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A spectral domain approach.
Understanding the
Shielding Efficiency of a
Faraday Grid Cage
Bernhard Jakoby , Roman Beigelbeck, and
Thomas Voglhuber-Brunnmaier
The static shielding properties of Faraday wire cages are
intuitive as for a small mesh size, the effect on the field
can be expected to approach that of an ideal Faraday
cage, i.e., a closed conductive surface. However, as it has
been recently pointed out, the shielding efficiency is somewhat
worse than one might expect and does not particularly conform
to the simple approximation of an exponentially decaying field,
as it is, e.g., described in The Feynman Lectures on Physics. In
the present contribution, we use the case of a circular 2D wire
cage to illustrate how the residual field inside such a cage can
be visualized in terms of a spatial spectral consideration of the
induced charge. It is shown how the residual field in the cage’s
center is related to a single Fourier coefficient of this spectral
expansion, and that the approximation of the induced charge
as a sampled version of the induced charge of a corresponding
ideal Faraday cage yields useful approximations for the residual
fields close to the cage boundary. The latter also turn out to jus-
tify the exponential decay approximation, at least in this region.
INTRODUCTION
The first description of the screening of electric (and in par-
ticular, electrostatic) fields by metallic, or more generally,
conductive, cages is generally attributed to Faraday’s observa-
tions, as reported in his seminal monograph [1]. Today, Faraday
cages and their effect can be considered common knowledge,
being taught in schools and known to the common public, par-
ticularly in connection with protection from lightning strikes,
as illustrated by demonstrations in science museums or the
protective effect of sitting in a car during a thunderstorm. In
2016, Rajeev Bansal reported in IEEE Antennas and Propaga-
tion Magazine’s “Turnstile” column [2] that Chapman et al.
investigated the shielding efficiency of Faraday cages composed
of grids devising and using rigorous methods, finding that the
shielding is not as powerful as may be intuitively expected [3].
For a 2D circular cage, an external line charge in a distance d
from the center of the cage creates a remaining field in the cen-
ter of the cage, which is found to be approximately proportional
to
(/ )/ ,logrr nd
0 where
/rr
0
denotes the radius of the wires
the cage is made of (normalized to some reference value
)r0
,
and n is the number of wires forming the 2D cage. They also
pointed out that apparently, the shielding efficiency before had
scarcely been analyzed in a rigorous mathematical manner; at
least such an analysis cannot be found in common textbooks or
accessible journals, which was the stimulus for their own work.
In particular, an (approximately) exponential decay of the fields
inside the cage, which is commonly assumed, was proven wrong.
As an example, for such a simplified treatment, The Feynman
Digital Object Identifier 10.1109/MAP.2022.3229287
Date of current version: 5 January 2023
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3IEEE AN TENNAS & PROPAGATI ON MAGAZIN E MONTH 2023
Lectures on Physics [4] are cited stating the origin of the flawed
treatment is that “Feynman considers equal charges rather than
equal potentials” on the wires. Later in this article, we discuss
the context of Feynman’s treatment, illustrating that Feynman’s
conclusions can indeed be justified for large or infinitely extend-
ed wire grids as has also been pointed out by other authors [5].
The original analysis provided in [3] considered the elec-
trostatic case, which was later extended to the dynamic case
[6], where in both cases the devised concept of homogenized
boundary conditions was successfully employed. In particular,
it is confirmed that the static case can be a useful approximation
if the wavelength is large compared to the spacing of the grid
(unless spurious resonances are excited in the cage). It is also
pointed out that the treatment of lightning and sparking involves
complex nonlinear effects such as ionization, which is also not
covered by an electrostatic approach. The interested reader
seeking a thorough treatment of the problem is particularly
referred to these works.
In this contribution, we do not aim at such a rigorous analy-
sis but rather at developing an illustrative view on the problem
by utilizing a tool well-known to electrical engineers, i.e., Fou-
rier analysis.
As in [3], for the sake of simplicity, we consider the case of a
circular case in two dimensions and treat the purely electrostatic
case. This consideration, to a certain extent, also particularly
applies to the dynamic cases where the wavelengths involved
are much larger than the spacing of the considered grid. For the
sake of completeness, we also note that shielding of static and
quasi-static magnetic fields is a different (and quite challenging)
topic, see, e.g., [7].
In the following, we first introduce a mathematical descrip-
tion of the problem, where the field is described in terms of a
Fourier series expansion of the induced charges. We particularly
consider the residual field in the center of the cage and close
to the cage boundary, where the approximate assumption that
the charges in the grid wires represent a sampled version of
the charge induced in an ideal (closed) Faraday cage is particu-
larly useful. These considerations are illustrated by means of an
example.
STATEMENT OF THE PROBLEM AND
SPECTRAL CONSIDERATIONS
We consider 2D electrostatics in free space, i.e., in 3D space
everything is considered uniformly extended to infinity with
respect to one spatial dimension, say z, such that we can sim-
plify the 3D equations by setting
/.z022 =
In the following,
we use polar coordinates with radial and angular coordinates
r and
,a
respectively, where the coordinate center is located in
the center of the cage. Figure 1 shows a Faraday cage composed
of
N12=
circular conductive wires with diameters (widths) w
that are uniformly distributed on a circle
,rR=
i.e., with radius
R. In the general case, the wires shall be considered located
at angles
/nN2
n
ar=
with
.nN01f=-
We consider the
important case of thin wires, referring to the assumption that the
diameter w of each wire is significantly smaller than the spacing
between them and the dimension of the cage, which essentially
means that
/.wRN2%r
As in [3], we consider the field gener-
ated by a 2D point charge (which is a line charge in 3D) located
in a distance d from the cage’s center and lying on the x-axis as
excitation in our problem. Even though we consider static fields,
in this article we use the term incident field, commonly used in
electromagnetic scattering, to address this field contribution.
In electrostatics, the electric field is conservative and the
field vector E can be expressed as the negative gradient of the
electric potential
,{
i.e.,
.Ed
{
=-
In charge-free regions, such
as the region within the cage, the potential
{
fulfills Laplace’s
equation
.0T
{
=
When using polar coordinates, the potential
can be written as a function
(, )r{a
and will be
2-periodic
r in
a
such that it can be expanded in a Fourier series with respect
to
:a
(Note that two types of indices are used in this article.
Whenever an index refers to a spectral Fourier component, we
use the letter k, whereas n is used as the index for the cage wires
.)nN01f=-
(, )()()
() () .
exp
co
ss
in
rrjk
rk
rk
2
,
,,
k
k
cck sk
k
0
1
{a {a
{{a
{a
=
=+ +
3
3
3
=-
+
=
u
u
uu
^^ ^
hh
h
/
/
(1)
Here the imaginary unit is denoted by .j1=- The
r-dependent Fourier coefficient
()r
n
{
u
is thereby given by
() (, )( )exprrjk d
2
1
k
0
2
{
r
{a
aa=-
r
u
#
(2)
which is related to the coefficients of the alternative sine and
cosine representations by
( )
( )
.
k
jk
0
1
for
for
,
,
ck kk
sk kk
$
$
{{{
{{{
=+
=-
-
-
uuu
uuu
^h
(3)
Inserting the expansion (1) into Laplace’s equation
0
{
D=
in polar coordinates leads to a simple general solution for the
r-dependent Fourier coefficients
(),r
k
{
u
or alternatively,
()r
,ck
{
u
and
(),r
,sk
{
u
which, inside the cage, yields an r-dependence
proportional to
rk;;
[see “Supplement A” of the supplementary
material (available at 10.1109/MAP.2022.3229287), where the
case
k0=
is also discussed]. Each spectral coefficient corre-
sponds to a particular contribution to the total field.
Wires
Forming
the Cage
(Diameter w)
External Line
Charge Creating
Incident Field
y
x
r
R
d
α
s
ng
ge
g
e
r
g
w
)
E
xterna
l
Li
n
e
Charge Creatin
g
I
nc
id
ent
Fi
e
ld
y
x
r
R
d
α
FIGURE 1. A 2D circular wire cage and a line charge in
some distance d, serving as a source for the external field
to be screened.
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4IEEE AN TENNAS & PROPAGATI ON MAGAZIN EMONTH 2023
This already gives a clue as to what we need to identify in the
remaining field in the cage’s center. Close to the center, i.e., for
vanishing r, out of all the terms proportional to
,r||k
the asymp-
totically dominant one will be proportional to r, i.e., the one for
.k1!=
Thus, the associated spectral coefficients essentially
determine the remaining field close to the center.
Close to the cage, i.e., when r approaches R, the situation is
not as clear and depends on amplitude distribution of the indi-
vidual spectral coefficients.
The remaining fields close to the cage boundary as well as
those close to the center can serve as particular measures for the
screening efficiency of the cage. As outlined in the next section,
considering the field in the cage’s center is in line with the con-
clusions made in [3], whereas the fields just behind the wire grid
happen to support Feynman’s conclusion [4] to a certain extent,
as is further discussed.
To discuss all of this, we move on to look into the utilization
of the Fourier series expansion in a bit more detail.
THE FIELD INSIDE THE CAGE
The field remaining inside the cage can be represented as a
superposition of the incident field and the field generated by the
induced charges on the wires of the cage, which ideally cancel
out the former as well as possible. To determine the residual
field, the first step is to solve for the induced charges, and there
are a variety of analytical and numerical as well as approximate
methods to obtain these. Later in this section, we show some
simple approaches for illustrational purposes but will not par-
ticularly stress this point as there are many suitable ways for
calculating electrostatic fields, which are extensively discussed
in the literature. For the treated circular Faraday cage, we par-
ticularly refer to [3], which provides a more profound insight into
these topics.
For the present, let us assume that the charge distribu-
tion in the cage wires is already known, and we consider how
the residual field can be obtained. The charge distribution on
each wire is located at the wire’s surface and, in general, is not
uniformly distributed across the surface. This is in contrast to
the simple case of a single isolated charged wire with a circular
cross section where the surface charge would be uniformly
distributed. In this case, the field generated outside of the wire
by these surface charges is identical to that of an equivalent line
charge located at the wire’s center (replacing the wire). This
is a consequence of the equivalence principle of electrostatics
(see, e.g., [8]). Similarly, even though in our case the charge on
a particular wire’s surface will not be uniformly distributed, in
distances sufficiently large compared to the wire’s diameter w,
the field originating from the charges on a particular wire will
approach that of an equivalent line charge
q
n (for wire number
n) placed instead of the wire at its center location at the angle
/,nN2
n
ar=
which can be used to approximately evaluate the
field within the cage.
These equivalent line charges, which are located at positions
rR=
and
/nN2
n
ar=
(with
),nN01f=-
can, in turn, be
conceptually described in terms of a surface charge distribution
()va
located at the circle (cylinder in 3D)
rR=
() R
qn
N
12
n
n
N
0
1
va da r
=-
=
-
`
j
/ (4)
using Dirac delta functions
()da
to represent the line charges,
where
q
n are the associated line charge densities. The factor
/R1
maintains proper scaling, i.e., integrating the surface charge
density around the entire circumference yields the sum of the
line charges:
() .Rd qn
n
N
0
1
0
2va a=
r
=
-
/
# (5)
This representation nicely connects to the case of the ideal
Faraday cage, where a distributed surface charge distribution on
the circle
rR=
is obtained. As this surface charge exactly com-
pensates the incident field inside the circle, it can also be used
to represent the incident field in terms of an equivalent charge
distribution, which is simply equal to the negative ideal surface
charge density.
As discussed in “Supplement A” of the supplementary mate-
rial (available at 10.1109/MAP.2022.3229287), a Fourier series
representation allows for a convenient and simple representation
of fields and charges. Also, the surface charge representing the
equivalent line charges from (4) can be expanded into a Fourier
series yielding
exp
R
qj
nk N2
12
kn
n
N
0
1
vr
r
=-
=
-
u
`
j
/ (6)
as Fourier series coefficients, which are periodic in the dis-
crete variable k. One period, e.g., from
k0=
to
,kN1=-
corresponds to the spectral representation of the discrete
Fourier transform (DFT), which can be efficiently computed
using the fast Fourier transform (FFT) algorithm. (Note that
a periodic but continuous function corresponds to a discrete
spectrum, as described by the Fourier series. A periodic and
discrete function corresponds to a periodic and discrete spec-
trum, as described by the DFT. The FFT is a numerically
efficient way to calculate the DFT, which is why these terms
are sometimes used alternately.) Although the spatial charge
distribution in terms of line charge is discrete, the fields gener-
ated by these line charges are, of course, continuous, and to
calculate their spectral representation we stick to the classical
Fourier series.
In the following, we consider that the external field excit-
ing the cage is provided by a line charge in a distance d from
the cage’s center, as shown in Figure 1, which yields symmetric
fields and charges with respect to
.a
In doing so, the real-valued
representation of the Fourier series (1) is particularly handy
as the coefficients of the sine terms vanish, e.g.,
0
,sk
{
=
u
for
the potential. Also, using the resulting cosine series, we have
to deal with only positive spectral indices k, which eases the
notation in the following considerations, yet they similarly apply
to asymmetric cases. For this special case, the Fourier cosine
coefficients, e.g., of the surface charge density, are related to the
general coefficients by
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5IEEE AN TENNAS & PROPAGATI ON MAGAZIN E MONTH 2023
().k22 0
,ck kk
$
vvv
==
-
uuu
(7)
Under these assumptions, analogous to (1), we therefore
have
(,
)(
)().cos
rr
k
2
,,
cck
k
0
1
va
vva
=+
3
=
u
u
/ (8)
The zeroth-order coefficient
,c0
v
u
corresponds to the total
charge of the cage, which has to vanish in the case of electro-
static induction because the wires are connected and initially
uncharged. When calculating the charges numerically, a non-
vanishing, zeroth-order coefficient can thus be an indicator of
inaccuracies. As outlined in “Supplement A” of supplementary
material (available at 10.1109/MAP.2022.3229287), the poten-
tial s
{
due to the charges on the wire can be similarly expanded
in a Fourier series with respect to
.a
(Note that the subscript “s”
stands for “scattered,” indicating that this field is related to the
induced charges as opposed to the “incident” field generated by
the exciting charge.)
As a result, a simple solution ansatz for the r-dependent coef-
ficients is obtained, i.e.,
() ( ),
() ( )
rA
rr
R
rB
rr
R
inside the cag
ea
nd
outside of the cage
,,
,,
sckk
k
sckk
k
1
2
{
{
=
=-
u
u (9)
with yet-to-be-determined coefficients
Ak
and
.Bk
By virtue
of the orthogonality of the expansion functions in the Fourier
series (cosine functions in the present case), the spatial domain
interface conditions across sheets of surface charge translate
into corresponding conditions for every spectral component. In
particular, by virtue of Gauss’s law, the surface-normal compo-
nent of the electric displacement (in free space
)E
0
e
experi-
ences a jump when passing through a sheet carrying a surface
charge, which yields the following condition for the spectral
coefficients of potential and charge at the surface
:rR=
() () .
r
r
r
r
,, ,, ,
sck
rR
sck
rR
ck
0
2
2
2
2
{{
v
e
-=
=+ =-
uu
u
(10)
Together with continuity of the potential at
,rR=
i.e.,
,, ,,sck
rR
sck
rR
{{
=
=+ =-
uu
(11)
one can solve for the coefficients
Ak
and
,Bk
and the following
solution of the Fourier coefficient for the potential inside the
cage is obtained:
()r
R
rk
R
2
,, ,sck
k
ck
0
{v
e
=
uu
`
j
(12)
and therefore, by inserting these coefficients in the Fourier
cosine series expansion, we obtain the desired representation of
the field in terms of the generating charges
(,
)(
).cosrR
r
k
R
k
2,s
k
k
ck
1
0
{a
va
e
=
3
=
u
`
j
/ (13)
It is particularly useful that this approach can not only be used
to represent the field generated by the induced charges (the
“scattered” fields) but also the incident field. As the charge
density
id
v
induced in an ideal circular Faraday cage exactly
cancels out the incident field inside the cage and as
id
v
repre-
sents a distribution of surface charges on the surface
,rR=
the
incident field can be expressed in terms of these charges and
their Fourier series coefficients. “Supplement B” of the supple-
mentary material (available at 10.1109/MAP.2022.3229287)
summarizes some fundamental properties for the ideal circular
Faraday cage and also provides a closed-form expression for
Fourier coefficients of the induced surface charge density
,,id ln
v
u
for the special case when the incident field stems from a line
charge, as depicted in Figure 1 [see (28)].
HOW WELL DOES IT SHIELD?
The answer to this question depends on the criterion used to
evaluate it. It is near at hand to start with considering the field at
the center of the cage as it can be expected that the remaining
field will be smaller close to the center. But what can also be of
interest is how fast the field decreases right behind the screen
when moving toward the center. Let us have a look at these
cases.
CENTER OF THE CAGE
The electric field at the center of the cage
()r0=
can be fully
characterized in terms of its radial component [note that at
,r0=
we have the following simple relation between the polar
components of the electric field
(, )(,/),EE002
raar=-
a
and thus, all the information is contained in one of these
components]:
.E
rr
E,rr
sr
0inc
2
2
2
2
{{
=- =- -
= (14)
Here, s
{
denotes the electric potential generated by the
induced charges on the wire cage, and
E,inc r
denotes the r com-
ponent of the electric field generated by external charges, i.e.,
the line charge at
xd=
(and
)y0=
in our example.
E,inc r
is
given simply by the field of a line charge in distance d and read-
ily obtained as
.
cos
E
d
q
2
,r
0
inc
inc
re
a
=
-
(15)
Utilizing the Fourier series expansion (13) for the scattered field,
the derivative with respect to r can be taken for every element
of the series (representing a power series with respect to r), and
upon setting
,r0=
we are left with a single nonzero element of
the series, i.e.,
,k1=
yielding for the total field
(,
).
Eco
sc
os
d
q
0
22
,
rc1
00
inc
a
e
v
a
re
a=- -
u
(16)
So the remaining field in the center is essentially determined
by the first-order Fourier series coefficient
,c1
v
u
of the induced
charge distribution! This equation is exact, i.e., not an approxi-
mation, and holds for an arbitrary charge distribution along the
circle
;rR=
it also holds for the continuous charge distribution
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6IEEE AN TENNAS & PROPAGATI ON MAGAZIN EMONTH 2023
induced in an ideal Faraday cage, which is considered in “Sup-
plement B” of the supplementary material available at 10.1109/
MAP.2022.3229287. As can be seen in (28), the corresponding
Fourier coefficient for
k1=
indeed exactly cancels the incident
field as it should be as the charges induced in the ideal cage
cancel the incident field everywhere inside the cage
().rR1
[Note that (28) in “Supplement B” of the supplementary mate-
rial (available at 10.1109/MAP.2022.3229287) provides the two-
sided Fourier coefficient, which has to be multiplied by two to
obtain the corresponding coefficient of the cosine series.]
CLOSE TO THE CAGE BOUNDARY
Again, we can use the spectral Fourier representation for the
induced charges to establish the residual field. At first sight, it
is near at hand to assume that the induced charges in the wire
cage will closely resemble a “sampled” version of the continuous
charge distribution of the ideal Faraday cage
()
id
va
[see “Sup-
plement B” of the supplementary material (available at 10.1109/
MAP.2022.3229287)]. Indeed, as illustrated in this section and
in the example, this turns out to be a useful approximation close
to the cage boundary, however, its accuracy deteriorates when
the residual fields in the cage’s center (i.e., for
)rR%
are calcu-
lated. Figure 2 illustrates the idea of sampling and the resulting
spectra. The continuous charge distribution (upper-left plot),
which, by definition, is periodic in
,a
corresponds to a spectrum
of discrete Fourier coefficients. The charge distribution can be
“sampled,” yielding a set of N line charges uniformly distributed
at angles
/nN2
n
ar=
around the circumference
.rR=
In the
sampled version, the strength of each line charge is proportional
to the value of the continuous charge distribution sampled at the
location of the respective line charge, i.e., at the corresponding
angle
n
a
(lower-left plot). As it is well known from the spectral
analysis of sampled signals (see, e.g., [9]), the spectrum, i.e.,
the Fourier coefficients representing this arrangement of line
charges, is given by a periodic repetition of the coefficients asso-
ciated with the sampled (continuous) function. The coefficients
contained within one spectral period can also be obtained by
applying an FFT to the original charge coefficients.
The line charge densities
q
n associated with the sampled
version in (4) are given by
.
q
N
R
n
N
22
nid
.
r
v
r
`
j
(17)
The factor
/RN2r
represents the length of one of N equally
sized segments of the circle and relates the surface charge
density at the sampling location to an associated (approxi-
mate) line charge density. Obviously, for sufficiently large N,
the compensating field generated by such a distribution of
line charges can be expected to come arbitrarily close to the
continuous charge distribution on the ideal Faraday cage, and
also, the sampled version cancels out the incident field fairly
well. However, particularly for small N, the sampled ideal
charge distribution does not represent the actually induced
charges very accurately, as also pointed out in [3]. Thus, the
actually induced charge distribution on the wires tends to
compensate the incident external field not as well as it would
be possible, in principle, for an arrangement of line charges.
Yet, on second thought, the physical requirement determining
the charge distribution among the wires is not the cancellation
of the incident field inside the cage but the boundary condi-
tion of a constant potential at the wire surfaces, which obvi-
ously represents a different requirement.
Still, approximation of the sampled ideal charge distribution
provides some very valuable insights and turns out to be useful,
particularly when considering fields close to the cage boundary,
as illustrated further in this section. Considering the corre-
sponding discrete charge distribution as given in (17) according
to the spectral properties of sampled signals (see, e.g., [9]), the
resulting Fourier coefficients are
,
kk
mN
m
id
.
vv
3
3
-
=-
+
uu
/ (18)
i.e., the spectrum for the sampled version is given by the sum of
the periodically repeated spectra of the underlying continuous
function
id
v
with a period N, as presented in Figure 3. Note
that here we use the complex Fourier series (rather than the
cosine series), which allows for representing the impact of sam-
pling in a simpler fashion.
The spectrum is now not only discrete but also periodic
and, as mentioned previously, the coefficients within one period
()nN01f=-
can be efficiently obtained using the FFT
algorithm, taking the FFT of the discrete charges
q
n in spatial
domain. Depending on the width of the ideal charge distribu-
tion’s spectrum
,
,kid
v
u
the shifted spectra will, in general, overlap
to a certain extent (see also the discussion later in this section).
If we now want to obtain the field generated by the induced
charges, we employ (13). [We note that (13) refers to the Fourier
cosine expansion; for symmetric situations, these are related to
the complex series coefficients by (7).] Note that even though the
charge spectrum is periodic, the spectrum of the potential is,
in general, not periodic as the potential is a continuous function
of
a
in spatial domain. At the same time, due to the intrinsic
periodicity in spatial
()a
domain, the spectrum remains discrete.
Hence, the associated spectral coefficients of the potential for a
given r are also not periodic, which is represented by the fact that
σ(α)
FFT
–N+N
k
k
α
qn
2π
2π
α,n
|σk,sample|
~
|σk|
~
FIGURE 2. Spectra (Fourier series coefficients) of continuous
and sampled charge distribution. The spectrum associated
with the sampled distribution is a periodic repetition of the
original one.
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7IEEE AN TENNAS & PROPAGATI ON MAGAZIN E MONTH 2023
the kth series coefficient in (13) features a factor
(/ )/,rR k
k
which
is obviously not periodic in k.
To represent the incident field inside the cage, we now use
the previously mentioned fact that, inside the cage, the incident
field can be represented simply as the field that would be cre-
ated inside the cage by a fictitious charge distribution
inc
v
on
,rR=
which is equal to the negative ideal charge distribution,
i.e.,
() ().
incid
va va=-
This is the case as
id
v
would exactly
compensate for the incident field inside the cage.
If we now adopt our preliminary assumption that the
induced charges on the wires are approximately equal to a
sampled version of the ideal continuous distribution
(),
id
va
the total field inside the cage is approximately related to an
“effective” charge distribution whose spectrum is that of the
periodically repeated spectrum of the ideal distribution (rep-
resenting the sampled ideal charge distribution) as given by
(18) but omitting the fundamental period
,m0=
which is
cancelled out by adding the negative ideal charge distribution
() (),
incid
va va=-
which effectively cancels the “baseband”
of the periodically repeated spectrum. This is illustrated in
the bottom plot of Figure 3(c). Therefore, using the “sampling
approximation,” the residual field is related to the entire spec-
trum of the sampled charge distribution, except for the “base-
band” contribution
()k0=
in the series in (18).
This effective charge distribution
,kkinc
vv+
uu
can now be
used to calculate the residual field using (13). As previously
pointed out, this series for the potential inside the cage
()rR1
features coefficients proportional
(/ )/rR k
k
which, close to the
cage boundary
,rR=
essentially impose an additional decay,
with
/k1
for increasing k.
We now introduce a further approximation by assuming that
the spectral width of the ideal charge distribution
,kid
v
u
is small
compared to the shift parameter N in (18). This also means that
neighboring shifted spectra overlap only moderately (which is
also assumed in Figures 3 and 4). This condition will often be
fulfilled as a Faraday cage will typically feature a mesh width,
within which the external field will only moderately change,
which in a spectral domain translates into a narrow spectral
width of
,kid
v
u
compared to the number of samples N, which
also means that the sampling is more “dense.” Consider now
that apart from constant factors, the kth spectral coefficient of
the potential is essentially given by the associated spectral coef-
ficient of the charge times a k-dependent factor
(/ )/,rR k
k
as
given in (12). In our current consideration, the relevant charge
spectrum (including an effective charge distribution account-
ing for the incident field) is approximately given the periodically
repeated spectrum of the ideal charge distribution where the
fundamental period around
k0=
is almost entirely canceled
out (see Figure 4). The envelope associated with the factor
(/ )/,rR k
k
which has to be applied to obtain the spectrum of
the potential, is also indicated as a dashed line in Figure 4. The
shifted charge spectra are centered around
kmN=
with inte-
ger m. If the spectral widths of these shifted spectra are small,
which means that they are essentially concentrated around
,kmN=
instead of the factor k-dependent factor
(/ )/,rR k
k
we
can use an approximately constant factor
(/ )/rR mN
mN
to be
applied to each of these partial, shifted spectra. This in turn
means that upon transformation back into spatial
()a
domain,
the mth partial spectrum (shifted by mN) is weighed with a con-
stant factor
(/ )/rR mN
mN
and, using the Fourier shift theorem
for the inverse transform, results in the ideal charge distribution
()
id
va
times an additional phase term
(),expjmN
a accounting
for the spectral shift by mN. Combining the terms for
,m!
the
phase terms yield a cosine function
()cos mNa
for each partial
spectrum such that the entire spectrum is given by
(, )(
)(
)
()
()
cos
ln cos
rRmN mN
R
N
N
2
12
mN
m
NN
1
0
2
0
id
id
.{a va ta
va
e
tt a
e
=-
+-
3
=
6
@
/
(19)
where
/rR
t
=
and
ln
denotes the natural logarithm. Note that
the series starts at
m1=
instead of zero, which accounts for
the canceled baseband contribution. The closed-form solution
for the series was obtained by considering a related geomet-
ric series for the complex variable
(),expZj
N
N
ta
= where
the original series can be represented as the real part of the
indefinite integral of the geometric series for Z with respect
to Z. This approximation is particularly appealing as it can
be established for arbitrary incident fields as the ideal charge
distribution
()
id
va
can be readily expressed in terms of the
–N+N
–N+N
k
k
k
(a)
(b)
(c)
–
–
–
–
–
–
N
+
+
+
+
+
+
N
k
k
k
k
k
k
k
k
k
|σid|
~
|σid,sample|
~
|σid,sample + σinc|
~~
FIGURE 3. (a) The spectrum of the charge distribution
induced in the ideal Faraday cage, (b) the sampled
ideal distribution, which may serve as an, albeit coarse,
approximation of the actually induced charges, and (c) the
spectral sum of the sampled distribution plus the equivalent
charge distribution
inc
v
u
generating the incident field
inside the cage. Note that the Fourier coefficient for
k0=
vanishes due to the charge neutrality, i.e., the total charge
on the cylinder vanishes. The spectrum of the sampled
charge distribution is given in terms of a periodic repetition
of the spectrum of the function being sampled. As can be
seen, when adding the distributions id,sample
v
u
and
,
inc
v
u
the
“baseband” contribution cancels out such that the residual
field is entirely given in terms of the shifted spectra in this
approximation.
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8IEEE AN TENNAS & PROPAGATI ON MAGAZIN EMONTH 2023
incident field at the cage boundary [see (26)]. The existence of
this closed-form solution shall, however, not cloud the main
point, i.e., that the residual field behind the cage boundary (and
its decay), by virtue of considering the sampling approximation,
can be directly related to the spectral repetitions of the ideal
charge distribution.
A simpler, yet less accurate approximation is obtained by
keeping only the first term of the series featuring the slowest
decay for decreasing
,t
yielding
(, )(
)(
).cosr
N
R
N
N
0
id
.{a va
ta
e
(20)
The electric field components can be readily obtained from
the potential by
E
r
r
2
2{
=- (21)
and
E
r
1
2
2
a
{
=-
a (22)
which shows that the field components show an r-dependence
.r()N1
- Note again that this approximation is particularly useful
to describe the fields close to the screen, while it becomes par-
ticularly inaccurate when
,r0"
where it predicts that the field
strength vanishes entirely.
As shown in the previous section, the field at the center is
essentially determined by the first spectral coefficient of the
induced charge [see (16)]. If the charges on the wires were
actually sampled versions of the ideal continuous charge distri-
bution, the field at the center would indeed almost completely
vanish as the field generated by the “baseband” contributions of
the periodically repeated charge spectrum, i.e., the term asso-
ciated with
m0=
in (18), is exactly canceled by the incident
field. Referring to (16), for vanishing
Er
at
,r0=
the first
Fourier coefficient of the induced charge density would have
to be
/.qd2
,c11inc
vv r
==-
uu
Referring to (28) in “Supple-
ment B” of the supplemental material (available at 10.1109/
MAP.2022.3229287), we find that this is exactly the first Fou-
rier coefficient of the charge distribution of the ideal Faraday
cage, i.e.,
.
,1id
v
u
Yet, the sampled ideal charge distribution would
have additional, albeit small, contributions to
,
1
v
u
stemming
from the shifted spectra, i.e., the terms associated with
m0!
in (18), which is why the field in the center would not exactly
vanish, even if the sampled ideal charge distribution were
induced in the cage wires. Hence, as discussed in the previous
section, if one wants to know the remaining field in the center
of the cage, the first spectral coefficient has to be determined
exactly, which is not feasible with the sampling approximation,
as will also be illustrated in the examples presented in the next
section. When moving close to the cage boundary
,rR
"
^h
however, the contributions of the spectral “baseband” become
less prominent. Therefore, the sampled ideal charge approxima-
tion and others based thereon can at least qualitatively describe
the field behavior. In particular, we can expect a dominating
r-dependence
(/ )rR N
for the potential, as shown in (20).
If we increase the cage size
R"3
but keep the distance
(or arc length) between the wires, i.e.,
/,aRN2r=
constant,
we can introduce a scaled-coordinate
()/,Rra
p
=-
which,
for increasing
,02
p leads from the cage boundary to the
inside to the cage. The dominant dependence
(/ )rR N
turns into
(/),
N
12 N
rp- which, in the limit
,N"3
yields an exponential
decay
()
lim exp
N
122
N
N
rp rp
-=-
"3
cm
(23)
which corresponds to the statement given in The Feynman Lec-
tures on Physics [4]. This means that Feynman’s approximation
–4N–3N–2N–N+N+2N+3N+4N
Small Residues
in “Base-Band”
Contribution Approximately
∝ cos (α)
Contribution Approximately
∝ cos (2α)
k
|σ
∼ + σ
∼inc||n|
r
R
|n|
1
–
4
N
–
3
N
–
2
N
–
N
+
N
+
2
N
+3
N
+
4
N
Small Residues
in
“
Base-Band”
Contribution Approximatel
y
∝
cos
(
α
)
α
k
|
σ
∼
+
σ
σ
∼
i
nc
|
|
n
|
r
R
|
n
|
1
FIGURE 4. The spectrum of the actually induced charge distribution
v
u
plus the equivalent charge distribution
inc
v
u
representing the incident field. As the real induced charge distribution does not exactly correspond to the sampled ideal
charge distribution (as considered in Figure 3), the “baseband” contribution is not entirely canceled out by
.
inc
v
u
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9IEEE AN TENNAS & PROPAGATI ON MAGAZIN E MONTH 2023
is still valid for a large cage if the cage’s wire spacing is small
enough for the sampling approximation to be made.
Still, deep inside the cage, the approximation is not as good
as there are substantial contributions from the remaining charg-
es in the “baseband” present (see Figure 4), which are small
but yield fields, which do not decay as strongly, as will also be
illustrated by the examples in the next section. In “Supple-
ment C” of the supplementary material (available at 10.1109/
MAP.2022.3229287), relevance of the approximations made is
briefly summarized.
EXAMPLES
In this section, we show the shielding efficiency and application
of the different approximations presented for a simple exam-
ple. As we are not so much interested in the shielding effi-
ciency itself (which has been studied in the references given
earlier) but rather demonstrate the degree of validity of the
approximations, and particularly, the underlying consideration
of periodically repeated spectra associated with the sampling
approximation, we refrain from extended parameter studies.
All the calculations were performed using scaled quantities,
where the scaled line charge density l
t
t
is related to the line
charge density l
t
by
/( )2
ll 0
ttre
=
t
with the vacuum permit-
tivity
.
0
e
For excitation by a single line charge, we set l
t
t
to
unity yielding associated units for the resulting fields (labeled
as “a.u.” in the plots).
In particular, we consider a cage consisting of
N30=
wires,
which will be excited by the field of a positive line charge in a
distance d from its center (see also Figure 1). Alternatively, we
also briefly consider a uniform external field afterward. The
radius of the wire cage amounts to
/,rd2=
and the diameter
of the wires is
. .wd001=
This means that the wire diameter
amounts to roughly 10% of the spacing between the wires. The
resulting potential landscape is shown in Figure 5 for the case
of an external uniform field (oriented in a –x direction) and for
the field generated by the external line charge. In both cases, it
can be seen that the potential inside the cage is fairly constant,
corresponding to a vanishing electric field. Also, the location of
the wires is clearly visible as they pin the potential to a constant
value, i.e., the potential of the cage.
Solving for the charges on the wire, we compare the resulting
Fourier coefficients of the equivalent line charge array with the
Fourier coefficients of the induced continuous charge distribution
in an ideal Faraday cage, which is shown in Figure 6. As discussed
previously in this section, the coefficients
k
v
u
approximately repre-
sent a superposition of spectra of
,id
vk
u
shifted by
.mN!
The field around the cage in terms of the potential illus-
trated in Figure 5 was obtained using the numerical method
outlined in “Supplement C” of the supplementary material
available at 10.1109/MAP.2022.3229287. In Figure 7, we show
the corresponding radial field strength
Er
according to the
numerical solution and the approximate closed-form solution
corresponding to (19). The field was obtained in polar
(, )ra
coordinates and is plotted in 3D surface plots using the polar
coordinates as independent axes. The figure qualitatively
suggests that the approximation is reasonable. To investigate
this in more detail, we plot
Er
versus
a
for three different r:
. ,. ,rRrR099093==
and
. .rR087=
It can be seen that
accuracy of the approximation deteriorates quite quickly by
increasing the distance
Rr-
from the cage boundary
,rR=
yet the magnitude of
Er
also decreases rapidly (consider the
y-axis scaling of the three plots) such that this relative error is
not shown as clearly in the plot given in Figure 7, i.e., the abso-
lute error is comparatively small everywhere.
This behavior can also be seen when plotting
Er
versus r; we
did this for two fixed angles
:a
0a=
corresponding to the axis
formed by the cage’s center and exciting the line charge (where
the strongest field
Er
can be expected), and
.()0 125 2#ar=
corresponding to a 45° angle to this axis. In the first case,
the approximation appears reasonable, but a closer inspection
reveals that the residual
Er
at the center of the cage
()r0=
is
not reproduced; here the approximation yields a zero field. For
.(),0 125 2#ar=
the approximation is qualitatively worse but
again, the fields are much smaller in absolute value.
Figure 8 also depicts the characteristics for
,0a=
again
with a magnified inset highlighting the behavior of small
1
0.5
0
ϕ (a.u.)
ϕ (a.u.)
y/dx/d
–0.5
–1
1
0
–1 –1
0
1
1
0.5
0
y/dx/d
–0.5
–1
1
0
–1 –1
0
1
y
/
y
d
0
0
y
/
y
d
0
0
FIGURE 5. The electrostatic potential around a 2D circular
wire cage
(N30=
wires) exposed 1) to a uniform electric
field oriented in the −x direction (the radius of the cage is 0.5
in used scaled units) and 2) to the field generated by a line
charge at
,xd=
where the radius of the cage is 0.5 d and the
diameter of the individual wires is
..wd001=
These results
were obtained using a numerical simulation, as described in
“Supplement C” of the supplementary material (available at
10.1109/MAP.2022.3229287). a.u.: arbitrary units.
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10 IEEE AN TENNAS & PROPAGATI ON MAGAZIN EMONTH 2023
field strengths at the origin. The closed-form approximation
approaches zero while the numerical result yields a small, yet
nonzero, residual field. The dashed-dotted line indicates the
field in the center according to (16), which corresponds to the
first spectral coefficient. The numerical result asymptotically
approaches this value for
.r0"
Note that this residual field is
related to the difference of the two first
()k1=
spectral coef-
ficients shown in Figure 6.
Figures 9–11 in “Supplement D” of the supplemental mate-
rial (available at 10.1109/MAP.2022.3229287) show similar
plots for the angular field component
Ea
conforming to the
same observations. Note that
0a=
was not selected as the
scan line for the plot versus r as
Ea
vanishes along this line;
instead,
.()0 017 2#ar=
was used, which corresponds to
the radial line, which is approximately between the first two
wires of the cage. Furthermore, Figures 12–17 in “Supple-
ment D” of the supplemental material (available at 10.1109/
MAP.2022.3229287) show corresponding plots for a uniform
incident field.
CONCLUSIONS
The shielding efficiency of a circular Faraday cage with
respect to arbitrary external (“incident”) electrostatic fields
was considered as a 2D problem, demonstrating the applica-
tion of a Fourier approach to represent the remaining field
Numerical Solution (Line Charge Excitation)
Closed Form Approximate Solution (Line Charge Excitation)
r/R = 0.99
r/R
r/R = 0.93
r/R = 0.87
10
5
0
–5
0.5
–0.5
0
0.1
–0.1
–0.2
–0.30 0.2 0.4 0.6 0.8 1
0
0 0.2 0.4 0.6
α/(2π)
α/(2π)
0.8 1
0 0.2 0.4 0.6
α/(2π)
α/(2π)
0.8 1
Er (a.u.)
Er (a.u.)
Er (a.u.)
10
5
0
–5
1
0.5
000.2 0.4 0.6 0.8 1
Er (a.u.)
r/R
α/(2π)
10
5
0
–5
1
0.5
00
(a) (b)
0.2 0.4 0.6 0.8 1
Er (a.u.)
Numerical Solution
Closed-Form Approximation
Numerical Solution
Closed-Form Approximation
Numerical Solution
Closed-Form Approximation
FIGURE 7. (a) The radial field component
Er
inside the cage upon excitation by an external line charge versus the polar
coordinates r and
a
and (b) versus
a
for different constant radii r. For the (a) plots, the upper plot gives the numerical solution
and the lower plot the approximation according to (19).
Ideal Charge Distribution |σ
∼id,k|
Grid of Equivalent Line Charges |σ
∼k|
1
0.8
0.6
Fourier Coefficients of Charge
0.4
0.2
0
051015
k
20 25 30
FIGURE 6. The Fourier coefficients
k
v
u
corresponding to the
equivalent line charges (representing the charges on the
cage wires) yield a periodic pattern of Fourier coefficients,
which can be obtained using the FFT algorithm. The Fourier
coefficients of the induced charge distribution
,kid
v
u
of the
corresponding ideal Faraday cage (a conducting cylinder)
are similar for low n. The coefficients
k
v
u
approximately
correspond to a superposition of periodically repeated
patterns of
,nid
v
u
as the induced charges on the wires
are approximately equal to the sampled ideal charge
distribution. (The coefficients are all real and negative; the
plot shows their absolute value.)
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11IEEE AN TENNAS & PROPAGATI ON MAGAZIN E MONTH 2023
in the cage, where fields and charges were expanded into
a Fourier series with respect to the angular coordinate. It
turns out that the remaining field in the center of the cage is
directly related to the first-order Fourier series coefficient of
the charge distribution induced in the cage wires. A simple,
approximate consideration where the induced field was
expressed as a sampled version of the continuous charge dis-
tribution of an equivalent ideal Faraday cage (a conducting
cylinder) showed that the field close to the cage boundaries
can be expressed in terms of the shifted spectra that were a
result of the sampling. Based on this consideration, a simple,
closed-form approximation was established. The approach
also explained to what extent the often-disputed claim that
the field decays exponentially behind the boundary can be
maintained.
ACKNOWLEDGMENT
Roman Beigelbeck is grateful for stimulating discussions
with Andreas Kainz and other researchers involved in grant
3619S92411 from the German Federal Office for Radiation
Protection. This article has supplementary downloadable mate-
rial available at 10.1109/MAP.2022.3229287.
AUTHOR INFORMATION
Bernhard Jakoby (bernhard.jakoby@jku.at) is a full professor
with the Institute for Microelectronics and Microsensors at the
Johannes Kepler University, A4040 Linz, Austria. He has
worked in academic research in Belgium and the Netherlands as
well as in the German automotive industry. He is a member of
the Austrian Academy of Sciences and a Fellow of the IEEE.
Roman Beigelbeck (roman.beigelbeck@donau-uni.ac.at) is
a senior scientist at the Department for Integrated Sensor Sys-
tems, the University for Continuing Education, A2700 Krems,
Austria. He has (co)-authored more than 150 peer-reviewed
papers. His current research comprises (semi)-analytical model-
ing techniques for sensing and thin-film characterization
applications.
Thomas Voglhuber-Brunnmaier (thomas.voglhuber
-brunnmaier@jku.at) is a senior researcher with the Institute for
Microelectronics and Microsensors at the Johannes Kepler
University Linz, A4040 Linz, Austria. Since 2008, he has been
working on the modeling of microsensors for liquid property
sensing.
REFERENCES
[1] M. Faraday, Experimental Researches in Electricity, vol. 1. Redditch, U.K.:
Read Books Ltd., 2016.
[2] R. Bansal, “Shielded no more? [Turnstile],” IEEE Antennas Propag. Mag.,
vol. 58, no. 4, pp. 88–89, Aug. 2016, doi: 10.1109/MAP.2016.2569409.
[3] S. J. Chapman, D. P. Hewett, and L. N. Trefethen, “Mathematics of
the Faraday cage,” SIAM Rev., vol. 57, no. 3, pp. 398–417, Sep. 2015, doi:
10.1137/140984452.
[4] R. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Phys-
ics, vol. 2. Reading, MA, USA: Addison-Wesley, 1964.
[5] K. Subbarao and S. Mahajan, “Is Feynman’s analysis of electrostatic screen-
ing correct?” 2016, arXiv:1609.05567.
[6] D. P. Hewett and I. J. Hewitt, “Homogenized boundary conditions and reso-
nance effects in Faraday cages,” Proc. Roy. Soc. A, Math., Phys. Eng. Sci., vol.
472, no. 2189, May 2016, Art. no. 20160062, doi: 10.1098/rspa.2016.0062.
[7] T. Sumner, J. Pendlebury, and K. Smith, “Convectional magnetic shielding,”
J. Phys. D, Appl. Phys., vol. 20, no. 9, p. 1095, Sep. 1987, doi: 10.1088/0022
-3727/20/9/001.
[8] K. Simonyi, Foundations of Electrical Engineering: Fields–Networks–Waves.
New York, NY, USA: Pergamon, 1963.
[9] J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication.
Berlin, Germany: Springer Science & Business Media, 2012.
[10] M. Zahn, Electromagnetic Field Theory: A Problem Solving Approach. New
York, NY, USA: Wiley, 1979.
[11] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products,
5th ed. San Diego, CA, USA: Academic, 1994.
[12] R. E. Collin, Antennas and Radiowave Propagation. New York, NY, USA:
McGraw-Hill, 1985.
12
10
8
6
0.2
0.20 0.4
0.2010.4
(a) (b)
0.6 0.8
–0.2
0
4
Er (a.u.)
Er (a.u.)
2
0
–2
0
–0.05
–0.1
–0.15
–0.2
–0.25
–0.3
Line Charge Incident Field, α/(2π) = 0 Line Charge Incident Field, α/(2π) = 0.125
r /R
0.2
01
0.4 0.6 0.8
r /R
Numerical Solution
Closed-Form Approximation
Numerical Solution
Closed-Form Approximation
Field in Center From
First Fourier Coefficient
FIGURE 8. The radial field component
Er
inside the cage upon excitation by an external line charge versus r for different
constant angles
.a
These plots again represent cross sections through the ones given in Figure 7(a). The inset in (a)
()0
a
=
shows a zoomed-in version illustrating the (limited) accuracy of the approximation close to the cage’s center, showing
the validity of the center field approximation (16) at the same time as indicated by the “Field in Center From First Fourier
Coefficient.”
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