Finite Element Analysis of Reverberation Chambers:
A Two-Dimensional Study at Cutoff
Charles F. Bunting,
Ph.D.
Old Dominion University
Dept. of Eng. Technology
Norfolk, VA 23529
e-mail: cfblOOf@et.odu.edu
Karl J. Moeller
C. J. Reddy, Ph.D.
Stephen A. Scearce
Electromagnetics Res. Br.
Mail Stop 490
NASA Langley Research
Center
Hampton, VA 23681
Hampton University
Dept. of Electrical
Engineering
Hampton, VA 23668
Old Dominion University
Dept. of Electrical
Engineering
Norfolk, VA 23529
Abstract: A two-dimensional analysis of reverberation
chambers is performed at cutoff. The structure considered is
lossless and corresponds to an infinite Q chamber. The
concept of frequency stirring is used to generate field data for
a discrete set of modes and the resulting statistics are
analyzed. The total field and power are examined for TE
modes. This analysis yields statistics similar to the expected
reverberation chamber statistics for the fields. Preliminary
data for mechanical stirring is also presented. The twodimensional model may yield considerable insight into the
nature of the loss experienced on real airframes, and also
provide an error measure on data in the non-ideal mode-stirred
chamber.
INTRODUCTION
+
Wu concludes that the amount of frequency shift for a given
stirrer essentially determines the random nature of the fields.
Wu also makes an important observation regarding the
Figure 1. Basic 2D Reverberation Chamber.
consequences of the shifting eigenfrequencies in noting that
the effect of a large stirrer is essentially a form of random
A reverberation chamber is an enclosure consisting of
metal walls with a metallic paddle wheel (denoted a “stirrer” modulation. This modulation is considered to have both
or “tuner”) essentially forming a high quality factor (Q) cavity amplitude and frequency modulation. Wu concludes that an
with continuously variable boundary conditions. The fields ineffective (electrically small) stirrer is unable to provide
inside the chamber for a given tuner position are completely adequate frequency modulation effects.
Hill [3] examined the concept of electronic mode stirring
deterministic. However, given the nature of the variable
initially
suggested by Loughry [4]. Hill’s analysis provides
boundary condition, the ability of a given source to couple
considerable insight into the connection between a twoenergy into certain modes, and the passband characteristic due
the chamber Q, the fields are typically characterized by dimensional versus and three-dimensional analysis. Electronic
statistical means.
Specifically the probability density mode stirring essentially implies that instead of simply stirring
functions for the in-phase and quadrature component of the via mechanical means to obtain a statistical field distnbution,
electric and magnetic fields are normally distributed [ 11. The the source can be swept over a narrow bandwidth to effect the
field magnitudes are Rayleigh distributed and the power is shift in eigenfiequencies as suggested by Wu [2]. Loughry
exponentially distributed [l]. The degree to which the actual uses bandwidth-limited white Gaussian noise as the
fields conform to the specific statistical description is modulation source. Hill indicates that there is freedom in the
dependent upon the number of modes that can simultaneously signal used, but constrains the signal to be spectrally flat over
exist at a given frequency. The number of modes that can the bandwidth. Hill’s analysis employs a two-dimensional
exist at a given frequency is a function of the cavity Q and the rectangular cavity with an electric line source.
By utilizing a numerical analysis scheme such as finite
operating frequency.
elements
the effects of introducing loss may be accurately
The transmission line matrix (TLM) method was
simulated,
however, for this paper a two-dimensional lossless
employed by Wu [ 2 ] to examine the characteristics of the
siniulation
will be performed. A two-dimensional model may
fields in the presence of a large stirrer in the reverberation
yield
insight
into the nature of the loss experienced on real
chamber. Wu used a 2-D cavity with a 1-D stirrer model to
airframes
and
also provide a n error measure on data in the
esainine the shifting of the eigenfrequencies and the effects on
ideal
mode-stirred
chamber.
the mode amplitudes for a variety of different stirrer sizes.
This paper will consider the basic 2D geometry as
depicted in Figure 1. An anti-symmetric tuner may be rotated
to provide mechanical mode stirring, however the focus of this
early work is to examine the use of a frequency stirring
approach. A FORTRAN code applying finite element
techniques has been written that provides the electric field at
any point within the chamber.
FINITE ELEMENTS FOR THE REVERBERATION
CHAMBER [5]
Consider Maxwell's curl equations in a source free,
homogeneous, isotropic, time harmonic form given by
and
(7)
The transverse components will be discretized using edge
elements and the longitudinal component used node-based
Lagrange fimctions. After integrating and summing over all
elements in the domain the following eigenvalue problem of
the form Ax = ABx for the cutoff frequency is obtained
v x E = -j w f i ,
V x H = jwEE,
Although analytical results are available for a large class of
problems for static, quasi-static, and dynamic conditions,
numerical methods typically must be applied whenever the
geometry does not coincide with a separable coordinate
system.
A full-field approach is required when the medium is
characterized by higher order modes, hybrid modes such as
for a general inhomogeneous media, or a waveguide or cavity
with imperfectly conducting walls. Other problems can be
expressed in a scalar form when there is a distinct separation
between TE and TM modes. In general this approximation
cannot be applied and a fill vector formulation is required.
The solution for the fields inside the reverberation chamber
will be required to support hybrid modes in the presence of
any metallic discontinuity or an inhomogeneity due lossy
objects. For the finite element approach the electric field in
terms of all three vector components is the desired field
quantity. A two dimensional representation for the fields may
be expanded in terms of the transverse field and the
longitudinal field where
RESULTS
A two-dimensional model of NASA Langley Research
Center's B chamber with dimension 3.96m x 7.10m was
generated and is depicted in Figure 2. The resulting system of
equations was on the order of 1O,OOOx1O,OOO and was solved
using a CONVEX 220 using a sparse eigenvalue solver in
FORTRAN. The data was then processed in MATLAB. The
solution of eq. (8) has on the order of 10,000 eigenvalues and
eigenvectors as possible values of cutoff frequency and
corresponding field configurations. Although there are a huge
number of solutions available, only those solutions
corresponding to a discretization of 10 samples per
wavelength were retained. This level of discretization allowed
accurate solutions up to 300 MHz, and resulted in a possible
171 valid cutoff frequencies. A typical result for the field
behavior is shown in Fig. 3 for the x-component of the electric
field, Ex. Note that the hghest field levels are in the
immediate vicinity of the stirrer as would be expected since
s and setting
Expressing the del operator as V = VI + 2
y = 0 for cutoff, it is possible to write two separate equations one for the transverse part and another for the z component in
terms of the cutoff frequency,
Figure 2. Discretization for the two-dimensional
geometry.
leading to the following weighted residual form:
the fields become infinite near the tuner. Additionally,
consider a contour plot of the s-component of the electric field
as given in Fig. 4. In this view it is possible to see the large the modal fields represented by the eigenvectors exist at
amplitude variations of the field near the tuner.
discrete frequencies and the chamber Q does not directly enter
Typical experimental work using a reverberation chamber into the formulation. The implication is that the computed
involves the rotation of the tuner and a measurement of the modal fields are for an infinite Q chamber, and that multireceived power to determine such characteristics as the stirring moding does not occur.
ratio, or a measurement of the VSWR based on reflected
power at the transmit port. The rotation of the tuner provides
a variable boundary condition that results in a deterministic Frequency Stirring
variation of the field by applying finite elements. One issue of
Motivated by the notion that the action of stirring excites
concern for the eigenvalue problem of eq. (8) involving the
cutoff frequency is that although all modes are found the modes weighted by a factor governed by the quality factor
corresponding to the particular geometrical configuration, the of the chamber, and noting that frequency stirring may be
weighting of the modes is not possible. In an actual used in place of mechanical stirring [3] an interesting
reverberation chamber the source does not couple energy experiment was performed. For a fixed tuner position
equally between the possible modes. Another concern is that computing all the eigensolutions, within the accuracy
constraint based on proper discretization, corresponds to
Interpolated Ex, Frequency (MHz) 4 0 8 1143, Date=K-Feb1998 chmbq
computing all the possible modes that could be excited in the
04
chamber. The solutions for cutoff result in a field that is either
03
TE or TM as the frequency is varied. Consequently, when the
solution is TE only Ex and Ey exist and when the solution is
02
TM only E, exists. Figure 5 depicts the amplitude of Ex at
01
one point in the chamber as the frequency is varied. The field
G o
variations
increase with frequency. This is related to the
01
increased
mode
density at higher frequencies. A histogram
-0 2
was constructed using MATLAB in Fig. 6 based on the data
-0 3
of Fig. 5. Any single component should have a Gaussian
04
distribution [ 11. The results are approximately Gaussian
A
considering the limited number of data points used (101 total
-2
.J
points
for TE).
y In m. Mode Num =21
x in m. Actual Grid Size -100
Figure 3. The x-component of the electric field for a
frequency of 108.1143 MHz.
Ex
m).Element Number =5021.
Dale=C+Mar-1998
0.5
InlerpOlatedEx. Frequency (MHz) =108.1143. Dale=25Fet-1998 chmbr2
0.4.
03
15
02
1
01
0.3.
0.2
c
N
'
fi0.10:
0
05
-,
U-
-01
-0.1
; o
1
-
.
-0.2
-02
E -05
-0.3.
-03
>
.
Jy
-0.4.
1
-04
-0.5
.o
15
3
2
1
0
1
2
0
5
3
x in m. Actual Gnd Size =lo0
Figure 4. Contour plot of the x-component of the electric
field for a frequency of 108.1143 MHz.
210
50
100
150
200
250
3CU
350
Figure 5. The x-component of the electric field for a
fixed point inside the chamber as a function of
frequency.
Hslogramb r E%
Element Number =502!.
Dale=Z5Febl998
Histogramfor Elo&.
Element Number =5GZ1. Dale=C&Mar-1998
14
5
1
12
10
-E
$ 8
.R
c
6
r
4
2
0
005
01
015
02
025
03
035
04
EloGE amplilude.20 bim. xbar =-O 83694. ybar =-I 617
0
Figure 6. The histogram of the data of Figure 5.
Figure 8. The histogram for Figure 7.
'(E). Element Number =5021.
E,,
EO
l l (E). Element Number =5021. Dale=C&Mar-IPB
0,45 I
-!
E 025
2 02.
0.05
-
0.1
u-008
0.06
.
0.04
0.02
0
01
0
1
0.12
~
0.1
I
0.14L
-
0.15
I
0.16
0 35 -
I
I
O.I8i
I
0.41
0.3
Dale=06Mar-1998
0.2
50
100
150
200
250
f i n MW. xbar=-O.@3694.
ybar=-l 617
300
0
1
50
350
100
150
200
250
f i n MM.xbar =4.83694. ybar=-1.617
350
3M
Figure 9. The power for a single point in the
reverberation chamber.
Figure 7. The total field for a fixed point in the
reverberation chamber.
It is also of interest to examine the total field.
Considering a field that is TE we can write Etot as,
Hislopramb r Elar,2.
25
,
,
,
Element Number =5021. D a l e W a r - 1 9 8 8
,
,
,
,
,
,
Examining the total field for the same point described earlier
in Figs. 4-5 results in the results of as shown in Fig. 7 and Fig.
8. The histogram of the data in Fig. 7 closely resembles the
expected Rayleigh distribution [ 11.
It is also possible to examine a term that is proportional to
the power where
2
EIotE2 amplilude.20 bins, xbar=-0.83694.
P CC Elf,l= Ef + E t
The data proportional to the power is depicted in Fig. 9 with
the corresponding histogram in Fig. 10. The data resembles
the expected exponential distribution [I].
21 1
ybar=-1.617
Figure 10. The histogram for Figure 9.
Mechanical Stirring
The action of mechanical stirring is an important aspect
of the reverberation chamber analysis. Applying finite
elements to a rotating stirrer requires a significant amount of
data processing since the field will need to be found for each
tuner position. For the purposes of this investigation the tuner
was rotated in 1.6” increments for a total of 225 steps. The
increment was chosen to match the step size used by NIST
when the NASA Langley chambers were characterized in
early 1997. The finite element discretization was reduced to a
system s u e of 3,000 unknowns resulting in an upper
frequency limit of 150 MHz. The eigenvalue problem of eq.
(8) was solved for each position of the tuner which resulted in
a total of 43 eigenvalues between the frequencies of 20 MHz
and 150 MHz.
Figure 11 depicts the total field at approximately 60 MHz
for at a point located far from the tuner and greater than half a
wavelength from any walls. The total field is periodic and
relatively smooth. Another interesting study not shown in this
paper is the change in frequency as a function of tuner
position. Additionally, movies of the field have been made in
which each frame corresponds to the solution for a particular
tuner position.
available random access memory (RAM) required for in-core
processing. Virtual storage, or out-of-core processing, of the
data can also be used, but greatly slows the solution time.
Taking advantage of sparsity in the resulting system of
equations helps significantly. In spite of the advantages
gained by using the sparse eigenvalue solver optimized for the
CONVEX the system of equations was limited to the value
used in this paper at around 1O,OOOx1O,OOO. The CONVEX
220 at NASA-Langley has 1 GB of random access memory.
By taking advantage of sparsity the dimensions of the matrices
A and B in eq. (8) are reduced to approximately 10,000~7.
However, the computation of the eigenvectors results in a full
matrix.
Work is continuing in the development of a high speed
sparse complex eigensystem solver that will allow the solution
of much larger systems (with on the order of 60,000
unknowns) to be analyzed. Other current work includes the
coupling of a source to the fields in the 2D geometry by
solving the eigenvalue problem for the propagation constant
and including a source. The resulting solution would contain
the proper mode weighting for the particular configuration
depending upon the stirrer position. Future work will examine
the three dimensional problem at low frequencies that will
include the source.
E tatal. mode number =7. I =-1.0975. y =a66
ACKNOWLEDGEMENTS
This work was sponsored by the American Society of
Engineering Education under an ASEE Faculty Fellowship at
NASA Langley during the Summer of 1997.
REFERENCES
[ l ] Lehman, T.H., “A statistical theory of electromagnetic
fields in complex cavities,” EMP Interaction Notes, No.
494, 1993.
Figure 11. The total field for a 225 tuner steps.
[2] D.L. Wu and D.C. Chang, “The effect of an electrically
large stirrer in a mode-stirred chamber,” IEEE Trans. on
Electromag. Compat., vol. 31, no. 2, pp. 164-170, 1989.
CONCLUSIONS
This paper has presented a two-dimensional analysis of
reverberation chambers using finite element techniques. The
results of the finite element analysis of the reverberation
chamber illustrate the potential utility of a 2D representation
for enhancing the basic statistical characteristics of the
chamber. One important advantage of employing a numerical
scheme such as finite elements to analyze reverberation
chambers is that the field values at virtually any point in the
structure can be easily obtained without perturbing the field as
can occur when using an antenna to monitor the fields.
The challenge of agplying finite element techniques to
reverberation chambers is that enormous computational
resources are required. The primary limitation is in the
[3] D.A. Hill, “Electronic mode stirring for reverberation
chambers,” IEEE Trans. on Electromag. Compat.? vol. 36,
pp. 294-299, 1994.
[4] T.A. Loughry, “Frequency-stirring: an alternative to
mechanical mode-stirring for the conduct of
electromagnetic testing,” Philipps Laboratory, Kirtland
AFB, NM PL-TR-91-1036, NOV.1991.
[5] Reddy, C.J., M.D. Deshpande, C.R. Cockrell. and F.B.
Beck, “Finite element method for eigenvalue problems in
electromagnetics,” NASA Technical Paper 3485. NASA
Langely Research Center, Dec. 1994.