952ISSN 1064-2269, Journal of Communications Technology and Electronics, 2016, Vol. 61 , No. 8, pp. 952956. Pleiades Publishing, Inc., 2016. [603007]

952ISSN 1064-2269, Journal of Communications Technology and Electronics, 2016, Vol. 61 , No. 8, pp. 952–956. © Pleiades Publishing, Inc., 2016.
Study on the Validity of the Formalism of Norton to Describe
the Propagation of Electromagnetic Waves on the Surface of the Body1
R. Barake, A. Harmouch, and M. Kenaan
Lebanese University; Faculty of Engineering, Doctoral Sc hool, El Miten Street, P.O. Box 210, Tripoli, Lebanon
e-mail: [anonimizat]
Received September 2, 2015
Abstract —In this paper, we focus on the study of electromagnetic waves that allow the communication
between two antennas located on the surface of the human body. The link between the antennas leads to theappearance of space wave, whose level increases with height above the skin, and a surface wave which, on thecontrary, decreases with the altitude. We use the wave propagation model of Norton with the notion ofSomerfield’s attenuation. We give the limits of validity of this formalism in the area containing mainly surfacewaves. To generate this type of waves, the excitation sources used are small elementary electric dipoles with
normal polarization.
Keywords : Norton wave, surface wave, wireless body links, body centric antenna, on-body propagation
DOI: 10.1134/S1064226916080039
INTRODUCTION
The body surface is not just the envelope that is the
interface between the outer and inner environments. Itis home to many sensory endings allows the collection
of useful information for controlling health, optimiz-
ing the physical activity, and monitor the chronic con-ditions. It covers the entire surface of the body which
can be viewed as source of information (potential,
temperature, cutaneous blood flow, etc.) [1, 2], whichp r o v i d e s g u i d a n c e o n m a n y o r g a n s . I t a l l o w s c o n –
trolling and constantly monitoring the vital criteria for
inpatients failures for the elderly and physiologicalparameters of stress for sports people [3].
1. ATTENUATION FUNCTION
OF SOMMERFELD
It seems essential to analyze the attenuation func-
tion of Sommerfeld that describes the behavior of sur-face of TM- and TE-modes. The attenuation function
of Sommerfeld is expressed formally
, ,
where N is the normalized wave number defined by
, is the complex wave num-
ber; is the free space wave number; f is the
operating frequency; c is the speed of light; ε
0 is the
free spase electric permittivity and εc is the absolutecomplex permittivity defined by ,
where is the material conductivity and ω = 2π f is the
angular frequency. It is also important to note that
“erf” is the complex error function, and r is the path
travelled by the surface wave on the human body
model [7].
The height of the elementary dipole antenna is very
small, the reflection coefficient approaches 1, and theonly propagation mechanism is proposed along the
surface [6]. Norton wave describe the difference
between the field of geometrical optics and the realfield. In our case, the elementary dipole is placed on
the body with a vertical polarization. The electric field
can be expressed by the formula:
,
where the z-component of the electric field is
radiated by the elementary dipole antenna as a func-
tion of distance r; and is the initial value of
the electric field component in the vicinity of the
antenna; and is the Somerfield’s attenuation
function.
The error function of a real number is implemented
with limited development to ten nonzero coefficients
of the polynomials. For < 0.3 the polynomial is
given by the Taylor approximation as
, where Z is consid-
ered to be In the case where 0.3< < 6, the
1The article is published in the original.() () () 1j π e x p 1e r f Fp p p jp ⎡⎤ =− −⎣⎦0
22jk rp
N−≈
0 c Nk k=00  cckk εε ≈
02 kf c=π()ωc j ε= ε + σ
 σ
() ( )() 0zzEr Er F p ≈=
() zEr
( )0zEr =
() Fp
  jp
()() 22
11 e x p 1  !n
n
nZZn∞
=−−= + ∑
()    .  jp   jpELECTRODYNAMICS
AND WAVE PROPAGATION

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 61 No. 8 2016STUDY ON THE VALIDITY OF THE FORMALISM OF NORTON 953
result of is given by the total of 57 polynomi-
als of the ninth order in an interval of length 0.1 to sat-
isfy a double precision. For > 6, this application
has a value of ±1 as . In addition, if
the support consists of biological tissues, the error
function is defined as a complex permittivity and each
partial function is evaluated only if it contributes to the
result [8]. The evolution of the field E(z) with the dis-
tance can be estimated in several attenuation forms.
This evolution depends on | p| which in turn
depends on the frequency, the permittivity, the con-
ductivity and the distance. The propagation studies
are done on equivalent models to the body parts (arm,thigh, abdomen, etc.) that have an average length of
35 cm.
2. APPLICATION ON BILAYER MODEL
Figure 1 show an elementary dipole located on the
surface of a dielectric body. The electric field propa-
gates on the surface by performing cylindrical or ellip-tical rotation in the direction of propagation. It is
found that the evolution module of the field E
z essen-
tially depends on the permittivity, the conductivity,
and the frequency of the wave used. It can be con-
cluded that the attenuation of the field function of the
distance depends mainly on the value of | p|.
In the case where | p(λ)| < 1, λ = 2π c/ω is a wave-
length, an approximation of the normal field, from
which the E-field attenuation is determined, is given
by the following correlation:
,( 1 )
where I is the current across the elementary dipole and
Δl is known as the length of the elementary dipole
antenna used on the body surface (see Fig. 1), so that
IΔl is understood as the differential current element
over the short dipole of length Δl, and is the free
space magnetic permeability.
In the case where | p(λ)| > 1, at the working wave-
length, the approximation of the normal field, from() erf   jp
  jp
( ) 1e r f     jp −< ε
()()2
00
3
0μω e x p
2πc
zkj k r I lEr
kr−Δ≈−
0μ which the E-field attenuation is given by the following
correlation:
(2)
In the case where , the approximation of
the normal field presents an attenuation model
between the formulas (1) and (2).
3. THEORETICAL STUDY
ON THE ATTENUATION MODEL
Various attenuation formulas have been proposed
in the previous paragraph. To determine the validity ofNorton formulas, electromagnetic simulations were
pe r f o r me d w i t h t h e s o ft w a r e C S T M WS , o n a n e l e –
mentary dipole placed on the surface of a dielectricbody. The physical properties of the dielectric were
chosen for all cases of desired | p|.
Attenuation models of (1) and (2) and the Norton
model, for | p| smaller and larger than 1 are shown in
Figs. 2 and 3 respectively.
Figure 2 shows the attenuation model of equation (1)
(triangular dot) with an attenuation 1/ r, while equa-
tion 2 (cross dot) shows an attenuation 1/ r
2. For values
of |p| < 1, the Norton formula (spherical dot) has an
attenuation similar to equation (1) (Fig. 2). In the case
where | p| > 1 Norton has an attenuation similar to that
of equation (2) (Fig. 3).
It can be concluded that the formula of Norton fol-
lows the model of equation (1) (attenuation 1/ r) for
values of | p| < 1 and it follows the model of equation (2)
(attenuation 1/ r2) for values of | p| > 1.
In cases where , the formula of Norton pres-
ents an attenuation model between the models pre-
sented by equation (1) and equation (2) as shown in
Fig. 4.()()2
00
32
0μω e x p r.
2πc
zkj k I lEr
kr−Δ≈−
()1 pλ/asymptequal
1p/asymptequalFig. 1. Elementary dipole on the surface of a dielectric.DielectricFree spaceSmall dipole
Ez
I Δldz
E
Fig. 2. Attenuation model of field on the surface for | p| <1.–20020406080100
00.05 0.10 0.15 0.20 0.25 0.30 0.35Ez(r), dB
r, mAttenuation in Eq. 1
Attenuation in Eq. 2
Attenuation in Norton
0.40

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JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 61 No. 8 2016BARAKE et al.
4. SIMULATION RESULTS
To validate the theoretical study presented in the
previous paragraph, we make numerical simulationsusing CST MWS. Figure 5 shows the elementary
dipole placed on the surface of the dielectric having a
parallelepiped form.
The value of | p| depends primarily on the frequency,
the permittivity and the conductivity. Table 1 showsthe selected values of the physical properties in order
to obtain values of | p| less than 1.
Figure 6 shows a comparison between equation (1),
the Norton model, and the simulation result obtained
by CST MWS (cross dot). The results, not shown, ofall cases listed in Table 1 but they gave similar attenu-
ation models as equation (1).
Now the models are compared for values of | p| > 1.
It is important to note that the layers of human tissue
match these conditions. Table 2 shows the physicalproperties of different layers on several frequencies [9].
Figure 7 shows a comparison between Eq. 1, the
simulation result, and the Norton model for | p| > 1.
The simulation result given by CST (cross dot) showsgood agreement with the Norton model. We can con-
clude that the Norton model follows the model of
equation (2) presented in Section III, and thus follows
an attenuation of 1/ r
2 in the case | p| > 1, which is not
the case for | p| < 1.
We note that all cases listed in Table 2 showed sim-
ilar results as in Fig. 7.
Table 3 present the physical properties that we have
chosen for values of | p| close to 1 for an arbitrary tissue
under specific conditions at two different frequencies.
Figure 8 shows a comparison between Eqs. (1) and
(2), the Norton model and the results of simulationperformed by CST MWS.
Note that the Norton model presents a good agree-
ment with the model obtained by simulation. It can be
concluded that the attenuation models presented in
Eqs. (1) and (2) cannot be generalized in the case
where | p | is larger or smaller than 1.Fig. 3. Attenuation model of field on the surface for | p|> 1.–40–20020406080100
00.05 0.10 0.15 0.20 0.25 0.30 0.35Ez(r), dB
r, mAttenuation in Eq. 1
Attenuation in Eq. 2
Attenuation in Norton
0.40
Fig. 4. Attenuation model of field on the surface for | p| .–40–20020406080100
00.05 0.10 0.15 0.20 0.25 0.30 0.35Ez(r), dB
r, mAttenuation in Eq. 1
Attenuation in Eq. 2
Attenuation in Norton
0.40
1/asymptequalFig. 5. Vertically polarized dipole on a body surface
(top view).
z
xElectric dipole
1
Table 2. Physical properties for | p| > 1
TissuesFrequency
(GHz)Permittivity
(ε)Conductivity
σ (S/m)
Fat 10 4.6 0.58
Skin 20 15.5 27
Muscle 60 10.2 39.5Table1. Physical properties for | p| < 1
TissuesFrequency
(GHz)Permittivity
(ε)Conductivity
σ (S/m)
Fat 4 5.12 0.18
Skin 3 37.4 1.74
Muscle 2.5 50 1.7

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 61 No. 8 2016STUDY ON THE VALIDITY OF THE FORMALISM OF NORTON 955
Finally, the attenuation model of Norton gave good
agreements with the simulation results for all values of | p|.
5 . DO M AIN O F V A LI D IY
OF THE NORTON MODEL
In this section we are interested to find the condi-
tions under which the model of Norton does notreflect good agreements with numerical simulation.
Recall that in the preceding paragraph, the Norton
model shows a great ability to estimate the attenuationmodel of the field on the dielectric surface whatever
the frequency, the permittivity and conductivity.
Table 4 shows the physical properties in which the
use of this model does not present a good agreementwith the simulation. This occurs when the conductiv-
ity of the tissue is equal to zero. This is surely not thecase of the human body tissue, however it has been
considered for theoretical reasons.
Figure 9 shows a comparison between the Norton
model (spherical dot) and the model of equations (1)
and (2) and the simulation result (cross line) for thephysical properties given in Table 4 (| p| > 1).
Norton model (spherical dot) shows an attenuation
of 1/ r
2 since | p| is greater than 1 as shown in Table 4.
However, and with the same physical properties, thesimulation result has an attenuation of 1/ r. The result of
Norton as well as that of the simulation is not similar.
We find that the Norton model is not applicable in
case the conductivity of the environment equal to zero.Fig. 7. Comparison between attenuations for | p| .–40–20020406080100
00.05 0.10 0.15 0.20 0.25 0.30 0.35Ez(r), dB
r, mAttenuation in Eq. 1
Attenuation in NortonAttenuation in Simulation
0.40
1> Fig. 6. Comparison between attenuations for | p| .–20020406080100
00.05 0.10 0.15 0.20 0.25 0.30 0.35Ez(r), dB
r, mAttenuation in Eq. 2
Attenuation in NortonAttenuation in Simulation
0.40
1<
Fig. 8. Comparison between attenuations for | p| .–40–20020406080100
00.05 0.10 0.15 0.20 0.25 0.30 0.35Ez(r), dB
r, mAttenuation in Eq. 2Attenuation in Eq. 1
Attenuation in Norton
Attenuation in Simulation
0.40
1/asymptequalFig. 9. Comparison of attenuations for conductivity equal
zero.–40–20020406080100
00.05 0.10 0.15 0.20 0.25 0.30 0.35Ez(r), dB
r, mAttenuation in Eq. 2Attenuation in Eq. 1
Attenuation in Norton
Attenuation in Simulation
0.40

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JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 61 No. 8 2016BARAKE et al.
CONCLUSIONS
In this paper we have presented studies on attenua-
tion models of propagation of electromagnetic waves
that propagate on the surface of the dielectric. Initiallywe studied the attenuation function of Somerfield andwe have developed through it two mathematical modelequations (1) and (2). The first equation shows a prop-agation loss as 1/ r while the second one shows a prop-
agation loss as 1/ r
2. In the second step, we studied the
Norton model to treat the surface waves. We foundthat the Norton model shows good agreement with thealready mentioned mathematical models based on pfactor, which connects proportionally the permittivity,the conductivity and the frequency. We noted that forvalues of | p| < 1, the Norton model follows a path
attenuation 1/ r. In case where | p| > 1 (muscle, skin, fat,
etc.) Norton model has a path attenuation 1/ r
2. For
values of | p| near 1, the Norton model follows a path
attenuation between 1/ r and 1/ r2. In order to validate
these conclusions we performed electromagnetic sim-ulations using CST MWS. Attenuation models
obtained by simulation are similar to those obtained bythe Norton model for all values of | p|. Nevertheless, the
model of Norton is not applicable to environments
with zero conductivity.
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Birmingham, 2010).Table 3. Physical properties for | p| considering arbitrary
ε and σ
Frequency (GHz) Permittivity (
ε)Conductivity
σ (S/m)
2 5 0.05
4261 /asymptequal
Table 4. Physical properties not compatible with Norton if
σ= 0
Frequency (GHz) Permittivity ( ε)Conductivity
σ (S/m)
2.5 1 0