Download Electromagnetic Coupling Through Perforated Shields due to Near

Electromagnetic Coupling Through Perforated Shields
due to Near Field Radiators
Shabista Ali, Daniel S. Weile
Thomas Clupper
Department of Electrical and Computer Engineering
University of Delaware
140 Evans Hall
Newark, DE 19716
[email protected], [email protected]
W. L. Gore and Associates
402 Vieve’s Way
Elkton, MD 21921
[email protected]
Abstract
The electromagnetic coupling through an infinite
conducting sheet perforated with a finite field of holes
excited by a metal radiator placed in its near field is
investigated. The coupling is analyzed numerically by
method of moments (MoM) with emphasis on
understanding the effect of the interactions between the
field of holes and the metal radiator placed in its proximity.
The radiation leakage through the field of holes is found to
be significant due to the proximity to the antenna, even if
the holes are electrically small. Numerical results confirm
that classical theory underestimates the electromagnetic
coupling through small apertures by antennas placed in
their near field.
GEOMETRY OF THE PROBLEM
The geometry analyzed here is shown in Figure 1. The x-y
plane is an infinite perfectly conducting sheet, perforated
with a finite array of holes. The coordinate origin is placed
in the center of the circular aperture at the corner of the
field of holes as shown. The holes are arranged in a regular
hexagonal manner on the sheet. The distance between
centers of two adjacent holes is a.
Keywords
Method of Moments, interference, shielding, small
apertures.
a
INTRODUCTION
Many components in modern electronic devices are
shielded from radiation leakage by using conductive
electromagnetic interference (EMI) caps that are placed
above the component to block the radiation produced by
the component. Most shielding caps contain small holes to
allow heat to flow when joining the cap to a board. It has
long been assumed that the coupling through these holes is
negligible since they are generally very small in
comparison with the wavelength of the radiation for which
the shield was designed to be effective. In this work, the
method of moments (MoM) is used to numerically
demonstrate that the coupling through small holes is not
always irrelevant, because the presence of the antenna in
the near field of the holes significantly alters the coupling.
There is some precedence for this idea. Reference [4]
shows that the shielding effectiveness of a perforated sheet
of small holes in the near field is greatly overestimated by
classical theory. However, [4] considers an infinite array
of holes and does not take into account the interactions
between the holes and the antenna. Moreover, because the
authors investigate near field antenna-to-antenna coupling,
evanescent fields contributed strongly to fields observed by
the receiving antenna. In contrast, this work analyzes the
behavior of an antenna near a finite array of holes and
shows that the interaction between the antenna and the
holes causes significant coupling, even to the far field
outside the shield.
z
a
y
d
dipole
antenna
l
shield
x
Figure 1. Geometry for analyzing coupling through
small holes
An antenna (shown in Figure 1 as a dipole of length l) is
positioned in the center of the holes array below the shield.
The distance from the wire to the sheet of holes is denoted
by d. All of the results shown here consider a 5 by 5
hexagonal array of holes.
NUMERICAL ANALYSIS
This section explains the numerical computation involved
in analyzing the geometry outlined above. The first
subsection describes the creation of integral equations
appropriate to the problem. The second subsection then
details how these equations can be solved using the method
of moments. Finally, the third subsection defines the
coupling coefficient, which is the quantity of interest in this
work.
Integral Equation Formulation
In order to analyze the problem, we model the wire antenna
and the holes lying on the infinite ground. We derive an
equivalent problem from the existing problem and then
determine the unknown currents [1].
Figure 2(a) shows the problem at hand. Consider a circular
aperture on the ground plane. The region above the plane
will be called Region I and that below it will be called
Region II. For Region I, the original problem is replaced
by an equivalent problem with current sources J1 and M1
lying in the plane of the conductor and defined as
J1 = nˆ × H1
(1)
M1 = E1 × nˆ
(2)
Since the electric field tangential to the conductor vanishes,
M1 is zero on the conductors as shown in Figure 2(b).
Since the fields in region II are zero in the equivalent
formulation, a perfect electric conductor can be placed in
the aperture. Image theory is then applied to get the
equivalent problem in Figure 2(c). Note that the original
electric currents in the holes cancel their images exactly.
J1
Source
n̂
a)
E1 , H1
Circular
Aperture
pec
Region I
Region II
E2 , H 2
J1
Source
J1
J1
b)
M1
J1
pec
Circular
Aperture
J1
Source
c)
2M1
J1
Image
2M 2
d)
J 2 = −nˆ × H 2
(3)
M 2 = E2 × (−nˆ )
(4)
Again, the aperture can be filled with perfect electric
conductor and image theory can be applied. The problem
is then reduced to that shown in Figure 2(d). Note that
because the antenna is assumed to be above the plane, it
does not appear in Figure 2(d) at all.
Electric Field Integral Equation
Now that equivalent problems have been created, integral
equations can be defined. The total electric field E t on the
wire can be written as the sum of an incident field Einc
imposed on the wire, a scattered field EEJ ( r ) generated by
the electric currents excited on the wire, and a scattered
M
field EM
( r ) generated by the equivalent magnetic currents
in the holes. The total electric field tangential to the wire
must vanish, so
M
(r) )
(5)
0 = tˆ ⋅ ( Einc (r) + EJE (r) + EM
on the wire, where t̂ is a tangent to the wire. The notations
M
EEJ (r) and EM
(r) thus represent operators mapping
electric and magnetic currents to the electric fields they
create.
The electric field due to the currents on the wires can be
written as
1
⎡
⎤
EEJ (r ) = − jkη ⎢ A(r ) + 2 ∇ ( ∇ ⋅ A(r ) ) ⎥
(6)
k
⎣
⎦
where η is the impedance of free space, k is the angular
wave number of the radiation, and j is the imaginary unit.
Recalling that the field due to the wires contains a
component due to the image of the wires, and denoting a
point on the wires as r ′ = x′xˆ + y ′yˆ + z ′zˆ , the magnetic
vector potential may be written as
− jk r − r ′
e
d ′−
A ( r ) = ∫ J (r′)
4π r − r ′
(7)
− jk r − r ′ + 2 z ′zˆ
e
∫ ⎣⎡J ( r ′) − ( 2zˆ ⋅ J ( r′) ) zˆ ⎦⎤ 4π r − r′ + 2 z ′zˆ d ′
Similarly, the electric field of the holes can be written in
terms of an electric vector potential as
M
EM
(r ) = −∇ × F (r )
(8)
where F may be computed from the equivalent magnetic
current M = M1 using the relation
F (r ) = ∫∫ M (r ′)
− jk r − r ′
e
dS ′
2π r − r ′
(9)
Figure. 2 (a) Original Problem (b) Equivalent problem in
region I (c) Final problem in region I after applying
image theory (d) Final problem in region II after
applying image theory.
Substituting Equations (6) through (9) into Equation (5)
results in a single integral equation for the two unknowns J
and M.
Similarly for the Region II, an equivalent problem is
constructed. In Region II, the equivalent sources are
defined as
Magnetic Field Integral Equation
To derive another integral equation, the condition that the
electric and magnetic fields are continuous in the holes is
S
applied. Setting M 2 = −M ensures that the total electric
field is continuous in the holes. To ensure magnetic field
continuity in the holes, we let
scat
scat
⎤ ˆ
nˆ × ⎡⎣ H inc (r ) + H scat
(10)
I,wires (r ) + H I,holes (r ) ⎦ = n × H II (r )
where H inc is the incident magnetic field, H scat
I,wires and
H scat
I,holes are the scattered fields produced by the electric and
magnetic currents in equivalent problem I, and H scat
is the
II
scattered field in Region II produced by its equivalent
magnetic current. Equation (10) is enforced in each of the
holes.
In Region I, the electric and magnetic vector potentials
derived above are unaltered for the derivation of the MFIE.
Thus, expressions for Region I fields are easily derived:
E
H scat
(11)
I,wires (r ) = H J (r ) = ∇ × A (r )
and
M
H scat
I,holes (r ) = H M (r ) = −
jk ⎡
⎤
1
F(r ) + 2 ∇ ( ∇ ⋅ F (r ) ) ⎥ . (12)
η ⎢⎣
k
⎦
Finally, the Region II magnetic field is produced only by
the current −2M radiating in free space, so
M
H scat
(13)
II (r ) = − H M (r ) .
Substituting Equations (7), (9), and (11) through (13) into
Equation (10) gives rise to another equation satisfied by the
unknown currents J and M.
Method of Moments
The frequency domain MoM can now be applied to solve
the integral Equations (5) and (10) [2]. The integral
equations are converted into a set of matrix equations,
which are then solved to determine the unknown current.
To apply the MoM the unknown currents are expanded in a
series of basis function as
M
J (r ) = ∑ am J m (r )
(14)
m =1
N
M (r ) = ∑ bn M n (r )
(15)
n =1
where the am and bn are unknown constant coefficients, the
J m (r ) are triangular basis functions representing wire
currents, and the M n (r ) are Rao-Wilton-Glisson basis
functions representing the magnetic currents in the holes
[3].
Upon substituting Equations (14) and (15) into Equations
(5) and (10), a set of algebraic equations may be derived for
the unknown weighting coefficients by multiplying
Equation (5) by the J m (r ) and integrating over the wires,
and multiplying Equation (10) by the M n (r ) and
integrating over the holes. This procedure gives rise to a
matrix equation that may be written as
⎡ Z ww Z wh ⎤ ⎡ a ⎤ ⎡ einc ⎤
(16)
⎢ hw
⎥⎢ ⎥ = ⎢ ⎥
Z hh ⎦ ⎣b ⎦ ⎣hinc ⎦
⎣Z
The elements of the vectors and matrices in Equation (16)
are given by
[a]m = am , [b ]n = bn ,
∫E
⎡⎣e ⎤⎦ =
p
inc
(17)
inc
(r ) ⋅ J p (r )d ,
wire
∫∫
⎡⎣hinc ⎤⎦ =
p
H inc (r ) ⋅ M q (r )dS ,
(18)
holes
and
⎡⎣ Z ww ⎤⎦ =
pm
∫
J p (r ) ⋅ EEJ m (r )d ,
wire
⎡⎣ Z wh ⎤⎦ =
pn
∫
M
J p (r ) ⋅ EM
(r )d ,
n
wire
⎡⎣ Z ⎤⎦ =
qm
hw
∫∫
M q (r ) ⋅ H JEm (r )dS ,
(19)
holes
⎡⎣ Z hh ⎤⎦ =
qn
∫∫ M
q
M
(r ) ⋅ H M
(r )dS ,
n
holes
where m, p = 1,… , M and n, q = 1,… , N . The wire-wire
interactions are computed using the standard thin-wire
kernel. Moreover, a delta-gap model is used to compute
the incident field, so hinc = 0 , and einc vanishes
everywhere except at the antenna feed, where it is
arbitrarily chosen to be unity. Once Equation (16) is
created, it can be solved for the unknown weighting
coefficients.
Coupling Coefficient
After the unknown current is determined, the coupling due
to the finite sheet of holes can be evaluated. Specifically,
the total power Prad radiated by the antenna can be
computed once the current at the feed is known. Moreover,
the far field radiated by the array of holes can be computed,
and used to determine Pcouple , the total energy that passes
through the holes to the far field. The coupling coefficient
K can the be defined as
Pcouple
K=
(20)
Prad
In addition to this straightforward computation, a coupling
coefficient must also be computed for the (imaginary) case
in which the antenna is imagined to radiate in free space,
and the incident field on the holes is taken as this radiation.
To accomplish this, a new system is created in which the
image term arising from the wire in the matrix Z ww is
ignored, and the matrix Z wh is set to zero. The coupling
coefficient can then be computed in the normal fashion.
Because the main hypothesis of this work is that leakage
through fields of holes occurs due to the coupling between
holes and radiators, comparison of the two types of
coupling coefficient serves to verify the claim.
RESULTS
In this section, the method of the previous sections is
applied to the analysis of coupling problems with variations
in hole sizes, hole spacing, antenna length, antenna type,
and distance of the antenna from the holes. The frequency
range considered is 1 GHz to 10 GHz, and the coupling
coefficient is computed over that band for each of the cases
analyzed. Each of the cases demonstrates the effect of a
given parameter on the coupling of antenna radiation
through finite fields of holes.
Case 1. Proximity of the Radiator to the Hole
Array
Figure 3 shows the coupling coefficient vs. frequency for a
1.2 cm dipole at various distances from a field of 2mmdiameter holes spaced 3mm apart. Figure 4 shows the
coupling coefficient for the same problem computed by
ignoring the coupling between the antenna and the holes.
These figures demonstrate that (i) the coupling increases
with proximity to the hole array, whether or not the
coupling between the antenna and holes is included in the
computation, and (ii) that the coupling between the antenna
and the hole array has a profound effect on the amount of
radiation that penetrates the shield.
The first of these observations is no surprise; no one would
imagine that increasing the distance between the antenna
and the holes would increase the coupling. The second
result, however, is somewhat surprising. In particular, it is
well known that plane waves do not significantly penetrate
arrays of holes with diameters much smaller than a
wavelength. Since any wave incident from free space can
be conceived as a superposition of plane waves, and since
the result shown in Figure 4 was computed by removing the
antenna from the problem once its radiation is known,
Figure 4 demonstrates this familiar behavior. Specifically,
the coupling coefficient decreases rapidly with increasing
wavelength. Because the results of Figure 3 do not display
this behavior, the coupling between the antenna and the
holes must be primarily responsible for the radiation into
Region II.
Case 2. Antenna Length
To further demonstrate that the conclusion is independent
of the length of the dipole, graphs of coupling coefficient
versus frequency for different dipole lengths are shown in
Figures 5 and 6. Both results pertain to 2mm-diameter
holes spaced, 3mm and the distance d is varied from 0.2mm
to 3mm. For Figure 5, the antenna length l is 0.6cm mm
and the antenna length is 2.4cm for Figure 6. Comparing
these graphs with Figure 3 clearly shows that there is no
significant effect on the coupling coefficient due to
variation in the length of antennas.
Case 3. Hole Spacing
Figures 7 and 8 show the coupling coefficient for antenna
of length 1.2cm, hole diameter 2mm and d varying from
0.2mm to 3mm. The hole spacing for the Figure 7 is 3.5mm
and that for Figure 8 is 4.5mm. These graphs also show
that while increasing hole spacing improves the shielding,
the effect of proximity of the radiator is much more
important.
Case 4. Type of Antenna
Figure 9 shows the coupling coefficient for a loop antenna
placed below the hole array for various d. The loop
antenna used to generate Figure 9 is 14mm in length and
10mm in width. It is placed such that 14 mm side is
parallel and the 10mm side is perpendicular to the hole
array. The hole diameter is 2mm and a is 3mm. It is
obvious from this figure that the coupling mainly depends
on proximity of the antenna to the shield, even at very low
frequencies.
This fundamental result appears to be
independent of antenna type as well.
Case 5. All Parameters
Results for Figures 10 and 11 were computed assuming a
2.4cm dipole antenna sitting in front of a screen of 4mmdiameter holes, spaced 5mm apart. Again the distance of
the antenna from the screen was varied, and Figure 10
shows the “coupled” result whereas Figure 11 shows the
“uncoupled” result. For Figure 12 everything is same as
for Figure 10, however the antenna type is different. The
antenna used here is a loop antenna that is 14mm in length
and 10mm in width. It is placed such that 14 mm side is
parallel and the 10mm side is perpendicular to the hole
array. The results shown in Figures 10, 11 and 12 when
compared to that of Figure 3 demonstrate that the coupling
increases with proximity to the hole array and that the
coupling between the hole array and the antenna has a
significant effect on the radiation leakage through the
shield. This conclusion is independent of antenna length,
hole diameter (as long as the holes are electrically small),
antenna type and hole spacing.
All of the above examples show that there is no significant
difference in low frequency coupling due to the differences
in antenna length or type, or due to the size and spacing of
the hole array so long as all of the above remain electrically
small. These results clearly demonstrate that proximity of
the antenna to the screen is the most significant factor in
creating coupling through an array of small holes, and that
significant energy can couple through even very small
holes.
CONCLUSIONS
The coupling of antenna radiation through a finite array of
holes placed on an infinite conducting sheet was
investigated. The problem was analyzed using the method
of moments, using both a straightforward, full-wave
approach, and by ignoring the interaction between the holes
and the antenna while it radiates. Results demonstrate that
the coupling through small holes is strongly affected by the
nearness of the metal radiator to the hole array, and that
significant energy can leak through even small holes if the
source of that energy is proximal.
Moreover, it was observed that factors like antenna size,
antenna type, hole spacing and hole sizes had negligible
effect on this conclusion. In short, plane wave analyses of
perforated shields can vastly underestimate the coupling of
antenna radiation to the far field, and this underestimation
results from ignoring the presence of the antenna in the
near field. Further results will be given in the presentation.
-5
-10
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
2mm
3mm
-15
-20
-25
-30
-35
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
2mm
3mm
-15
Coupling Coefficient (dB)
Coupling Coefficient (dB)
-10
-20
-25
-30
-35
-40
-40
-45
0
2
4
6
8
10
12
0
2
Frequency (GHz)
Figure 3. Coupling coefficient for hole diameter 2mm,
l=1.2cm, a=3mm for various d
8
10
12
-15
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
2mm
3mm
-40
-45
-50
-55
-60
Coupling Coefficient (dB)
Coupling Coefficient (dB)
6
Figure 6. Coupling coefficient for hole diameter 2mm,
l=2.4cm, a=3mm for various d
-35
-65
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
2mm
3mm
-20
-25
-30
-35
-40
-45
0
2
4
6
8
10
12
0
2
Frequency (GHz)
4
6
8
10
12
Frequency (GHz)
Figure 4. Data for the same problem as Figure 3 but
without coupling between antenna and holes
Figure 7. Coupling coefficient for hole diameter 2mm,
l=1.2cm, a=3.5mm for various d
-15
-10
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
2mm
3mm
-15
-20
-25
-30
-35
-40
Coupling Coefficient (dB)
-5
Coupling Coefficient (dB)
4
Frequency (GHz)
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
2mm
3mm
-20
-25
-30
-35
-40
-45
0
2
4
6
8
10
12
Frequency (GHz)
Figure 5. Coupling coefficient for hole diameter 2mm,
l=0.6cm, a=3mm for various d
0
2
4
6
8
10
12
Frequency (GHz)
Figure 8. Coupling coefficient for hole diameter 2mm,
l=1.2cm, a=4.5mm for various d
-10
-5
-30
0.2mm
0.6mm
1mm
1.4mm
1.8mm
-40
-50
-60
Coupling Coefficient (dB)
Coupling Coefficient (dB)
-10
-20
-15
-20
0.2mm
0.6mm
1mm
1.4mm
1.8mm
-25
-30
-35
-40
-45
0
2
4
6
8
Frequency (GHz)
10
12
0
2
4
6
8
10
12
Frequency (GHz)
Figure 9. Coupling coefficient for vertical, 14 X 10mm
loop antenna for various d, 2mm hole diameter, a=3mm
Figure 12. Coupling coefficient for vertical, 14 X 10mm
loop antenna for various d, 4mm hole diameter, a=5mm
ACKNOWLEGMENTS
The authors are grateful to W. L. Gore & Associates for
providing us with resources for this research.
Coupling Coefficient (dB)
-5
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
2mm
3mm
-10
-15
-20
-25
-30
0
2
4
6
8
10
12
Frequency (GHz)
Figure 10. Coupling coefficient for hole diameter 4mm,
l=2.4cm, a=5mm for various d
Coupling Coefficient (dB)
-25
0.2mm
0.4mm
0.6mm
0.8mm
1mm
1.2mm
1.4mm
1.6mm
1.8mm
3mm
2mm
-30
-35
-40
-45
-50
0
2
4
6
8
10
12
Frequency (GHz)
Figure 11. Data for the same problem as Figure 10 but
without coupling between antenna and holes
REFERENCES
[1] A. F. Peterson, S. L. Lay and R. Mittra, Computational
Methods for Electromagnetics, ed. D.G. Dudley. 1997:
IEEE Press.
[2] R. F. Harrington, Field Computation by Method of
Moment. 1968.
[3] S. M. Rao, D. R. Wilton, and A.W. Glisson,
“Electromagnetic Scattering by Surfaces of Arbitrary
Shape”. IEEE Transaction, Antennas and Propagation, May
1982. vol. AP-30, No. 3, pp. 409-418.
[4] S. Criel, L. Martens and D. De Zutter, “Theoretical and
experimental near-field characterization of perforated
shields”.
IEEE
transactions
on
Electromagnetic
Compatibility, August 1994. vol. 36, No. 3, pp. 161-168.