41: 1349-1364; 1993.

[52] Camell, D.G.; Larsen, E.B.; Cruz, I.E. NBS calibration procedures for horizontal dipole

antennas (25 to 1000 MHz). Nat. Bur. Stand. (U.S.) Technical Note 1309; 1987.

[53] Schelkunofif, S.A.; Friis, H.T. Antennas, Theory and Practice . New York: Wiley; 1952.

[54] Kanda, M.; Orr, R.D. Near-field gain of a horn and an open-ended waveguide:

comparison theory and experiment. IEEE Trans. Antennas Propagat., 35: 33-44; 1987.

[55] Wu, D.I.; Kanda, M. Comparison of theoretical and experimental data for the near field of

an open-ended rectangular waveguide. IEEE Trans. Electromag. Compat., 31: 353-358;

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reference fields in the NIST anechoic chamber. Natl. Inst. Stand. Technol. Tech. Note

1335; 1990.

[58] Camell, D.G.; Larsen, E.B.; Cruz, J.E.; Hill, D.A. NIST calibration procedure for

vertically polarized monopole antennas. Natl. Inst. Stand. Technol. Tech. Note 1347;

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Trans. Electromag. Compat, 35: 94-96; 1993.

21
[60] Harris, I. A.; Warner, F.L. Re-examination of mismatch uncertainty when measuring

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70C: 263-273; 1966.

APPENDIX A. PROPAGATION OF UNCERTAINTY

In many cases a radiated emission Y is not measured directly, but is determined from a

functional relationship/of Another quantities A';,X, . . . ,Xj^:

7 -fiXr,X^,.-.,X^). (A-1)

Here the function/ should represent the entire measurement process and should include all quantities that contribute to the measurement uncertainty. The measurement estimate >» is obtained from eq (A-1) using estimates Xj, x^, . . . , x^:

y - Ax^, Xj, . .. , x^). (A-2)

If we expand eq (A-2) in a first-order Taylor series, we can evaluate the mean of (y -YY to obtain

the combined standard uncertainty u^(y) in the following form [5]:

dx

N-\ N

"" '«^(x,).2E E i^|^.(x„^;, (A-3)

i-i j<.\ OX. ox.

'7

where the partial derivatives dfldx^ (often referred to as sensitivity coefficients) are equal to

dfldX^ evaluated at X, = x„ w(x,) is the standard uncertainty associated with the input estimate x„ and w(x„ x^) is the estimated covariance associated with x, and x^. Because eq (A-3) is derived from the first-order Taylor series expansion of/, it contains

only first-order derivatives of/ with respect to x,. Even so, it does not fit the RSS form in eq (1)

because of the covariance terms in the double summation. The result of the double sum in eq

(A-3) can be either positive or negative because the partial derivatives and the covariances can be

positive or negative. In the very worst case where the uncertainties are perfectly correlated.

22
u(x^,x) = u(x^)«(xp,the right side of eq (A-3) is a perfect square. Then the square root of eq

(A-3) yields

dx.

N

u{x) - Y, «,, (A-4)

i-\

where we have assumed that all dfldx^ factors are positive (or negative) to yield the worst case. The linear sum in eq (4) clearly gives a larger combined uncertainty than the RSS sum in eq (1),

and this is the recommended form in reference [2] for cases of "adverse correlation."

If the covariances are zero (x,and x, are uncorrected for / *j), then eq (A-3) reduces to

N

«>) = Ei-\

2 N

(A-5)

i-l

where we have defined u^ = (dfldx) u(x). Equation (A-5) now fits the RSS form in eq (1).

However, it is important to recognize that the derivation of eq (A-5) requires that terms higher

than first order in the Taylor series expansion for^- are negligible and that x, and Xj are uncorrelated.

To examine the requirement that higher-order terms must be negligible for the validity of

eq (A-5), we start by writing a Taylor series of eq (A-2) including terms up to second order:

i.\ dx^ 2 i.i 5jc/

E E

dV

(A-6)

i.\ j^A dx^dXj

(x^ - X^) (x - X) + higher order terms

Following the mathematics for the evaluation [62] of the mean of (y -Y)^, we find that the first correction to the right side of eq (A-5) is a summation of cubic terms of the following form

NE^f^f 3/ \

M dxi dXf

(A-7)

where Yi is a measure of skewness given by y, = [mean (x^ - X)^]/u ^(x.) . If all of the x's have a symmetrical distribution, then all of the y/s are zero and the summation of cubic terms in eq (A-7)

23
vanishes.

The next higher-order corrections to the right side of eq (A-5) involve summations of

fourth power terms of the following form [62]:

N

4 i.i

N-l N

i-i y-«.i dxdx.^

u\x)u\x). (A-8)

where F, is related to the elongation [62] and is given by T^ = [mean (x^-Xy]/u \x^). For a normal distribution, F,= 3. In this case the higher-order correction terms in eq (A-8) reduce to

1 "" 'bV'

dx

2 N-\ N f ay

dxpx.^

u\x)u\x). (A-9)

If/ is a linear function of the Jf„ then the second-order derivatives are zero, and the correction

terms in eq (A-9) are zero regardless of the values of w(x,). For general functions/, the correction

terms in eq (A-9) are not zero and are small only if w(x,) or the partial derivatives are small.

APPENDIX B. UNCERTAINTY OF A PRODUCT OF QUANTITIES

In general the propagation of uncertainty requires the evaluation of partial derivatives with

respect to estimated quantities, and the expressions in Appendix A cannot be simplified flirther

without specifying the function/ The special case where/ is a product is interesting because all the partial derivatives are easily evaluated and the results are conveniently cast in terms of relative

uncertainties of the individual estimates [2]. A simple example is the case where a radiated

electric field is the product of the antenna factor and the received voltage.

For a product of N quantities, Y is given by

7 = X^X^. . .Xj^. (B-1)

The measurement estimate y is similarly given by a product of the estimates Xj, x^, .. . , X;^:

y = x^x

2

• • • ^N- (B-2)

The individual standard uncertainties u. can be written

24
ax. u{x) - uix) -

1/1 u{x)

(B-3)

Hence w, is proportional to the relative standard uncertainty w(x,)/|x,|.

For the case of perfectly correlated uncertainties, we can substitute eq (B-3) into eq (A-4)

to obtain the combined relative uncertainty:

u(x)

(B-4)

Here the relative combined relative is seen to be the linear sum of the individual relative

uncertainties. For the case of uncorrected uncertainties, we can substitute eq (B-3) into eq (A-5)

to obtain

".0)

. ^1 , \x\

(B-5)

In this case the relative combined uncertainty fits the RSS form. If we attempt to write eq (A-5)

in terms of relative uncertainties for a general fijnction/, then the result is

[ \y\ I {dx,y]

\2 \2

(B-6)

When/has the product form as in eq (B-2), then the first factor in the summation is equal to 1 and eq (B-6) reduces to (B-5).

Following Appendix A, we can evaluate the higher correction terms to eq (B-5) for the

special case of the product fianction. The cubic correction term in eq (A-7) is zero because the

second derivative of/with respect to x, is zero. The first summation in eq (A-8) is also zero because the same second partical derivative is zero. The second partial derivative in the double

summation reduces to

_aV_ __ _y_

dxBx. XX. t J 'J

(B-7)

Thus the first nonzero correction to eq (B-5) is

25
N-\ N

E E

[

^,

J

2 (B-8)

which involves a product of the squares of relative uncertainties.

Many EMC measurements are expressed in decibels relative to some convenient reference

level. If Y in eq (B-1) represents power, then we can rewrite eq (B-1) as

>' (dB) = 101og,,(x,x,. . .x^) = 5: x^ (dB),

i-l (B-9)

If we make the convenient assumption of a log normal distribution for>^ and x„ then^- (dB) and x, (dB) are normally distributed, and we can write the combined uncertainty in the RSS form:

u(y) (dB) = ^ y: u\x) (dB). (B-10)

Since all of the higher-order partial derivatives are zero, all of the higher-order correction terms

for eq (B-10) are zero, and eq (B-10) is exact. (This assumes the original convenient log-normal

assumption.)

26
Table 1. Measurement uncertainty of vertically polarized electric field strength over the

fi^equency range 30 MHz to 1 GHz on an open area test site at 3 m and 10 m [2].

Contribution

Probability

Distribution

Uncertainty (dB)

Biconical Antenna Log periodic

Antenna

3m 10m 3m 10m

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 directivity rectangular +0.5 +3.0 +0.5

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

Antenna phase center variation rectangular ±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 ±0.4

Site imperfections rectangular ±2.0 ±2.0 ±2.0 ±2.0

Mismatch

Receiver VRC:r|=0.2

Antenna VRC: rg=0.67 (Bi) 0.3 (Lp)

Uncertainty limits 20Log (1 ±r, rj

U-shaped ±1.1 -1.25

±1.1 -1.25±0.5 ±0.5

System repeatability (previous assessment of s(qi() from 5 repeats, 1 reading on EUT)std. deviation ±0.5 ±0.5 ±0.5 ±0.5

Repeatability of EUT - - - -

Combined standard uncertainty ufy) normal +2.19

-2.21

+2.16

-2.20

+2.52

-1.82

^1.74

-1.72

Expanded uncertainty U normal (k = 2) +4.38

-4.42

+4.32

-4.40

+5.04

-3.64

+3.48

-3.44

27
Cables

Measurement Facility

Receiver

yf / \

Receiving

Antenna

aX

EUT

^

Figure 1 . Generic setup for a radiated emissions measurement.

£o/ M-o

Figure 2. Geometry for radiation from a surface surrounding an EUT.

28
Bq/ M-o

Figure 3. Geometry for radiation from surface currents, cable currents, and apertures.

D

Receiving

Antenna

EUT Height

Scan

/////////////////////////////////// /////^//

Figure 4. Setup for radiated emissions measurement on an OATS.

29
Stirrer

EUT

Transmitting

Antenna

Receiving

Antenna

Figure 5. Setup for radiated emissions measurement in a reverberation chamber.

30
Outer Conductor

Portl

EUT

Inner Conductor

Port 2

Outer Conductor

Figure 6. Setup for radiated emissions measurement in a TEM cell.

Figure 7. Setup for radiated emissions measurement with a three-loop antenna system.

31

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