1. Source Stirring Analysis in a Reverberation
Chamber Based on Modal Expansion of the
Electric Field
E. Amador1, P. Besnier2
1EDF Lab, LME. Moret sur Loing, France
emmanuel.amador@edf.fr
2IETR-UMR CNRS 6164, INSA Rennes, France
philippe.besnier@insa-rennes.fr
Joint IEEE International Symposium on Electromagnetic Compatibility and EMC Europe, Dresden 2015
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2. Content
Motivations
Modal field distribution of a rectangular cavity and its response
Waveguide-Cavity equations
Accounting for losses
E-field computation
Response to an isotropic EM source
Source Stirring
Number of modes and number of independent positions
Different source stirring methods
Conclusion
References
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3. Motivations
Motivations
Reverberation chamber simulations based on numerical solutions of
Maxwell’s equations (FDTD, FEM, image theory,...) :
are convenient to provide general trends (Ex. statistical
uniformity vs losses)
provide more data than measurements,
do not allow a much better understanding of the cavity physics.
By using an analytical approach at the mode level we expect to :
get a more precise view of the multi-modal field distribution in
the chamber,
understand how the number of independent observations is
related to the successive states of multi-modal field distribution
according to source stirring.
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4. Modal field distribution of a rectangular cavity and its response Waveguide-Cavity equations
Rectangular cavity
a
b
d
x
y
z
fmnp =
c
2
m
a
2
+
n
b
2
+
p
d
2
, (1)
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5. Modal field distribution of a rectangular cavity and its response Waveguide-Cavity equations
E-field wave guide equations [1, 2]
(similar equations may be derived for the H-field) For a propagation
along z :
TM mode :
ETMz
xmnp
=
−kx kzE0
k2
mnp − k2
z
cos kx x sin ky y sin kzz (2)
ETMz
ymnp
=
−ky kzE0
k2
mnp − k2
z
sin kx x cos ky y sin kzz (3)
ETMz
zmnp
= E0 sin kx x sin ky y cos kzz (4)
TE mode :
ETEz
xmnp
=
−jωmnpµky H0
k2
mnp − k2
z
cos kx x sin ky y sin kzz (5)
ETEz
ymnp
=
−jωmnpµkx H0
k2
mnp − k2
z
sin kx x cos ky y sin kzz (6)
ETEz
zmnp
= 0 (7)
(kx = mπ
a , ky = nπ
b , kz = pπ
d , k2
mnp = k2
x + k2
y + k2
z )
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6. Modal field distribution of a rectangular cavity and its response Accounting for losses
Universal resonant curve
At a frequency f, the amplitude of a mode with frequency , fmnp can be
computed from the effective/composite Q factor [3] :
I(f, fmnp) =
1
1 + Q2(f) f
fmnp
−
fmnp
f
2
, (8)
10-2 10-1 100 101 102
f/fmnp
0.0
0.2
0.4
0.6
0.8
1.0
I(f,fmnp)
Q=1
Q=10
Q=100
Q=1000
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7. Modal field distribution of a rectangular cavity and its response E-field computation
E-field (1/2)
The E-field observed at the frequency f is approximated by summing
the E-fields produced by all modes in an adequate bandwidth f ± ∆f.
The E-field along the x axis at the frequency f at the position (x, y, z)
is given by :
Ex (f, x, y, z) =
fmnp∈f±∆f
I(f, fmnp) ETMi
xmnp
+ ETEi
xmnp
, (9)
with i = x, y, z if the TM or TE modes exist for the given triplet
(m, n, p).
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8. Modal field distribution of a rectangular cavity and its response E-field computation
E-field (2/2)
268 270 272 274 276 278 28010-3
10-2
10-1
100
101
A.U
|Ex |
268 270 272 274 276 278 28010-3
10-2
10-1
100
101
A.U
|Ey|
268 270 272 274 276 278 280
f in MHz
10-3
10-2
10-1
100
101
A.U
|Ez|
FIGURE: E-field computed at M(3.5, 1.7, 1).
However, the modal response of the cavity is bounded to the source
excitation.
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9. Modal field distribution of a rectangular cavity and its response Response to an isotropic EM source
Response to an isotropic and infinitely small EM
source
A infinitely small transmitter is placed at the position A(xe, ye, ze)
The mode amplitudes/coefficients of each TM or TE mode
excited at the frequency f at this position A are given by : (For
the E-field components on the x axis)
CTMz
xmnp
= I(f, fmnp)
−kx kz
k2
mnp − k2
z
cos kx xe sin ky ye sin kzze (10)
CTEz
xmnp
= I(f, fmnp)
−jωmnpµky
k2
mnp − k2
z
cos kx xe sin ky ye sin kzze (11)
By reciprocity, the Ex component of the E-field at the frequency f
produced by this transmitter at a given position M(x, y, z) is
given by :
Ex (f, x, y, z) =
fmnp∈f±∆f
CTMi
xmnp
ETMi
xmnp
+CTEi
xmnp
ETEi
xmnp
, (12)
with i = x, y, z.
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10. Source Stirring
Source stirring
Source stirring is then performed analytically.
FIGURE: |Ex | in V/m at f = 274 MHz, with a source moving along a
circle in the plane x = 1 m.
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11. Source Stirring Number of modes and number of independent positions
Source stirring : linear path
a=2.9m
b = 3.7 m
d = 8.7 m
x
y
z
(1,1,0.5)
(1,1,8.2)
Ns = 201 steps, linear movement along the z axis
The complex Ex field is computed at N = 150 arbitrary positions
in the working volume for every step and every frequency
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12. Source Stirring Number of modes and number of independent positions
Counting the modes
By moving the source in the working volume, different modes
are excited.
We compute the mode coefficients of every mode excited.
We want to estimate the number of modes Nxmodes
that have a
significant impact on the E-field in the chamber.
We define as a significant mode, a mode whose maximum magnitude
over the source positions reaches at least 30% of the maximum
magnitude of the mode with maximum magnitude.
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13. Source Stirring Number of modes and number of independent positions
Counting the modes - Example at 50 MHz (1/2)
0 1 2 3 4 5 6 7 8
z/m
0.00
0.05
0.10
0.15
0.20
0.25
V/m
FIGURE: |Ex | vs. transmitter position at N = 150 positions (50 MHz).
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14. Source Stirring Number of modes and number of independent positions
Counting the modes - Example at 50 MHz (2/2)
0 1 2 3 4 5 6 7 8
z/m
1.0
0.5
0.0
0.5
1.0
A.U.
TEyz
0 1 1
TMxyz
0 1 1
TMxyz
0 1 2
TEyz
0 2 1
FIGURE: Modes coefficients vs. transmitter position at 50 MHz.
4 main modes. 3 of them with index p = 1 are clearly
correlated. Number of independently excited modes :
Nxmodes
= 2
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15. Source Stirring Number of modes and number of independent positions
Counting the modes - Example at 425 MHz (1/2)
0 1 2 3 4 5 6 7 8
z/m
0.0
0.5
1.0
1.5
2.0
2.5
3.0
V/m
FIGURE: |Ex | vs. transmitter position at N = 150 positions (425 MHz).
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16. Source Stirring Number of modes and number of independent positions
Counting the modes - Example at 425 MHz (2/2)
0 1 2 3 4 5 6 7 8
z/m
1.0
0.5
0.0
0.5
1.0A.U. TEyz
0 1 24
TEyz
0 1 25
TEyz
0 2 24
TMxyz
0 8 16
TEyz
0 10 1
TEyz
0 10 7
TEyz
1 2 24
TMxyz
1 2 24
TMxyz
1 6 20
TEyz
1 10 7
TMxyz
3 9 9
TMxyz
4 9 4
TMxyz
8 2 3
FIGURE: Modes coefficients vs. transmitter position at 425 MHz.
Groups of modes with same p indexes are correlated,
Nxmodes
= 9 (with distinct p indexes)
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17. Source Stirring Number of modes and number of independent positions
Number of modes and number of independent
positions
Entropy gives the number of independent positions Nxeff
along the
transmitter path (among Ns positions) the correlation matrix of
complex Ex values [4] :
Nxeff
=
N2
s
Ns
i,j=1 |rxij
|2
, (13)
where rx is the correlation matrix computed from the complex
Ex values.
How Nxeff
and Nxmodes
are related ?
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18. Source Stirring Number of modes and number of independent positions
Nxeff
and Nxmodes
0 200 400 600 800 1000
f/MHz
0
2
4
6
8
10
12
14
16
Nxeff
Nxmodes
FIGURE: Nxeff
and Nxmodes
vs. frequency
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19. Source Stirring Number of modes and number of independent positions
Partial conclusion
Nxeff
and Nxmodes
are in good agreement for an isotropic source
stirring.
The number of modes that are excited independently by moving
linearly the transmitter is close to the number of independent
positions derived from the entropy estimation [4].
Source stirring implies independent mode excitations but also
correlated excitation of groups of mode (linear path along z
direction)
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20. Source Stirring Different source stirring methods
Different methods
Linear movement along the z axis (as previously presented).
Along a smoothed random curve in the whole working volume.
Source(s) randomly placed in a cylinder volume (1 m radius) and
rotating around a vertical axis :
1 source
6 sources
100 sources
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a=2.9m
b = 3.7 m
d = 8.7 m
x
y
z

21. Source Stirring Different source stirring methods
Linear path vs. random path
200 400 600 800
f/MHz
2
4
6
8
10
12
14
Nxeff
, 1 source linear path
Moving av. 1 source linear path
Nxeff
, 1 source random path
Moving av. 1 source random path
FIGURE: Independent source stirring positions vs. frequency for a
linear source movement or a random movement.
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22. Source Stirring Different source stirring methods
Sources rotating around one axis
200 400 600 800
f/MHz
1
2
3
4
5
6
7
1 source rotating
6 sources rotating
100 sources rotating
1 source linear path
1 source random path
FIGURE: Independent source stirring positions vs. frequency for
different numbers of sources rotating around one axis.
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23. Source Stirring Different source stirring methods
Partial conclusion
At low frequencies moving a large number of sources reduce the
number of independent positions, the modes are not selected
individually
At high frequencies, increasing the number of sources allows to
get more independent positions.
Moving one source randomly in the volume gives the best
results.
Efficient source stirring is achieved by selecting a large number of
modes individually.
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24. Conclusion
Conclusion
We presented a very simple and fast analytical model
based on a modal combination of 3 wave guides field
equations.
This model allows an analysis of the behavior of a
rectangular cavity at the mode level.
The number of modes independently excited by a
moving source along a path is related to the number of
independent positions observed.
The model allows to benchmark different source(s) stirring
approaches.
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25. Conclusion References
[1] R. Harrington, Time-Harmonic Electromagnetic Fields.
New York : McGraw-Hill Book Company, 1961 (repr. 2001).
[2] D. A. Hill, Electromagnetic Fields in Cavities : Deterministic and
Statistical Theories.
Hoboken, NJ, USA : John Wiley & Sons, Inc., 2009.
[3] F. E. Terman, Radio Engineer’s Handbook.
McGraw-Hill Book Company, 1943.
[4] R. Pirkl, K. Remley, and C. Patane, “Reverberation chamber
measurement correlation,” Electromagnetic Compatibility, IEEE
Transactions on, vol. 54, pp. 533–545, June 2012.
Presentation, source code and data available at :
http://github.com/manuamador/IEEE-EMC2015
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26. Statistical Validation
Anderson Darling, GoF test :
0 200 400 600 800 1000 1200
f/MHz
0.0
0.2
0.4
0.6
0.8
1.0Rejectionrate
Ex
Ey
Ez
ER , Meas.
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