Canavero, Flavio

Serra, R.

English

PORTO Publications Open Repository TOrino

June 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P83
Introduction of Randomness in Deterministic
Descriptions of Reverberation Chambers
Ramiro Serra∗ Flavio Canavero∗
Abstract — The main purpose of this work is to
present some simple yet complete models that are
able to describe the statistical behavior of Rever-
beration Chambers (RC) as well as the inﬂuence of
their main factors and their overall functioning. We
start by building a simpliﬁed two-dimensional model
which is able to reproduce the statistical behavior of
real RCs without renouncing the advantages of de-
terministic descriptions. The statistical properties
of the ﬁelds inside the cavity are introduced either
by varying the width or the relative dielectric con-
stant of a perturbing lossless layer inside a 2D cav-
ity. The ﬁeld statistics derived with this process
are shown to correspond to the statistics found in
properly functioning RCs in the overmoded regime.
With the use of a suitable coordinate transforma-
tion, we assume our non-homogeneous rectangular
cavity to be a virtual mathematical domain, and ﬁnd
an analogous homogeneous non-rectangular cavity
as our physical domain. With this simple process
we are able to link our 2D ”dielectric” stirrer with
actual geometrical discontinuities. By means of this
procedure, a novel design for RCs appears as feasi-
ble.
1 INTRODUCTION
Reverberation Chambers (RC) are gaining signiﬁ-
cant conﬁdence in use for radiated emissions and
immunity measurements. RC users need to fully
understand its working principles in order to cor-
rectly interpret the measurement results and to op-
timize the performance for various measurement
tasks.
Reverberation chamber knowledge to date results
from a partial juxtaposition of four diﬀerent ap-
proaches: the deterministic models, the statisti-
cal models, the empirical techniques, and the com-
puter/numerical methods. It is not possible to leave
one of these approaches behind, as each one of them
behaves as a non-exhaustive, non-excluding part of
a RC description. Furthermore, they mutually col-
laborate to give fairly successful answers in ﬁelds
where the other one fails, and viceversa. Therefore,
there is an obvious gap which makes us change our
methodology depending on what kind of result we
seek. In this sense it is to point out that a statis-
tical description is meaningful only if the chamber
is working in an overmoded regime and only on
∗The authors are with the Dipartimento di Elet-
tronica, Politecnico di Torino, C.so Duca degli Abruzzi
24, 10129 Torino, Italy, tel.: +39 011 5644000, fax:
+39 011 5644099. E-mails: ramiro.serra@polito.it -
flavio.canavero@polito.it
special chamber geometries. On the other hand,
a deterministic model’s success is intimately linked
to the speciﬁcity of the chamber geometry and it
compels us to pay no attention to the mode stirring
process, which is an essential constituent of the RC
performance. Consequently, a call for ﬁlling this
gap and linking the two approximations is needed.
This necessity is supported by the aim of having
a better understanding, to manage a simpler yet
complete model, and to reduce up to a reasonable
minimum the empirical techniques.
2 THE 1D RC MODEL
An attempt at ﬁlling this gap was presented in [1],
where a one-dimensional RC model was shown to
have a statistical behavior equal to real RCs. It
simulates the electromagnetic ﬁeld distribution in-
side a theoretical vacuum-ﬁlled 1D segment with
the presence of a 1D ”stirrer” (a perturbing loss-
less dielectric layer) and of losses in the walls. In
this model, the statistically uniform ﬁeld can be ob-
tained by means of diﬀerent stirring processes, each
one of them ﬁnding a strong analogy with real RCs.
2.1 Size stirring and dielectric stirring re-
sults
For reference, we show the results of 500 indepen-
dent calculations of diﬀerent stirrer sizes (e.g., ”size
stirring”) and diﬀerent relative dielectric constants
(e.g., ”dielectric stirring”). The electromagnetic
ﬁeld was calculated at a ﬁxed position inside the
test volume. Histograms of the real and imaginary
parts of the electric ﬁeld with their ﬁtted normal
distributions for every stirring process are shown
in Fig. 1.
The Anderson-Darling Normality Test (A-D) [2]
was applied to these values to determine whether
the data of the sample are nonnormal. The re-
sults largely justify the hypothesis that they fol-
low the normal distribution as reported in [1], thus
reproducing the literature ﬁndings, i.e., that the
ﬁeld-components distributions match the probabil-
ity density functions ([3], [4]).
3 THE 2D RC MODEL
We will next start the description of the 2D RC
inspired by [1] and conceived as a 2D simple exten-
sion of the basic conﬁguration of its analogous 1D
2008 Asia-Pacific Sympsoium on Electromagnetic Compatibility &
38719th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, Singapore
June 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P83
19th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, Singapore
−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02
0
20
40
60
80
100
120
Real(E) [V/m]
Fr
e
qu
e
n
cy
−0.015 −0.01 −0.005 0 0.005 0.01 0.015
0
20
40
60
80
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120
Imag(E) [V/m]
Fr
e
qu
e
n
cy
Size Stirring
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0
20
40
60
80
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140
Real(E) [V/m]
Fr
e
qu
e
n
cy
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0
50
100
150
200
Imag(E) [V/m]
Fr
e
qu
e
n
cy
Dielectric Stirring
Figure 1: Probability distribution of the real and
imaginary parts of the electric ﬁeld measured at a
given position after 500 iterations of the stirrer size
and of the relative dielectric constant with their
ﬁtted normal distributions.
model, and afterwards some statistical results will
be shown.
3.1 The 2D Cavity Model
The description of our chamber (see Fig. 2 for a
schematic diagram) starts as a 2D cavity including
a layer of a perturbing medium with relative dielec-
tric constant κ inside the vacuum-ﬁlled space and
a continuous-wave source located at (x0, y0). The
dimensions of the chamber are a and b.
x
y
MS ; f0
x0
y0
a
b
),(
),(
yx
yx
H
E
r
r
x1 x2
κ
Figure 2: Deﬁnition of the two-dimensional cavity
under study.
The electromagnetic ﬁeld inside this chamber
obeys the wave equation:
∇2 E(x, y) + κ(x, y)k2 E(x, y) = g(x, y) , (1)
where
κ(x, y) =
{
1 0 ≤ x < x1 and x2 ≤ x ≤ a
κ x1 ≤ x < x2
and k = ω
√
μ is the free-space wavenumber; μ, 
are the free-space permeability and permittivity,
respectively. The e−jωt time dependence is sup-
pressed. One possible set of eigensolutions ([7], [8])
is:
Emn(x, y) = Amnψm(x)sin
(nπ
b
y
)
, (2)
where Amn and ψm(x) are found by means of the
process described in [1] and subindexes m,n are
the modal indexes. The proposed solution auto-
matically satisﬁes the boundary conditions at the
walls of the chamber at x = 0, a and y = 0, b.
Losses in the walls are introduced according to
the method described in [7] and shown in [1].
Figure 3 shows the modiﬁcation of the ﬁeld distri-
bution inside the chamber, due to a change of the κ
value in the dielectric region, assumed to maintain
a constant ratio t/a = 0.1, being t = x2 − x1 the
width of the stirrer. From the observation of Fig.
3, where the real part of the electric ﬁeld inside the
chamber for κ = 1 (i.e., absence of dielectric) and
κ = 1.2, it is evident that the main eﬀect of the di-
electric layer inside the chamber is to appreciably
change the ﬁeld distribution inside the ”Test Vol-
ume” region. Thus, an analogy with the stirrer in
real RCs can be established. This result matches
what was shown in [1].
Figure 3: The real part of electric ﬁeld E inside the
2D cavity model for κ = 1 (top panel) and κ = 1.2
(bottom panel).
3.2 The 2D Reverberation Chamber Model
Up to now, we have not been solving an RC but a
cavity resonator. As shown previously, and in co-
herence with ergodicity, we can reproduce the sta-
388
June 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P83
2008 Asia-Pacific Sympsoium on Electromagnetic Compatibility, 19–22 May 2008, Singapore
tistical behavior of an RC if we randomly vary some
selected parameters (in this work, the stirrer size
and the stirrer relative dielectric constant).
The results of 500 independent calculations of the
electromagnetic ﬁeld at a ﬁxed measurement posi-
tion inside the test volume are shown in Fig. 4, that
presents the histograms of the real and imaginary
parts of the electric ﬁeld with their ﬁtted normal
distributions for every stirring process.
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02
0
20
40
60
80
100
120
Real(E) [V/m]
F
re
q
u
e
n
c
y
Size Stirring
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0
20
40
60
80
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120
Imag(E) [V/m]
F
re
q
u
e
n
c
y
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0
20
40
60
80
100
120
Real(E) [V/m]
F
re
q
u
e
n
c
y
Dielectric Stirring
-0.015 -0.01 -0.005 0 0.005 0.01 0.015
0
20
40
60
80
100
120
140
Imag(E) [V/m]
F
re
q
u
e
n
c
y
Figure 4: Probability distribution of the real and
imaginary parts of the electric ﬁeld calculated at
position (x, y) = (8.5, 2.5) m after 500 iterations for
size stirring and dielectric stirring with their ﬁtted
normal distributions.
The A-D Test again justiﬁed the hypothesis of
normality giving p-values of 0.462 and 0.092 for the
real and imaginary part of the electric ﬁeld for size
stirring and 0.069 and 0.058 for dielectric stirring,
respectively. Even though the p-value of 0.462 is
largely better than the other three, all of them are
above the commonly accepted level of signiﬁcance
of 0.05.
3.3 Validity of the Physical Analogy of the
Dielectric Stirrers
A question arises: what is the link and the cor-
relation between real stirrers and our virtual ”di-
electric” stirrers? Diﬀerent interpretations could
be given to explain the inﬂuence of our dielectric
stirrer inside the chamber:
The eﬀect of placing a material with a consider-
able dielectric constant inside an electric ﬁeld as in
our cavity is mainly to change the conﬁguration of
the eigenvalues. It is analogous to adding a capac-
itance to a circuit, or a reactive load into a trans-
mission line. As we are assuming that no losses
are into the dielectric, what happens is that this
new material inserted has stored some energy from
the system. This would result in a particular ﬁeld
conﬁguration. It is then intuitively easy to imagine
that a little change in size or in its relative dielec-
tric constant, could generate a fairly diﬀerent dis-
tribution of the ﬁeld. The present analogy could be
justiﬁed by the fact that usually, ”complex-shaped”
stirrers outperforms simpler ones (i.e., see [5], [6]).
Nevertheless, it is true that real stirrers do not
”store” energy and so the problem remains unan-
swered in a closed manner. A conformal mapping
technique will be presented in the next section to
solve this problem.
4 CONFORMAL MAPPING
The technique of conformal mapping will be ap-
plied to (1), resulting in a successful strategy for
the study of our two-dimensional RC model. This
process will help us identify one irregularly-shaped
homogeneously-ﬁlled cavity, that maps into the 2D
cavity of Section 3.1.
4.1 Basic Theory
Let us consider the inhomogeneous Helmholtz equa-
tion in two dimensions:
[∇2xy + k2
]
f(x, y) = g(x, y) . (3)
Let us apply the conformal mapping w = w(z)
between the complex variable z = x + jy and
w = u + jv to (3) as was shown in [9] to obtain
a slightly more complicated equation in the trans-
formed domain:
[∇2uv + k2/Juvxy
]
F (u, v) = G(u, v) , (4)
where ∇2uv = ∂2/∂u2 + ∂2/∂v2 and Juvxy is the
Jacobian of the transformation (x, y) ⇔ (u, v) as-
sociated with the conformal mapping
Juvxy =
∣∣∣∣
∂u/∂x ∂u/∂y
∂v/∂x ∂v/∂y
∣∣∣∣ . (5)
Equation (4) corresponds to an inhomogeneous
medium situation, where the wavenumber k would
be a function of the coordinates.
4.2 Changing Domains
The cavity described in Section 3.1 is indeed of
the same class of (4) if we assume that κ(u, v) =
1/Juvxy. Let us now suppose it as if it were a cer-
tain cavity in the transformed domain and ﬁnd out
a geometry in the physical domain that maps into
it. In other words, we would like to ﬁnd out a con-
formal mapping
w = w(z)⇔ {u = u(x, y); v = v(x, y)} , (6)
whose Jacobian corresponds with the inverse of the
κ(u, v) function. One possible solution was found
to use the following transformation,
u(x, y) = x
v(x, y) =
⎧
⎨
⎩
y + b/2 0 ≤ x < x1
y/κ + b/2 x1 ≤ x < x2
y + b/2 x2 ≤ x ≤ a
(7)
389
June 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P83
19th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, Singapore
in the cavity depicted in Figure 5. It is quite
straightforward to see that the conformal mapping
in 7 will have a Jacobian as the one we search for.
x
y
MS ; f0
x0
a
b/2
00
),(
),(
,εμ
yx
yx
H
E
r
r
-b/2
κb/2
−κb/2
x1 x2
Figure 5: The cavity of Fig. 2 in the assumed phys-
ical domain.
4.3 Highlighting the Analogy
The cavity geometry conceived as our new physical
domain has two discontinuous steps in the horizon-
tal walls of the cavity. The diﬀerence of height of
the steps and the walls is (κ − 1)b/2. Pushing our
problem to some simple limits, we can make clearer
the eﬀect of our transformation and correlate the
dielectric stirrer with a geometrical discontinuity.
Let us, as a ﬁrst example, make κ = 1. In the
physical domain, this means that no discontinuity
steps are present. On the transformed domain, this
means that the perturbing layer is made of vacuum.
So the conformal mapping will transform a rectan-
gular cavity into itself.
As a second example, we change instead of κ,
the width of the discontinuity steps so as to make
x1 = 0 and x2 = a. In this case, the physical do-
main turns again into a rectangular geometry, with-
out discontinuities. The transformed domain re-
sults to be homogeneously ﬁlled, because the layer
occupies all the space. Both of them are regular
rectangles, but with a diﬀerence: their sizes. The
solutions inside these two cavities are equal, but one
of them is smaller than the other one and ﬁlled with
a dielectric material. This is in perfect coherence
of what is known in waveguides theory.
It is again clear to see that all the statistical re-
sults pointed out in Section 3.2 will be reproduced
in our new physical domain.
5 CONCLUSIONS
This paper describes a 2D RC model inspired by [1]
that presents a strong behavioral analogy with 3D
RCs. In this model, the statistically uniform ﬁeld
can be obtained in two diﬀerent ways: either by
varying the size of the stirrer, or its relative dielec-
tric constant. Both processes show reliable normal-
ity conditions. The main convenience of this model
consists in providing a complete understanding of
RCs, without leaving a gap in the theoretical de-
velopment. With the use of the conformal mapping
technique, we can interpret the role of a dielectric
stirrer as a geometrical discontinuity. Changing the
width of the stirrer is equivalent to changing the
width of the discontinuity steps. Changing the rel-
ative dielectric constant is equal to changing the
height of the discontinuity steps. From this analy-
sis, a novel RC design arises. A chamber with a pro-
ﬁle as Fig. 5 appears to be feasibly constructible.
Further work (currently under way) should involve
both the development of a correlation between the
real stirrer and its 1D parameters and a 3D exten-
sion of this model.
Acknowledgments
The authors are grateful to F. Leferink from Uni-
versity of Twente for the inspiring discussions and
to G. Ghione and M. Goano from Politecnico di
Torino for the useful hints given.
References
[1] R. Serra, F. Canavero: ”A one-dimensional in-
terpretation of the statistical behavior of rever-
beration chambers”, International Conference
on Electromagnetics in Advanced Applications,
2007, pp. 221-224.
[2] L. C. Wolstenholme: Reliability Modelling:
A Statistical Approach, CRC Press, London,
1999.
[3] J. G. Kostas and B. Boverie: ”Statistical
model for a mode-stirred chamber”, IEEE
Trans. on EMC., vol. 33, pp. 366-370, 1991.
[4] D. Hill: ”Plane wave integral representation
for ﬁelds in reverberation chambers”, IEEE
Trans. on EMC, vol. 40, pp. 209-217, 1998.
[5] C. Bruns, ”Three-dimensional simulation and
experimental veriﬁcation of a Reverberation
Chamber”, PhD Thesis, Swiss Federal Insti-
tute of Technology, 2005.
[6] J. Clegg, A. C. Marvin, J. F. Dawson, S. J.
Porter: ”Optimization of stirrer design in a re-
verberation chamber”, IEEE Trans. on EMC,
November 2005.
[7] R. E. Collin: Field Theory of Guided Waves,
IEEE-Press, New York,1991.
[8] P. M. Morse, H. Feshbach: Methods of The-
oretical Physics, Parts I and II, New York:
McGraw-Hill, 1953.
[9] J. R. Mosig: ”Static Green’s functions with
conformal mapping and MATLAB”, IEEE An-
tennas and Propagation Magazine, vol. 45, No.
5, pp. 123-135, October 2003.
390


C o l l i n :Field Theory of Guided Waves, IEEE-Press,

Canavero: ”A one-dimensional interpretation of the statistical behavior of reverberation chambers”,

Feshbach: Methods of Theoretical Physics,P a r t sIa n dI I ,N e wY o r k :

K o s t a sa n dB .B o v e r i e : ” S t a t i s t i c a l model for a mode-stirred chamber”,

Optimization of stirrer design in a reverberation chamber”,

Plane wave integral representation for ﬁelds in reverberation chambers”,

Reliability Modelling: A Statistical Approach,C R CP r e s s ,L o n d o n ,

Static Green’s functions with conformal mapping and MATLAB”,

Three-dimensional simulation and experimental veriﬁcation of a Reverberation Chamber”,

Introduction of Randomness in Deterministic Descriptions of Reverberation Chambers

Ramiro Serra

Flavio Canavero

Crossref

Introduction of randomness in deterministic descriptions of Reverberation Chambers

Serra R.

CANAVERO, Flavio

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEIntroduction of Randomness in Deterministic Descriptions of Reverberation Chambers / Serra R.; Canavero F.G.. -STAMPA. - (2008), pp. 387-390. ((Intervento presentato al convegno 2008 Asia-Pacific Symposium on ElectromagneticCompatibility tenutosi a Singapore (Singapore) nel May 19-23, 2008.OriginalIntroduction of Randomness in Deterministic Descriptions of Reverberation ChambersPublisher:PublishedDOI:10.1109/APEMC.2008.4559893Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1815096 since:IEEEJune 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P83Introduction of Randomness in DeterministicDescriptions of Reverberation ChambersRamiro Serra∗ Flavio Canavero∗Abstract — The main purpose of this work is topresent some simple yet complete models that areable to describe the statistical behavior of Rever-beration Chambers (RC) as well as the influence oftheir main factors and their overall functioning. Westart by building a simplified two-dimensional modelwhich is able to reproduce the statistical behavior ofreal RCs without renouncing the advantages of de-terministic descriptions. The statistical propertiesof the fields inside the cavity are introduced eitherby varying the width or the relative dielectric con-stant of a perturbing lossless layer inside a 2D cav-ity. The field statistics derived with this processare shown to correspond to the statistics found inproperly functioning RCs in the overmoded regime.With the use of a suitable coordinate transforma-tion, we assume our non-homogeneous rectangularcavity to be a virtual mathematical domain, and findan analogous homogeneous non-rectangular cavityas our physical domain. With this simple processwe are able to link our 2D ”dielectric” stirrer withactual geometrical discontinuities. By means of thisprocedure, a novel design for RCs appears as feasi-ble.1 INTRODUCTIONReverberation Chambers (RC) are gaining signifi-cant confidence in use for radiated emissions andimmunity measurements. RC users need to fullyunderstand its working principles in order to cor-rectly interpret the measurement results and to op-timize the performance for various measurementtasks.Reverberation chamber knowledge to date resultsfrom a partial juxtaposition of four different ap-proaches: the deterministic models, the statisti-cal models, the empirical techniques, and the com-puter/numerical methods. It is not possible to leaveone of these approaches behind, as each one of thembehaves as a non-exhaustive, non-excluding part ofa RC description. Furthermore, they mutually col-laborate to give fairly successful answers in fieldswhere the other one fails, and viceversa. Therefore,there is an obvious gap which makes us change ourmethodology depending on what kind of result weseek. In this sense it is to point out that a statis-tical description is meaningful only if the chamberis working in an overmoded regime and only on∗The authors are with the Dipartimento di Elet-tronica, Politecnico di Torino, C.so Duca degli Abruzzi24, 10129 Torino, Italy, tel.: +39 011 5644000, fax:+39 011 5644099. E-mails: ramiro.serra@polito.it -flavio.canavero@polito.itspecial chamber geometries. On the other hand,a deterministic model’s success is intimately linkedto the specificity of the chamber geometry and itcompels us to pay no attention to the mode stirringprocess, which is an essential constituent of the RCperformance. Consequently, a call for filling thisgap and linking the two approximations is needed.This necessity is supported by the aim of havinga better understanding, to manage a simpler yetcomplete model, and to reduce up to a reasonableminimum the empirical techniques.2 THE 1D RC MODELAn attempt at filling this gap was presented in [1],where a one-dimensional RC model was shown tohave a statistical behavior equal to real RCs. Itsimulates the electromagnetic field distribution in-side a theoretical vacuum-filled 1D segment withthe presence of a 1D ”stirrer” (a perturbing loss-less dielectric layer) and of losses in the walls. Inthis model, the statistically uniform field can be ob-tained by means of different stirring processes, eachone of them finding a strong analogy with real RCs.2.1 Size stirring and dielectric stirring re-sultsFor reference, we show the results of 500 indepen-dent calculations of different stirrer sizes (e.g., ”sizestirring”) and different relative dielectric constants(e.g., ”dielectric stirring”). The electromagneticfield was calculated at a fixed position inside thetest volume. Histograms of the real and imaginaryparts of the electric field with their fitted normaldistributions for every stirring process are shownin Fig. 1.The Anderson-Darling Normality Test (A-D) [2]was applied to these values to determine whetherthe data of the sample are nonnormal. The re-sults largely justify the hypothesis that they fol-low the normal distribution as reported in [1], thusreproducing the literature findings, i.e., that thefield-components distributions match the probabil-ity density functions ([3], [4]).3 THE 2D RC MODELWe will next start the description of the 2D RCinspired by [1] and conceived as a 2D simple exten-sion of the basic configuration of its analogous 1D2008 Asia-Pacific Sympsoium on Electromagnetic Compatibility &38719th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, SingaporeJune 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P8319th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, Singapore−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02020406080100120Real(E) [V/m]Frequency−0.015 −0.01 −0.005 0 0.005 0.01 0.015020406080100120Imag(E) [V/m]FrequencySize Stirring−0.05 0 0.05020406080100120140Real(E) [V/m]Frequency−0.02 −0.015 −0.01 −0.005 0 0.005 0.01 0.015 0.02050100150200Imag(E) [V/m]FrequencyDielectric StirringFigure 1: Probability distribution of the real andimaginary parts of the electric field measured at agiven position after 500 iterations of the stirrer sizeand of the relative dielectric constant with theirfitted normal distributions.model, and afterwards some statistical results willbe shown.3.1 The 2D Cavity ModelThe description of our chamber (see Fig. 2 for aschematic diagram) starts as a 2D cavity includinga layer of a perturbing medium with relative dielec-tric constant κ inside the vacuum-filled space anda continuous-wave source located at (x0, y0). Thedimensions of the chamber are a and b.xyMS ; f0x0y0ab),(),(yxyxHErrx1 x2κFigure 2: Definition of the two-dimensional cavityunder study.The electromagnetic field inside this chamberobeys the wave equation:∇2 E(x, y) + κ(x, y)k2 E(x, y) = g(x, y) , (1)whereκ(x, y) ={1 0 ≤ x < x1 and x2 ≤ x ≤ aκ x1 ≤ x < x2and k = ω√μ is the free-space wavenumber; μ, are the free-space permeability and permittivity,respectively. The e−jωt time dependence is sup-pressed. One possible set of eigensolutions ([7], [8])is:Emn(x, y) = Amnψm(x)sin(nπby), (2)where Amn and ψm(x) are found by means of theprocess described in [1] and subindexes m,n arethe modal indexes. The proposed solution auto-matically satisfies the boundary conditions at thewalls of the chamber at x = 0, a and y = 0, b.Losses in the walls are introduced according tothe method described in [7] and shown in [1].Figure 3 shows the modification of the field distri-bution inside the chamber, due to a change of the κvalue in the dielectric region, assumed to maintaina constant ratio t/a = 0.1, being t = x2 − x1 thewidth of the stirrer. From the observation of Fig.3, where the real part of the electric field inside thechamber for κ = 1 (i.e., absence of dielectric) andκ = 1.2, it is evident that the main effect of the di-electric layer inside the chamber is to appreciablychange the field distribution inside the ”Test Vol-ume” region. Thus, an analogy with the stirrer inreal RCs can be established. This result matcheswhat was shown in [1].Figure 3: The real part of electric field E inside the2D cavity model for κ = 1 (top panel) and κ = 1.2(bottom panel).3.2 The 2D Reverberation Chamber ModelUp to now, we have not been solving an RC but acavity resonator. As shown previously, and in co-herence with ergodicity, we can reproduce the sta-388June 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P832008 Asia-Pacific Sympsoium on Electromagnetic Compatibility, 19–22 May 2008, Singaporetistical behavior of an RC if we randomly vary someselected parameters (in this work, the stirrer sizeand the stirrer relative dielectric constant).The results of 500 independent calculations of theelectromagnetic field at a fixed measurement posi-tion inside the test volume are shown in Fig. 4, thatpresents the histograms of the real and imaginaryparts of the electric field with their fitted normaldistributions for every stirring process.-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02020406080100120Real(E) [V/m]FrequencySize Stirring-0.015 -0.01 -0.005 0 0.005 0.01 0.015020406080100120Imag(E) [V/m]Frequency-0.02 -0.01 0 0.01020406080100120Real(E) [V/m]FrequencyDielectric Stirring-0.015 -0.01 -0.005 0 0.005 0.01 0.015020406080100120140Imag(E) [V/m]FrequencyFigure 4: Probability distribution of the real andimaginary parts of the electric field calculated atposition (x, y) = (8.5, 2.5) m after 500 iterations forsize stirring and dielectric stirring with their fittednormal distributions.The A-D Test again justified the hypothesis ofnormality giving p-values of 0.462 and 0.092 for thereal and imaginary part of the electric field for sizestirring and 0.069 and 0.058 for dielectric stirring,respectively. Even though the p-value of 0.462 islargely better than the other three, all of them areabove the commonly accepted level of significanceof 0.05.3.3 Validity of the Physical Analogy of theDielectric StirrersA question arises: what is the link and the cor-relation between real stirrers and our virtual ”di-electric” stirrers? Different interpretations couldbe given to explain the influence of our dielectricstirrer inside the chamber:The effect of placing a material with a consider-able dielectric constant inside an electric field as inour cavity is mainly to change the configuration ofthe eigenvalues. It is analogous to adding a capac-itance to a circuit, or a reactive load into a trans-mission line. As we are assuming that no lossesare into the dielectric, what happens is that thisnew material inserted has stored some energy fromthe system. This would result in a particular fieldconfiguration. It is then intuitively easy to imaginethat a little change in size or in its relative dielec-tric constant, could generate a fairly different dis-tribution of the field. The present analogy could bejustified by the fact that usually, ”complex-shaped”stirrers outperforms simpler ones (i.e., see [5], [6]).Nevertheless, it is true that real stirrers do not”store” energy and so the problem remains unan-swered in a closed manner. A conformal mappingtechnique will be presented in the next section tosolve this problem.4 CONFORMAL MAPPINGThe technique of conformal mapping will be ap-plied to (1), resulting in a successful strategy forthe study of our two-dimensional RC model. Thisprocess will help us identify one irregularly-shapedhomogeneously-filled cavity, that maps into the 2Dcavity of Section 3.1.4.1 Basic TheoryLet us consider the inhomogeneous Helmholtz equa-tion in two dimensions:[∇2xy + k2]f(x, y) = g(x, y) . (3)Let us apply the conformal mapping w = w(z)between the complex variable z = x + jy andw = u + jv to (3) as was shown in [9] to obtaina slightly more complicated equation in the trans-formed domain:[∇2uv + k2/Juvxy]F (u, v) = G(u, v) , (4)where ∇2uv = ∂2/∂u2 + ∂2/∂v2 and Juvxy is theJacobian of the transformation (x, y) ⇔ (u, v) as-sociated with the conformal mappingJuvxy =∣∣∣∣∂u/∂x ∂u/∂y∂v/∂x ∂v/∂y∣∣∣∣ . (5)Equation (4) corresponds to an inhomogeneousmedium situation, where the wavenumber k wouldbe a function of the coordinates.4.2 Changing DomainsThe cavity described in Section 3.1 is indeed ofthe same class of (4) if we assume that κ(u, v) =1/Juvxy. Let us now suppose it as if it were a cer-tain cavity in the transformed domain and find outa geometry in the physical domain that maps intoit. In other words, we would like to find out a con-formal mappingw = w(z)⇔ {u = u(x, y); v = v(x, y)} , (6)whose Jacobian corresponds with the inverse of theκ(u, v) function. One possible solution was foundto use the following transformation,u(x, y) = xv(x, y) =⎧⎨⎩y + b/2 0 ≤ x < x1y/κ + b/2 x1 ≤ x < x2y + b/2 x2 ≤ x ≤ a(7)389June 17, 2004 19:46 Research Publishing: Trim Size: 8.50in x 11.00in (IEEE proceedings) ieee-emc08:P8319th International Zurich Symposium on Electromagnetic Compatibility, 19–22 May 2008, Singaporein the cavity depicted in Figure 5. It is quitestraightforward to see that the conformal mappingin 7 will have a Jacobian as the one we search for.xyMS ; f0x0ab/200),(),(,εμyxyxHErr-b/2κb/2−κb/2x1 x2Figure 5: The cavity of Fig. 2 in the assumed phys-ical domain.4.3 Highlighting the AnalogyThe cavity geometry conceived as our new physicaldomain has two discontinuous steps in the horizon-tal walls of the cavity. The difference of height ofthe steps and the walls is (κ − 1)b/2. Pushing ourproblem to some simple limits, we can make clearerthe effect of our transformation and correlate thedielectric stirrer with a geometrical discontinuity.Let us, as a first example, make κ = 1. In thephysical domain, this means that no discontinuitysteps are present. On the transformed domain, thismeans that the perturbing layer is made of vacuum.So the conformal mapping will transform a rectan-gular cavity into itself.As a second example, we change instead of κ,the width of the discontinuity steps so as to makex1 = 0 and x2 = a. In this case, the physical do-main turns again into a rectangular geometry, with-out discontinuities. The transformed domain re-sults to be homogeneously filled, because the layeroccupies all the space. Both of them are regularrectangles, but with a difference: their sizes. Thesolutions inside these two cavities are equal, but oneof them is smaller than the other one and filled witha dielectric material. This is in perfect coherenceof what is known in waveguides theory.It is again clear to see that all the statistical re-sults pointed out in Section 3.2 will be reproducedin our new physical domain.5 CONCLUSIONSThis paper describes a 2D RC model inspired by [1]that presents a strong behavioral analogy with 3DRCs. In this model, the statistically uniform fieldcan be obtained in two different ways: either byvarying the size of the stirrer, or its relative dielec-tric constant. Both processes show reliable normal-ity conditions. The main convenience of this modelconsists in providing a complete understanding ofRCs, without leaving a gap in the theoretical de-velopment. With the use of the conformal mappingtechnique, we can interpret the role of a dielectricstirrer as a geometrical discontinuity. Changing thewidth of the stirrer is equivalent to changing thewidth of the discontinuity steps. Changing the rel-ative dielectric constant is equal to changing theheight of the discontinuity steps. From this analy-sis, a novel RC design arises. A chamber with a pro-file as Fig. 5 appears to be feasibly constructible.Further work (currently under way) should involveboth the development of a correlation between thereal stirrer and its 1D parameters and a 3D exten-sion of this model.AcknowledgmentsThe authors are grateful to F. Leferink from Uni-versity of Twente for the inspiring discussions andto G. Ghione and M. Goano from Politecnico diTorino for the useful hints given.References[1] R. Serra, F. Canavero: ”A one-dimensional in-terpretation of the statistical behavior of rever-beration chambers”, International Conferenceon Electromagnetics in Advanced Applications,2007, pp. 221-224.[2] L. C. Wolstenholme: Reliability Modelling:A Statistical Approach, CRC Press, London,1999.[3] J. G. Kostas and B. Boverie: ”Statisticalmodel for a mode-stirred chamber”, IEEETrans. on EMC., vol. 33, pp. 366-370, 1991.[4] D. Hill: ”Plane wave integral representationfor fields in reverberation chambers”, IEEETrans. on EMC, vol. 40, pp. 209-217, 1998.[5] C. Bruns, ”Three-dimensional simulation andexperimental verification of a ReverberationChamber”, PhD Thesis, Swiss Federal Insti-tute of Technology, 2005.[6] J. Clegg, A. C. Marvin, J. F. Dawson, S. J.Porter: ”Optimization of stirrer design in a re-verberation chamber”, IEEE Trans. on EMC,November 2005.[7] R. E. Collin: Field Theory of Guided Waves,IEEE-Press, New York,1991.[8] P. M. Morse, H. Feshbach: Methods of The-oretical Physics, Parts I and II, New York:McGraw-Hill, 1953.[9] J. R. Mosig: ”Static Green’s functions withconformal mapping and MATLAB”, IEEE An-tennas and Propagation Magazine, vol. 45, No.5, pp. 123-135, October 2003.390

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Introduction of Randomness in Deterministic Descriptions of Reverberation Chambers

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