Characterization of the Maximum Test Level
in a Reverberation Chamber
T. H. Lehman
G. J. Freyer
Science& EngineeringAssociates,Inc
6100 Uptown Blvd., NE
Albuquerque,NM 87110,USA
Science& EngineeringAssociates,Inc.
6100 Uptown Blvd., NE.
Albuquerque,NM 87110,USA
Abstract: Among the significant characteristics of the
electromagneticenvironment in a reverberationchamber are
isotropy, random polarization, and uniformity.
These
characteristicspermit robust immunity testing without moving
the equipment-under-test
or the field generatingantenna.These
characteristics have been repeatedly demonstrated for
frequencieswhere the chamber is highly overmoded. In this
case, the statistical electromagnetictest environmentcan be
describedin terms of a chi-squaredistribution. However, the
maximum value of the test environmentwhich is generallythe
parameterof most interest for immunity testing is a random
variable. Thus thereis a potentialfor an underor over test of the
equipment-under-test.
distribution for experimentaldata. The two DOF chi-square
distribution is a one parameterdistribution totally characterized
by its mean. This makesit possibleto directly comparetheory
to experimentaldatawith no adjustableparameters.The theory
hasbeenvalidatedby an extensiveexperimentaldatabase[7].
However for immunity testing the maximum EME is usually
most interesting. Since the maximum EME is a random
variable,there is the potential for under or over testing of the
equipment-under-test(EUT).
This paper discusses the
distribution of the maximum to mean ratio, its expectedvalue,
and its uncertaintyrange.
APPROACH
This paper discussesthe distribution function for the maximum
to mean ratio for an overmodedreverberationchamber. The
paper addressesthe expectedvalue of the maximum to mean
ratio and the uncertainty in the ratio for a given level of
confidence. The paperalso comparesthe theoreticalpredictions
to experimentaldata.
INTRODUCTION
Over the past severalyears,reverberationchambershave grown
in popularity for radiated immunity testing. Currently several
test standardsare being revised to include the option of using
reverberationchambers. Unlike other radiated immunity test
techniques,a reverberationchamberprovidesan electromagnetic
environment(EME) which is isotropic,randomlypolarized,and
spatiallyuniform. References[l-4] provide a descriptionof the
conceptsof operationof a reverberationchamberand its use for
immunity testing.
The probability density function (PDF) for a two DOF chisquare distribution, expressedin terms of mean referenced
logarithmicdata(dB), is shownin Figure 1. The relatively sharp
cut-off at the high end of the distribution is apparent.
Neverthelessit is difficult to quantify the behavior of the
maximum using the &i-square distribution. However extreme
value theory for orderedstatistics can be used to examinethe
distributionfunction for the maximumEME.
1.00
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- cm-mumJ*
DQ
“”
Theory exists for predicting the statistical properties of the
randomized EME [5,6]. It predicts that the statistical
distribution function for the power densityin a suitably excited,
overmoded reverberation chamber should be a chi-square
Rmalvod Powor Ratoronced to Mrm (de)
distribution with six degreesof freedom (DOF). This is based
Figure 1. ProbabilityDensityFunctionfor a Two DOF
on two polarizations of three field componentsbeing random
Chi-squareDistribution.
variableswith normal distributionswith zero meanand the same
standarddeviation. This is the requirementfor the EME to be
A limited databaseis available from measurementsin a large
isotropic and randomlypolarized.
shieldedenclosure.The datawere collectedin the final stagesof
Typical monitor antennasor probeswill sampletwo of the six constructionof a semi-anechoicchamberbefore installation of
independentfield componentsyielding a two DOF chi-square the absorbingmaterial.
0-7803-4140-6/97/$10.00
44
Figure 3 also shows that at the 95% confidence level the
uncertainty
in the M/M of a singlemeasurementwould be about
Let f,(x) and F,(x) be the PDF and cumulative distribution
3.5 dB.
function (CDF) respectively for a random variable (as for
examplethe distributionof Figure 1) and xi be a samplefrom the
distribution.Let w be the maximum of N independentsamples
xi, Then extremevaluetheory [ 81yields the probability
RESULTS
P(xN<w)= FN(w)= [F,(w)]~
(1)
This is the CDF for the maximum or in the presentcase the
maximum to mean ratio. The PDF for the maximum to mean
ratio is
= N E4w)l”’ fx(w>
fN(w)
(2)
s
For the specialcaseof an exponentialdistribution(2 DOF chisquare distribution), the PDF and CDF (equations2 and 1
respectively)can be evaluatedanalytically.
Figure 2 showsthe PDF of the maximumto meanratio for 200
independentsamplesin addition to the PDF of the cavity EME
from Figure 1 (on a different scale). It showsthe limited range
for the maximumto meanratio for the commonguidelineof 200
samples[ 11. It is the “tightness”of this distributionwhich limits
under or over test extremeswhen using reverberationchambers
for immunity testing.
0.60
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-
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QI-8wMw8
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45
-10
-5
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8,.
0
”
0,
S
1 ”
e ”
10
-
llmwv
I
DATA
6
Mulmum
180
(188
r
w4P6NDolr8AwL88I
llam
6
to Maw
:
Pwlrwla)
6
10
11
(de)
R8tlo
Figure3. CumulativeDensityFunctionfor 200 Samples.
A limited experimental data base is available for theory
verification. Figure 4 shows data from a large shielded
enclosure.Thesedataprovided60 independentsamplesat a 151
frequencies.
”
1s
4
S
6
7
6
0
Maxlmum to Moan Rxtlo (dB)
Rmcolved Powor Roloronood to Moan (de)
10
11
*
Figure 4. ExperimentalVerification of the Theory.
Figure 2. Probability Density Function for 200
IndependentSamples.
Additional appreciationof the value of this distribution can be
gainedby consideringthe CDF of the maximumto mean(M/M)
ratio for 200 samplesshownin Figure 3. The expectedvalue of
7.4 dB is consistentwith the empiricalvalueof 7 - 8 dB [l].
45
CONCLUSIONS
Theory provides an estimate of the expected value of the
maximum to mean ratio and the uncertaintyin the ratio for a
given confidencelevel. The maximumto meanratio distribution
function provides the basis for a statistical estimate for the
maximumEME during an immunity test. It alsopermitsa
Figure 3 also shows that at the 95% confidence level the
Let f,(x) and F,(x) be the PDF and cumulative distribution uncertaintyin the M/M of a singlemeasurementwould be about
3.5 dB.
function (CDF) respectively for a random variable (as for
examplethe distributionof Figure 1) and xr be a samplefrom the
distribution.Let w be the maximum of N independentsamples
xi. Then extremevaluetheory [8] yields the probability
RESULTS
P(x,,,<w)= FN(w)= [FX(w)IN
(1)
This is the CDF for the maximum or in the presentcase the
maximum to mean ratio. The PDF for the maximum to mean
ratio is
= N UW)l”’ f,(w)
fN(w)
(2)
For the specialcase of an exponentialdistribution(2 DOF chisquare distribution), the PDF and CDF (equations2 and 1
respectively)can be evaluatedanalytically.
Figure 2 showsthe PDF of the maximumto meanratio for 200
independentsamplesin addition to the PDF of the cavity EME
from Figure 1 (on a different scale). It showsthe limited range
for the maximumto meanratio for the commonguidelineof 200
samples[ 11. It is the “tightness”of this distributionwhich limits
underor over test extremeswhen using reverberationchambers
for immunity testing.
0.50
‘.
-
0.50
E
0.50
-
0.10
-
-00
I
* ” 3 “‘1
al--1
ow
uAxJnNllATm
-16
40
“,
-6
,’
I”“(
0
”
5
”
8 ”
10
Yulmum to Ymn Rotk (dB)
Figure3. CumulativeDensityFunctionfor 200 Samples.
A limited experimental data base is available for theory
verification. Figure 4 shows data from a large shielded
enclosure.Thesedataprovided60 independentsamplesat a 151
frequencies.
”
I5
4
5
5
Yulmum
Roooivod Power Rofonneod to Yoan (dB)
7
to Man
5
0
Ratlo (dB)
10
11
l
Figure 4. ExperimentalVerification of the Theory.
Figure 2. Probability Density Functionfor 200
IndependentSamples.
CONCLUSIONS
Theory provides an estimate of the expected value of the
Additional appreciationof the value of this distribution can be maximum to mean ratio and the uncertaintyin the ratio for a
gainedby consideringthe CDF of the maximumto mean(M/M) givenconfidencelevel. The maximumto meanratio distribution
ratio for 200 samplesshown in Figure 3. The expectedvalue of function provides the basis for a statistical estimate for the
maximumEME during an immunity test. It also permitsa
7.4 dB is consistentwith the empiricalvalueof 7 - 8 dB [ 11.
46
tradeoff betweenacceptableuncertaintyand test time. The
theoreticaldistribution function for the maximum to mean ratio
comparesfavorably with a limited set of experimentaldata.
ACKNOWLEDGMENTS
This work was partially funded by the Naval Surface Warfare
Center, Dahlgren Division, Dahlgren, VA and the Naval Air
Warfare Center, Patuxent River, MD. The authors also
acknowledge the cooperation of the Hewlett Packard Co,
Roseville, CA and Lindgren RF Enclosures,Inc for permitting
access to the shielded enclosure during the final stages of
constructionand the use of the data in this paper.
REFERENCES
[l] Crawford, M. L. and Koepke, G. .H., Design, Evaluation,
and Use of a Reverberation Chamber fou Performing
Electromagnetic SusceptibilityNulnerability Measurements,
NBS TechnicalNote 1092,April 1986.
[2] Liu, B. H., Chang, D .C., and Ma, M. T., Design
Considerationsof ReverberatingChambersfor Electromagnetic
Inteflerence Measurements,presentedat the IEEE International
Svmnosiumon EMC, Washington,DC, August, 1983.
[3] Crawford, M. L. and Koepke, G. .H., Operational
Considerationsof a ReverberationChamberfor EMC Immunity
Measurements- Some Experimental Results, presentedat the
IEEE International Svmuosium on EMC, San Antonio, TX,
August 1984.
[4] Bean,J. L. and Hall, R. A,. ElectromagneticSusceptibility
MeasurementsUsing a Mode-Stirred Chamber,presentedat the
IEEE InternationalSvmnosiumon EMC, Atlanta, GA, August
1978.
[S] Lehman, T. H,. Statistics of MSCs, presented at the
Anechoic Chamber and Reverberation Chamber Operators
Group Meeting, Naval SurfaceWarfare Center, Dahlgren, VA,
November 1992.
[6] Lehman,T. H,. Statistical Theory of ElectromagneticFields
in Complex Cavities, Interaction Notes, Note 494, USAF
Phillips Laboratory,May 1993.
[7] Freyer, G. J., Hatfield, M. O., Johnson,D. M., and Slocum,
M. B., Comparisonof Measuredand TheoreticalStatistical
Parametersof ComplexCavities,presentedat the IEEE
InternationalSvmuosiumon EMC, SantaClara, CA, August
1996.
[8] Galambos, J., The Asymptotic Theory of Extreme Order
Statistics,JohnWiley & Sons,New York, NY, 1978
47