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Some critical parameters for the statistical characterization of power density within a microwave reverberation chamber

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SOME CRITICAL PARAMETERS FOR THE STATISTICAL
CHARACTERIZATION OF POWER DENSITY WITHIN
A MICROWAVE REVERBERATION CHAMBER
by
ATINDRA KUMAR MITRA, B.S.E .E ., M.S.E.E.
A DISSERTATION
IN
ELECTRICAL ENGINEERING
Submitted to the Graduate Faculty
o f Texas Tech U n iv e rs ity in
P a rtia l F u lfillm e n t o f
the Requirements fo r
the Degree o f
DOCTOR OF PHILOSOPHY
Approved
Chairperson o f the Committee
£
■
____________________________
Accepted
Dean o f the Graduate/School
May, 1996
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UMI Number: 9623845
Copyright 1996 by
Mitra, Atindra Kumar
All rights reserved.
UMI Microform 9623845
Copyright 1996, by UMI Company. AH rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
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ACK NOW LEDGM ENTS
I would like to thank Professor Thom as T rost for his guidance on this project. I have
found our frequent discussions with regard to the specifics of this project, as well as a
num ber o f other discussions that were directed tow ards broader issues concerning EMC
testing and microwave engineering, to be very insightful and enjoyable. I would like to
thank Professor Thom as Krile, Professor H erm ann Krom pholtz, and Professor David
Mehrl for their insights and suggestions. I would like to thank Professor Victor Shubov
for serving as the Graduate Dean's Representative at my dissertation defense.
I would like to express my gratitude and appreciation to m y parents for their support
and encouragem ent throughout the course of my educational experiences.
I would like to acknowledge Adriel Alvarado and Jam es Ledbetter for their
contributions to the developm ent o f the m icrowave reverberation cham ber m easurem ent
system that was used in this study. I would like to acknowledge Lonnie Stephenson for
machining a num ber o f parts that were integrated into the m easurem ent system. I would
like to thank Judy and Steve Patterson for their help in preparing this and related
manuscripts and for their advice with regard to the com puter facilities at Texas Tech.
ii
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T A B L E OF CONTENTS
ACKN O W LEDGM EN TS .......................................................................................................
ii
A BSTRA CT .................................................................................................................................
v
LIST O F TABLES .....................................................................................................................
vi
LIST O F FIGURES ...................................................................................................................
vii
CHAPTER
I.
INTRO D U CTIO N ..................................................................................................
1
II.
CH A M BER PARA M ETERS AND TH EO RETICA L FIELD M ODELS
7
Cham ber Quality Factor ................................................................................
7
Transfer Function M odels ............................................................................
10
Probability Density Functions ......................................................................
13
Correlation Functions ....................................................................................
15
Sum m ary o f Critical Param eters ..................................................................
24
III.
SIM U LA TIO N M ETH O D O LO G Y ..................................................................
27
IV.
M EA SU REM EN T APPA RA TU S .....................................................................
43
System Specifications ....................................................................................
43
Mechanical Design C onsiderations .............................................................
50
RESU LTS A N D CO N C LU SIO N S ....................................................................
58
Power Transfer C haracteristics ...................................................................
58
V.
iii
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PDF o f Pow er Density ..................................................................................
66
SCF o f Pow er Density ..................................................................................
72
Summary and Discussion .............................................................................
90
REFERENCES ...........................................................................................................................
95
A P P E N D K : M ATLAB SIM ULATION M-FELES ..........................................................
97
iv
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ABSTRACT
Statistical m odels for a com ponent of the power density inside a microwave
reverberation cham ber are studied. These models include average chamber gain
characteristics, an ideal probability density function (PDF), and an ideal spatial correlation
function (SCF). The SCF model is extracted from a m ore general theory of complex
cavities. Emphasis is placed on observing the applicability of these models as the
conditions in the chamber are varied from non-ideal, multi-moded operation to ideal,
"overmoded" operation in the high-frequency limit. The observations are made with
m easurements and simulations. Deviations from the models are characterized by and
related to a num ber of critical cham ber parameters such as the Q, the cham ber electrical
size, and the frequency range o f chamber operation. The introduction o f a chamber
param eter M, the number o f modes in a 3-dB bandwidth, provides useful information with
regard to the behavior of the models in the low frequency limits of chamber operation.
The introduction o f SCF measurements to microwave reverberation cham ber systems
provides useful insights into the physical behavior o f the fields inside the chamber.
v
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LIST OF TABLES
5.1
Tabulation o f Calulated M Values from M easured G ain Values ..........................
65
5.2 M easured M and a Values From the Q M easurem ent of Figure 5.3 Alongside
M and a Values From the Corresponding Simulation Output ..............................
70
vi
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LIST OF FIGURES
3.1
Plots o f the M inimum, M aximum, and Average o f the Pow er Density, in dB,
from a Sample Cham ber Simulation .............................................................................
35
3.2
Sam ple O utput File K l.O U T from Cham ber Sim ulation Program .......................
37
3.3
Sample O utput File K 2 .0 U T from Cham ber Sim ulation Program .......................
38
3.4
H istogram s o f Pow er D ensity Data for a Sim ulation Frequency o f 10 G H z and
a C ham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m ........................................
39
H istogram s o f Pow er D ensity Data for a Sim ulation Frequency o f 10 G H z and
a C ham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m ........................................
40
3.6
SC F Sim ulation Output for Two D ifferent Q V alues ..............................................
42
4.1
Block Diagram o f the Basic Reverberation Cham ber System that was Designed
and C onstructed for this Study .....................................................................................
44
D ata Sheet for the Prodyn A D -S 10 ( R ) W all-M ounted D -dot Sensors that
are used to M easure the SC F in this Study ...............................................................
47
Reverberation Cham ber System that is used with HP8719A M icrow ave
Netw ork A nalyzer to m ake SCF M easurem ents from 1 to 13.5 G H z ................
49
4.4
Draw ing o f the Cham ber Door .....................................................................................
54
4.5
D raw ing that Shows Cross-Section o f the Cham ber D oor G roove and
E lastom er Seal ..................................................................................................................
55
4.6
Photograph o f Sensor Plate for C lose Spacing SCF M easurem ents ....................
57
5.1
C ham ber Gain versus Frequency in Configuration 1 ...............................................
59
5.2
Cham ber Gain versus Frequency in Configuration 2 ...............................................
60
5.3
C ham ber Q M easurem ent via Cham ber Gain M easurem ent with Two
Receiving (Prodyn) D -dot Sensors (configuration 3) .............................................
64
3.5
4.2
4.3
vii
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5.4 Probability Density Function for EM Pow er Density in Configuration 1 ............
67
5.5 Probability Density Function for EM Pow er Density in Configuration 2 ............
68
5.6 M easured, Simulated, and Theoretical SCF o f the Pow er Density, versus
Spacing, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or
Figure 4.1 .........................................................................................................................
73
5.7
5.8
M easured, Simulated, and Theoretical SCF o f the Pow er Density, versus
Frequency, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or
Figure 4.1 ..........................................................................................................................
82
Plots o f M easured SCF Response with 50 Paddle W heel Positions and 200
Paddle W heel Positions ..................................................................................................
89
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CHAPTER I
INTRODUCTION
The increasing popularity o f microwave reverberation chambers [1] for
electromagnetic im munity testing applications has, in the recent past, motivated a number
of investigations [2 - 8 ] with regard to obtaining accurate models for the electromagnetic
fields within the chamber. These chambers are generally associated with a number of
desirable features such as statistically uniform (or homogeneous) fields [ 1] as well as high
field strengths in relation to the input power level. The statistical uniformity o f the field
allows a very large cross-section o f the test object to be illuminated with a uniform
(average) power level and is typically accomplished by varying the chamber boundary
conditions with a rotating mechanical tuner (or paddle wheel).
Standard microwave reverberation chambers are comprised o f a metallic rectangular
enclosure (cavity) with a m otor-controlled paddle wheel installed in the vicinity of the top
boundary. Statistical field samples are typically acquired at various points in the chamber
for a large (usually about 200) number o f incrementally spaced paddle wheel positions. A
primary chamber design objective is to obtain a spatial field uniformity such that the spatial
variance o f the average field, calculated by averaging over all incremental paddle wheel
positions, is minimized.
As far as detailed information with regard to these chambers is concerned, NBS
Technical Note 1092 [4], entitled “Design, Evaluation, and Use of a Reverberation
1
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Cham ber for Performing Electromagnetic Susceptibility/Vulnerability Measurements,” is
currently perhaps the m ost commonly referenced guide to standard (m echanically stirred)
microwave reverberation cham ber construction and evaluation. Three significant chamber
quantities that are defined and described in this article are:
(1) M ode Density: A form ula for the total num ber of m odes that can propagate “inside
an unperturbed, lossless, rectangular cham ber” [4] at a particular frequency, f, is
approximately given by Eq. 1.1.
( 1. 1)
where a,b,d are rectangular chamber dimensions, c is the speed o f light, and f is the
chamber source frequency under consideration.
For an overmoded cavity, where the
source wavelength is small in relation to the chamber dimensions, N(f) can be further
approximated by the first term in Eq. 1.1. The derivative o f this term with respect to
frequency is known as the mode density and can be expressed by Eq. 1.2.
dN _ 8jtabd 2
d f~
( 1.2)
c3
This term can be evaluated at various frequencies within the range o f interest and
considered with other significant cham ber quantities to determ ine the effectiveness o f a
reverberation chamber. In other words, the lower frequency lim it o f a particular cham ber
is typically associated with insufficient m ode density.
(2) Composite Quality Factor: Eq. 1.3 is recom m ended for purposes of calculating the
com posite (theoretical) Q o f the chamber.
2
C
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where V is the cham ber volum e, S is the internal surface area o f the cham ber, 8 is the skin
depth, X is the wavelength, and a,b,d are cham ber dim ensions. This com posite Q was
originally developed in [3] and can be viewed as an average o f the Q ’s corresponding to
the resonant modes about a particular frequency. Some o f the effects of lowering the Q
by adding absorbing m aterial are reported to be
(i) “decreases the effectiveness o f tuner” [4, p. 12],
(ii) “increases the cham ber loss and hence increases the r f pow er required to obtain
test fields o f the sam e level” [4, p. 12],
(iii) “decreases the spatial statistical E-field uniformity” [4, p. 12],
(3) Minimum Tuning R atio: “A reasonable guideline for proper operation o f the tuner is
a m inim um tuning ratio o f 20dB ” [4, p. 7]. In other words, the ratio o f the maximum
power to the m inim um pow er (at a selected test point and over all tuner positions) should
be at least 20dB. Apparently, im plem entation of this criteria (though not theoretically
derived) assures acceptable levels of field uniformity, or spatial hom ogeneity, for a large
number o f applications.
Careful consideration o f the discussion in this report seem s to indicate a trend tow ards
applying a set o f very broad em pirically based guidelines towards the operation and
analysis o f m icrowave reverberation chambers. In fact, the m ost concise set o f guidelines
3
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available for the design o f these cham bers (to the author’s knowledge) is as follows ([3],
also repeated in [4]):
(1) “The volum e abd o f the cham ber should be as large as possible in order to have large
values o f N(f) for stirring or tuning purposes.” [3, p. 20]
(2) The ratios a2 : b2 : c 2 o f edge lengths squared should not be too rational in order to
reduce fluctuations in N(f) (avoid m ode degeneracy) “ and, hence, to increase the
uniformity in mode distribution.” [3, p. 20]
While these relatively straightforward guidelines for choosing cham ber dim ensions [34] and evaluating proper cham ber operation (in the frequency range o f interest) are
available, theoretical m odels for the fields inside the chamber are more difficult to obtain.
These difficulties are prim arily due to the fact that, in order to achieve the “m ode stirring”
required for field uniformity, a boundary is perturbed with an irregularly shaped
m echanical tuner. Other m odeling problem s such as determ ining accurate wall
conductivities and additional geom etrical constraints due to small excitation wavelength
(or multi-mode excitation) make the possibility of obtaining a closed-form deterministic
solution unrealistic. Thus, attempts to model these electromagnetic fields have lead to
calculations o f [5-7] statistical models for field quantities under a variety o f idealized
assumptions. Further investigations with regard to the validity o f these m odels might
provide additional insight about the general operation of a m icrowave reverberation
cham ber and could lead to the definition o f useful test parameters.
4
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This dissertation includes an evaluation o f the probability density function (PDF)
m odel for the power density under certain limiting, non-ideal conditions as well as the
derivation of a new transfer function model for the gain of a microwave reverberation
chamber. The evaluation process is perform ed with an elaborate m easurem ent apparatus
and via numerical simulations. The deviation o f the chamber measurements from the
corresponding theoretical models is considered as a function o f some critical param eters
such as chamber quality factor, electrical size, and m ode density.
The primary focus of this study is the investigation o f a possible second-order
statistical model for the fields inside the chamber. The mathematical form o f this model is
extracted from a m ore general theory of “com plex cavities” and is termed the spatial
correlation function (SCF) of the pow er density. The study is conducted by making actual
SCF measurements and developing a computer SCF experim ent (simulation).
The presentation of the m aterial is initiated in Chapter II with a detailed discussion of
some relevant mathematical models. The first section o f this chapter outlines a more
elaborate analysis o f cham ber Q calculations [9] that includes adjustm ents due to power
liberated to any receiving sensors and antennas that may be included in a chamber
m easurem ent configuration. The next section of Chapter II describes an approximate
method for analyzing the gain versus frequency of a m icrowave reverberation chamber
[10]. This method utilizes the overall Q models of the previous section. The third section
o f this chapter describes an ideal first-order statistical model for the power density in a
microwave reverberation, or mode-stirred, cham ber (MSC) while the fourth section
5
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provides a concise sum m ary o f Lehm an’s [7] statistical theory o f “com plex cavities.” This
fourth section includes a sketch o f the derivation of an ideal spatial correlation function in
a “complex cavity.” As mentioned above, the applicability o f this type o f function to
M SC ’s is the prim ary topic under consideration in this dissertation. The fifth section o f
this chapter is a description and tabulation o f the various cham ber param eters that are to
be considered in the subsequent analysis o f cham ber simulations and measurements.
Chapter i n outlines the approach that is applied towards obtaining a statistical
simulation o f an MSC. This approach, which uses a “m oving wall” [8 ] [12] to model the
paddle wheel, incorporates a number of param eters such as chamber excitation frequency,
size, and Q models.
Chapter IV describes the entire m icrowave reverberation chamber m easurement
system that is used in this study. Some details with regard to the significant mechanical
design considerations along with the selected electrical properties o f the cham ber are
included. Also, a block diagram o f the m easurement apparatus along with a description of
the electronic control m echanisms are provided.
The final chapter includes a set o f m easurements and simulation outputs. The
measurements and sim ulations are analyzed in a variety of ways including via comparison
with the appropriate theoretical model. Conclusions with regard to properties of the
average cham ber pow er gain, the PD F of the pow er density, and the spatial correlation
function o f the pow er density are presented. Emphasis is placed on observing trends in
these functions due to variations, over preselected ranges, in a number o f critical chamber
parameters.
6
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CH A PTER II
C HA M B ER PARAM ETERS A N D THEO RETICA L
H E L D M ODELS
Cham ber Quality Factor
A significant elaboration of the theoretical Q analysis that is described in the
introduction is contained in Phillips Laboratory Technical Report 91-1036 [9]. Here, an
equation that accounts for the loading o f the receiving antennas and sensors is postulated
based on an energy distribution argument. This overall Q is specified as a net Q and given
by Eq. 2.1.
net
1
1 ■+ N -------1 +N
..
Q eqv
a Q
^ a n t*
(2 . 1)
1
b Q-
where Q„,v is the com posite Q o f the cham ber (Eq. 1.3), Qant is the Q of a standard
receiving antenna, Q g is the Q o f a standard B -dot probe, and Na ,Nb are the num ber o f
standard antennas and B -dot probes, respectively.
An expression for Qant is derived [9] from an expression for the average effective area
of a receiving antenna with an incident signal that is randomly polarized [17]. This
expression is given by Eq. 2.2.
7
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where A is the average effective area, A, is the microwave wavelength, and D is the
average directivity o f the antenna. Ideally, the average directivity o f an antenna in a
m icrowave reverberation cham ber is unity [9] since a high degree o f statistical
homogeneity o f the field can be assumed when the cham ber size is (ideally) much larger
than the m icrowave source wavelength. W ith this assumption, the average power
delivered to the antenna, P<j, can be calculated as follows:
(2.3)
where W is the average energy stored in the chamber, V is the volum e o f the chamber, c is
the speed o f light and the quantity in the second parenthesis is the average pow er density
in the chamber. This equation (Eq. 2.3) can be manipulated to yield an expression for
Qant *
(2.4)
where co is the microwave radian frequency and the relation X =
2nc
has been used.
co
A sim ilar approach is also applied in the report [9] to derive an expression for Q 6 .
However, this derivation is not outlined here since the m easurem ents in this particular
study are conducted with antennas and D-dot sensors. The final section of this report
includes an analysis o f these Qnet models in comparison with a set o f m easured Q values.
8
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The results indicate that the Qnct calculations match experimentally measured values
significantly more closely than the QeqVmodel o f Eq. 1.3.
Finally, an expression for the Q o f a wall-mounted D-dot sensor, denoted
, can be
derived by applying the above-mentioned approach to a first-order model for the operation
o f a D -dot sensor [10]. This model [11] relates the voltage at the sensor output terminals
to the electric field at the cham ber wall as follows:
v . = R A e coE
D
o
n
(2.5)
where v r) is the average o f the m agnitude o f the sensor output voltage, E n is the average
of the m agnitude o f the normal electric field at the chamber wall, R is the sensor load
resistance, A is the sensor equivalent area, and e 0 is the permittivity of free space. Also, a
relationship between the average normal field magnitude at a cham ber wall and the
average energy density in the cham ber is derived in [5] and is presented here as Eq. 2.6:
£ E 2 =—
o n
3
(2.6)
where U is the average energy density in the chamber. These two expressions (Eq. 2.5
and Eq. 2.6) can be combined to obtain the following relationship for Q 6 :
ooW
coUVR
Pd
v j )2
3
V
(2.7)
2 R A 2 e o co
Eq. 2.1 can be modified to accom m odate D-dot sensors as follows:
( 2 .8 )
9
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where Nd is the num ber o f D -dot sensors.
Transfer Function M odels
A param eter that provides a considerable am ount o f insight with regard to the overall
operation of a cham ber is the pow er gain, as a function o f frequency, between a
transmitting antenna inside the cham ber and a receiving antenna/sensor inside the chamber.
In this section, two first-order theoretical m odels for the chamber gain versus frequency
are derived with the aide o f the Q„e,m odels o f the previous
section.
A calculation for the pow er transfer characteristic, or gain, of a chamber with a
receiving antenna can be initiated from the definition o f the gain in Eq. 2.9.
G
P.
=—
ant p
o
(2.9)
y ’
where Gant is the gain o f the cham ber with receiving antenna, P 0 is the power delivered to
the cham ber from the transm itting antenna, and Pd is the power available to the receiving
antenna from the chamber.
Next, Eq. 2.9 can be manipulated as follows:
(2.10)
where co is the m icrowave radian frequency, W is the average energy stored in the
chamber, Qnet is the overall Q o f the cham ber, and Q ^t is the contribution to the overall Q
due to the receiving antenna.
10
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The parallel loading effect o f the chamber walls and the antenna can be observed by
evaluating Eq. 2.8 with Na= l, Nb=0, and Nd=0 (Eq. 2.11).
1
1
^net
^ant
(2 -ID
^eqv
where Q ^v is the contribution to the overall Q due to the walls.
Substitution of Eq. 2.11 into Eq. 2.10 yields the following simplified expression for
the gain.
f
„
G
ant
i
i
r 1
1----------
, Q ant* Q eqv ,
1
=-2-------------- -2 -^!— = ----o
ant
Q
(2.12)
*
j_^___ a n t.
Q
^eqv
An ideal expression for QcqVcan be obtained by letting X «
Q
eqv
=co
W
3 V
= --------P
2 li S5
eqv
^r
a,b,d in Eq. 1.3.
(2.13)
where PeqVis the total pow er lost to the cham ber walls, V is the volume o f the chamber, |i r
is the relative permeability o f the cham ber walls, 8 is the skin depth o f the cham ber walls,
and S is the surface area of the cham ber walls. T his limiting case corresponds to the case
of a highly “overmoded” cavity where the source wavelength is infinitesimally small in
relation to the chamber dim ensions and is an approximation that is frequently applied in
the analysis of microwave reverberation cham bers [4],
11
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Substitution o f the expression for QMt (Eq. 2.4) and Eq. 2.13 for QcqVinto Eq. 2.12
yields the desired expression for the gain o f the cham ber with a receiving antenna:
°a n r
<2-14>
. lair1 ' „-------
1+i p i4 -a > 2-5
3 y
c Jt
where the relations ^ =
a°d lt = g 0 ltr have been applied, with |i 0 the permeability
o f free space and o the conductivity o f the cham ber walls.
A calculation for the gain o f a cham ber with a receiving D-dot sensor can be
perform ed in the same m anner as the previous gain calculation. The initial steps are
identical, and Eq. 2.12 is modified as follows:
Gc r — 5 7 i+ — y Q
eqv
(2 I5 )
where G ^ is the gain of the cham ber with receiving D -dot sensor.
T he desired gain expression is obtained by substituting Eq. 2.7 for
and Eq. 2.13
for Qcqv into Eq. 2.15 to yield:
G ■ = ------ p = ^ _ —-----------------,
2^ r
S
-is
1 + J — ------- 9— ®
(2.16)
p 0° RA 2eo
where the relations 5 = ,/—^ — and u ^ u ^ n have been applied.
\c o p a
0 r
12
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T he significance o f these relationships is discussed in detail in the results and
conclusions chapter (Chapter V) where two sets o f gain m easurem ents are presented
alongside plots o f the theoretical gain curves derived in this section.
Probability Density Functions
As m entioned in the introduction, investigations into possible statistical models for the
fields in a reverberation chamber are motivated by the difficulties associated with finding
closed-form determ inistic field solutions for chamber geometries. The statistical models
that are presented in [ 6 ], entitled “ Statistical M odel for a M ode-Stirred Cham ber,” are
representative o f current statistical treatments for the analysis o f reverberation chambers.
Here, the PDF for the pow er density, at a point in the chamber, is given by the exponential
distribution (chi square with two degrees o f freedom) in Eq. 2.17.
f(p) = A e ' p/2(j2
2a
(2.17)
where p is the pow er density for one field com ponent (Ex, Ey, or E2) and is therefore
proportional to the received pow er for m ost (linearly polarized) sensor/antenna
configurations. In this treatment, the two possible polarization’s o f each field com ponent
seem to be referred to as “in-phase and quadrature com ponents o f the field in each
dim ension” [6 ]. This leads to the definition o f six “field com ponents” : components in
three orthogonal directions each with an in-phase and quadrature com ponent cr, in Eq.
2.17, is defined as “the variance o f each of the six field com ponents” [6 ]. PD F’s for the
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magnitude of the total field and for one dimension o f the total field are represented as chi
distributions with six degrees of freedom and chi distributions with two degrees o f
freedom, respectively. All three o f these PD F’s are “derived” based on an heuristic
argument that is initiated by observing that, when a large num ber o f m odes are present in
the chamber, each o f the six field components can be thought o f as a sum o f a large
number of random variables, where the random nature o f the sum m ing elem ents (modes)
is introduced by the paddle wheel. Then, by the central limit theorem , each o f these six
components should be Gaussian. W hen points away from the wall are considered, a
“reasonable assum ption” [6 ] is to consider these field com ponents to be independent and
identically distributed. The forms of the PD F’s are specified by applying existing
statistical theories for sums o f squared-Gaussian random variables.
Histograms o f power density measurements from a highly overmoded (or electrically
large) cavity along with goodness-of-fit-test results are also provided in [6 ]. The
distributions o f this measured data are in close agreem ent with the theoretical exponential
distribution of Eq. 2.17. Similar histograms are presented in chapter V for the case o f a
chamber system with a range of excitation frequencies such that, in the low er end of the
frequency spectrum, the chamber is electrically small. Also, histograms for measurements
taken on a cham ber wall are presented. For these particular applications (i.e., PDF
measurements), the pow er m easurem ents are processed in log m agnitude (or dB) and it is
convenient, for comparison purposes, to perform the following change of variables on the
PDF o f Eq. 2.17.
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P = 101og(p)
(2.18)
This change of variables leads to the following distribution for the power (power density)
in dB [7].
f p (P) = ( l/p /> e - z
where z = e
(2.19)
, p ' = (10/ln 10)=4.343, and P is the average measured power in dB. The
standard deviation for this distribution is always 5.57 dB [7]. Above the mean, the form
o f this curve (Eq. 2.19) is such that the distribution falls towards zero rapidly, form ing a
theoretical “c u t-o ff’ for the power m easurem ents.
Correlation Functions
The possibility o f extending the current statistical theory for the fields in the chamber
to include a second-order statistical m odel is addressed in this section. As is the case in
many practical statistical applications, the investigation of complicated, and perhaps
unrealistic, joint PDF models is bypassed with the consideration o f more compact, and
perhaps more easily interpretable, correlation function models.
The theoretical fram ework for this study is based on Lehm an’s [7] “A Statistical
Theory of Electromagnetic Fields in Complex Cavities.” In this eighty-page Phillips
Laboratory Interaction Note, Lehm an initiates a lengthy discussion with regard to
developing a general theory for the fields in irregularly shaped, or unsymmetrical, cavities.
Here, Lehm an’s treatm ent o f the first-order statistics o f com plex cavities can be
15
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inteipreted as a generalized and detailed m athem atical expression o f the heuristic
treatm ent for reverberation cham bers in [6].
The first half o f this work [7] is dedicated to finding a determ inistic solution for the Efield in an arbitrarily-shaped cavity. This solution is deem ed unsuitable for computations
and is used to find volum e averages that are shown to be equivalent to expectation
operations in the second half o f the paper. T he form o f this solution, for the electric field,
is known as an eigenvector, or m odal, expansion and is derived from an expansion
solution to an inhom ogenous w ave equation for the m agnetic vector potential. The
excitation is a filam entary source and a damping term (to m odel wall losses) is also
incorporated into the wave equation. A m ajor portion o f this half o f the treatm ent is
dedicated towards specifying the eigenvectors that are contained in the m odal expansion.
Here, a seemingly novel approach is applied in the sense that an arbitrarily-shaped cavity is
split into a large number, N, o f rectangular elem ents that are individually enlarged and
defined as “virtual cavities.” The dim ensions o f these virtual rectangular cavities are
chosen by extending each boundary of the corresponding cavity elem ent such that the
extended boundary intersects a surface of the original arbitrarily shaped cavity. Thus, the
dim ensions o f N virtual cavities that are derived from N cavity elements are specified.
This formulation allows the real com plex cavity eigenvectors to be expressed as a
superposition of known virtual cavity eigenvectors. These resultant eigenvectors can be
used in the m odal expansion solution provided that an orthogonality condition is satisfied.
This required orthogonality condition is, in turn, satisfied if Eq. 2.20 is satisfied.
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2 Jco s[K n ( r - r n )]dV = 0
(2.20)
n=l V
where N is the num ber o f cavity elements, in is the position vector to a selected com er o f
the nth virtual cavity, K , is the wavevector for the nth virtual cavity, and V is the com plex
cavity volume. This equation is Lehm an’s mathematical definition o f a com plex cavity and
can be physically interpreted as follows: If the shape o f the cavity is sm ooth compared to
a wavelength, the phase factor in Eq. 2.20 “will be slowly varying because the position
vectors to the com ers o f the virtual cavities will all be highly correlated” [7, p. 20] and the
left side o f the equation will be large. Similarly, “if the shape o f the cavity is very
irregular compared to a wavelength” [7, p. 20], the left side o f Eq. 2.20 will be small.
With regard to the general applicability of this criterion, Lehman states that “it will not
be possible to a priori validate the complex cavity assumption for every cavity o f interest.
The only practical way to proceed is to assume com plexity based on experience with other
cavities and then use experim ental data to justify the assum ption” [7, p. 21]. However,
with regard to the present development, this criterion is repeatedly applied during the
calculation o f “the volum e averages o f all the powers of a single com ponent of the
eigenvectors” [7, p. 23]. These calculations, along with the “ volume averages of products
o f different eigenvector com ponents” [7, p. 28], are used in the developm ent o f the
statistical models.
The PD F’s for the field variables are calculated by treating r, the position vector, as a
uniformly distributed random variable. This “postulate” leads to the equivalence of
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volume averaging and statistical expectation. Also, though not directly stated, the
distributions o f the eigenvector com ponents can be assum ed to be Gaussian since the large
number of virtual cavity eigenvectors are a function o f the “random ” position vector and
the actual eigenvector com ponents are a sum o f a large num ber o f virtual cavity
eigenvector components. The developm ent proceeds in a relatively straightforward
manner by building upon these basic results. The following ideal assumptions, in addition
to the complex cavity requirement, are em bedded in the calculations:
(1) Equal Energy Assumption: In a 3 dB bandwidth about a particular excitation
frequency, the m odal Q ’s are approximately equal to the average Q o f the cavity and the
modal frequencies are approxim ately equal to the excitation frequency.
(2) The num ber o f m odes within a 3 dB bandwidth is large.
The resulting PD F’s, from these com plex cavity calculations, are identical to the PD F’s
derived in [6 ] for reverberation chambers.
Clearly, the most significant simplifying assumption that is em bedded in the
mathematical derivation of these P D F ’s is assumption (2) in the previous paragraph. This
assumption is applied rather early in the treatment and is implemented by defining a
variable, denoted as M, for the num ber o f m odes in a 3 dB bandwidth. An approximate
expression for M can be derived from the m ode density form ula of Eq. 1.2 and is given by
Eq. 2.21.
M = -JtyVT q
"
(2'21)
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where Q is the average o f the m odal Q ’s about a particular excitation frequency, to. The
actual limiting process is applied during the derivation o f the PDF for the am plitude o f a
com ponent of the “partial fields” where the term partial field is used to denote two
m utually orthogonal electric field solutions each corresponding to a different polarization.
This PDF calculation is a preliminary calculation that facilitates the straightforward
derivation o f the rem ainder o f the PD F’s and correlation functions and allows the
(deterministic) infinite modal expansions to be converted to the following statistical form:
E us(r,t) = X us sin(cot)
E ws(r,t) = X ws sin(cot)
(2.22)
where u and w denote polarizations and s represents a Cartesian com ponent (x,y, or z).
Here, a fairly intricate m athematical procedure is applied to show that the characteristic
function for all the X ’s is in the form o f Eq. 2.23.
<i>x(u)~[l + | 3 V r M/2
(2.23)
where |3 is the standard deviation o f a Gaussian eigenvector com ponent, u ks (r), and u is
the “dum m y” variable in the transformed domain of the characteristic function. A form of
the determ inistic solution (in terms o f eigenvector expansions) [7] for the partial fields is
included here, in Eq. 2.24, for completeness.
k(l+M /2)
E us(r,t) = a 0
I
u k,x( r 0 ) i k,s(r)sin(cot)
(2.24)
k'=k(l-M /2)
where k=oVc is the wavenumber, r 0 is a position vector that specifies the location of a
filamentary source oriented along the x-axis and a 0 is a constant that depends on
param eters such as the length o f the filam entary source, the am plitude o f the source
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current, the excitation (source) frequency, and the cavity Q. As m entioned above, u ^ r )
can also be expressed (in general) as the sum o f a large num ber of (virtual cavity)
eigenvectors and is therefore assum ed Gaussian. (The solution for the w-polarization is
identical to Eq. 2.24.)
T he result of letting M go to infinity in Eq. 2.23 leads to the follow ing ideal form for
the characteristic function o f the X ’s.
--M 0V
< M u )~ e 2
(2.25)
The PDF for the X ’s is obtained by incorporating the constant, a 0 , into Eq. 2.25 and then
taking the inverse Fourier transform to yield:
f (x) = - = = L ----- - e 2Ma°p<
xv
(2.26)
V2itMaop
Thus, Lehm an has applied a rather elegant m athem atical procedure to show that the ideal
PD F o f the partial field am plitudes are Gaussian. He then proceeds with a large sequence
o f short, but detailed, calculations to derive expressions for the PD F’s and correlation
functions for the field variables of interest.
Two sets o f correlation functions are calculated for all the field variables. The first
type o f correlation is denoted as spatial correlation and [7] “results because the fields are
continuous functions o f position.” The second type o f correlation is denoted as temporal
correlation and [7] “ will occur when the bandw idths o f two driving frequencies, say
co and c o ', overlap.” These correlation functions are calculated for the [7, p. 1] “am plitude
o f the com ponents o f the partial cavity fields, the magnitude o f a com ponent o f the tim e
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averaged electric field, the square o f a com ponent o f the tim e average field, the time
averaged com ponent o f the pow er density and the total energy density.” Inspection o f
these functions indicates that the concept o f spatial correlation o f a com ponent of the
power density is readily applicable to microwave reverberation chambers.
A sketch o f the derivation of the spatial correlation function (SCF) for a com ponent of
the pow er density can be initiated by noting the fact that the spatial correlation coefficient
for the eigenvector com ponents is given by Eq. 2.27.
r
/ x
m
sinfklrj - r , | )
K [u ks ( r ,), Uks ( r 2 )] =
^
™ = K s( r , , r 2 )
(2.27)
where k is the wavenum ber as a function o f frequency and ri , 12 are position vectors.
This coefficient is derived by applying the standard definition o f a correlation coefficient,
evaluating the expectation operators in this definition with the formulas for the volume
averages that are derived in the first part o f the paper [7], and then sim plifying the result
by applying the com plex cavity definition in Eq. 2.20. The next critical step towards
obtaining the SCF o f the pow er density is the calculation o f the SCF for a com ponent of
the partial fields. This is accomplished by once again using the definition o f the correlation
coefficient, expanding the Xs(r,) and Xs(r2) terms with eigenvectors, applying the
appropriate volum e average form ulas, and then substituting Eq. 2.27. The result turns out
to be identical to the result in Eq. 2.27.
4 X u s ( l i ) X us( r 2) ] = K s (r i , r 2)
k [ X ws ( t j ), X ws ( r 2 )] = K s ( r , , r 2 )
(2.28)
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The desired result for the SCF of a com ponent o f the time averaged power density is
obtained by observing the definition in Eq. 2.29 and further evaluating the individual
expectation operators on the right-hand side of Eq. 2.30 by substituting Eq. 2.28 and the
appropriate param eters (mean and variance) from the exponential distribution for a
com ponent of the power density.
{E . 2( l ,) E ! 2 ( r 2)> = i ( x „ ! (£ l) X „ ! (r 2 ) ) + i ( x „ 2 ( i 1)X „ . 2 (r! )) +
(2.30)
i(X „ .2(r,)X„2(r2)) + i { x wl2(r,)Xw.2(r2))
The final form o f this result is expressed in Eq. 2.31.
(2.31)
In the concluding section o f this paper, Lehm an’s com m ents with regard to the
application of these results to reverberation cham bers are as follows [7, p. 65]:
Although the statistical model developed herein is based on the
assum ption that the position vector r in the cavity is a random variable,
it is easy to dem onstrate that these statistical m odels also apply to the
fields in mode-stirred chambers. The probability density functions and
the correlation functions are shown to be independent of the shape of
the cavity as long it satisfies the definition o f a complex cavity.
Therefore, the statistical model is valid for the set o f all complex
cavities with constant volume V and constant Q. All o f the cavities
belonging to this set o f constant V and constant Q com plex shaped
cavities is called an ensemble and the volum e averages are replaced by
ensemble averages. For each stirrer position, a mode-stirred chamber
represents a different shaped complex cavity having the same volume
and Q as any other stirrer position. Therefore, m easurem ents performed
at different stirrer positions correspond to measurem ents performed for
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different members o f the ensem ble o f cavities. As a result, averages over
stirrer positions are equivalent to ensemble averages which in turn are
equivalent to volum e averages and the statistical m odels are applicable to
the analysis o f mode-stirred cham ber test data.
Closing discussions with regard to the com plex cavity, equal energy, and large m ode
assum ptions are also included in the paper [7] with the general conclusion being that the
relative validity of these assumptions is unresolved and requires further study.
A m ajor segment o f chapter V o f this dissertation (Results and Conclusions) is
dedicated to the analysis o f the applicability o f this SCF function to m icrowave
reverberation chambers. Selected sets o f em pirical and com puter simulation outputs are
included. Chapter ID provides a description of the simulation algorithm that is developed
and implemented for the computer simulation o f an SCF experim ent in a m icrowave
reverberation chamber. Chapter IV provides a detailed description of the measurement
apparatus that is designed and implemented for purposes o f measuring the spatial
correlation o f the average power density in an actual microwave reverberation chamber.
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Sum m ary of Critical Parameters
A brief description o f seven critical param eters that effect the electrom agnetic
response o f any standard m icrow ave reverberation cham ber is presented in this section.
The objective is to introduce the reader to the significance o f these param eters
im mediately following the presentation o f the theoretical m odels o f in tere st This type of
approach should give the reader an increased awareness of the dom inant them e in this
dissertation o f investigating the effects o f varying cham ber param eters on the applicability
of a few relatively simple, idealized, m athematical (statistical) models. The param eters are
described individually in the following numbered segments.
(1) Excitation Frequency, f : The frequency o f the m icrow ave source is critical in the
sense that large source w avelengths, in relation to the cham ber dim ensions, generate small
mode densities and vice versa. In this study, the phrase “ideal cham ber operation” implies
that the source w avelengths are small in relation to the cham ber dimensions.
(2) Cham ber Size: The cham ber size, including the volum e and the surface area, is
related to the quality factor, Q, and the Q is, in turn, directly related to the num ber of
modes in the 3-dB bandw idth, M.
(3) C ham ber Quality Factor. O : A t the present time, the Q is perhaps the m ost frequently
discussed, m easured, and theoretically studied param eter o f a m icrowave reverberation
chamber. H ow ever, w hile the Q net m odels are m ore accurate than the previous Qa,v
models, there do not seem to be any sets o f docum ented m easurem ents available that do
not deviate from the corresponding theoretical calculations by less than a factor o f two
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[9]. M ost studies, for exam ple [4] and [6], report m easured Q ’s that are low er than values
from the given theoretical m odels by a factor of 10 or more.
(4) N um ber o f M odes in 3-dB Bandwidth. M : This param eter is one of the m ost
significant param eters that is embedded in Lehm an’s theory o f complex cavities. As
described in chapter V, a new approach for m easuring the average cham ber Q by applying
the gain m odels that are derived in the first section o f this chapter is investigated in this
study. These Q m easurem ents are used to determ ine M values, which are im portant with
regard to the accuracy o f the PD F and SCF m odels that are presented in this chapter.
(5) Paddle W heel Size: This param eter is not investigated in detail in this particular
study. W u and Chang, in [18], have examined a hypothetical [18, p. 164] “2-D cavity
with a 1-D perturbing body.” T he results from their sim ulations indicate that the
dimension o f the perturbing body should be [18, p. 169] “ of the order o f two wavelengths
or longer.” For the m easurem ents that are included in this dissertation, the effectiveness
of the paddle wheel is evaluated from an analysis o f chamber gain measurements (Chapter
V).
( 6) Number o f Paddle W heel Positions: As mentioned in the introduction, a typical
microwave reverberation cham ber experim ent is usually designed to accommodate
approximately 200 increm entally spaced paddle wheel positions. The effect of selectively
excluding som e of these samples in a spatial correlation m easurem ent is documented in
Chapter V.
(7) Sensor and Antenna Critical Frequencies: Sensor and antenna critical frequencies
occur due to electrom agnetic energy being stored in stray distributed capacitances (and
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sometimes inductances) between structures within and in the vicinity o f the sensor or
antenna. These frequencies affect the net, or overall, Q o f the cham ber and in certain
extreme cases can effect PDF as well as SCF measurements. A discussion o f these effects
is included in Chapter V.
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CHAPTER HI
SIM ULATION M ETHODOLOGY
A com puter simulation o f a spatial correlation function (SCF) m easurement inside a
microwave reverberation chamber is developed by modeling the cham ber as a rectangular
cavity with one moving, or “perturbed,” wall and no paddle wheel. In this approach, field
m easurements inside the chamber are simulated with values calculated from field equations
for a set o f rectangular cavities with heights that vary in equally spaced increments from
the height o f the paddle wheel to the height of the top o f the actual cham ber o f interest.
Before proceeding with im plementing an algorithm based on the above-mentioned
approach, a num ber o f authorities, including those cited in [14] and [15], have been
consulted with regard to the possibility o f applying a standard numerical technique to this
specific simulation problem. The general consensus is that the finite-difference timedomain (FD-TD) method [16], based on relatively straightforward second-order centraldifference approxim ations for the space and time derivatives in M axw ell’s equations, is
m ore suited to this application than the m ore traditional ffequency-dom ain integral
equation approaches. However, Y. Rahm at-Sam ii [15], a recognized authority on
com putational electrom agnetics, pointed out, during a detailed discussion, that while the
chamber under consideration may be considered electrically small for reverberation
cham ber applications, it is considered electrically large for simulation applications. His
com m ents referred to the fact that, while detailed FD-TD sim ulations have been
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successfully conducted with m ultim ode cavities [13], a successfully completed detailed
simulation (including paddle wheel and antennas) of a reverberation-chamber-sized
apparatus has not been done. H e w as generally not optim istic about the possibility o f
obtaining a accurate sim ulation (using FD-TD) o f the cham ber with reasonable execution
times and reasonable com puter storage requirements.
In another consultation [14], K.C. Chen , a authority on m icrowave reverberation
chambers, expressed the view that sim ulation techniques such as FD-TD and the
frequency domain m ethods would not lead to useful results in this application due to the
fact that a successful simulation, after a lengthy developm ent process, w ould not yield a
simulation that could be easily linked to cham ber parameters.
After considering the com m ents provided by the consulted authorities and evaluating
the results o f a literature search, a viable approach to the sim ulation is obtained by
modifying the basic approach in [8 ], entitled “ An Investigation o f the Electromagnetic
Field inside a M oving-W all M ode-Stirred Cham ber.” This technique, in [8] and [11], is
based on obtaining the G reen’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 o f a rectangular cavity is perturbed in order to simulate a
“m ode-stirred” cham ber response. T he direct application of this m ethod to the present
simulation problem is not necessarily preferable since im portant cham ber parameters, such
as the cham ber Q, are not directly em bedded in the G reen’s function approach to the
solution for a rectangular cavity. This approach is modified by applying the “moving wall”
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cham ber concept to a “m ode selection” procedure instead o f using the G reen’s function
solution. This procedure leads to a simulation that is directly related to cham ber
param eters such as Q and the num ber o f m odes in a bandwidth. A skeletal description o f
this approach is as follows:
—>A large num ber o f resonant frequencies for each rectangular cavity associated with a
“moving wall” chamber is calculated.
—>The frequency response of each resonant mode is approxim ated by a simple secondorder curve that is derived directly ffom Qnet. The shape of this curve is sim ilar to the
frequency response characteristic of a high-Q narrow-band filter with the resonant
frequency being analogous to the filter center frequency.
—» The second-order curve is used to determ ine which m odes 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 o f interest.
This “ moving wall” algorithm is implemented with certain pre-processing steps in
order to shorten overall execution times. The pre-processing steps involve the calculation,
sorting, and archiving o f a large num ber o f resonant frequencies for all o f the rectangular
cavities that are to be considered in the calculations. In this case, a simulation o f a
chamber experim ent with 200 distinct paddle wheel positions is desired, and resonant
frequency arrays are calculated, using Eq. 3.1, for 200 rectangular cavities.
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where c = speed o f light, m, n, and p are integers, and a, b, and d are cavity dimensions.
The heights o f the cavities are separated by increments o f (dmax - d,nin)/200, where dmin is
the minimum cavity simulation height (paddle wheel height) and dmax is the maximum
simulation height (height o f cham ber to be simulated). For a given height, d, a resonant
frequency array is evaluated by varying (m,n,p) from ( 1, 1,0 ) to some “acceptably large”
numbers. The “acceptably large” final values o f (m,n,p) are determined from an analysis
of a mode density formula [2] combined with consideration of the desired simulation
frequency range. (The number o f resonant modes in a particular frequency interval
converges to a finite number as m, n, and p become sufficiently large.) The initial values
of n and m are unity, instead o f zero, since in this particular study the wall-mounted Ddot sensors are designed to detect the z-component o f the field which, from an analysis of
the field equations [3], is identically zero when n or m is zero.
The simulation program processes the sorted resonant frequency data generated by the
pre-processing program. It first determines a set of significant resonances, or modes, at a
particular simulation frequency for a specific cavity height. These modes are selected by
modeling the frequency response o f each cavity resonance, in the neighborhood of the
desired simulation frequency, with a second-order curve, Eq. 3.2, that is analogous to the
transfer function of an ideal RLC circuit
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where cq, = a cavity m odal resonance in radians/s and Q net = overall Q o f the chamber.'
The m agnitudes o f these second-order responses at the sim ulation frequency are used as a
measure of the relative significance o f each mode. Specifically, the program scans the
appropriate resonant frequency array for the resonance that is closest to the simulation
frequency. The m agnitude o f the response associated with this nearest resonance is
evaluated at the sim ulation frequency and stored as a variable called m ag i. T he responses
associated with other neighboring resonances are evaluated successively, at the simulation
frequency, in ascending and descending frequency order until resonances are encountered
that generate a m agnitude that is less than . 1*mag 1 and the mode selection process is
terminated. The results o f this mode selection process are sets o f m, n, and p indices and
amplitudes that correspond to m odes that significantly contribute to the simulated chamber
response at the chosen simulation frequency.
Q nct, the overall theoretical Q of the cham ber with receiving D-dot sensors/sensor, is
calculated from Eq. 2.8 and is given by Eq. 3.3 .
(3.3)
where, as defined in chapter 2 , QgqV
, N = num ber o f D-dot
sensors, V = volum e o f chamber, |ir = relative permeability o f chamber walls, 5 = skin
depth o f chamber walls, S = surface area o f chamber walls, R = sensor load resistance, A
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= sensor equivalent area, and £o = perm ittivity o f free space. The QcqVterm in this
equation expresses the loading due to the cham ber walls while the
term gives the
loading due to a sensor. The sensor loading term is derived, in Chapter II, from an ideal
first-order m odel for a D-dot sensor. One objective of this sim ulation is to study the
simulated SC F response as Qnct is divided by various scale factors (i.e. Q = Qaet / 5 , Q =
Qact / 10 ,...), corresponding to possible additional cham ber losses.
The electric field, due to each m np mode, is calculated from the z-com ponent o f the Efield solution for an ideal rectangular cavity (Eq. 3.4) [3].
(3.4)
where k
k r = ^ k x2 + k y2 + k z2 , and Eo , Ej are am plitude constants in volts per m eter. Here, the
first term represents the TE solution whereas the second term represents the TM solution.
This equation is evaluated for all o f the selected mnp sets where each resulting term is
m ultiplied by the corresponding m agnitude factor from the selection step. The total
electric field, at a point on the bottom o f the chamber, is calculated by summing all o f the
these individual terms. A pow er density sam ple is obtained by taking the magnitudesquared o f the total field.
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In the sim ulation program , the aforementioned process o f calculating pow er density
samples is em bedded inside tw o nested loops. The outer loop is the “ height perturbation
loop” where a resonance frequency array corresponding to a particular cavity height is
loaded into m em ory from disk storage. The inner loop is the “frequency loop” where a
pre-determined initial simulation frequency is incremented by a specified amount. Two
power density calculations, for two different spatial locations on the bottom o f the
simulated cham ber, are perform ed within the inner loop. This procedure generates two
power density samples for each simulation frequency and each rectangular cavity height
The pow er density data is stored in two large m em ory segments. O ne segm ent contains
all the data for one spatial location, while the other segm ent contains all the data for the
other spatial location. A com puter-generated spatial correlation function output is
obtained by correlating the data in the two memory segm ents using the usual productm om ent formula. In addition to calculating the SCF for a given spacing and frequency
range, the sim ulation program generates a num ber o f outputs that allow for com parisons
of this simulation approach to available theoretical and experim ental models for
m icrowave reverberation chambers.
The implementation of the “ moving w all” algorithm is a set of M atlab m-files that
execute on a SUN10 SPA RC workstation. Two main m -file program s perform the actual
calculations w hile a set of small m-files, that are called by the main programs, perform
auxiliary calculations such as evaluation of the Qnet form ulas (Eq. 3.3), evaluation o f
(second-order response) m agnitudes (Eq. 3.2), and generation o f mnp integers. The first
33
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
main program calculates, processes, and stores, on disk, approximately 150 megabytes of
resonant frequency data. The second main program processes this data in segments and
generates a simulated SCF o utput The most significant array sizes in this program, at any
given time during the execution process, are three approximately 1,000,000 elem ent mnp
integer arrays, one 50,000 - 75,000 elem ent integer array segm ent of resonant frequency
data, one 50,000 - 75,000 elem ent double-precision array segm ent o f resonant frequency
data, and two 51 x 200 elem ent double-precision arrays with calculated power density
samples.
In [8], the simulation output is evaluated by choosing 20 incrementally-spaced moving
wall positions, over a wavelength, and then taking 20-sample E-field averages at several
different points in the chamber. The algorithm is said to simulate a microwave
reverberation chamber response since two of the observation points yield an E-field tuning
ratio o f at least 40 dB and a homogeneity o f 2 dB. The results o f perform ing these tests
on a sample output from the simulation program that is developed for this study yield
sim ilar results: (power) tuning ratios o f well above 20 dB as well as a pow er density
homogeneity o f better than 1 dB for the chosen cham ber size and at the highest simulation
frequency o f 13.5 Ghz..
Plots o f the maximum, average, and minim um of the power density data from this
sample simulation are shown in Figure 3.1. Inspection of this figure indicates that the
average curve is generally 5-10 dB below the maximum curve. In an actual microwave
reverberation chamber, shifts in the 5-10 dB range between the corresponding measured
34
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
Simulated Chamber Responce
0
-20
S' -40
2
co -60
-80
-100
2
4
8
6
10
12
-20
m
2,
CVJ
-40
„
co -60
-80
-100
Frequency (GHz)
Figure 3.1. Plots o f the M inimum, M aximum, and Average of the Power Density, in dB,
from a Sample C ham ber Simulation. The two selected points, with power
densities SI and S2, are spaced 17.5 cm apart and are located in the
vicinity o f the center o f the bottom wall. The dimensions o f the simulated
cham ber are 1.0342 m X 0.8087 m X 0.5812 m. T he Q values are from a
2 D-dot sensor Q net model that is divided by 10.
35
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
cham ber attenuation curves can be considered as another indicator o f efficient tuner
operation [4].
Figures 3.2 and 3.3 are the two output text files that are generated by this sample run.
In Figure 3.2, the colum ns from left to right represent the sim ulation frequency, the total
num ber o f inodes processed by the program , the Q values from the chosen Q model, the
average num ber o f m odes processed within the 3 dB bandwidth, the theoretical num ber o f
modes in the 3 dB bandwidth, and the calculated spatial correlation coefficient. In Figure
3.3, the colum ns from left to right represent the simulation frequency, the minimum
contribution to the overall cham ber response from a selected m ode, the average
contribution to the overall chamber response from the selected modes, the minimum
bandwidth o f the selected modes from the 200 height perturbations, the average
bandwidth o f the selected modes from the 200 height perturbations, and the maximum
bandwidth o f the selected modes from the 200 height perturbations. All these parameters
are used for diagnostic purposes and are inspected periodically for unusual trends. M ost
o f the param eters can be applied tow ards com paring simulations to m easurements and
theoretical values. A third output data file contains all the calculated pow er density
samples in compressed form for future analysis applications such as the plots in Figures
3.1, 3.4, and 3.5.
An additional statistical test is perform ed to evaluate the quality o f the simulation
output. This test involves tabulating the standard deviation o f the output pow er density
samples and plotting them in histogram form. A sam ple test plot is shown in Figure 3.4.
36
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
f
1.000e+009
1.250e+009
1.500e+009
1.750e+009
2.000e+009
2.250e+009
2.500e+009
2.750e+009
3.000e+009
3.250e+009
3.500e+009
3.750e+009
4.000e+009
4.250e+009
4.500e+009
4.750e+009
5.000e+009
5.250e+009
5.500e+009
5.750e+009
6.000e+009
6.250c+009
6.500e+009
6.750e+009
7.000e+009
7.250e+009
7.500e+009
7.750e+009
8.000e+009
8.250e+009
8.500e+009
8.750e+009
9.000e+009
9.250e+009
9.500e+009
9.750e+009
1.000e+010
1.025e+010
1.050e+010
1.075e+010
1.100e+010
1.125e+010
1.150e+010
1.175e+010
1.200e+010
1.225e+010
1.250e+010
1.275e+010
1.300e+010
1.325e+010
1.350e+010
modes
Q
modes3
N3
K
2741
2092
790
1543
1963
2231
2373
2642
2474
2665
2845
3137
2491
3627
3846
3994
4591
5017
5342
6073
6542
7130
8038
8530
9243
10823
11276
11079
13587
14116
14762
16529
18048
19091
20909
21286
23462
25001
26514
28145
30223
31443
34035
36086
38498
40393
42580
44806
46933
50589
52792
5.759e+003
6.427e+003
7.027e+003
7.574e+003
8.079e+003
8.549e+003
8.989e+003
9.403e+003
9.794e+003
1.016e+004
1.052e+004
1.085e+004
1.117e+004
1.148e+004
1.177e+004
1.206e+004
1.233e+004
1.259e+004
1.283e+004
1.307e+004
1.331e+004
1.353e+004
1.374e+004
1.395e+004
1.415e+004
1.434e+004
1.452e+004
1.470e+004
1.487e+004
1.504e+004
1.520e+004
1.536e+004
1.551e+004
1.565e+004
1.579e+004
1.592e+004
1.605e+004
1.618e+004
1.630e+004
1.641e+004
1.652e+004
1.663e+004
1.674e+004
1.683e+004
1.693e+004
1.702e+004
1.711e+004
1.720e+004
1.728c+004
1.736e+004
1.744e+004
0.040
0.070
0.140
0.220
0.280
0.500
0.530
0.730
0.840
1.120
1.370
1.440
4.360
2.470
2.970
2.880
3.340
4.010
4.370
4.740
5.320
8.230
9.250
7.730
8.540
10.000
10.350
10.820
14.320
13.150
14.230
18.800
17.210
19.040
19.680
20.960
22.990
27.160
25.920
28.150
29.250
32.840
35.890
37.180
40.100
40.100
44.260
43.170
48.680
49.920
52.430
0.079
0.138
0.218
0.321
0.449
0.604
0.788
1.003
1.250
1.531
1.848
2.203
2.597
3.032
3.509
4.031
4.598
5.213
5.878
6.593
7.361
8.183
9.061
9.997
10.993
12.050
13.170
14.355
15.607
16.927
18.318
19.781
21.318
22.931
24.623
26.394
28.248
30.185
32.209
34.321
36.523
38.817
41.206
43.691
46.276
48.961
51.749
54.643
57.645
60.757
63.981
0.741
0.618
0.339
0.610
0.303
0.456
0.409
0.485
0.388
0.445
0.153
0.030
0.171
0.130
0.285
0.065
0.102
0.278
0.039
0.086
0.205
0.012
0.013
0.104
0.000
0.010
0.061
0.070
0.060
-0.021
-0.083
0.155
-0.007
0.090
0.011
0.060
0.039
0.112
0.125
0.036
-0.013
-0.083
-0.025
0.061
0.075
-0.060
-0.052
0.063
-0.029
-0.041
0.130
Figure 3.2. Sam ple Output File K 1 .0 U T from Chamber Simulation Program
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
f
magmin
1.000e+09
1.250e+09
1.500e+09
t.750c+09
2.000e+09
2.250e+09
2.500e+09
2.750e+09
3.000e+09
3.250e+09
3.500e+09
3.750e+09
4.000e+09
4.250e+09
4.500e+09
4.750e+09
5.000e+09
5.250e+09
5.500e+09
5.750c+09
6.000e+09
6.250e+09
6.500e+09
6.750e+09
7.000e+09
7.250e+09
7.500e+09
7.750e+09
8.000e+09
8.250e+09
8.500e+09
8.750e+09
9.000e+09
9.250e+09
9.500e+09
9.750e+09
1.000e+10
1.025e+10
I.050e+10
1.075e+10
1.100e+10
1.125e+10
1.150e+10
1.175e+10
1.200e+10
1.225e+10
1.250e+10
1.275e+10
1.300e+10
1.325e+10
1.350e+10
0.008
0.023
0.239
0.130
0.038
0.040
0.088
0.085
0.174
0.108
0.152
0.128
0.997
0.462
0.262
0.239
0.292
0.278
0.382
0.439
0.452
0.917
0.997
0.665
0.624
0.646
0.703
0.716
0.995
0.714
0.872
1.000
0.898
0.961
0.872
0.894
0.896
0.898
0.843
0.935
0.953
0.984
0.972
0.996
0.993
0.972
0.941
0.968
0.995
0.978
0.979
magave minbw
0.098
0.133
0.313
0.293
0.338
0.418
0.456
0.500
0.568
0.617
0.670
0.695
0.997
0.804
0.835
0.835
0.861
0.892
0.900
0.902
0.924
0.963
0.997
0.952
0.960
0.968
0.972
0.977
0.996
0.984
0.985
1.000
0.989
0.992
0.988
0.990
0.994
0.994
0.993
0.995
0.997
0.997
0.998
0.998
0.998
0.998
0.998
0.998
0.999
0.998
0.999
0.000e+00
0.000e+00
1.319e+05
6.600e+05
0.000e+00
0.000e+00
5.945e+05
0.000e+00
8.393e+05
7.603e+05
7.720e+05
1.162e+06
1.751e+06
2.543e+06
2.118e+06
2.157e+06
2.600e+06
3.107e+06
2.646e+06
3.683e+06
3.393e+06
4.155e+06
3.554e+06
4.173e+06
3.879e+06
4.377e+06
4.522e+06
4.598e+06
4.890e+06
4.677e+06
4.977e+06
5.235e+06
5.351e+06
5.376e+06
5.603e+06
5.642e+06
5.736e+06
6.006e+06
6.093e+06
6.234e+06
6.316e+06
6.327e+06
6.466c+06
6.673e+06
6.788e+06
6.925e+06
7.021 e+06
7.121c+06
7.263e+06
7.248e+06
7.521 e+06
avebw
6.318e+07
3.232e+07
4.967e+06
9.519e+06
1.193e+07
1.062e+07
8.533e+06
8.046e+06
6.105e+06
5.989e+06
5.436e+06
5.420e+06
2.916e+06
4.390e+06
4.395e+06
4.498e+06
4.645e+06
4.514e+06
4.535e+06
4.766e+06
4.600e+06
4.523e+06
4.557e+06
4.864e+06
4.944e+06
5.025e+06
5.122e+06
5.165e+06
5.264e+06
5.385e+06
5.507e+06
5.536e+06
5.710e+06
5.809e+06
5.946e+06
6.040e+06
6.148e+06
6.247e+06
6.352e+06
6.457e+06
6.556e+06
6.665e+06
6.773e+06
6.871 e+06
6.999e+06
7.103e+06
7.215e+06
7.328e+06
7.425e+06
7.541e+06
7.653e+06
maxbw
2.179e+08
8.057e+07
8.409e+06
1.747e+07
6.266e+07
6.419e+07
3.105e+07
3.373e+07
1.717e+07
2.920e+07
2.150e+07
2.602e+07
3.540e+06
7.961e+06
1.437e+07
1.628e+07
1.341e+07
1.375e+07
1.094e+07
9.734e+06
9.669e+06
5.004e+06
4.719e+06
6.692e+06
7.876e+06
7.65 le+06
7.247e+06
7.317e+06
5.376e+06
7.383e+06
6.220e+06
5.662e+06
6.343e+06
6.090e+06
6.83 le+06
6.732e+06
6.838e+06
6.978e+06
7.320e+06
6.900e+06
6.927e+06
6.835e+06
6.993e+06
6.958e+06
7.092e+06
7.308e+06
7.680e+06
7.590c+06
7.520e+06
7.748e+06
7.775e+06
Figure 3.3. Sample O utput File K 2.0U T From Chamber Simulation Program
38
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
SIMULATED PDF FOR EM POWER DENSITY
Relative Q xurence
0.2
i
i
i
»
’T "
i
i--------------- r
Location 1, Std. Dev. = 10.2 dB
0.15
0.1
0.05
. LL
0
Relative Ocarrence
10
20
,ll
30
1 11III I lit
40
50
60
|S1| (dB)
70
80
70
80
90
Location 2, Std. Dev. = 10.3 dB
30
40
50
60
|S2| (dB)
Figure 3.4. Histograms o f Power Density D ata for a Simulation Frequency of 10 GHz and
a Cham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m. The standard
deviations are well above the theoretical value of 5.57 dB.
39
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
SIMULATED PDF FOR EM POWER DENSFTY
0.3
Location 1, Std. Dev. = 6.0 dB
p
O.2
-50
-40
-30
-20
|S1| (dB)
-10
0.3
Location 2, Std. Dev. =5.8dB
1 0.2
g>0.1
1
4
0
-50
-40
-30
-20
|S2| (dB)
Figure 3.5. Histogram s o f Power Density Data for a Simulation Frequency o f 10 GHz and
a Cham ber Size o f 1.0342 m X 0.8087 m X 0.5812 m. T he standard
deviations approach the theoretical value of 5.57 dB after adding a
random phase term between the TE and TM com ponents o f the E-field.
40
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Observation o f a num ber o f test plots indicates that the spread, or standard deviation, of
this data is significantly above the theoretical value o f 5.57 dB (Chapter II). A careful
analysis of the basic physical principles that are embedded in this original implementation
indicate that the large standard deviations m ay be due to a relatively large number of
occurrences o f the TE and TM term s in Eq. 3.4 adding in phase. The inclusion o f a
random phase term between these TE and TM term s generates standard deviations that
are in the neighborhood o f the theoretical standard deviations for the higher frequencies in
the simulated frequency range. For the particular cham ber size that is considered in these
sim ulations, these high frequencies correspond to the ideal case of large chamber
electrical size. Figure 3.5 is a sam ple histogram that is generated by using data from the
final form o f the simulation program . Figure 3.6 show s plots o f sim ulated SCF outputs
for two different Q values with a spacing o f 2.5 cm. A selected set o f simulated
correlation outputs will be discussed in detail in chapter V. The matlab M -file
im plementation o f the simulation m ethodology that is docum ented in this chapter is
included in the appendix.
41
with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Sinxiated SCF of EM Power Density
2
4
2
4
6
8
10
12
6
8
10
12
£ 0.5
Frequency (GHz)
Figure 3.6. SCF Simulation O utput for Two Different Q Values. The spacing, r, is equal
to 2.5 cm. The solid curves are theoretical values from Eq. 2.31 while the dotted
curves are plots of the simulation output.
42
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
CHAPTER IV
M EASUREM ENT APPARATUS
System Specifications
A block diagram o f the experim ental apparatus that was designed and constructed for
this study is shown in Figure 4.1 [2]. The welded aluminum cham ber dimensions are
1.034 m by 0.809 m by 0.581 m while the microwave source frequency, from a network
analyzer, varies from 1 G H z 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 cham ber dim ensions in the lower end o f the test spectrum. Here the
apparatus allows for the accentuated observation o f non-ideal phenom ena to be contrasted
with ideal theoretical calculations. A cham ber with these moderate dimensions is often
referred to as an “electrically small” chamber.
The aluminum alloy used to construct the cham ber is known as 6061 T6 and is found,
in [19], to have a conductivity of o = 2.32 x 107 S/m. This param eter is em bedded in the
theoretical QDCt formulas and is, in turn, also em bedded in the theoretical gain formulas o f
chapter II as well as in the simulation program (appendix).
Antenna 1 (Figure 4.1) is connected to the source o f a network analyzer and is
denoted the transmitting antenna. This antenna is a log-periodic dipole-array with dipoles
that allow for efficient transmission in the 1-18 GHz range.
A ntenna 2 (Figure 4.1), which is also a (1-18 GHz) log-periodic dipole array, is
denoted as a receiving antenna and is placed in the cham ber w hen it is desirable to
43
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
NETW ORK
ANALYZER
DIGITAL BUSES
COM PUTER
/\
MICROWAVE
TRANSMISSION
LINES
MOTOR
SENSORS
PADDLE WHEEL
ANTENNAS
R E V E R B E R A T IO N C H A M B E R
Figure 4.1. Block Diagram o f the Basic Reverberation C ham ber System that was
Designed and Constructed for this Study. A ntenna 1 is the transmitting
antenna and is connected to a netw ork analyzer source. T h e other antenna
and the two D-dot sensors are selectively placed in the cham ber and
connected to the receiving port/ports o f a netw ork analyzer in accordance
with type o f m easurem ent that is desired.
44
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
evaluate the cham ber in a standard (documented) test configuration. A significant amount
o f effort has been dedicated tow ards gathering and evaluating experim ental data in this
(antenna-to-antenna) configuration in order to insure that this system is a functional
reverberation cham ber apparatus and is acceptable for further experimental investigations.
Some o f this prelim inary data is included in chapter V alongside plots of the theoretical
model in Eq. 2.14. One noteworthy characteristic o f this antenna is that it cannot be
m odeled as a point sensor since the locations o f the dipoles are at varying distances from
the wall and the dimensions o f the larger dipoles are not infinitesimal in relation to the
cham ber dim ensions. In other words, the power liberated to the receiving port o f the
analyzer is not representative o f the pow er at a “point” in the chamber.
In order to study in detail certain statistical characteristics o f the fields in the chamber
(e.g. the spatial correlation o f the pow er density between two points), it is necessary to
m odify the apparatus by replacing the receiving antenna with either one or two point
sensors. This problem has been studied in detail with the decision being to replace the
antenna with two ($2400/sensor) wall-m ounted D-dot sensors [4], These sensors measure
the “tim e derivative o f the electric displacement, D, norm al to the w all” [2] and liberate
an am ount o f pow er to an analyzer receiving port that is proportional to the pow er density
present at the sensor tip. In C hapter V, the approxim ate theoretical model o f Eq. 2.16,
developed for the cham ber response with receiving sensors, is com pared with a
corresponding set of m easured data. Acceptable levels o f agreem ent have been observed
and this basic experim ental apparatus is deemed suitable for further study o f the fields in
45
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
the chamber. The data sheet for these sensors, with all the critical dimensions, is shown in
Figure 4.2.
Alternatives to this wall-mounted D-dot sensor approach to making “point”
measurements which em ploy a "free field" ( not wall mounted) probe are prohibitively
expensive in term s o f the budgetary constraints for this project. For example, a system
that is centered around an EM CO (The Electromechanics Company) M odel 7121
Broadband Probe could be used to measure the SCF in a cham ber if the following items
were acquired:
(1) One EM CO Broadband Probe - $5845
(2) Tw o or Three Hughes TW T (traveling wave tube) Am plifiers - $10,000 -$15,000 each
(3) One Gigatronics Source - $27,450
Items (2) and (3) are necessary due to the sensitivity rating o f the probe which requires the
presence o f higher field levels inside the cham ber than can be provided by the sources in
the analyzer block o f Figure 4.1.
The analyzer block in Figure 4.1 is implemented with either the HP 8753C or the HP
8719A m icrowave network analyzer. The 8753C has a source frequency range o f 300
kH z - 3 GHz and a m axim um output power level of 20 dBm, while the 8719A has a
source frequency range of 130 M H z - 13.5 G H z and a m axim um output power level o f 10 dBm. In addition, the 8753C can be physically configured with two receiving ports to
accept two D-dot sensor outputs while the 8719A has only one receiving port.
Initially, the 8753C was used, along with a set of software control/data-acquisition
46
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
H 0 DEL
C l i e t r s e i l I b ic
A O -S IO (R )
AO -IO (A )
C qu I v . A r n l A i q l
U IB -V
F ro o . I b i m a i i
tld b p o in t 1
K U e tiO O
( t r i e -1 0 1
> I1 0 H |
H IG H ,
AO~Ct10(A )
Io l0 * * « *
A O -1 1 0 (A )
> 2 .SON,
L ll-V
H .S Q H ,
A D -B 20(A )
AO-SO(A)
A O -S O O (ft)t
»IOHz
>IOH,
moow x
(.) ) m
< 1 .0 no
Ix lO ^ a *
IM IB -V
( .H n i
< .l ) m
t .I Q n o
t.Ilm
< .) S n i
Mb i I m b O u tp u t
(h ik t
• 1S0V
•ISOV
• 1KV
• IKV
ikKV
IkKV
•IK V
O u tp u t C o n n e c to r
(F o n o lo l
OSSN
OSSH
BMA
SKA
SNA
BMA
T ypo *
F ti v c te o l t o a c
(I n .) N
.2 2
.22
.1 0
.0 1
t.tt
1.11
1 .1 2
0
1 .0 0
1 .0 0
i.a o
2 .1 0
-
S . 10
S.SO
1 1 .1 2
L
1 ,1 0
T
.M
-
.M
A
-
.S I
•0 0
-
C
.k i
*
.0 0
1 .0 0
1 2 .0
.0 0
• 01
.7 )
-
.0 0
22.0
-
.11
.72
- ■
.1 2
1 .0 2
AXIAL (A]
Figure 4.2. D ata Sheet for the Prodyn A D -S 10 ( R ) W all-M ounted D-dot Sensors that
are used to M easure the SCF in this Study. (Provided by Prodyn
Technologies)
47
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
routines, to m easure the spatial correlation o f the pow er density within the chamber. The
8753C was chosen because o f its ability to accept both sensor outputs. It had the
drawback, however, o f limiting the highest frequency of m easurem ent to 3.0 GHz. The
values for the SCF were obtained by calculating a correlation coefficient for each analyzer
frequency with a set sensor spacing. These calculations are perform ed by the
control/data-acquisition software as a post-processing step by using the usual productm om ent estim ate [20] for the correlation in Eq. 4.1.
where
Pj = N
1 N
X P jn
n= 1
1
> *2 = ^
N
^ ^ 2n
n= 1
’ *>ln = Power d e n sity at
location 1 for paddle wheel position n, P2n = power density at location 2 for paddle wheel
position n, and N = total num ber o f paddle wheel positions.
After carrying out these prelim inary 8753C measurements, a set of enhancem ents were
m ade to the apparatus with the 8719A in the system. These enhancements included the
integration o f a m icrowave switch (TA2F31 from DB Products) between the analyzer
receiving port and the tw o cham ber output ports, shown in Figure 4.3, along with a
com puter interface card designed for this switch. This modification allows for 1 -1 3 .5
G H z two-sensor SCF m easurem ents via the single receiving port on the 8719A. M ajor
48
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
MOTOR
ANTENNA
PADDLE WHEEL
R E V E R B E R A T IO N
CHAM BER
SENSORS
RF
LINES
SWITCH
NETW ORK
ANALYZER
DIGITAL
LINES
COM PUTER
Figure 4.3. Reverberation C ham ber System that is used with HP8719A M icrow ave
Netw ork A nalyzer to m ake SCF M easurem ents from 1 to 13.5 GHz.
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
structural enhancem ents, that are discussed in the following section, included the
integration o f a “low-backlash” gearbox that reduces paddle wheel vibrations.
The basic m easurem ent param eters that these analyzers m easure are known as Sparam eters (or scattering param eters). Sn is the voltage reflection coefficient at analyzer
p o rt 1, S22 is the voltage reflection coefficient at port 2 , S21 is the voltage transmission
coefficient from port 1 to port 2 , and S i2 is the voltage transmission coefficient from port
2 to port 1. For reverberation cham ber applications, S 21 (or equivalently S 12) is of
prim ary im portance since it gives the ratio o f the pow er at the receiving port to the power
at the transm itting port and is therefore an indication o f the pow er level that is detected
by an antenna/sensor. These S 21 values are proportional to the average pow er density in
the neighborhood o f the antenna/sensor in the cham ber and are used directly as statistical
sam ples for both the PDF and the SCF m easurements. Sn and S 22 m easurements can be
used as correction factors for S2i measurem ents to obtain cham ber gain param eters
between the transmitting antenna and the receiving sensor/antenna. These correction
factors provide some compensation for antenna/sensor reflections and are discussed in
m ore detail in C hapter V.
M echanical Design Considerations
Three significant structural upgrades that are implemented on the system o f Figure 4.3
are the placem ent of a “low-backlash” gearbox on the output shaft o f the stepper motor, a
new door design for the cham ber enclosure with an elastomer seal, and a sensor plate
50
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
design that allows for close-spacing SCF measurements. These upgrades are discussed
briefly in the following numbered segments.
(1) Low-Backlash G earbox: T he gearbox (speed reducer) on the original m otor assembly
was a standard worm-gear type. With this gearbox in place, a m axim um free-play o f
approximately 0.9 degrees was m easured at the paddle wheel. For m ost of the
measurements in this study, the paddle wheel is rotated in 200 1. 8-degree incremental
steps. Therefore, the relatively large amount o f free-play could lead to a situation where
the paddle wheel does not rotate to a unique position at each step. This inherent
inaccuracy in the system was corrected by replacing the worm-type gearbox with a
precision low-backlash planetary gearbox from Sterling Instruments. This particular
gearbox (S9123A-PG010) has a m aximum backlash rating o f 6 arc m inutes (or 0.1
degrees).
(2) D oor and Seal: The electromagnetic seal on the original cham ber system was a wire
mesh gasket. The gasket was not attached to the door but was attached around the
circumference of the chamber opening. This was accomplished by gluing a piece o f foam
rubber weather stripping around the cham ber opening and then by gluing the wire mesh
gasket to the edge o f the w eather stripping. The door w as fastened, with bolts, to the
cham ber and the wire mesh gasket was compressed between the door and the chamber.
Leakage tests were perform ed to evaluate the quality of this seal by connecting an
external broadband log-periodic antenna to an analyzer source and placing it in the close
vicinity o f the door. This allowed for an S 12 (transfer pow er ratio from the source to the
51
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
receiving port) analyzer m easurem ent to be used as an indicator o f the leakage levels into
the cham ber when the analyzer receiving port was connected to an antenna in the
chamber. The results o f perform ing several o f these leakage tests in the 1-10 GHz range
indicated that leakage levels at a num ber o f frequencies were, on the average, about 10-15
dB above the average system noise level whereas m ost o f the leakage signals for the
selected sweep frequencies were, on the average, only 0-5 dB above the system noise
level. This problem, o f small segmented frequency ranges with leakage, was temporarily
addressed by taping the edges o f the door to the cham ber with conductive tape. The wiremesh gasket and tape com bination yielded leakage levels that were only 0-5 dB above the
system noise level for all sweep frequencies in the 1-10 GHz range. The S21 values in
these tests were in the -90 to -80 dB range. This shielding system was used to conduct a
small set of prelim inary SCF measurem ents but is not practical for obtaining a complete
set o f SCF m easurem ents since adjusting the sensor spacing inside the cham ber would
require repeatedly removing, cleaning, and applying strips of conductive tape.
The developm ent o f a perm anent door and seal design was initiated by soliciting some
m ajor vendors o f EM I products for technical literature with regard to state-of-the-art
electromagnetic shielding design methodologies. In particular, handbooks from
Chom erics [21] and Tecknit [22] provided detailed and fairly com prehensive material on
this su b ject After careful consideration of the m aterial in these handbooks, a decision was
made to design a door with a groove that can be fitted with a conductive elastom er gasket.
The dim ensions o f the door along with the bolt specifications that w ere selected to m eet
52
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
certain m inim um torque requirem ents are shown in Figure 4.4. The dim ensions o f the
groove and gasket are shown in Figure 4.5. The (nom inal) groove dim ensions were
calculated from a set o f algebraic form ulas in [22 ] by substituting values for the m inim um
and m axim um allowable gasket deflections along with the fabrication tolerance o f the
gasket diam eter. Leakage tests perform ed on the im plementation of this design also yield
leakage levels that are 0-5 dB above the system noise level and S 21 values that are in the 90 to -80 dB range.
(3) Sensor P late: W hile it is possible to fasten the D-dot sensors to the bottom wall of
the cham ber with conductive tape, it is preferable to have a thin, flat, rectangular plate
with sm all threaded holes that allow for the sensors to be easily fastened to the plate with
sm all screws. This type of plate was designed and fabricated for the Prodyn sensors with
holes that allow for 6 sensor spacings ranging from 2.5 to 17.5 cm . SCF m easurem ents in
this spacing range were conducted by fastening this plate to the bottom wall o f the
chamber.
Sensor spacings below 2.5 cm cannot be obtained with this sensor and plate
m echanism due to the 1 inch diam eter o f the sensor ground disk (see Figure 4.2). Thus, a
plate and sensor design that allows for close-spacing measurem ents was designed by
fabricating two D-dot sensors from type 141 semi-rigid cables. The sensors were m ade by
stripping about 1 inch of the outer conductor, along with the dielectric material, from the
end of the cables and then bending the inner conductor 90 degrees. The resulting
m onopoles w ere cut to a length o f 0.125 inches. These 141 cable-sensors are placed on a
53
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
C H A M B E R
D O O R
1 .5 0 . e a c h s id e
G ro o v e ra d iu s :
2 .0 0 in n e r. 2 .2 6 o u te r,
each co rn er
2 1 .5 0
H o le fo r 1 /4 -2 8 b o lt. 5 p e r sid e
,G ro o v e f o r e la s to m e r s a s k e t
/s e e d e ta il!
2 1 .5 0
M a te r ia l: a lu m in u m . 3 /8 th ic k
/A ll d im e n s io n s in in ch es')
Figure 4.4. Drawing of the C ham ber D oor
54
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
C R O S S -S E C T IO N D E T A IL
O F G R O O V E A N D G A SK ET
Gasket: 1/4 diameter metal-filled
\
silicone elastomer
3/8
Groove: 0.205 deep. 0.260 wide
("All dimensions in inches)
Figure 4.5. Drawing that Show s Cross-Section of the Cham ber Door Groove and
Elastomer Seal.
55
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
plate with a semicircular slit in the middle. After the cables are placed in the slit, clamps
are placed on the cables to insure that the outer conductor is in contact with the plate and
the spacing between the monopoles rem ains fixed. Figure 4.6 shows a photograph of this
plate with the 141 cable-sensors attached. The sensor spacing is 0.5 cm.
A m ajor feature o f this structure is the conductive tape, with two holes, that is placed
over the ends o f the cables. The tape covers the region between the sensors and provides
a m ore sym m etric current flow around the m onopoles. M easurem ents, at 2.5 cm , without
the conductive tape yield lower m easured correlation coefficients than m easurements
taken with the Prodyn sensors with the same spacing. M easurem ents with the tape yield
correlations that agree, on the average, with those from the Prodyn sensor at a spacing of
2.5 cm. The results are not identical, however, since random fluctuations are observed in
all the SCF measurements in this study. This effect is discussed in m ore detail in the
following chapter along with SCF m easurements for close sensor spacings o f 0.5 cm , 1.0
cm, and 1.5 cm and SCF measurem ents for six larger spacings ranging from 2.5 cm to
17.5 cm.
56
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.6. Photograph o f Sensor Plate for C lose Spacing SCF M easurem ents. The
sensors in this design are m ade from 141 semi-rigid cables.
57
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER V
RESULTS AND CONCLUSIONS
Pow er Transfer Characteristics
T he apparatus shown schematically in Figure 4.1 was used to m easure cham ber gain
for com parison with the theoretical m odels of Eq. 2.14 and Eq. 2.16. T he H P 8719A was
selected as the network analyzer since it allows for com parisons over a larger frequency
range. This system is referred to as configuration 1 when the receiving antenna is in place
and as configuration 2 when the receiving antenna is replaced with a receiving D -dot
sensor. As mentioned in chapter IV, the antennas are linearly polarized log-periodic
dipole arrays (W atkins-Johnson W J-48195, 1.0 to 18.0 G H z) and are m ounted well apart
and with perpendicular polarizations in order to minim ize direct coupling between them.
The D-dot sensor is a surface-m ounted asymptotic conical dipole from Prodyn
Technologies (Figure 4.2) and is mounted to a cham ber wall where the electric field is
perpendicular to the polarization o f the transmitting antenna to m inim ize coupling. The
sensor is used at frequencies up to its 3-dB point of 10 G H z, where its response has fallen
to 3 dB below the first-order m odel o f Eq. 2.16, by correcting the sensor output values
for this fall-off during data analysis.
Plots o f m easured cham ber gain for configuration 1 and configuration 2 are shown in
Figure 5.1 and Figure 5.2, respectively [10]. The corresponding theoretical m odels are
also plotted in these Figures, Eq. 2.14 in Figure 5.1 and Eq. 2.16 in Figure 5.2. The
measurements presented in these Figures were conducted with a 21-point frequency sweep
58
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
(transmitting antenna, receiving antenna)
theoretical
-10
exptl. max.
G21 (dB)
exptl. avg.
-20
-30
-40
-50
-60
FREQUENCY (GHz)
Figure 5.1. Chamber Gain versus Frequency in Configuration 1.
59
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
(transmitting antenna, receiving sen sor)
-10
theoretical
-20
.
exptl. max.
exptl. avg.
CM
0 -4 0
exptl. min.
-50
-60
-70
FREQUENCY (GHz)
Figure 5.2. Cham ber Gain versus Frequency in Configuration 2.
60
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
from 1 to 10 GHz. A total o f 153 separate gain m easurem ents w ere accum ulated for each
frequency point. Each o f these m easurem ents was taken with the paddle wheel adjusted
to a unique angular position controlled by the stepper m otor, which was program m ed to
turn the paddle wheel one com plete revolution in 153 equal angular increm ents.
The gain values in Figures 5.1 and 5.2 w ere obtained from S21 and Sn measurements
as follows:
2
S21
G 21 ~
1-
r
(5-1)
S11
where G 21 is the m easured pow er gain from the transm itting port (port 1 in Figure 4.1) to
the receiving port (port 2 in Figure 4.1), |S2 i| is the m agnitude o f the voltage transmission
coefficient from port 1 to port 2, and |Sj j| is the m agnitude of the voltage reflection
coefficient at port 1. The 1—|Si j |2 term is included to account for pow er returned to the
source from the transmitting antenna. An im portant characteristic o f the experim ental
curves in both Figure 5.1 and Figure 5.2 is the large, greater than 20 dB, difference
between the m axim um and m inim um values o f G 21 over the entire frequency interval. This
large difference is generally desirable with regard to proper reverberation chamber
operation and is an indication o f a properly functioning paddle wheel.
As m ight be expected, the theoretical curves in both Figure 5.1 and Figure 5.2 are only
approxim ate representations o f the actual response o f the chamber. T he slopes of the
theoretical curves match the general trends o f the slopes o f the experim ental m axim um and
61
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
average curves, but the theoretical values tend to be larger than the experim ental. This
discrepancy can perhaps be attributed to factors such as losses in the paddle wheel blades,
antenna internal and m ismatch losses, and losses through the access panel (door) gasket,
which were not modeled and included in the calculations. Regarding cham ber wall losses,
which were m odeled, the value used for the wall conductivity, c , w as 2.32107 S/m.
Although this is the handbook value [19] for our particular alum inum alloy (6061T6), it
may in fact be too high [23]; and this would add to the discrepancy.
Som e insight into the physical phenom ena that determ ine the shape o f the cham ber
response can be obtained from an examination o f equations used in the models. For
example, for configuration 1 the GT5dependence in the second term o f the denom inator of
Eq. 2.14 can be attributed to the loading o f the receiving antenna decreasing with ci/
combined with the chamber wall loss increasing with co1/2. Specifically, the ratio o f Q ’s in
Eq. 2.12 can be rewritten as a pow er ratio as follows:
(5.2)
Here, Pd
co2 from Eq 2.3 and P«,v °= (01/2 from Eq. 2.13 since 8
to 1/2.
A t low frequencies, however, the first term in the denom inator o f Eq. 2.14
dom inates. Thus there are two distinct frequency regions. The transition point between
these tw o regions can be calculated by setting the second term in the denom inator equal to
unity and solving for the corresponding value o f ox T he result is a transition point for our
cham ber at 3.34 G Hz. B elow this transition point, Pd > Pcqv in Eq. 5.2, so that pow er
62
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
extraction from the chamber by the receiving antenna dominates over pow er loss to the
chamber walls.
Similarly, for configuration 2 the cd3/z dependence in the second term o f the
denom inator o f Eq. 2.16 is due to the cd2variation in the D-dot sensor loading combined
with the col/2 variation in the cham ber wall loss. (The frequency dependence o f the sensor
loading can be observed by solving for Pd in Eq. 2.7.)
The first term of the denom inator o f Eq. 2.16 becomes im portant only for high
frequencies- well above 10 G H z for our particular cham ber and sensor. This is the range
where the pow er extracted by the sensor is dominant. Over the interval o f our
measurements, 1 - 1 0 GHz, the cham ber wall loss dominates.
An immediate application o f the theoretical gain models of Eq. 2.14 and Eq. 2.16 can
be developed from observation of Eq. 2.10. This equation expresses the chamber gain as
a ratio of the cham ber Q , Qnet, over the antenna or sensor Q. Therefore, a frequency
domain approach to measuring the Q o f a microwave reverberation chamber, in a given
configuration, is to measure the cham ber gain and then calculate Q nct values by applying
the appropriate m odel for the sensor/antenna Q. This procedure w as carried out for the
chamber system o f Figure 4.3, referred to as configuration 3, with the two Prodyn D-dot
sensors installed as the receiving sensors. Eq. 2.7 was used as the sensor model with R =
50 ohms and A = 10^ m2. A plot o f the resulting Q measurements versus frequency is
shown in Figure 5.3 where the smooth curve is Qnet, given by Eq. 2.8, divided by seven
since this gives the best fit to the measurements. Table 5.1 is the tabulation o f a series of
63
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
6
Measured Chamber Q Versus Frequency
x 10
5
4
O
3
2
1
0
2
4
6
8
10
12
Frequency (GHz)
Figure 5.3. Cham ber Q M easurement via Cham ber Gain M easurem ent with Two
Receiving (Prodyn) D-dot Sensors (configuration 3). The jagged curve
represents the m easured values whereas the smooth curve is QDCt, from Eq.
2.8, divided by seven. The gain m easurem ent was taken with a 21-point
frequency sweep and 200 incremental paddle wheel positions.
64
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Table 5.1. Tabulation o f Calculated M Values from M easured G ain Values. Each
colum n, from left to right (not including f), represents a step in the
calculation process.
f
M easured
Gain
M easured
q d
1.0000e+009
1.6250e+009
2.2500e+009
2.8750e+009
3.5000e+009
4.1250e+009
4.7500e+009
5.3750e+009
6.0000e+009
6.6250e+009
7.2500e+009
7.8750e+009
8.5000e+009
9.1250e+009
9.7500e+009
1.0375e+010
1. 1000e +010
1.1625e+010
1.2250e+010
1.2875e+010
1.3500e+010
-45.56
-35.19
-33.20
-27.88
-28.29
-28.96
-24.60
-26.33
-24.30
-21.35
-19.89
-22.46
-19.53
-19.45
-21.62
-19.96
-23.64
-16.42
-17.52
-20.57
-19.77
2.62e+007
1.61e+007
1.17e+007
9.12e+006
7.49e+006
6.35e+006
5.52e+006
4.88e+006
4.37e+006
3.96e+006
3.62e+006
3.33e+006
3.08e+006
2.87e+006
2.69e+006
2.53e+006
2.38e+006
2.25e+006
2.14e+006
2.04e+006
1.94e+006
Q
3-dB
BW
7.28e+002
4.88e+003
5.58e+003
1.48e+004
1.1 le+ 004
8.08e+003
1.9 le+ 0 0 4
1.13e+004
1.62e+004
2.90e+004
3.7 le+ 0 0 4
1.89e+004
3.43e+004
3.26e+004
1.85e+004
2.55e+004
1.03e+004
5.14e+004
3.79e+004
1.79e+004
2.05e+004
1.37e+006
3.33e+005
4.03e+005
1.94e+005
3.15e+005
5.10e+005
2.48e+005
4.74e+005
3.70e+005
2.29e+005
1.96e+005
4.17e+005
2.48e+005
2.80e+005
5.27e+005
4.07e+005
1.07e+006
2.26e+005
3.24e+005
7.21e+005
6.59e+005
M
0.62
0.40
0.93
0.73
1.75
3.94
2.54
6.21
6.03
4.55
4.66
11.72
8.11
10.57
22.72
19.84
58.58
13.86
22.02
54.21
54.43
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
calculations that were perform ed, including the calculations for the Q m easurem ent, that
lead to a set o f values for the num ber o f m odes in a 3-dB bandwidth, M. These
calculations were perform ed sequentially from left to right by processing the gain
m easurem ents (second colum n) first and arriving at the values for M by substituting the
results from the 3-dB bandwidth calculation (fifth column) for d f in the mode density
formula o f Eq. 1.2. The resulting dN values are the M values. The 3-dB bandw idth was
obtained simply by observing the relationship BW = f/Q where the Q's are the m easured Q
values of column four.
PDF o f Power Density
A study of the probability density function for a com ponent o f the pow er density in the
special (small, non-ideal) cham ber o f Figure 4.1 was initiated by exam ining the sample
distributions, at each frequency, for the gain measurem ents of Figures 5.1 and 5.2. This
involved plotting and analyzing the form o f the histograms for the 153 m easured samples
at each frequency. Figure 5.4 shows these histograms [2] at the low est and the highest
m easurem ent frequencies, 1 and 10 GHz, for the receiving antenna configuration
(configuration 1) and Figure 5.5 show s the histogram s [2] at 1 and 10 G H z for the
receiving D -dot sensor configuration (configuration 2). Inspection o f these figures
indicates that there is good agreem ent at 10 GHz between the m easured distribution and
the theoretical PDF (Eq. 2.19) both aw ay and on the wall. At 1 GHz, poor levels of
agreem ent are observed since the distributions o f the m easured values are much "flatter"
66
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
CD
O
c
£
w
3
O 0.2
O
O
CD
> 0.1
05
(1 GHz, aw ay from wall)
curve = theoretical bars = experim ental
CD
CE
L
■Win
0
-50
-40
-30
-20
|S211 (dB)
CD
O
c
(10 GHz, aw ay from wall)
CD
w
w
3
O 0.2
O
O
>
'■g
05
-
CD
cc
-30
-20
|S211 (dB)
-10
Figure 5.4. Probability Density Function for EM Power Density in Configuration 1. The
bin sizes for the histogram s are 0.25 dB. T he standard deviation for the 1
GHz data is 6.61 dB and the standard deviation for the 10 GHz data is 5.35
dB.
67
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
CD
O
c
f£
o 0.2 ■
CJ
O
.1 0.1 -
(1 GHz, on the wall)
curve = theoretical bars = experim ental
0)
DC
-80
|S211 (dB)
CD
o
c
2t _>
13 0.2
O
o
(10 GHz, on the wall)
o
.1 0.1
ja
CD
DC
0
-60
-5 0
-40
-30
|S211 (dB)
-20
-10
Figure 5.5. Probability Density Function for EM Power Density in Configuration 2. The
bin sizes for the histograms are 0.25 dB. The standard deviation for the 1
G H z data is 8.57 dB and the standard deviation for the 10 G H z data is 5.36
dB.
68
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
than the shapes o f the corresponding theoretical curves. In configuration 1, the standard
deviation o f the m easured data varies from 6.61 dB, at 1 GHz, to 5.35 dB at 10 GHz. In
configuration 2, the standard deviation o f the m easured data varies from 8.57 dB , at 1
GHz, to 5.36 dB at 10 GHz. As m entioned in chapter II, the theoretical value for this
standard deviation is 5.57 dB. The upward shifts in the m easured standard deviations, as a
function of decreasing frequency, indicate that the distribution w idens as the num ber of
m odes in a 3-dB bandwidth, M, decreases.
Inspection of the sam ple standard deviations of the m easurem ents used to obtain the Q
o f Figure 5.3 can provide m ore insight into the relationship between the transition of the
non-ideal m easured PDF's at 1 GHz to the m ore ideal forms at higher frequencies and M.
For example, inspection o f the standard deviation values in the seventh column o f Table
5.2 indicates that the first sample standard deviation that is between 5 and 6 dB occurs at
about 4.5 or 5 GHz. Observing the corresponding M values in the sixth column o f this
table indicates that the m easured distribution could converge to an ideal one for M values
as low as 3 or 4. This m eans that Lehm an's assumption o f an infinite M may not restrict
the applicability o f his theory as much as one might have supposed.
A sim ilar type o f analysis can be perform ed on the sim ulation output. Here, standard
deviation values, from a post-processing program that calculates a set o f statistics (and
also generates histograms) for the sim ulation samples, are tabulated and observed in the
fourth colum n o f Table 5.2. These particular sample standard deviations result from a
simulation run with Q = Qnct / 10 and a distance of 8 cm between calculated field points
(or samples) and correspond to the sim ulation run, from the overall set o f simulations, that
69
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
Table 5.2. Measured M and o Values From the Q Measurement of Figure 5.3 Alongside M and a
Values From the Corresponding Simulation Output
Simulation
f
1.0000e+009
1.2500e+009
1.5000e+009
1.7500e+009
2.0000e+009
2.2500e+009
2.5000e+009
2.7500c+009
3.0000e+009
3.2500e+009
3.5000e+009
3.7500e+009
4.0000e+009
4.2500e+009
4.5000e+009
4.7500e+009
5.0000e+009
5.2500e+009
5.5000e+009
5.7500e+009
6.0000e+009
6.2500e+009
6.5000e+009
6.7500e+009
7.0000e+009
7.2500e+009
7.5000e+009
7.7500c+009
8.0000e+009
8.2500e+009
8.5000e+009
8.7500e+009
9.0000e+009
9.2500e+009
9.5000e+009
9.7500e+009
1.0000e+010
1.0250e+010
1.0500e+010
1.0750e+010
l.lOOOe+OlO
1.1250e+010
1.1500e+010
1.1750e+010
1.2000e+010
1.2250e+010
Mave
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
4.01
4.37
4.74
5.32
8.23
9.25
7.73
8.54
10.00
10.35
10.82
14.32
13.15
14.23
18.80
17.21
19.04
19.68
20.96
22.99
27.16
25.92
28.15
29.25
32.84
35.89
37.18
40.10
40.10
Measurement
M*
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
5.21
5.88
6.59
7.36
8.18
9.06
10.00
10.99
12.05
13.17
14.36
15.61
16.93
18.32
19.78
21.32
22.93
24.62
26.39
28.25
30.18
32.21
34.32
36.52
38.82
41.21
43.69
46.28
48.96
a
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
5.64
6.90
6.36
6.29
6.27
6.67
6.14
6.34
6.09
6.10
6.01
5.77
5.96
6.31
6.44
5.62
5.25
4.85
5.18
5.42
5.46
5.44
5.96
6.18
6.03
6.48
6.54
5.17
5.93
f
M
a
1.0000e+009
0.62
6.99
1.6250e+009
0.40
8.37
2.2500e+009
0.93
6.72
2.8750e+009
0.73
7.46
3.5000e+009
1.75
6.25
4.1250e+009
3.94
6.42
4.7500e+009
2.54
5.25
5.3750e+009
6.21
5.77
6.0000e+009
6.03
5.51
6.6250e+009
4.55
5.24
7.2500e+009
4.66
5.63
7.8750e+009
11.72
6.66
8.5000e+009
8.11
6.28
9.1250e+009
10.57
5.69
9.7500e+009
22.72
5.25
1.0375e+010
19.84
6.05
1.1000e+010
58.58
5.62
1.1625c+010
13.86
5.95
1.2250e+010
22.02
5.37
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Table 5.2. Continued.
Simulation
f
1.2500e+010
1.2750e+010
1.3000e+010
1.3250e+010
1.3500e+010
Mave
44.26
43.17
48.68
49.92
52.43
Measurement
M*
51.75
54.64
57.64
60.76
63.98
a
f
5.54
5.56
6.24
5.73
5.60
M
a
1.2875e+010
54.21
5.04
1.3500e+010
54.43
5.75
most closely matches the conditions in the chamber for the m easurem ent o f Figure 5.3.
Inspection o f these standard deviation values, which are for one o f the two field points,
also indicates that the first sample standard deviation that is between 5 and 6 dB occurs at
about 4.5 or 5 GHz. One o f the output files (in the form of Figure 3.2) from this
simulation run also shows that, in the 4.5 - 5 GHz range, the average num ber o f selected
modes that fall within the 3-dB bandwidth is approximately 3 or 4. These average M
values are shown in the second colum n o f Table 5.2. Also, the theoretical values o f M
(from the output file) are included in the third column of Table 5.2. These values range
from about 3.5 to 4.5 in the 4.5 - 5.0 G H z range.
71
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
SCF o f Power Density
Figure 5.6 is a set o f plots that show m easured and simulated spatial correlations
versus spacing w hile Figure 5.7 is a set o f plots that show m easured and simulated spatial
correlations versus frequency. Lehm an's theoretical curve, Eq. 2.31, is plotted alongside
all o f these results. All of the measurements were taken with the m easurem ent system of
o f Figure 4.3 (denoted as configuration 3). M easurem ents with sensor spacings o f 2.5,
5.5, 8.0, 11.5, 13.5, and 17.5 cm were taken with the Prodyn sensors (Figure 4.2) in place
whereas m easurem ents with sensor spacings of 0 .5 ,1 .0 , and 1.5 cm were taken with the
141 cable-sensor plate (Figure 4.6) in place. Fifty-one discrete frequency points between
1 and 13.5 GHz, with 0.25 G H z increments, were used for both the m easurem ents and
the simulations.
The com plete set o f simulation runs include three runs, with Q = Qn(:t , Q = Qnct /10 ,
and Q = Qnct / 100 , for each o f the sensor spacings o f the above-mentioned measurements.
(Qnci is given by Eq. 2.8 for the case of two receiving D -dot sensors.) The SCF values
from the Q = Qn<:t /10 runs were chosen for the plots o f Figures 5.6 and 5.7 since Q„ct /10
is o f the same order o f m agnitude as the measured cham ber Q's o f Figure 5.3. The SCF
outputs from the sim ulations with Q = QDet/100 were found to be in close agrrem ent with
Lehman's theoretical curve in the upper half o f the selected frequency range, where large
values of M occur. Slightly low er levels o f agreem ent with the theoretical curve are
observed in the S C F outputs from the Q = Qnet /10 runs in the upper half o f the frequency
range. In contrast, the SCF outputs from the Q = Qnet runs are significantly higher than
72
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
SCF of EM Power Density
f = 1.00 GHz
f= 1.25 GHz
® 0.5
® 0.5
0
5
10
15
20
0
5
f = 1.50 GHz
10
15
20
f = 1.75 GHz
® 0.5
® 0.5
0
5
10
15
20
0
5
f = 2.00 GHz
10
15
20
f = 2.25 GHz
® 0.5
0
5
10
15
20
0
Spacing (cm)
5
10
15
20
Spacing (cm)
Theo. = Solid M eas. = Dashed
Sim. = D otted
Figure 5.6. M easured, Simulated, and Theoretical SC F o f the Pow er Density, versus
Spacing, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or
Figure 4.1. The measurem ents were taken with the m easurem ent system of
Figure 4.3. (a) Plots for frequencies o f 1, 1.25, 1.50, 1.75, 2.00, and 2.25
GHz.
73
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
SCF of EM Power Density
f = 2.50 GHz
f = 2.75 GHz
<u 0.5
a) 0.5
-4/- -
0
5
10
15
20
0
5
1
f = 3.00 GHz
® 0.5
10
15
20
f = 3.25 GHz
a> 0.5
0
0
5
10
15
20
0
1
f = 3.50 GHz
10
5
15
20
f = 3.75 GHz
o 0.5
V '
o
0
5
10
15
20
0
Spacing (cm)
5
10
15
20
Spacing (cm)
Theo. = Solid M eas. = Dashed Sim. = Dotted
Figure 5.6. Continued, (b) Plots for frequencies o f 2 .5 ,2 .7 5 , 3.00, 3.25, 3.50, and 3.75
GHz.
74
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
SCF of EM Power Density
1
f = 4.00 GHz
f = 4.25 GHz
cu 0.5
a> 0.5
0
0
5
10
15
20
0
5
1
f = 4.50 GHz
® 0.5
cd
10
15
20
f = 4.75 GHz
0.5
0
0
5
10
15
20
0
1
f = 5.00 GHz
cd
5
0.5
cd
10
15
20
f = 5.25 GHz
0.5
0
0
5
10
15
0
20
Spacing (cm)
5
10
15
20
Spacing (cm)
Theo. = Solid M eas. = Dashed Sim. = Dotted
Figure 5.6. Continued, (c) Plots for frequencies o f 4 .0 0 ,4 .2 5 ,4 .5 0 ,4 .7 5 , 5.00, and 5.25
GHz.
75
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Correlation
SCF of EM Power Density
1
0.5
g> 0.5
0
0
0
5
Correlation
1
10
15
0
a> 0.5
0
0
1
10
15
5
1
0.5
5
f = 5.75 GHz
20
f = 6.00 GHz
0
Correlation
1
f = 5.50 GHz
0
20
15
20
f = 6.25 GHz
5
1
f = 6.50 GHz
10
10
15
20
f= 6.75 GHz
0.5
0
0
0
5
10
20
0
Spacing (cm)
5
10
15
20
Spacing (cm)
Theo. = Solid Meas. = Dashed Sim. = Dotted
Figure 5.6. Continued, (d) Plots for frequencies o f 5.50, 5 .7 5 ,6 .0 0 ,6 .2 5 , 6.50 and 6.75
GHz.
76
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Correlation
SCF of EM Power Density
f = 7.00 GHz
0.5
a> 0.5
Correlation
0
5
10
15
0
20
5
f = 7.50 GHz
10
15
20
f = 7.75 GHz
0.5
a> 0.5
0
Correlation
f = 7.25 GHz
5
10
15
20
0
5
f = 8.00 GHz
10
15
20
f = 8.25 GHz
0.5
0
5
10
15
0
20
Spacing (cm)
5
10
15
20
Spacing (cm)
Theo. = Solid Meas. = Dashed Sim. = Dotted
Figure 5.6. Continued, (e) Plots for frequencies o f 7.0 0 ,7 .2 5 , 7 .5 0 ,7 .7 5 , 8.00, and 8.25
GHz.
77
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
Correlation
SCF of EM Power Density
f = 8.50 GHz
0.5
® 0.5
Correlation
0
5
10
15
20
0
5
f = 9.00 GHz
10
15
20
f= 9.25 GHz
0.5
® 0.5
0
Correlation
f= 8.75 GHz
5
10
15
0
20
5
f = 9.50 GHz
10
15
20
f= 9.75 GHz
0.5
® 0.5
0
5
10
15
20
0
Spacing (cm)
5
10
15
20
Spacing (cm)
Theo. = Solid M eas. = D ashed Sim. = D otted
Figure 5.6. Continued, (f) Plots for frequencies o f 8.50, 8.75, 9.00, 9.25, 9.50, and 9.75
GHz.
78
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Correlation
SCF of EM Power Density
f= 10.00 GHz
0.5
<d 0.5
Correlation
0
5
10
15
20
0
5
f = 10.50 GHz
10
15
20
f= 10.75 GHz
0.5
a) 0.5
0
5
1
Correlation
f= 10.25 GHz
10
15
20
0
5
f= 11.00 GHz
10
15
20
f= 11.25 GHz
0.5
0
0
5
10
15
20
0
Spacing (cm)
5
10
15
Spacing (cm)
20
Theo. = Solid M eas. = Dashed Sim. = Dotted
Figure 5.6. Continued, (g) Plots for frequencies o f 10.00, 10.25, 10.50, 10.75, 11.00, and
11.25 GHz.
79
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SCF of EM Power Density
1
f= 11.50 GHz
f= 11.75 GHz
® 0.5
0
0
5
1
10
15
20
0
5
f= 12.00 GHz
10
15
20
f = 12.25 GHz
a> 0.5
0
0
5
1
10
15
0
20
5
f = 12.50 GHz
10
15
20
f = 12.75 GHz
a> 0.5
0.5
0
0
5
15
10
Spacing (cm)
0
20
5
10
15
Spacing (cm)
20
Theo. = Solid Meas. = Dashed Sim. = Dotted
Figure 5.6. Continued, (h) Plots for frequencies o f 11.50,11.75, 12.00, 12.25, 12.50, and
12.75 GHz.
80
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SCF of EM Power Density
1
f= 13.00 GHz
f= 13.25 GHz
P 0.5
__I
0
0
5
10
15
20
f= 13.50 GHz
c
o
f = 1.00 GHz
s
1 0.5
p 0.5
o
O
0L
15
10
1
0
20
5
10
15
20
{=1.50 GHz
f = 1.25 GHz
A'
p 0.5
0
0
5
10
15
Spacing (cm)
0
20
5
10
15
Spacing (cm)
20
Theo. = Solid Meas. = Dashed Sim. = Dotted
Figure 5.6. Continued, (i) Plots for frequencies o f 13.00, 13.25, and 13.50 GHz. Plots for
frequencies of 1.00,1.25, and 1.50 G H z are repeated for convenience.
81
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
SCF of EM Power Density
r = 0.5 cm
c 0.8
v \
O
5 0.6
CD
g 0.4
0.2
°
-
0.2
r = 1.0 cm
c 0.8
o
5 0.6
CD
g 0.4
\ \
0.2
°
v
-
0.2
Frequency (GHz)
Theo. = Solid Meas. = D ashed
Sim. = Dotted
Figure 5.7. M easured, Sim ulated, and Theoretical SC F o f the Pow er Density, versus
Frequency, Inside the M icrow ave Reverberation Cham ber o f Figure 4.3 or
Figure 4.1. The m easurem ents were taken with the m easurem ent system o f
Figure 4.3. (a) Plots for spacings o f .5 and 1.0 cm.
82
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SCF of EM Power Density
r= 1.5 cm
Correlation
0.8
0.6
0.4
0.2
-
> \ , ' ' 't — •
rV
+
V
0.2
1
r = 2.5 cm
Correlation
0.8
0.6
0.4
0.2
0
-
0.2
2
4
6
8
Frequency (GHz)
10
12
Theo. = Solid Meas. = Dashed Sim. = Dotted
Figure 5.7. Continued, (b) Plots for spacings o f 1.5 and 2.5 cm.
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
SCF of EM Power Density
r = 5.5 cm
C 0.8
o
5 0-6
0)
fc 0.4
0.2
°
-
0.2
r = 8.0cm
c
0 .8
o
'•S 0 .6
0.2
-
0.2
Frequency (GHz)
Theo. = Solid Meas. = Dashed Sim. = Dotted
Figure 5.7. Continued, (c) Plots for spacings o f 5.5 and 8.0 cm.
84
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SCF of EM Power Density
Correlation
0.8
0.6 -A’,
0.4 /
0.2
-
■ ^
0.2
T
T
T
T
T
r = 13.5 cm
Correlation
0.8
-
0 6 -.
0.4
0.2
-
0.2
Frequency (GHz)
Theo. = Solid M eas. = Dashed Sim. = Dotted
Figure 5.7. Continued, (d) Plots for spacings o f 11.5 and 13.5 cm.
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
SCF of EM Power Density
r= 17.5 cm
C 0.8
o
5 0-6
CD
* 0.4
0.2 - \ /
-
0.2
r = 0.5cm
c 0.8
o
V \
5 0.6
a>
g 0.4
0.2
°
-
0.2
Frequency (GHz)
Theo. = Solid Meas. = Dashed
Sim. = Dotted
Figure 5.7. Continued, (e) Plots for spacing o f 17.5 cm. Plots for spacing o f 0.5 cm are
repeated for convenience.
86
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
the corresponding theoretical values in the upper half of the frequency range. Here,
inspection of the output files shows that even at the highest sim ulation frequency o f 13.5
GHz, the (average) num ber o f selected m odes that are within the 3-dB bandwidth is only
about 5 whereas in the Q = Qnet /10 and Q = Q„et/100 cases the values for this param eter
are about 50 and 500, respectively.
Inspection o f the forms o f the m easured SCF response in Figures 5.6 and 5.7 reveals
the following three trends.
(1) High Spatial Correlations at Large Spacings and Low Frequencies: This effect can be
observed in Figure 5.6(a)-(c) and Figure 5.7(c)-(d). The measurem ents seem to
"converge," on the average, to Lehm an's theoretical curve at frequencies above the 4.5 to
5.0 G H z range. The frequency range below 4.5 G H z w as cited, in the previous section, as
a region that can be associated with insufficient m ode densities for the chamber in this
study.
(2) Random (Jagged) Fluctuations in the ('Locally') M onotonic N ature of the R esponse:
This lack of "smoothness" can be observed in all the o f measurements as well as in all o f
the simulation outputs. An effort to analyze this effect by taking the Fourier transform
(FFT) o f a num ber o f m easured SCF sam ple outputs did not reveal a "signature" in the
transform dom ain. As shown in Figure 5.8, this effect can be related to the num ber of
data acquisition samples, or paddle w heel positions, that are program m ed into the m otor
control algorithm o f the m easurem ent system. All o f the SCF measurements in this study
were taken with 200 paddle wheel positions. Selectively removing every fourth sample
87
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from a selected m easurem ent set (Figure 5.8) indicates that the peak-to-peak levels of
these fluctuations decrease as the number o f samples increases.
(31 Low Correlation at Close Sensor Spacings and at High Frequencies: This effect can
be observed in Figure 5.7(a)-(b). Since the presence o f undesirable signals is a relatively
common problem in microwave circuits, a number of possibilities were carefully
investigated to determ ine the physical nature o f these low correlations. Some o f the
possibilities that were eliminated by inspection o f some additional sets of measurements
and additional simulations along with some rough hand calculations were:
(i) decorrelation due to electrom agnetic scattering at close sensor spacings,
(ii) the existence o f slowly propagating or heavily attenuated surface waves,
(iii) large sensor size in relation to the signal wavelengths, and
(iv) the existence o f non-standard waveguide modes that do not fall into the realm
of standard theories for the operation o f an overmoded rectangular cavity.
Further consideration o f this effect led to the hypothesis that the decorrelation may be due
to the unsymmetric nature of the tape in the vicinity of the 141-cable sensors. In other
words, the "bump" in the tape between the tw o sensors may be causing the sensors to
operate in a non-ideal manner. As mentioned in chapter IV, this was not considered a
problem atic structural flaw prior to conducting the SCF measurements since trial
m easurements at a spacing of 2.5 cm yielded acceptable levels of agreem ent with trial
measurements taken with the Prodyn sensors in place. However, since a 2.5 cm SCF
m easurem ent with the Prodyn sensors requires placing them in direct contact with each
88
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SCF of EM Power Density
Correlation
0.8
0.6
0.4
N = 50
0.2
-
0.2
0.8
Correlation
0.6
0.4
N = 200
0.2
-
0.2
Frequency (GHz)
Figure 5.8. Plots o f M easured SCF Response with 50 Paddle W heel Positions and 200
Paddle W heel Positions. The data for the N = 50 plot was obtained from
the N = 200 data by selecting every forth sam ple from the 200 sample
measurement. The sensor spacing for this m easurem ent was 0.5 cm.
Reproduced with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
other, it is conceivable that the 2.5 cm m easurem ents with these sensors also are
associated with structural dissymm etries. This hypothesis o f dissym m etries in the 141
cable-sensor plate was verified by m aking a final SC F m easurem ent with a modified sensor
structure. This structure was im plemented by drilling two sm all holes in the bottom of the
cham ber that are spaced 1 cm a p a rt Then, two 141 cable-sensors w ere press-fitted into
the holes from the outside o f the cham ber such that only the inner conductor o f the cables
was protruding into the cham ber with the flat cham ber wall between the outer conductors
of the cables. Significandy higher values o f spatial correlation, at the higher frequencies,
were m easured with this structure in relation to the 1 cm spacing m easurem ent with the
141 cable-sensor plate.
Summary and Discussion
The follow ing outline serves as a sum m ary o f the m ajor developm ents and results that
are docum ented in this dissertation project.
—> Design and Construction o f M icrow ave Reverberation C ham ber M easurem ent
Apparatus with Source W avelengths such that the Cham ber is Electrically Small at
the Low er End o f the Source Frequency Spectrum.
—» Development o f Algorithm that Allows for Com puter Simulation o f SCF Experim ent
by Generating Suitable Statistical Pow er Density Samples.
—» M atlab Im plem entation o f SCF Algorithm.
-»
Derivation o f Average Cham ber Gain Models.
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—> M easurements o f Cham ber Gains.
—> Developm ent o f Frequency-Dom ain M ethod to M easure Cham ber Q U sing Chamber
Gain Models.
—> Calculation o f M Values for Chamber.
—» Evaluation o f M easured PDF's Under Non-Ideal (Electrically Small) Cham ber
Conditions Using Histogram s and Sample Standard Deviations.
-»
Evaluation o f M easured PDF's Under Non-Ideal Conditions Via Consideration of
Cham ber M V alues and O utput from SCF Simulation Program.
-»
M easurem ent o f the SCF in a M icrow ave Reverberation Chamber.
—» Comparison o f SCF M easurem ents and Simulations to Lehm an's Theoretical SCF
Curve for Complex Cavities.
T he introduction and investigation o f M as a microwave reverberation cham ber
param eter is perhaps the m ost significant o f these developments. Further study o f this
param eter could lead to a sim ple criterion that provides an accurate low-frequency bound
for the operation o f a microwave reverberation chamber. Form ulas that are presently
under consideration [25] by other investigators include a 60-modes criterion and a 6 x f0
criterion, where fo is the fundamental resonant frequency of the cham ber cavity.
T he 60-modes criterion can be applied by either finding the 60th mode o f the chamber
cavity by com puter counting or by setting N equal to 60 in Eq. 1.1 and solving for f. A
com puter counting calculation for the cham ber in this study (Figures 4.1 and 4.3) yields a
low frequency cut-off o f 844 M Hz. This criterion seems to have originated from NBS
91
with perm ission of the copyright owner. Further reproduction prohibited without perm ission.
Technical Note 1092 [4, p. 21] where the authors state that the "practical lower frequency
limit for using the NBS enclosure as a reverberation chamber is approximately 200 MHz.
This lower limit is due to a num ber of factors including insufficient mode density, limited
tuner effectiveness, and ability to uniformly excite all modes in the chamber." The
dimensions o f this NBS cham ber are such that a frequency o f 200 M Hz corresponds to the
existence o f approxim ately 60 distinct modes from 0 to 200 MHz. However, the 60modes criterion does not seem to have been specified or generally recom m ended within
the body o f this paper.
The fundamental frequency for the cham ber cavity (Figures 4.1 and 4.3) 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 analysis of a sample set o f chamber data [25]. In this
case, plots o f the cham ber VSW R 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
response is approximately flat. Literature with regard to the underlying physical principles
behind this criterion has not been cited at the present time.
The M equals 3 or 4 criterion that was proposed in the second section of this chapter
as a possible criterion for the low frequency limit o f the chamber in this study corresponds
to a low frequency cut-off o f 4.5 - 5.0 GHz. W hether or not this criterion develops into
an acceptable bound for the low-frequency operation of a m icrowave reverberation
chamber is an open question since this criterion, like the other two that are described
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above, was not developed via a formal m athem atical procedure but is proposed based on
the observation o f a lim ited set of data.
Additional analytical methods that are currently under investigation [25] and could
lead to criteria for the low-frequency cut-off of a m icrowave reverberation cham ber are
m ethods based on applying Kolm ogorov-Sm irnov goodness-of-fit tests to the m easured
PDF's and Lehm an's "unstirred energy" calculation for a m icrowave reverberation
chamber. Published literature with regard to these investigations are not presently
available.
Possible applications for the SCF inside a m icrowave reverberation cham ber include
the consideration of "correlation lengths" as part o f the overall testing procedures [24].
This concept is particularly well-suited to applications that use a m icrowave reverberation
cham ber test environm ent to simulate the actual EM I environm ent of interest. For
example, inspection o f Lehm an's theoretical SCF curve (Eq. 2.31) for null points yields
that the first null occurs at:
|l .- l 2 | = R «r = V 2 .
(5.3)
This X/2 quantity can be used to investigate the concept of correlation length in a
m icrowave reverberation cham ber in the sense that, under ideal conditions, the correlation
between the power density at two points becomes negligible at and beyond this spacing.
However, interesting cases such as the m easured SCF response at 2 GHz in Figure 5.6a
exist that do not conform to this straightforward formulation. Accurate characterization
o f the correlation length could lead to situations w here the com prehensive testing of a
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particular device in a cham ber will not also require the placem ent o f all the surrounding
equipm ent from the device's intented operational environment. Only equipm ent that falls
within a particular correlation length would need to be placed in the chamber.
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REFERENCES
[ 1]
D. A. Hill, “Electronic M ode Stirring for Reverberation Cham bers,” IE E E Trans.
Electromagn. Compat., vol. EM C-36, no. 4 , pp. 294-299, Nov. 1994.
[2]
T.F. Trost, A.K. Mitra, and A.M. Alvarado, “Characterization o f a Sm all M icrow ave
Reverberation Cham ber,” Proceedings o f the 11th International Z urich Sym posium
and Technical Exhibition on EMC, pp. 583-586, M arch 1995.
[3]
B.H. Liu and D.C. Chang, “Eigenmodes and the Com posite Quality Factor o f a
Reverberating Cham ber,” N B S Tech. N ote 1066, Aug. 1983.
[4]
M.L. Crawford and G.H. Koepke, “Design, Evaluation, and Use of a Reverberation
Chamber for Performing Electromagnetic Susceptibility/Vulnerability
M easurements,” N B S Tech. N o t e 1092, Apr. 1986.
[5]
J.M. Dunn, “L o c a l, High-Frequency Analysis o f the Fields in a M ode-Stirred
Chamber,” IE E E Traits. Electromagn. Compat., vol. EM C-32, no. 1, pp. 53-58,
Feb. 1990.
[6]
J.G. Kostas and B. Boverie, “ Statistical M odel for a M ode-Stirred Cham ber,” IE E E
Trans. Electromagn. Compat., vol. EM C-33, no. 4, pp. 366-370, Nov. 1991.
[7]
T.H. Lehman, “A Statistical Theory o f Electromagnetic Fields in Com plex Cavities,”
Phillips Laboratory Interaction N ote 494, M ay 1993.
[8]
Y. Huang and D.J. Edwards, “An Investigation o f the Electrom agnetic Field inside a
M oving-W all M ode-Stirred Chamber,” The 8th IE E Int. Conf. on EM C, Edinburgh,
UK, pp. 115-119, Sept. 1992.
[9]
T.A. Loughry, “Frequency Stirring: An Alternate Approach To M echanical M odeStirring For The Conduct O f Electromagnetic Susceptibility Testing,” Phillips
Laboratory Tech. R eport 91-1036, Nov. 1991.
[10] A.K. M itra, T.F. Trost, “ Pow er Transfer Characteristics o f a M icrow ave
Reverberation Cham ber,” to appear in IE E E Transactions on EM C, May, 1996.
[11] C.E. Baum et al., “ Sensors for Electromagnetic Pulse M easurem ents Both Inside
and Away from N uclear Source Regions,” IE E E Trans. Electrom agn. Compat., vol.
EM C-20, no. 1, pp. 22-35, Feb. 1978.
95
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[12] Y. Huang, “The Investigation of Chambers for Electromagnetic Systems,” Ph.D
Dissertation, University o f Oxford, 1994.
[13] M. Iskander et al., “FDTD Simulation o f M icrowave Sintering o f Ceramics in
M ultimode Cavities,” IE E E Trans. M icrow ave, vol. M TT-42, no. 5, pp. 793-800,
M ay 1994.
[14] K.C. Chen, private communication, Sandia N ational Laboratory, Sept. 1994.
[15] Y. Rahmat-Samii, private com munication, UCLA, Sept. 1994.
[16] A. Taflove and K.R. Umashankar, “The finite-difference time-domain method for
numerical modeling of electromagnetic wave interactions with arbitrary structures,”
in M.A. M organ, Ed., Finite-Elem ent and Finite-D ifference M ethods in
Electrom agnetic Scatter. Amsterdam: Elsevier, 1990.
[17] C.T. Tai, “On the Definition o f the Effective Aperture of Antennas,” IEEE Trans.
Antennas Propagat., vol. AP-9, pp.224-225, M arch 1961.
[18] D. I. Wu and D.C. Chang, “The Effect o f an Electrically Large Stirrer in a ModeStirred Cham ber,” IE E E Trans. Electromagn. Compat., vol. EM C-31, no. 2, pp.
164-169, M ay 1989.
[19] J. H. Potter, H andbook o f the Engineering Sciences, Vol. II, Van Nostrand, 1967.
[20] R. N. Rodriguez, “Correlation,” Encyclopedia o f Statistical Sciences, vol. 2, pp.
193-204, W iley, 1982-.
[21] EM I Sheilding Engineering Handbook, Chom erics Inc., W oburn, M A, 1989.
[22] Sheilding D esign Guide, Tecknit EMI Sheilding Products, Cranford, NJ, 1991.
[23] D.A. Hill et al., “Aperture Excitation o f Electrically Large, Lossy Cavities,” IEEE
Trans. Electromagn. Compat., vol. EM C-36, pp. 169-178, Aug. 1994.
[24] T.F. Trost and A.K. M itra, "Eectromagnetic Compatibility Testing Studies," Final
Technical R eport on G rant NAG-1-1510, NASA Langley Research Center, January
15,1996.
[25] G. Freyer, Consultant, M onument, CO, private com m unication, M arch 1996.
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APPENDIX
M ATLAB SIM ULATION M-FDLES
(1)
B W R E S1.M (Used to permanently store resonant frequency data for simulation)
%***** Calculate, Sort, and Select Cavity Resonances for Reverberation Chamber
%***** Simulation
clkl=clock;
c=2.99792458E8;
a= 1.0342;
b=.8087;
d=.5812;
lpnum=95;
simres
lc=zeros(l,51);
m g l(l)= l;
mg l(2:52)=zeros( 1,51);
bwhl(l)=300e6;
bwhl(2)=75e6;
bwh 1(3:9)=50e6*ones( 1,7);
bwh 1(10:21 )=25e6*ones( 1,12);
bwhl(22:5 l)=50e6*ones(l,30);
ddmax=.2;
ddl=ddmax/200;
dl=d-ddmax-ddl;
df=.25e9;
for h= 1:200, %Perturbation Loop
lenl=0;
save stat.out h -ascii
dl= dl+ ddl;
fres = (c/2) * ( (fresl + fres2 + ((pl/dl).A2)) .A .5 );
[fres,k]=sort(fres);
ff=.75e9;
for i= 1:51,
%Frequency Loop
ff=ff+df;
f(i)=ff;
[fpt0,10] = min( abs( fres - ( (f(i)-bwhl(i)) * ones(size(fres))) ) ) ;
[fpt 1,11] = min( abs( fres - ( (f(i)+bwhl(i)) * ones(size(fres))) ) ) ;
Ien2=len 1+11-10+1;
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fbw 1(len 1+1 :len2)=fres(10:l 1);
kbw 1(len 1+1 :len2)=k(10:l 1);
[fptc,lc(i)] = min( abs( fbwl - ( f(i) * ones(size(fbwl))) ) ) ;
lenl=len2;
m gl(i+l)=lenl;
end
datak = ['data' int2str(h)];
fill = [datak ’.mat'];
eval(['save' f i l l ' fbwl kbwl m gl lc']);
end
clk2=clock;
save et.mat clkl clk2
(2) SIM RES.M
(Used by BW RES1.M )
%*** Integers For Cavity Resonances Starting ***
%*** With m=l n=l p=0 **************************
lp=lpnum;
len=(lp*2)*(lp+1);
<^5|cs|c:jcsf:sfej|csfc:jcijej}cj(c}(cji<:}c:fcjjc5|ejj<5j<s{eijc!}ca|csie:ic:$cs|cjj<sjcjfc
ml=zeros(l,len);
mm=l;
il= l;
i2=(lp)*(lp+l);
del=(lp)*Gp+l);
fori 100= l:(lp),
m 1(i 1:i2)=mm*ones( 1,del);
mm=mm+l;
il=il+del;
i2=i2+del;
end
nl=zeros(l,2*len);
nn=l;
il= l;
i2=(lp+l);
del=0p+l);
fori 100= I:(lp).
nl (i 1:i2)=nn*ones( 1,del);
nn=nn+l;
il=il+del;
i2=i2+del;
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end
i2=i2-del;
while i2 < le n ,
il= i2+ l;
i2=2*i2;
n l(il:i2 )= n l(l:il-l);
end
nl=nl(l:len);
%*********%********************
pl=zeros(l,2*len);
pl(l:lp+l)=0:l:lp;
i2=lp+l;
while i2 < le n ,
il=i2+l;
i2=2*i2;
p l(il:i2 )= p l(l:il-l);
end
pl=pl(l:len);
fresl = (m l/a).A2;
fres2 = (nl/b).A2;
(3) SIM 57.M
(This is the actual simulation program)
%***** Mode Selection in Microwave Reverberation *************
%***** Chamber By Perturbing Height **************************
%***** AND Correlation Calculation ***************************
%***** From Loughry's Field Equations ***********************
%***** Resonant Frequency Arrays Generated By BWRES1.M *******
r=.175;
rot=(21/180)*pi;
x0=.5;
y0=.325;
x l=x0+(r*cos(rot)/2);
x2=x0-(r*cos(rot)/2);
y l=y0-(r*sin(rot)/2);
y2=y0+(r*sin(rot)/2);
j= (-l)A5;
c=2.99792458E8;
sig=2.32E7;
u=4*pi*lE-7;
e=8.8542E-12;
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a= 1.0342;
b=.8087;
d=.5812;
S=2*a*b + 2*b*d + 2*a*d;
V=a*b*d;
AA=lE-4;
R=50;
eta=377;
rand(’seed',suni( 100*clock));
^**********
clkl=clock;
f=le9:.25e9:13.5e9;
eqv50
BD3 = (f./Q3);
N3 = ((8*pi*V)/(cA3)) * ((f.A2).*BD3);
fll=f-(.5*BD3);
ful=f+(.5*BD3);
% **********
lpnum=95;
simresl
modes=zeros(l,51);
modes3=zeros( 1,51);
mag 1=zeros(51,200);
fll0=zeros(51,200);
fu 10=zeros(51,200);
bw 10=zeros(51,200);
fid 10='e l.out';
fopen(fidlO,'w');
% **********
ddmax=.2;
ddl=ddm ax/200;
dl=d-ddmax-ddl;
for h= 1:200,
%Perturbation Loop
save stat.out h -ascii
dl=dl+ddl;
datak = ['data' int2str(h)];
fhl = [datak'.mat'];
eval(['load' fill]);
fres = fbwl;
k = kbwl;
clear fbwl kbwl
for i= 1:51, %Frequency Loop
%Model Excitation Resonances
inc=0;
l=lc(i);
ii=k(l);
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evml
inc=inc+l;
modes(i)=modes(i)+1;
if (ffes(l)>fll(i)) & (fres(l)<ful(i)), modes3(i)=modes3(i)+l;, end
data(inc,l)=ml(ii);
data(inc,2)=nl(ii);
data(inc,3)=pl(ii);
data(inc,4)=mag;
magl(i,h)=mag;
10=1;
1= 1+ 1 ;
evml
while mag > (.l*magl(i,h))
ii=k(l);
inc=inc+l;
modes(i)=modes(i)+1;
if (fres(l)>fl 1(i)) & (fres(l)<ful(i)), modes3(i)=modes3(i)+l;, end
data(inc,l)=ml(ii);
data(inc,2)=nl(ii);
data(inc,3)=pl(ii);
data(inc,4)=mag;
1=1+ 1 ;
if l==mg 1(i+1)
fprintf(fid 10,'Insufficient Modes At f=% 8.3g h = %4.0f\n',f(i),h)
break
end
evml
end
fulO(i,h)=fres(l-l);
1= 10;
1= 1 - 1 ;
evml
wliile mag > (.l*magl(i,h))
ii=k(l);
inc=inc+l;
modes(i)=modes(i)+1;
if (fres(l)>fll(i)) & (fres(l)<ful(i)), modes3(i)=modes3(i)+l;, end
data(inc,l)=ml(ii);
data(inc,2)=nl(ii);
data(inc,3)=pl(ii);
data(inc,4)=mag;
1= 1- 1 ;
if l==mgl(i)
fprintf(fid 10,'Insufficient Modes At f=% 8.3g h = %4.0f\n',f(i),h)
break
end
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evml
end
fllO(i,h)=fres(l+l);
^Calculate Field and Power Density At Two Points
kx=(pi/a)*data(l :inc,l);
ky=(pi/b)*data(l :inc,2);
%kz=(pi/dl)*data(l :inc,3);
%kxy=((kx.A2)+(ky.A2)).A(.5);
%kr=((kx.A2)+(ky.A2)+(kz.A2)).A(.5);
ph 1=exp(j*2*pi*rand(inc, 1));
erl = data( 1:inc,4) .* sin(kx*xl) .* sin(ky*yl);
eil = data(l:inc,4) .* sin(kx*xl) .* sin(ky*yl) .* phi;
er2 = data(l:inc,4) .* sin(kx*x2) .* sin(ky*y2);
ei2 = data(l:inc,4) .* sin(kx*x2) .* sin(ky*y2) .* phi;
emagl=abs(sum(erl)+sum(eil));
emag2=abs(sum(er2)+sum(ei2));
s 1(i,h)=(emag 1A2)/(2*eta);
s2(i,h)=(emag2A2)/(2*eta);
d ea r erl ei 1 er2 ei2 data phi
end
end
%***************************
save pden.m atfsl s2
<^s|<5|es|csJcsjcsfc:Jcsfcaie:Jc5jcsjcsJesjcjjc:fc}jc5jes|«:fc:jes|{}fcs|cj}c:}cs|c
fidl='kl.out';
fid2='k2.out';
fopen(fidl,'w’);
fopen(fid2,'w');
modes3=modes3/200;
bwlO=fulO-fllO;
for i=l:51
cc=corrcoef(s l(i, 1:200),s2(i, 1;200));
ccc(i)=cc(l,2);
fprintf(fidl,'%8.3e %10.0f %14.3e %10.3f %10.3f
%10.3Nt',f(i),modes(i),Q3(i),modes3(i),N3(i),ccc(i))
magmin=min(mag 1(i,:));
magave=mean(mag 1(i,:));
maxbw=max(bw 10(i,:));
minbw=min(bw 10(i
avebw=mean(bwlO(i,:));
fprintf(fid2,'%8.3e %10.3f %10.3f %10.3e %10.3e
%10.3eVi',f(i),magmin,magave,minbw,avebw,maxbw)
end
fclose('alT);
clk2=dock;
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save etsim.mat clkl clk2
(4) EQV50.M
(Used by SIM 57.M )
%**calc
qfac=10;
sd = (pi*u*sig*f) .A (-.5);
Q1 = ( (3/2)*V*ones(size(sd))) . / (S*sd);
Q I 1 = 2*Q1; %** 2-sensor model
Q2 = (3/2) * ( (V*ones(size(f)))./ (R*(AAA2)*e*2*pi*f));
Q3 = (1/qfac) * ( (Q1.*Q2)./(Q11+Q2) );
(5) S1MRES1.M (Used by SIM 57.M )
%*** Integers For Cavity Resonances Starting ***
%*** With m=l n=l
lp=lpnum;
Ien=(lpA2)*(lp+l);
ml=zeros(l,len);
mm=l;
il= l;
i2=(lp)*(lp+l);
del=(Ip)*0p+l);
fo ri 100= 1:(lp),
m 1(i 1:i2)=mm*ones( 1,del);
mm=mm+l;
il=il+del;
i2=i2+del;
end
nl=zeros(l,2*len);
nn=l;
il= l;
i2=(lp+l);
del=(lp+l);
fo ri 100= l:(lp),
n 1(i 1:i2)=nn*ones( 1,del);
nn=nn+I;
il=il+del;
i2=i2+del;
end
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i2=i2-del;
while i2 < le n ,
il=i2+l;
i2=2*i2;
n l(il:i2 )= n l(l:il-l);
end
nl=nl(l:len);
pl=zeros(l,2*len);
pl(l:lp+l)=0:l:lp;
i2=lpf 1;
wliile i2 < le n ,
il=i2+l;
i2=2*i2;
p l(il:i2 )= p l(l:il-l);
end
pl=pl(l:len);
(6) EVM 1.M
(Used by SIM57.M)
for SIM57.M to calc, cavity response************
%**calc. magnitude of second order response********
fO=fres(l);
num=fOA2;
den l=-(f(i)A2)+(fOA2);
den2=f(i)*fO/Q3(i);
den=(denlA2 + den2A2 ) A .5;
mag=( l/Q3(i))*num/den;
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