Statistical Simulations and Measurements Inside a
Microwave Reverberation Chamber
Thomas F. Trost
Atindra K. Mitra
Department of Electrical Engineering
University of Nevada,Reno
Reno, NV 89557
Departmentof Electrical Engineering
Texas Tech University
Lubbock, TX 79409
Abslraci: A set of measurementsand a set of simulations for two
types of statistical models that describe a component of the power
density inside a microwave reverberation chamber are presented.
The first model is a probability density function (PDF) whereas the
secondmodelis a spatialcorrelationfunction(SCF). Both types of
measurements and simulations are analyzed with respect to the
corresponding ideal theoretical model. The functions are observed
over a frequency range that allows for the conditions in the
(experimentaland simulated)chamberto vary from non-ideal,
multi-moded operation to ideal, overmoded operation in the highfrequency limit. A criterion for the low-frequency limit of chamber
operation is proposed from these observations.
the distribution falls towards zero rapidly, forming a theoretical
“cut-off’ for the power measurements.
The statistical theory for the fields in the chamber can be
extendedto include a second-orderstatistical model. As is the case
in many practical statistical applications, this is accomplished by
calculating correlation functions instead of considering PDF models
that could lead to complicated and perhaps unrealistic equations that
are not easily interpretable. The particular correlation function that
is considered and investigated as a ideal second-ordermodel in this
study is extracted from Lehman’s [l] “‘A Statistical Theory of
Electromagnetic Fields in Complex Cavities.” Lehman presents a
general statistical theory for the fields in irregularly shaped, or
unsymmetrical, cavities. Here, the treatment of the first-order
statistics of complex cavities can be interpreted as a generalized and
detailed mathematical expression of the heuristic treatment for
reverberation chambers in [2]. (The PDF’s from the idealized
complex cavity calculations, in [l], are identical to the PDF’s
derived in [2] for reverberation chambers.)
Two sets of correlation functions are calculated in [l] for a
number of field variables in a “complex cavity”. The first type of
correlation is denotedas spatial correlation
THEORETICALMODELS
In this paper ideal statistical models from two sources [ 1,2] are
compared with simulations and measurements. The statistical
models that are presented in [2], entitled “Statistical Model for a
Mode-Stirred Chamber,” are representative of current statistical
treatments for the analysis of reverberation chambers. Here, the
PDF for the power density, at a point in the chamber away from the
wall, is given by the exponential distribution (chi square with two
degreesof freedom) in (1).
f(p)=-e 1
2cr2
-p/2cT2
where p is the power density for one field component(Ex, Ey, or &)
and is therefore proportional to the received power for most (linearly
polarized) sensor/antennaconfigurations.
Histograms of power density measurements from a highly
ovetmoded (or electrically large) cavity along with goodness-of-tittest results are also provided in [2]. The distributions of this
measured data are in close agreement with the theoretical
exponential distribution of (1). Similar histograms are studied in
this investigation for the case of a chamber system with a range of
excitation frequencies such that, in the lower end of the frequency
spectrum, the chamber is electrically small. For these particular
applications (i.e., PDF measurements),the power measurementsare
processed in log magnitude (or dB) and it is convenient, for
comparison purposes, to perform the following change of variables
onthePDFof(1).
P = 1Olog(p)
(2)
This change of variables leads to the following distribution for the
power (power density) in dB [ 11.
fP (P) = (l/pl)ze-”
where k is the wavenumber as a function of tiequency and ~1, ~2are
position vectors.
Equation (4) can be applied towards the analysis of microwave
reverberation chambers by considering a reverberation chamber to
be an ensembleof complex cavities with volume averagesreplaced
by ensembleaverages[ 1, p. 651.
The following ideal assumptions are embedded in all of the
correlation calculations:
(1) Equal Energy Assumption: In a 3 dB bandwidth about a
particular excitation frequency, the modal Q’s are approximately
equal to the average Q of the cavity and the modal frequencies are
approximately equal to the excitation frequency.
(2) The number of modes, M, within a 3 dB bandwidth is
large. An approximate expression [ 1] for M is given by (5).
(5)
where Qnet is the average of the modal Q’s about a particular
excitation frequency, 0.
(3)
where z = e@p)/p , ,0’ = (1O/in 10) = 4.343, and P is the average
measuredpower in dB. The standard deviation for this distribution
is always 5.57 dB [l]. Above the mean, the form of (3) is such that
O-7803-4140-6/97/$1 0.00
(4)
(1)
A block diagram of the experimental apparatus that was designed
and constructed for this study is shown in Figure 1. The welded
aluminum chamber dimensions are 1.034 m by 0.809 m by 0.581 m
48
while the microwave source frequency, from a network analyzer,
varies from 1 GHz to 13.5 GHz. This frequency range corresponds
to a wavelength range of 30 cm to 2.22 cm. Thus, the source
wavelength is not negligible in relation to the chamber dimensions
in the lower end of the test spectrum. Here the apparatusallows for
the accentuatedobservation of non-ideal phenomenato be contrasted
with ideal theoretical calculations.
ANTENNA
where
F1 = N-’ $5,
,
p2 = N-’ $ P,,
n=l
,
PI,
=
n=l
power density at location 1 for paddle wheel position n, Pzn= power
density at location 2 for paddle wheel position n, and N = total
number of paddle wheel positions.
The basic measurementparameters that the analyzers measure
are known as S-parameters (or scattering parameters). For
reverberation chamber applications, SZI, the voltage transmission
coefficient from port 1 to port 2 is of primary importance since it
gives the ratio of the power at the receiving port to the power at the
transmitting port and is therefore an indication of the power level
that is detectedby an antenna/sensor.These St1 values also give the
averagepower density in the neighborhood of the antenna/sensorin
the chamberand are used directly as statistical samples for both the
PDF and the SCF measurements.
=k
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&NULATION
REVERBERATION
CHAMBER
;
;
b!tETHODOLOGY
A computer simulation of a spatial correlation function (SCF)
measurement inside a microwave reverberation chamber is
developedby modeling the chamber as a rectangular cavity with one
moving, or “perturbed,” wall and no paddle wheel. In this approach,
i
field measurementsinside the chamber are simulated with values
P b SENSORS
calculated from field equations for a set of rectangular cavities with
RF
LINES
heights that vary in equally spaced increments from the height of
the paddle wheel to the height of the top of the actual chamber of
SWITCH interest.
.
A viable approach to this type of simulation is obtained by
NETWORK
modifying the basic approach in [6], entitled ‘An Investigation of
ANALYZER
the Electromagnetic Field inside a Moving-Wall Mode-Stirred
DIGITAL
LINES
Chamber.” This technique, in [6] and [7], is based on obtaining the
Green’s function solution for a rectangular cavity and then
repeatedly applying this solution to a cavity with a perturbed
boundary. In other words, the location of an entire wall of a
rectangularcavity is perturbed in order to simulate a “mode-stirred’
chamber response. The direct application of this method to the
present
simulation problem is not necessarily preferable since
Figure 1. Reverberation Chamber System That Is Used to Make
important chamber parameters, such as the chamber Q, are not
SCF Measurements from 1 to 13.5 GHz.
directly embeddedin the Green’s function approach to the solution
The antenna is connected to the source of the network analyzer for a rectangular cavity. This approach is modified by applying the
and is used for transmitting. This antenna is a log-periodic dipole- “moving wall” chamber concept to a “mode selection” procedure
array with dipoles that allow for efficient transmission in the 1-18 instead of using the Green’s function solution. This procedure leads
GHz range. The two wall-mounted D-dot sensors [3] are used to to a simulation that is directly related to chamber parameterssuch as
study the spatial correlation of the power density between two the Q and the number of modes in a bandwidth. A skeletal
points. These sensorsmeasure the “time derivative of the electric description of this approachis as follows:
+ A large number of resonant frequencies for each
displacement,D, normal to the wall” [4] and liberate an amount of
power to the analyzer receiving port that is proportional to the power rectangular cavity associated with a “moving wall” chamber is
calculated.
density present at the sensortip.
-+ The frequency response of each resonant mode is
The values for the SCF are obtained by calculating a correlation
coefficient for each analyzer frequency with a set sensor spacing. approximated by a simple second- order curve that is derived
These calculations are performed by the control/data-acquisition directly from Qnet . The shape of this curve is similar to the
software as a post-processing step by using the usual product- frequency response characteristic of a high-Q narrow-band filter
with the resonant frequency being analogous to the filter center
moment estimate [5] for the correlation in (9).
frequency.
+ The second-ordercurve is used to determine which modes
are significant at a frequency of interest.
+ A statistical field sample is obtained by summing the fields
due to all significant modes at the frequency of interest.
This “moving wall” algorithm is implemented with certain preprocessing steps in order to shorten overall execution times. The
pre-processingsteps involve the calculation, sorting, and archiving
I
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49
of a large number of resonant frequencies for all of the rectangular
cavities that are to be considered in the calculations. In this case,a
simulation of a chamber experiment with 200 distinct paddle wheel
positions is desired, and resonant frequency arrays are calculated,
using (6), for 200 rectangular cavities.
fm* =;/@Y($@
(6)
phase term between these TE and TM terms generates standard
deviations that are in the neighborhood of the theoretical standard
deviations for the higher frequencies in the simulated frequency
range. For the particular chamber size that is considered in these
simulations, these high frequencies correspond to the ideal case of
large chamber electrical size. A selected set of simulated SCF
outputs are presented in the results and conclusions section.
where c = speedof light, m, n, and p are integers, and a, b, and d are
cavity dimensions.
The simulation program processesthe sorted resonant frequency
data generatedby the pre-processingprogram. It first determines a
set of significant resonances,or modes, at a particular simulation
frequency for a specific cavity height. These modes are selectedby
modeling the frequency response of each cavity resonance,in the
neighborhood of the desired simulation frequency, with a secondorder curve that is analogousto the transfer function of an ideal RLC
circuit.
Qnet, the overall theoretical Q of the chamber with receiving Ddot sensors/sensor,is discussedin detail in [8], as well as in [9], and
is given by (7).
I
02
LDc3th1,stci.Dev.=6.0
9 0.1
ii! 0
60
-40
-30
iscatim2,Std.Dehl.=5.8
-20
-10
0
di3
Qnet= (N/Q~ +1/Q,)'
, N = number of Ddot sensors,V = volume of chamber, h = relative permeability of
chamber walls, 6 = skin depth of chamber walls, S = surface area of
chamber walls, R = sensor load resistance,A = sensor equivalent
area, and EO= pennittivity of free space. The QeqVterm in this
equation expressesthe loading due to the chamber walls while the
Qb term gives the loading due to a sensor. The sensor loading
term is derived [8] [9] from an ideal first-order model for a D-dot
sensor. One objective of this simulation is to study the simulated
SCF response as Qnetis divided by various scale factors (i.e. Q =
Qneti 5 , Q = QnetI 10 , ...)I correspondingto possible additional
chamberlosses.
The electric field, due to each mnp mode, is calculated from the
z-componentof the E-field solution for an ideal rectangular cavity
in (8) [IO].
E, = A,
+A,
where k, =F
sin(k,x)
sin(k,, y) cos(kZz)
exp(j8)sin(k,x)sin(k,y)
, k, =F,
cos(k=z)
(8’
“-50
40
a
-20
-10
0
PA WI
Figure 2. Histograms of Power Density for a Simulation
Frequency of 10 GHz. The Smooth Curves Are Theoretical
Values from (3).
RESULTS AND CONCLUSIONS
Preliminary Measurements
A frequencydomain approachto measuring the Q of a microwave
reverberation chamber can be developed from an analysis of the
theoretical gain models derived in [S] and [9]. These derivations
were initiated by expressing the chamber gain as a ratio of the
chamberQ , Qnet, over the antennaor sensorQ (IO).
G
(10)
k, ~7,
and 8 is a random where G is the gain of the chamber, P, is the power delivered to the
chamberfrom the transmitting antenna,Pd is the power available to
angle between 0 and 360 degrees. Here, the frst term represents
a receiving antenna/sensorfrom the chamber, w is the microwave
the TE solution whereas the second term represents the TM
radian frequency, W is the average energy stored in the chamber,
solution. This equation is evaluated for all of the selected mnp sets
Qnetis the overall Q of the chamber, and Q is the contribution to the
where each resulting term is multiplied by the corresponding
overall Q due to the receiving antennas/sensors. This approach to
magnitude factor from the selection step. The total electric field, at
measuring the Q of a microwave reverberation chamber is to
a point on the bottom of the chamber at the location of a D-dot
measurethe averagechamber gain and then calculate Qnetvalues by
sensor, is calculated by summing all of the these individual terms.
applying the appropriate model for the sensor/antennaQ. This
A power density sample is obtained by taking the magnitudeprocedure was carried out for the chamber system of Figure 1.
squaredof the total field.
Equation (11) was used as the sensormodel with R = 50 ohms and
One statistical test performed to evaluate the quality of the
A= 1U4m2.
simulation output involves tabulating the standard deviation of the
output power density samples and plotting them in histogram form.
A sample test plot is shown in Figure 2. The inclusion of a random
50
PI.
the theoretical values of M (from (5)) are included in the third
cohunn of Table I. These show 3.5 at 4.5 GHz.
PDF of Power Den.@
SCF of Power Dens@
In an earlier study [4] it was found that there is good agreement
at 10 GHz between the measured distribution and the theoretical
PDF, (6), but at 1 GHz poor levels of agreement were observed
since the distributions of the measured values are much “flatter”
than the shapesof the correspondingtheoretical curves.
Inspection of the standard deviation values in the seventhcolumn
of Table I indicates that the first sample standard deviation that is
between 5 and 6 dB occurs at 4.75 GHz. Observing the
corresponding M values in the sixth column of this table indicates
that the measured distribution could convergeto an ideal one for M
values as low as 2.5. This means that Lehman’s assumption of an
infinite M may not restrict the applicability of his theory as much as
one might have supposed.
A similar type of analysis can be performed on the simulation
output. Here, standard deviation values, from a post-processing
program that calculates a set of statistics (and also generates
histograms) for the simulation samples, are tabulated and observed
in the fourth column of Table I. These particular sample standard
deviations result from a simulation run with Q = QnetI 10 and a
distance of 8 cm between calculated field points (or samples) and
correspond to the simulation runs that most closely match the
conditions in the chamber [S]. Inspection of these standard
deviation values, which are for one of the two field points, indicates
that the first sample standard deviation that is between 5 and 6 dl3
occurs at 4.5 GHz. One of the output files from this simulation run
also shows that, in the 4.5 - 5 GHz range, the average number of
selected modes that fall within the 3-dEI bandwidth is 3. These
averageM values are shown in the secondcolumn of Table I. Also
SCF measurementswith sensor spacings of 2.5, 5.5, 8.0, 11.5,
13.5, and 17.5 cm were taken with two commercial D-dot sensors,
whereasmeasurementswith sensor spacings of 0.5, 1.O, and 1.5 cm
were taken with sensorsthat were fabricated from type 141 semirigid cables. All of the measurements were taken with the
measurementsystem of Fig. 1. Fifty-one discrete frequency points
between 1 and 13.5 GHz, with 0.25 GHz increments, were used for
both the measurements and a corresponding set of simulations.
Figure 3 is a set of plots that show measured and simulated spatial
correlations versus spacing for the frequency points from 2.5 to 5.25
GHz. Lehman’s theoretical curve, (4), is plotted alongside these
results. A larger set of SCF measurements and simulations,
including a set of plots that show measured and simulated spatial
correlations versus frequency, are included in [S].
The complete set of simulation runs include three runs, with Q =
Qnet, Q = Qnet110 , and Q = Qnet1100 , for each of the sensor
spacingsof the above-mentionedmeasurements. (Qnetis given by
(7) for the case of two receiving D-dot sensors.) The SCF values
from only the Q = Q,,d/lO runs were chosenfor the plots of Figure 3
since Qnet00 is of the same order of magnitude as the measured
chamber Q’s [8].
Inspection of the forms of the measured SCF responsein Figure
3 reveals high spatial correlations at large spacings and low
frequencies. The measurementsseem to converge, on the average,
to Lehman’stheoretical curve at 3 GHz and above. From Table I at
3.0 GHz M is approximately 1.O. Thus, taken together, the PDF and
SCF measurements suggest that the low-frequency limit of the
theory lies in the range 3.0 to 4.75 GHZ with M from 1.0 to 2.5.
Plots of sample Q measurementsversus frequency are available in
Table I
Partial List of Measured M and c Values from a Q Measurement [9]
Alongside Values from the Corresponding Simulation Output.
Simulation
5-l
Measurement
f (Gm)
Iawe
Mth
0 (W
f (G=9
M
= (dw
1.0000
1.2500
1.5000
1.7500
2.0000
2.2500
2.5000
2.7500
3.0000
3.2500
3.5000
3.7500
4.0000
4.2500
4.5000
4.7500
5.0000
0.04
0.07
0.14
0.22
0.28
0.50
0.53
0.73
0.84
1.12
1.37
1.44
4.36
2.47
2.97
2.88
3.34
0.08
0.14
0.22
0.32
0.45
0.60
0.79
1.00
1.25
1.53
1.85
2.20
2.60
3.03
3.51
4.03
4.60
9.15
9.49
6.74
7.27
8.72
9.26
8.49
7.84
7.98
7.15
6.96
6.98
6.88
6.82
5.88
7.02
6.87
1.oooo
0.62
6.99
1.6250
0.40
8.37
2.2500
0.93
6.72
2.8750
0.73
7.46
3.5000
1.75
6.25
4.1250
3.94
6.42
4.7500
2.54
5.25
SCF
5
Spacing
IO
<cm)
Theo.
15
of EM
Power
20
Density
0
5
10
Spacing
- Solid
MCM.
- Dashed
Sim.
15
20
<cm)
- Dotted
(a) Plots for Frequencies of 2.5,2.75,3.00,3.25,3.50, and 3.75 GE&.
SCF
0
5
10
$5
20
5
10
15
20
I
0
of EM
Sower
Dcnsily
0
5
i0
15
20
5
10
15
20
J
Spacing
4
(cm)
Theo.
Spacing
= Solid
Meas.
= Dashed
Sim.
(cm)
= Dotted
(h) Plots for Frequencies of 4.0,4.25, 4.50,4.75,5.00, and 5.25 GHz
Figure 3. Measured, Simulated, and Theoretical SCF of the Power Density, Versus Spacing, Inside Microwave Reverberation
Chamber of Figure 1.
52
ACKNOWLEDGMENTS
Summary and Diwussion
The introduction and investigation of M as a microwave
reverberation chamber parameter is perhaps the most significant
result of this paper. Further study of this parametercould lead to a
simple criterion that provides an accuratelow-frequency bound for
the operation of a microwave reverberation chamber. Formulas that
are presently under consideration [l I] by other investigatorsinclude
a 60-modes criterion and a 6 x fo criterion, where fo is the
fundamentalresonantfrequencyof the chambercavity.
The 60-modescriterion can be applied by either finding the 60th
mode of the chamber cavity by computer counting or by setting N
equal to 60 in (1) and solving for f. A computercounting calculation
for the chamber in this study (Fig. 4) yields a low frequencycut-off
of 844 MHz. This criterion seems to have originated from NBS
Technical Note 1092 [ 12, p. 211 where the authors state that the
“practical lower frequency limit for using the NBS enclosure as a
reverberationchamber is approximately 200 MHz. This lower limit
is due to a number of factors including insufficient mode density,
limited tuner effectiveness,and ability to uniformly excite all modes
in the chamber.” The dimensionsof this NBS chamberare such that
a frequency of 200 MHz corresponds to the existence of
approximately 60 distinct modes from 0 to 200 MHz.
The fundamental frequency for the chamber cavity in this study
is 235 MHz. Six times this value yields a chamber low frequency
cut-off of 1.41 GHz. This criterion originated from the analysisof a
sample set of chamber data [l 11. In this case,plots of the chamber
VSWR versus frequency were analyzed, and the 6 x fo term was
found to correspond to the low-frequency edge of a region on the
plot where the VSWR responseis approximatelyflat.
The M equals 1.0 to 2.5 criterion that is proposedas a possible
criterion for the low frequency limit of the chamber in this study
correspondsto a low frequencycut-off of 3.0 to 4.75 GHz. Whether
or not this criterion develops into an acceptablebound for the lowfrequency operation of a microwave reverberation chamber is an
open question since this criterion, like the other two that are
described above, was not developed via a formal mathematical
procedurebut is proposedbased on the observationof a limited set
of data.
Additional analytical methods that are currently under
investigation [l l] and could lead to criteria for the low-frequency
cut-off of a microwave reverberation chamber are methodsbasedon
applying Kolmogorov-Smirnov goodness-of-tittests to the measured
PDF’s,
Possible applications for the SCF inside a microwave
reverberation chamber include the consideration of “correlation
length” as part of the overall testing procedures[ 131. This concept
is particularly well-suited to applications that use a microwave
reverberation chamber to simulate the actual EMI environment of
interest. Inspection of Lehman’s theoretical SCF curve (4) for null
points yields that the first null occursat:
(12)
The correlation between the power density at two points becomes
negligible at and beyond this spacing. Thus testing of a particular
device in a chamber will not also require the placement in the
chamberof all the surrounding equipment from the device’sintended
operational environment. Only equipment that falls within the
correlation length would be needed.
53
This research was supported by the National Aeronautics and
SpaceAdministration through Langley ResearchCenter grant NAGl-1510.
The authors would like to thank Dana McPherson at the
University of Nevada, Reno for his assistance in preparing this
document.
REFERENCES
[l] T.H. Lehman, “A Statistical Theory of Electromagnetic Fields
in Complex Cavities,” Phillips Laboratory Interaction Note 494,
May 1993.
[2] J.G. Kostas and B. Boverie, “Statistical Model for a ModeStirred Chamber,”IEEE Trans. Electromagn. Compat., vol. EMC33, no. 4, pp. 366-370,Nov. 1991.
[3] C.E. Baum et al., “Sensors for Electromagnetic Pulse
Measurements Both Inside and Away fIom Nuclear Source
Regions,”IEEE Trans. Electromagn. Compat., vol. EMC-20, no. 1,
pp. 22-35, Feb. 1978.
[4] T.F. Trost, A.K. Mitra, and A.M. Alvarado, “Characterization
of a Small Microwave ReverberationChamber,”Proceedings of the
11th international Zurich Symposiumand Technical Exhibition on
EMC, pp. 583-586,March 1995.
[5] R. N. Rodriguez, “Correlation,” Encyclopedia of StatisticaZ
Sciences,vol. 2, pp. 193-204,Wiley, 1982.
[6] Y. Huang and D.J. Edwards, “An Investigation of the
Electromagnetic Field inside a Moving-Wall
Mode-Stirred
Chamber,” The 8th IEE Int. Co& on EMC, Edinburgh, UK, pp.
115-119,Sept. 1992.
[7] Y. Huang, “The Investigation of Chambers for Electromagnetic
Systems,”Ph.D Dissertation,University of Oxford, 1994.
[8] A. Mitra, “Some Critical Parameters for the Statistical
Characterization of Power Density witbin a Microwave
ReverberationChamber,”Ph.D Dissertation, Texas Tech University,
1996.
[9] A.K. Mitra, T.F. Trost, “Power Transfer Characteristics of a
Microwave Reverberation Chamber,” IEEE Trans. Electromagn.
Compat., vol. EMC-38, no. 2, pp. 197-200,May 1996.
[lo] T.A. Lou&, “Frequency Stirring: An Alternate Approach To
Mechanical Mode-Sting For The Conduct Of Electromagnetic
Susceptibility Testing,” Phillips Laboratory Tech. Report 91-1036,
Nov. 1991.
[l 11 G. Freyer, Consultant,Monument, CO, private communication,
March 1996.
[12] M.L. Crawford and G.H. Koepke, “Design, Evaluation, and
Use of a Reverberation Chamber for Performing Electromagnetic
SusceptibilityNulnerability Measurements,”hrBS Tech. Note 1092,
Apr. 1986.
[13] T.F. Trost and A.K. Mitra, “Eectromagnetic Compatibility
Testing Studies,”Final Technical Report on Grant NAG-l-1510,
NASA Langley ResearchCenter, January 15, 1996.