Incompatibilities of the Shielding Theory

Incompatibilities of the Shielding
Theory
Abstract — The paper deals with highlighting the
incongruities appearing both in Schelkunoff's equation
as well as in the isomorphism attached to this model.
These things are important because many experimental
methods for determining shielding effectiveness are
based on this model. It has been observed, in a large
number of tests, that results do not match theory.,
especially in the area of high frequencies, where
wavelength ? becomes comparable to the shield
thickness d, as well as correct determination of
absorption and reflection loss.
I. INTRODUCTION
At present, there are many analytical and computing
models to solve, at least theoretically, the problem of
electromagnetic shi elding, meaning the determination
of the influence of a shield
(conductor/semiconductor, with/without magnetic
properties) on electromagnetic radiation. Mainly, two
have been used more often: Kaden model for
symmetrical structures and Schelkunoff model for the
infinite plane shield.
Schelkunoff model, the subject of this paper, is
considered the most accurate from the viewpoint of
the initial conditions and algorithm. Based on it many
testing methods have been issued, for different
materials, some of them generating civilian (ASTM)
or military (MIL) standards.
Recent research and tests performed in different
laboratories lead, more and more, to the conclusion
that even apparently correct, this model is not so
interesting for applications, due to a certain number
of problems:
- basic incongruities derived directly from the
theoretical background;
- limitations in practical applications;
- deficiencies of Schelkunoff isomorphism.
The paper aims to highlight all these three types of
problems and to revise the model where possible.
II. BASIC INCONGRUITIES
As known, Schelkunoff model gives a general
explanation for the attenuation introduced by an
infinite plane shield placed in free space, illuminated
on one face by an electromagnetic field (normal
incidence) in Fresnel or Fraunhofer zones. The model
Authors are with the Research Institute for Electrical Engineering
- ICPE,Splaiul Unirii 313, Bucharest 74204, ROMANIA, phone: +40 21
346 49 33, fax: +40 21 346 72 68, e-mail:mbadic@icpe.ro, mjm@icpe.ro.
Mihai Badic and Mihai-Jo Marinescu
is applicable from industrial frequency up to optical
domain, meaning the whole range of non-ionizing
radiation:
R
SE =A + R + B
dB
dB
dB
A = 20 log e
dB 10
ad
dB
( + Z )
2
Z 0 m
dB=
20 log
(1)
10
4 Z 0 Z m
Z m-Z
0
B dB=
10 - e
Z m+
Z 0
1 log 20 ⎟ ⎛ ⎞
⎜
⎝ ⎠
2
-2?d
In equation (1) [1], [2], SEdB is shielding
effectiveness, defined as the ratio between intensities
of electric/magnetic/electromagnetic fields without
respectively with shield, at normal incidence. The
other magnitudes are:
-Z0=F(µ0,e0) wave impedance in free space;
-Zm=F(µ,e,s,?) wave impedance in material
-?=a+jß=F(µ,e,s,?) propagation constant
-d=material thickness
According to the algorithm generated by the theory
given by Schelkunoff in 1943, SEdB is the sum of
three terms: AdB-absorption loss, RdB-reflection loss
and BdB-re-reflection correction. This mathematical
formulation does not represent accurately the
phenomenon. Thus, the reflection rules, at normal
incidence, applied on the incident surface, may be
written as:
WE I + WE R = WE T (for electric field) (2)
WH I + WH R = WH T (for magnetic field) (3)
meaning that on every boundary between
environments the sum of incident and reflected waves
amplitudes is equal to the transmitted wave amplitude.
Also, considering now the whole configuration,
following relationship must be satisfied:
WE I + WE R = WE T +WE A (for electric field) (4)
WH I + WH R = WH T + W H A (for magnetic field) (5)
containing, obviously, the wave absorbed in shield.
The way to deduce the correct relationship for
reflected, absorbed and transmitted waves,
17

considering multiple re-reflection phenomena on
surfaces separating the two environments, is done as
shown in Fig.1.
Adding the terms representing the three phenomena
in discussion, it is finally obtained:
18
2γd
R e −1
W = ± Γ 2γd
2
e − Γ
−γd
( ± Γ)(
1 − e )
(6)
γd
A
e
W = 1 (7)
γd
e m Γ
2γd
T
−γ
d e
W = ( 1−
Γ)
e
(8)
2γd
2
e − Γ
PHENOMENA :
1+ΓAM
e -γd
ΓAM Γ MA ΓMA
1+ΓMA
INCIDENT WAVE
1
ΓAM
(1+ΓAM) (1+ΓMA)ΓMA e -2γd
(1+ΓAM) (1+ΓMA) Γ 3 MA e -4γd
In the above equations we assumed incident wave
W I ≡1, G=GAM=-GMA representing the reflection
factor. The first algebraic sign is valid for the
electrical component and the second one for the
magnetic component. It can be easily observed that
above equations differ from the ones obtained using
the classical Schelkunoff algorithm, being identical
only in the case of transmitted wave. At the same
time, it can be verified that these expressions check
equations (4-5) - the laws of reflection and
transmission) - unlike the expressions from the
Schelkunoff model used now.
Equations (6-8) offer the possibility to check
experimentally the reflection and absorption
phenomena for the ideal case of infinite plane shield
placed in free space.
1+ΓMA
I AIR (A) II MATERIAL (M) III AIR (A)
1+ΓAM (1+ΓAM) e -γd (1+ΓAM) (1+ΓMA) e -γd
(1+ΓAM) ΓMA e -γd
(1+ΓAM) ΓMA e -2γd
(1+ΓAM) Γ 2 MA e -2γd
(1) (2)
(1+ΓAM) Γ 2 MA e -3γd (1+ΓAM) (1+ΓMA)Γ 2 MA e -3γd
(1+ΓAM) Γ 3 MA e -3γd
etc.
d
(1+ΓAM) Γ 3 MA e -4γd
Fig. 1. Algorithm for determining reflection and absorption loss in plane shield - far field zone.

III. EXTENSION OF PRACTICAL APPLICATIONS
Equation (1) was applied, up to present, only for
conductors/semiconductors with or without magnetic
properties. These considerations imply the following
condition:
σ ωε ⇔ Z >> Z
(9)
>> 0
As consequence, equation (1) becomes:
SE
dB
m
⎛
⎞
αd Z
⎜ 0
−2γd ≅ 20 log
⎟
⎜
e ⋅ 1−
e
(10)
⎟
⎝ 4 Z m ⎠
Considering that for metals
1 ωμσ
α ≅ β = =
(11)
δ 2
and detailing the respective modules results an
expression equivalent to (10), that is [3]:
⎧ d δ Z0σδ
−2d δ
−4d
δ ⎫
SEdB
= 20 log⎨e
⋅ ⋅ 1−
2e
cos2d
δ + e ⎬
⎩ 4 2
⎭
(12)
Equations (10) and (12) are used currently in the
experimental methods for determining shielding
effectiveness by means of coaxial TEM cells
according to ASTM ES7-83 and ASTM D4935-89. If
the shield material presents strong resonance
absorption regions so that loss becomes comparable
to the one characteristic to conduction, at least within
certain frequency range, then assumption s >>?e is no
longer valid and therefore all the above
approximations are not correct. In this case, modules
from equation (1) must be detailed, implying an
elementary but laborious calculation.
If we accept the separation of phenomena into
absorption, reflection and respectively re-reflection
correction, according to classical theory, then the
complete expression of (1) becomes as follows:
αd
ωε
A = 20 log e ; α = ωμ Δ ⋅ 0,
5 − 0,
5 (13)
dB
Δ
where
Δ =
σ
2
2
+ ω ε
2
⎡ Δ
ωε 1 ωμ ⎤
R = 20 log 0,
25⎢377
⋅ + 2 0,
5 + +
dB
⎥ (14)
⎢ ωμ
⎣
2 Δ 377 Δ ⎥⎦
B dB
2 2
A + B
= 20 log
(15)
2 ωμ ωε ωμ
377 + 2 ⋅ 377 ⋅ ⋅ 0,
5 + +
Δ 2 Δ Δ
Thus, we obtained expressions for reflection
absorption and respectively re-reflection loss for the
most general case of conductive dielectrics, all
macroscopic parameters e µ s having comparable
weights. In these equations A and B have the following
expressions:
2
2
ω με ω με −2αd
ωμσ −2
αd
A = − e cos 2βd
− e sin 2βd
+
Δ Δ
Δ
2 −2αd
+ 377 ( 1 − e cos2βd
) + 2 ⋅ 377
ωμ
⋅
Δ
ωε
0,
5 +
2 Δ
+
+ 2 ⋅ 377 ⋅
ωμ
⋅
Δ
ωε −2αd
0,
5 + e cos2βd
+
2 Δ
(16)
+ 2 ⋅ 377 ⋅
− 2 ⋅ 377 ⋅
+ 2 ⋅377
⋅
+ 2 ⋅377
⋅
ωμ
⋅
Δ
ωμ
⋅
Δ
ωμ
⋅
Δ
ωμ
⋅
Δ
ωε
0,
5 − e
2 Δ
−2αd
ωε
0,
5 + ⋅ e
2 Δ
ωε
0,
5 − e
2 Δ
ωε
0,
5 − e
2 Δ
sin 2βd
ωμσ ωμσ −2αd
B = − ⋅ e ⋅ cos 2βd
+
Δ Δ
2
ω με −2αd
2 −2αd
+ e sin 2βd
+ 377 ⋅ e sin 2βd
−
Δ
where a was given in (13) and :
−2αd
−2αd
−2αd
sin 2βd
+
+
cos 2βd
(17)
ωε
β = ωμ Δ ⋅ 0 , 5 + 0,
5
(18)
Δ
IV. SCHELKUNOFF ISOMORPHISM FAILURE
Besides its theoretical importance, the infinite
plane shield theory has generated the respective
isomorphism. Thus, it may be easily shown that
equation (1) is identical to the one describing the
functioning of a transmission line with impedance Z0,
having an insertion of a line segment with impedance
Zm and length d. This way, electric field E and
magnetic field H are replaced by voltage U
respectively current I. To macroscopic parameters of
shield (e, µ, s) correspond the parameters of the
transmission line. Shielding effectiveness SEdB is
replaced by the insertion attenuation IAdB given by the
Zm segment.
This isomorphism has generated, in its turn, an
experimental measuring method for shielding
capabilities of materials. The model simulates the
conditions of infinite plane shield in free space by
means of a washer shaped material introduced in a
TEM mode coaxial cell that is practically an expanded
coaxial cable. [4], [5], [6]. This system, according to
the respective ASTM standards, should function up to
frequency 1-1.5GHz, this being the limit
characterized by occurrence of higher modes (H11,
E01) in the TEM cell. The limit is determined by the
usual dimensions of the coaxial TEM cells.
By means of this kind of test, a typical theoretical
curve of shielding effectiveness should be obtained
for conductors/semiconductors (Fig.2)
17

Fig.2 Area A-verification of test equipment; Area B-verification of
Schelkunoff isomorphism.
The flat part of the curve (area A) correspond to
electrically thin samples (dδ).
Behavior of materials in area A is described by the
approximate equation (characteristic to electrically
thin samples, d