IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998
19
Ray Analysis of Electromagnetic
Field Build-Up and Quality Factor of
Electrically Large Shielded Enclosures
Do-Hoon Kwon, Student Member, IEEE, Robert J. Burkholder, Senior Member, IEEE,
and Prabhakar H. Pathak, Fellow, IEEE
Abstract— A high-frequency asymptotic ray solution is investigated for predicting the electromagnetic field build up and
steady-state parameters of shielded enclosures or cavities, which
are large with respect to wavelength. It is found that the ray
solution can deterministically predict the early-time field build
up after the source is switched on, but cannot predict the steadystate fields of high- enclosures because of the intractably large
number of ray reflections required for convergence. However, it
factor may be predicted
is demonstrated that the steady-state
from the early-time energy density build up at a point by
factor is
coherently summing the power in each ray. The
obtained via its relationship to the cavity time constant, which
may be extracted from the early-time energy density curve. A
clear indication of polarization diversity throughout the enclosure
may also be obtained by plotting the polarization components
of the early-time fields and energy density build up at different
points. The advantage of the ray method is that it can be used to
treat large closed cavities of relatively arbitrary shape.
Q
Q
Q
Index Terms—Arbitrary geometry, cavity, high frequency, ray
analysis, shielded enclosure.
Fig. 1. Reverberating cavity of arbitrary shape with metallic wall having
conductivity . The cavity has a transmitter and a mechanical mode stirrer
inside.
I. INTRODUCTION
T
HE electromagnetic (EM) analysis of electrically large
closed cavities is important for understanding the field
build-up and steady-state behavior of high- shielded enclosures and reverberation chambers. The procedure presented
here has been specifically applied to a completely enclosed
high- chamber, although the same approach could be used
in the analysis of other electrically large closed or shielded
EM environments, such as below decks of a ship or inside
aircraft cabins.
The EM reverberation chamber has become a useful tool
for testing EM shielding effectiveness of electronic enclosures and for emissions measurements and other EM interference/compatibility applications [1], [2]. Fig. 1 shows an
electrically large closed cavity. The cavity has metallic walls
with very high conductivity and is excited by a transmitting
antenna or aperture. (For emissions testing the antenna or
aperture would receive radiation from an active test object
in the chamber.) Since very little power is dissipated when
Manuscript received March 17, 1997; revised November 7, 1997. This work
was supported by the Joint Services Electronics Program, Grant N00014-89J-1007.
The authors are with the ElectroScience Laboratory, Department of Electrical Engineering, The Ohio State University, Columbus, OH 43212 USA.
Publisher Item Identifier S 0018-9375(98)01715-3.
a wave reflects from the highly conducting walls, the field
in the cavity can build up to a very large value a short time
after a steady-state source is turned on. For the case of the
reverberation chamber, it is desired that the field inside be
uniformly distributed in amplitude, polarization, and -space
spectral content so that a device under test is exposed to
a highly diverse electromagnetic environment. As shown in
Fig. 1, a mechanical “mode stirrer” is sometimes placed so that
it can stir up the steady-state fields and obtain time-averaged
uniformity for the interior fields as it rotates [3]. Alternatively,
“frequency stirring” may be used for producing the same effect
[4]. The analysis described here is applied to a large static
chamber, although it may be easily extended to the analysis
of time-varying reverberation chambers.
At microwave frequencies room-size cavities are very large
with respect to wavelength, so numerical analysis of the
interior fields becomes intractable. For these cases, most theoretical approaches for predicting cavity characteristics have
been based on cavity modal analysis for uniform geometries
[5], [6] or statistical methods which assume at the outset that
the average power density is uniform throughout the cavity
[7]–[9]. An alternative high-frequency asymptotic ray-based
0018–9375/98$10.00  1998 IEEE
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998
approach is presented here for analyzing electrically large
cavities of relatively arbitrary configuration. To within the
accuracy of the high-frequency asymptotic approximation, this
method can deterministically predict the fields at any point in
the cavity during the early part of the transient field build-up
after the source is turned on. The steady-state fields cannot
be directly computed for realistic high- chambers using this
method because the rays would have to be tracked through
thousands of reflections from the chamber walls to reach
a convergent solution. However, a method is presented for
predicting the steady-state behavior from the early-time energy
density build-up, which allows the cavity time constant
to be extracted, and from that the cavity quality factor
may be computed. Furthermore, the early-time ray solution
allows one to plot the individual polarization components at
different points throughout the cavity in order to visualize
the polarization diversity. The ray approach presented here
is intended to be used as a diagnostic tool for evaluating the
performance of shielded enclosure and test chamber designs.
In the following, it is assumed that the primary loss mechanism in a well-designed high- enclosure such as a reverberation chamber is due to the finite conductivity of the
walls. Power loss due to signal feedback into the source
antenna may also be computed using the method presented
here, but it has been shown that this is much less significant than the wall loss at microwave frequencies [9]. In
Section II, the high-frequency ray method is described and it
is demonstrated with numerical results for a static rectangular
chamber in Section III. Section IV introduces the approach
for calculating from the early-time behavior and compares
the results with the formulas derived in [8]. Conclusions and
suggestions for further extensions of the method are discussed
in Section V. An
harmonic time convention is assumed
and suppressed for the frequency-domain fields throughout.
The medium filling the cavity is free-space with impedance
and wavenumber
where is the steady-state
wavelength.
II. HIGH-FREQUENCY DESCRIPTION
OF
CAVITY FIELDS
In the high-frequency asymptotic sense, the field at a
point inside a cavity can be described as a superposition of
ray fields originating from the transmitting antenna. Since
the fields associated with the multiple-reflection of rays are
dominant compared with the fields associated with the effects
of diffraction (assuming the mode stirrer is electrically large),
one can represent the cavity fields by the geometrical optics
(GO) approximation [10], i.e., by the sum of the fields of
the direct ray and multiply-reflected rays. Unlike the cavity
mode expansion, which has been conventionally used for PEC
cavities of canonical shapes [11], [12] the ray method can be
applied to relatively arbitrarily shaped cavities and it can also
very easily take into account the effect of slight wall loss. The
latter is achieved by incorporating the finite conductivity
into the reflection coefficients associated with high-frequency
ray reflections [13]. Thus, inside a general cavity the ray
description of the electric field at an observation point can
m m+1 and direct and mul-
Fig. 2. Two consecutive reflection points Q
tiple-bounce rays contributing at an observer.
;Q
be written as
(1)
is the th ray originating from the transmitting
where
antenna and arriving at
after experiencing a particular
number of reflections along its unique ray path to . Several
rays contributing to the total field at an observer are depicted
in Fig. 2. Note that (1) is in general an infinite sum for a closed
cavity and it must be truncated for computational purposes as
discussed later.
Each individual ray field
can be found by ray tracing
according to the laws of GO. The complete path a particular
ray goes through needs to be found and if a ray experiences
multiple reflections from the walls, the reflection points and
the sequential order of reflections should also be determined.
One can then march along the ray path and compute the field
along the ray throughout a sequence of reflections.
After
reflections, the th ray field at the point is given
in general by
(2)
denotes the total ray path length from the source to
where
and
is the field incident at the first reflection
point
, but with its phase propagation to
extracted
(and absorbed into
). At each reflection point
,
the incident ray field is decomposed into two components
transverse to the ray propagation direction which are parallel
and perpendicular to the plane of incidence at
. The
incident field may then be written as
(3)
where the parallel and perpendicular unit vectors are denoted
and
.
is in the plane of incidence and
by
is perpendicular to it while both of these unit vectors are
KWON et al.: EM FIELD BUILD-UP AND QUALITY OF ELECTRICALLY LARGE SHIELDED ENCLOSURES
21
denotes the geometrical optics spatial divergence
factor which accounts for the amplitude variation of the ray
field. For an electrically small transmitting antenna and planar
walls, the divergence factor is given by
(9)
Fig. 3. Reflection coefficients
0? and 0k
is the distance from the transmitter to
and
where
denotes the total path length of the th ray from
all the way back to the transmitting antenna. For curved
walls, the divergence factor may be computed using the
method presented in [10] and [14]. Finally, it is noted that
(2) is a product of matrix multiplications so the order of
the multiplications is important and should be interpreted as
.
The magnitude of the plane wave reflection coefficients
and
are plotted in Fig. 3 as a function of incidence
for three highly conducting nonmagnetic metallic
angle
surfaces. Note that
and
are very close to unity
for high
as expected, indicating that very little energy is
absorbed at each reflection. (It is noted that both reflection
coefficients
and
go to one in magnitude for grazing
, although the rise of
occurs
incidence, i.e., as
to be visible on the plot scale. This
too close to
behavior of reflection coefficients for plane wave incidence on
planar interfaces is well documented as, for example, in [15].)
Fig. 3 also shows the aforementioned transverse unit vectors
associated with the incident
,
and the reflected
,
fields.
versus incidence angle i .
transverse to the propagation direction of the incident field.
Snell’s law of reflection applies, which states that the reflected
ray lies in the plane of incidence and the angle of reflection
is equal to the angle of incidence [10].
is the reflection
matrix at
, which relates the reflected field components to
the incident components and is defined by
(4)
where
given by
and
are plane wave reflection coefficients
(5)
(6)
(7)
for a highly conductive nonmagnetic metallic surface. The
incidence angle
is defined in Fig. 3.
accounts for the coordinate transformation from
and
to
and
and is defined by
(8)
III. RAY FIELDS IN A RECTANGULAR
CAVITY WITH MONOPOLE EXCITATION
Consider a rectangular cavity with a small monopole antenna radiating on the bottom surface, as shown in Fig. 4.
For this geometry, finding ray paths and reflection points is
greatly facilitated by employing image theory [11]. One can
remove the walls and place image sources of the monopole
at appropriate locations in space. Then, the field value at
any observation point
is determined by summing up the
contributions from the original source and all the image
sources. Some of the image sources and rays are illustrated
in Fig. 5. For more complicated chamber geometries, which
may also have a mode stirrer present, a ray shooting approach
[13], [16] is recommended to find the ray paths and reflection
points. A static rectangular chamber with a simple monopole
excitation is chosen for the results presented here to illustrate
the field build-up as more and more image rays add to the
field at the observation point.
The current on the original monopole antenna is determined
from the method of moments [17] using the electric field
integral equation for a monopole on a lossy ground plane. The
excitation of the antenna is a magnetic frill generator, modeling
a coaxial probe opening into the cavity. Once the current
on the monopole is found in the presence of a lossy ground
plane of infinite extent, image theory gives the corresponding
currents on all the image sources. It is noted that (2) assumes
the observation point in the cavity is in the far zone of the
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IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998
with the size of the cavity. Finally, it is noted that the moment
method analysis of the monopole can also be used to compute
received fields which may be generated by an active or passive
test object and also be caused by feedback from the cavity.
Now assume that the input antenna is excited by a step
function modulated signal operating at the frequency
(radian frequency
). The time domain electric field at
point
may then be written as
(10)
is the frequency response of the stepped
where
input source given by
(11)
From (2),
may be written as
(12)
where
(13)
Fig. 4. A rectangular cavity with a monopole antenna. The conductivity of
the wall is .
is band limited around
and we can
Since
safely assume that
is a slowly varying function of
(where
) around
, (10) is well approximated by
(14)
where
(15)
denotes the total ray path length from to the th image
source and
is the unit step function that accounts
for the delayed time of arrival of each ray. is the speed of
light in free-space with permittivity
and permeability .
Also,
with
being the applied frequency. The
envelope
of
is given in the high-frequency
limit by
(16)
Fig. 5. Image sources and rays contributing at
P.
source antenna. While small monopoles easily satisfy this
requirement, it is possible that some antennas may be too
large to employ ray tracing in the simplified format of (2).
In this case, a generalized ray expansion may used instead
[14]; however, it is expected that most antennas employed in
practice as sources of EM waves would be small compared
Fig. 6 shows the envelope of the early time fields at a point
obtained via (16) for three values of conductivity
,
, and
S/m (Siemens per meter). The chamber is a 1.75m cube and the frequency of the step-modulated signal is
10 GHz. Along with the time axis, the number of included
images and the maximum number of ray reflections are also
shown in Fig. 6. Since the image sources are evenly placed in
space, the number of images contributing to (16) at time is
KWON et al.: EM FIELD BUILD-UP AND QUALITY OF ELECTRICALLY LARGE SHIELDED ENCLOSURES
23
Fig. 6. The envelope of early time fields versus time: The chamber is a cube
with a = b = c = 1:75 m and the modulation frequency is 10 GHz. The
observation point P is at (x; y; z ) = (0:6; 0:7; 0:8).
Fig. 7. Envelope function of e(P; t) in each coordinate direction: the observation point P is at (x; y; z ) = (0:6; 0:7; 0:8). The chamber is a cube
with a = b = c = 1:75 m, = 106 S/m, and the modulation frequency
is 10 GHz.
TABLE I
COMPARISON OF LOSS PER REFLECTION, NUMBER OF IMAGES AND NUMBER OF
REFLECTIONS REQUIRED FOR STEADY-STATE CONVERGENCE AND SETTLE TIMES
FOR FIVE VALUES OF : THE SETTLE TIMES ARE BASED ON A CUBIC CHAMBER OF
SIDE 1.75 M AND THE NUMBER OF IMAGES AND REFLECTIONS ARE BASED ON (17)
magnitude. The minimum number of reflections required for
convergence may then be calculated approximately from
roughly
where
is the volume of the chamber.
The number of reflections denotes the maximum number of
by time . The
reflections any ray goes through to reach
jagged appearance of the field plots is due to the ideal step
input being used here; in reality, the limited bandwidth of the
source antenna and transmission line feed would cause the
source to have a less abrupt turn on and provide small scale
smoothing of the field plots without affecting the general shape
of the curves.
S/m,
As Fig. 6 illustrates, in the lossiest case with
the field reaches steady-state relatively early. This is because
have undergone a
the rays that take a long time to reach
large number of attenuating reflections and become too weak
to contribute significantly to the total field. As expected, it
S/m to settle
takes a longer time for the field with
down because there is less attenuation per reflection. For the
S/m, which corresponds to a
more realistic case of
good metallic conductor, convergence could not be reached
within a reasonable amount of computer time.
It was observed computationally that the field reaches its
steady-state value after a time when all the subsequent rays are
attenuated by reflections to less than 6/10 of their free-space
(17)
and are given in Table I for each
along with the corresponding number of images and the time to convergence (settle
time). The table indicates that a realistic chamber (
S/m) requires an intractably large number of images and
reflections to be computed for convergence.
Fig. 7 shows the envelope functions in each coordinate
direction as a function of time for the
S/m case
of Fig. 6. Plotting these field components at several different
locations allows one to visualize the polarization diversity
throughout the chamber.
IV. SEMIDETERMINISTIC PREDICTION
OF
QUALITY FACTOR
The most commonly used parameter for characterizing
closed cavities such as reverberation chambers is the quality
factor . As with other types of resonant systems, the of a
closed cavity will tell how effective the cavity is in building
up and storing energy. With a very large , the field in the
chamber can build up to a very large value even with small
input power. This is because upon turning on the excitation,
the energy in the cavity will build up to a steady-state level
where the total input power is exactly compensated by the
dissipated power from all the loss mechanisms combined.
The classical definition of
for a single frequency of
operation at steady state is given by
(18)
is the total stored energy and
is dissipated
where
power. A simple formula for the
of a closed cavity was
obtained in [8] by treating the cavity field as a uniformly
24
IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 40, NO. 1, FEBRUARY 1998
Fig. 8. Coherent energy density build-up for five values of : a =
sum of ray power and exponential curve fit overlay each other.
b
=
distributed spectrum of plane waves and calculating average
power dissipation statistically, arriving at
c
= 1:75 m. Observation pointis at (x; y; z ) = (0:6; 0:7; 0:8). Coherent
if one sums the power in the rays using magnitude only, the
energy density build-up at a point may be written as
(19)
(22)
where
is the relative permeability of the cavity
wall,
is the volume, and is the inner surface area of the
cavity. A time constant was suggested in [9] defined by
which is expected to have a behavior similar to (21). This
function is plotted in Fig. 8 for various cases of
values.
Also plotted for each is a best curve-fit of the function
(20)
(23)
Skin depth
which determines the exponential decay rate of the stored
energy when the source is suddenly turned off. Since the same
applies to the case of energy build-up when the source is
suddenly turned on, it can be written that
(21)
is the steady-state energy.
where
It is of interest to extract numerically using the ray method
presented here. However, to extract directly using (21), one
would have to compute the total stored energy as a function
of time, which would require integrating the field over the
entire volume of the chamber at each time step. Supposing
that the numerical sampling is on the order of a fraction
of a wavelength, a numerical integration scheme becomes
impractical for electrically large cavities. Therefore, the energy
build-up at a single point or a small set of sample points is
investigated instead. Sampling a small set of points randomly
distributed throughout the chamber should indicate whether
the energy density build-up at a point has a form similar to
the total energy build-up in the cavity.
As Figs. 6 and 7 show, there is no simple exponential buildup of the fields at a point because phase interference between
the various rays creates oscillations in the curves. However,
where the time delay
has been introduced because unlike
,
stays at zero until the first ray arrives at
. However, these two curves overlay each other to within
graphical resolution [except for the small fluctuations near zero
caused by the ideal step function in (22)], which supports the
hypothesis that the point-wise energy density build-up defined
by (22) has the same functional behavior as the total energy
build-up in the rectangular chamber.
It remains to be seen if the time constant which gives the
best numerical curve-fit in (23) agrees with the time constant
in (21). Using (20) to define the -factor in terms of the curvefit time constant in (23), Table II lists for the five cases of
values and compares it with the statistical formula of (19).
The agreement is very good between these two very different
approaches. The physical explanation of this result is that the
large set of rays involved in the evaluation of (16) and (22)
experience reflections nearly everywhere on the chamber walls
so that the field at any point is directly influenced by the wall
loss and chamber geometry.
For the static rectangular chamber considered here, it has
been found that the curves of Fig. 8 are nearly independent of
the chosen point of observation in the chamber. It is known
that for a low-loss cavity, the steady-state field distribution
due to a single frequency excitation is highly oscillatory as
KWON et al.: EM FIELD BUILD-UP AND QUALITY OF ELECTRICALLY LARGE SHIELDED ENCLOSURES
TABLE II
QUALITY FACTOR Q: A COMPARISON BETWEEN THE STATISTICAL
METHOD AND THE CURVE FIT. THE FREQUENCY IS 10 GHz
25
by ray methods, but it is not practical to compute the steadystate fields of high- cavities because rays typically need to
be tracked through thousands of reflections to reach a convergent solution. However, the ray solution appears to contain
sufficient information to predict the steady-state parameters
and pointwise from the early-time coherent energy density
curve well before reaching convergence. Furthermore, plotting
the individual polarization components of the early-time fields
and energy density build-up serve as excellent indicators of the
polarization diversity at different points throughout the cavity.
One of the main advantages of the ray method is that it can
be applied to relatively arbitrarily shaped cavity geometries.
It is, therefore, of further interest to apply the ray method to
more complex time-varying chamber geometries, which may
contain a movable mode stirrer and/or a test object in the
target zone, which will, of course, perturb the chamber fields.
This preliminary study of the static rectangular chamber has
shown that the ray method can provide considerable insight
into the operating characteristics of realistic shielded enclosure
and test chamber designs without relying on measurements or
statistical analysis.
ACKNOWLEDGMENT
The authors would like to thank Dr. D. A. Hill of the
National Institute of Standards and Technology, Boulder, CO,
for his technical assistance.
REFERENCES
Fig. 9. Coherent energy density build-up for each polarization component of
the = 106 (S/m) case: Observation point is at (x; y; z ) = (0:6; 0:7; 0:8).
Coherent sum of ray power and exponential curve fit overlay each other.
a function of location. This oscillatory behavior is caused by
constructive and destructive phase interference among rays.
However, once the phase information is taken out as in (22)
so that the power of individual rays are summed to form a
monotonically increasing function with time, (22) is expected
to be a slowly varying function of position. More complex
cavity geometries are expected to show more spatial variation
in energy density than the simple rectangular cavity.
Fig. 9 shows the energy density curves for each polarization
component of the
S/m case. They also have the
form of (23) and are expected to have similar time constants.
(However, it has been found that the time constant for the
cavity is most accurately computed using total field values.)
These curves will change for different observation points and,
therefore, may be used to characterize the polarization diversity throughout the cavity more clearly than the fluctuating
field plots of Fig. 7. For example, it is clear from Fig. 9 that
the component dominates at the given observation point.
V. CONCLUSION
It has been shown that the deterministic early-time field
build-up at points within a closed cavity may be computed
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Do-Hoon Kwon (S’94) was born in Seoul, Korea. He received the B.S. degree from the Korea Advanced Institute of Science and Technology
(KAIST), Korea, in 1994, and the M.S. degree from
The Ohio State University, Columbus, in 1995. He
is currently working toward the Ph.D. degree at The
Ohio State University.
Since April 1994, he has been a Graduate Research Associate with the ElectroScience Laboratory, Department of Electrical Engineering, The
Ohio State University. His main research interests
are high-frequency electromagnetic scattering and computational electromagnetics.
Robert J. Burkholder (S’85–M’89–SM’97)
received the B.S., M.S., and Ph.D. degrees
in electrical engineering from The Ohio State
University, Columbus, in 1984, 1985, and 1989,
respectively.
From 1989 to 1994, he was a Postdoctoral
Research Associate at The Ohio State University
ElectroScience Laboratory, where he is currently a
Senior Research Associate. His research specialties
are high-frequency asymptotic techniques and their
hybrid combination with numerical techniques for
solving electromagnetic radiation and scattering problems. He has contributed
extensively to the electromagnetic analysis of large cavities (such as jet
inlets/exhausts) and is currently working on the more general problem
of antenna radiation, propagation, and coupling in complex multibounce
environments.
Dr. Burkholder is currently serving as Associate Editor for the IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION and is Chairman of the
Columbus Joint Chapter of the IEEE Antennas and Propagation and
Microwave Theory and Techniques Societies. He is also a full member
of URSI Commission B, and a member of the Applied Computational
Electromagnetics Society (ACES).
Prabhakar H. Pathak (M’76–SM’81–F’86)
received the B.Sc. degree in physics from the
University of Bombay, India, in 1962, the B.S.
degree in electrical engineering from the Louisiana
State University, Baton Rouge, in 1965, and the
M.Sc. and Ph.D. degrees in electrical engineering
from The Ohio State University, Columbus, in 1970
and 1973, respectively.
He has been with The Ohio State University
since 1973 and is currently a Professor there. He
has participated in invited lectures and several short
courses on the uniform geometrical theory of diffraction and other highfrequency methods, both in the United States and abroad. He has authored
and co-authored chapters on the subject of high-frequency diffraction for five
books. Currently, he is serving as a member of the editorial board of the
International Series of Monographs on Advanced Electromagnetics (Tokyo,
Japan: Sci. House). He has dealt primarily with the development of uniform
asymptotic solutions that improve and extend the geometrical theory of
diffraction solutions for solving antenna and scattering problems associated
with complex structures, such as aircraft and spacecraft. In addition, he has
been involved with the development of efficient hybrid methods of analysis
for reflector and microstrip-type antennas and, more recently, for dealing
with electromagnetic wave propagation in the presence of complex radiating
structures such as those involved in shipboard and urban environments.
His work also includes the areas of geometrical theory of diffraction and
asymptotic methods and the analytical inversion of the solutions obtained
therefrom into the time domain to arrive at a progressing wave picture for
transient radiation and scattering and he is involved with the analysis of
electromagnetic penetration into and scattering by deep as well as shallow
open-ended cavities and the development of Gaussian beam techniques for
antennas and other applications. His research interests include electromagnetic
theory, mathematical methods, antennas, and scattering.
Dr. Pathak is a member of Sigma Xi and a member of the U.S. Commission
B of URSI. He was named an IEEE AP-S Distinguished Lecturer for a threeyear term beginning in 1991. He is a former Associate Editor of the IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION.