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In this paper a three dimensional Finite Difference Time domain(FDTD) method is used to analyze communication mobile handsets at 900-1800MHz. In the first part, the antenna input impedance and the reflection coefficient( | S11| ) and the voltage standing wave ratio(VSWR) are studied. In the second part we report the calculation of the near and far irradiated electromagnetic fields, the Specific Absorption Rate(SAR) distribution(included in a simple computational model of the human head) and the effect of changing the distance between the head and the antenna. Finally we study the effect of frequency on the radiation pattern.
Nowadays because of the rapid evolution of the wireless communication technology the widespread use of mobile phone and handy computers and the recent needs for wireless computer networks[1,2], there have been increasing public concern about the influence on the human body. Besides the SAR has been recognized as one of the most significant parameters that describes the electromagnetic field interaction with the human body[3,6]. Since it is very difficult to quantify the SAR directly in the living human body, the dosimetry is compelled to rely mainly on a computer simulation with high resolution numerical Human models. Many techniques have been developed to simulate all these situations. The FDTD algorithm is currently the most widely suitable means for numerical[5,6] because it descretizes space into a number of cells and assigns each cell a Corresponding permittivity and conductivity . This algorithm offers a great flexibility in modeling the heterogeneous structures of anatomical tissues and organs.[7-8]. The aim of this analysis is to study the influence of the following parameters: the distance between head and phone, the model of the head(homogeneous and heterogeneous),and the effect of frequency on both the absorption and the distribution of electromagnetic fields by the human head. The characteristics of the model of the head studied were obtained from . We used in the first case a homogeneous spherical head; and in the second case we have studied a spherical multilayer head. To model the radiofrequency, we use a dipole excited at its center.
The remainder of the paper is outlined as follows: Section (2) focuses on the method of calculations FDTD. Section (3) discusses the modeling of a dipole antenna in a free space. In Section(4) and Section(5) , the interaction between the mobile handset and the human head has been studied as well as Electric Field and Specific absorption rate are evaluated. Section (6) Near to far field transformation is discussed. Section (7) illustrates the effect of frequency on the distribution of the SAR. Finally we present our conclusions in Section (8).
2. FDTD formulation
In the FDTD formulation both space and time are divided in to discrete segments. space is segmented in to box shaped cells which are small in comparison with the wavelength The electric fields (Ex(i,j,k), Ey(i,j,k) and Ez(i,j,k)) are located on the edges of the box, and the magnetic fields (Hx(i,j,k), Hy(i,j,k) and Hz(i,j,k)) are positioned on the faces as shown in Figure 1. This orientation of the fields is known as the Yee cell  and is the basis for FDTD. The time is divided into small lapses where each step represents the time required for the field to travel from one cell to the next. Given an offset in space of the magnetic fields in relation to the electric fields, the values of the field in respect to time are also offset. The electric and magnetic fields are updated using a leapfrog scheme where the electric fields come first, then the magnetic ones are computed at each step in time. When many FDTD cells are combined together to form a three-dimensional volume, the result is an FDTD grid or mesh. Each FDTD cell will overlap the edges and faces with their neighbors. Therefore each cell will have three electric fields that begin at a common node associated with it. The electric fields at the other nine edges of the FDTD cell will belong to other adjacent cells. Each cell will also have three magnetic fields originating on the faces of the cell adjacent to the common node of the electric field as shown in fig 1.
Fig 1. Positions of the electric and magnetic
Field components in a Yee cell
This knowledge of field values associated with the characteristics of the tissue help to determine the SAR in the tissues without requiring an invasive measure. Now we present the Maxwell's equations in three dimensions. We suppose the absence of magnetic or electric current sources, and the existence of absorbing materials in the space.
Where the displacement vector is related to the electric field through the complex permittivity
The various components of the fields are evaluated on the basis of neighboring components of each lapse of time and each cell in the modeling area. This method works in the time domain and allows direct visualization of Electromagnetic fields.
3. Modeling dipole antenna in free space:
A simple dipole is illustrated in Figure (2), consists of two metal arms. A dipole antenna functions with a current flow through the arms, which results in radiation. FDTD simulates a dipole in the following way. The metal of the Arms are specified by setting the Ez parameters to zero in the cells corresponding to the metal; except in place where the source is placed. This insures that the corresponding Ez field at this point remains zero as well as it would if that point were inside the metal. The antenna length was held constant at each simulation. Perfectly Matched Layer (PML) boundary conditions were employed. The source is specified by setting the Ez field in the gap to a certain value. For the FDTD simulation, dipole is fed at the center (x = ic Î”x, y=jc Î”y, z=kc Î”z) gap of length Î”z with a Gaussian pulse . So, the electric field in the gap of the dipole is:
Figure 2. Geometry of the dipole antenna model.
The current in the antenna at the feed point is obtained by applying Ampere's law to the surface S with the bounding contour C on the wire at (ic, jc, kc +3/2):
- (8 & 9 &10)
Figure 3 shows the current flowing through the center of the dipole in the time domain.
The input impedance calculation:
The input impedance of an antenna is a very important parameter. After the final time domain results are obtained, the current and voltage are transformed to those in the Fourier domain. The input impedance was calculated in the centre fed dipole over a range of frequencies. It is determined from the ratio of the Fourier transform of the voltage wave and that of the input current wave:
It should be noted that the time difference Î”t/2 between voltage wave and current wave is ignored since its effect is very small.
Figure (3).Input current in terms of time I(t)
Figure4.Input impedance of the dipole antenna
The input impedance of the dipole antenna is shown in Figure4.
The input impedance is well matched at 75.48+j1.12 at the resonance frequency of 1800MHz.
The input return loss (S11) and the
voltage standing wave ratio (VSWR) :
The results of input impedance are then used to obtain the return loss characteristics of the antenna. So the reflection coefficient S11 of the half-wavelength dipole antenna is:
From the calculated reflection coefficient, the voltage standing wave ratio (VSWR) can be calculated as follows:
The bandwidth of the antenna, which was determined by the impedance data, is the frequencies corresponding to a reflection coefficient of the antenna (less than or equal to
1/3) that corresponds to VSWRâ‰¤2.
In Figure 4, the resonant frequency which is around 1.8 GHz. was chosen as a frequency through the whole study.
Interaction between the handset
and the human head :
In this section, the interaction between the mobile handset and the human head has been studied. A simplified homogeneous spherical head model is used. The sphere has a radius of r = 10 cm and the tissue it contains has a relative permittivity of Îµr =43.5 and conductivity of Ïƒ =1.15 S/m. These tissue equivalent dielectric = 10 cm and the tissue it contains has a relative permittivity of Îµr =43.5 and conductivity of Ïƒ =1.15 S/m. These tissue equivalent dielectric parameters were chosen according to  to simulate the brain tissue at 1.8GHz. For the computation of SAR, the head tissue density is assumed to be 1030 kg/m3. The relative position of the dipole antenna relative to human head model is illustrated in Figure6. The interaction between the mobile handset and the human head is studied from two viewpoints: first the impact of the distance between head and phone; second the effect of head type (homogeneous, heterogeneous) on the absorption and distribution of electromagnetic fields in the human head and on the radiation pattern.
The following table summarizes the dielectric constant, the conductivity Ïƒ and the mass density of the tissues used for the calculations at 900 & 1800 MHz .
The near-fields have been simulated in the plane defined by z=0.0mm. The results are viewed in Figure 7 and Figure 8 for the simulations. The origin of the plane wave has been aligned in the xz-plane with the feeding point of the handset model.
The calculations were made at a frequency of 1.8 GHz. for a homogeneous spherical head of dielectric permittivity of 51.8 and a conductivity of 1.5 S/m. We calculated the distribution of the electric field in the near head near the antenna. This latter is located at a distance of 5mm at the side of the spherical head. To see the effect of the position of the antenna on the radiation pattern we have traced the appearance of the reflection coefficient S11 for several distances between the header and the phone.
Figure 7. The transversed electromagnetic field distribution of the dipole antenna in the homogeneous head z =00mm
Figure8. The totally simulated electromagnetic field distribution of the dipole antenna in the homogeneous head z =00mm.
Figure 9.Input return loss (S11) of the antenna for different values of d.
From Figure9. we can say that the radiation of the antenna depends on the distance between the phone and the header. Hence, we can conclude that there is a coupling between head and the antenna. In table2, the results of the calculated driving point input impedance Zin of the dipole antenna are presented for each position of the handset in front of the homogeneous human head phantom.
Table 2. Results pf the driving point input impedance, VSWR and the input return loss (S11) at each distance d(cm) between axis of dipole antenna and the outer surface of the homogeneous human head phantom.
The VSWR for each case is then determined in respect to the free space input impedance. In this case the antenna input impedance changes drastically and the input power to antenna decreases considerably. In the presence of the human head, the resonance frequency is detuned approximately 5% at GSM frequencies. The presence of the human head also increases the input impedance of the dipole. Hence the impedance behavior of the GSM dipole shows quite a strong dependence on the surroundings.
A spherical heterogeneous Human head model was utilized. This phantom consisted of three layers with materials simulating the human head structure whose outer diameter is identical to that of homogeneous sphere previously used. In Tble1, the type of each layer and its corresponding relative permittivity , conductivity and density are depicted according to . In Figures 7,8,and 10 we see that the highest values of the electric field occur near the antenna. The magnetic field reaches its maximum value above the start of the antenna wire, near the feeding point, where the current reaches its maximum value.
Figure10.The simulated electromagnetic field distribution of the dipole antenna in the heterogenous Head for z=00mm.
The radiation source of the cellular phone was modeled by an equivalent dipole antenna. After having obtained the induced electric field by the FDTD method, the local SAR in W/Kg for
E is the electric field magnitude in V/m, Ïƒ is the material conductivity in S/m and Ï is the mass density in kg/cubic metres.
An half wave dipole of 77mm irradiating at 1800MHz is placed 10mm away from the outer surface of the spherical human head. The values of SAR( or electric fields) are the highest in tissues around the antenna. These SAR values decrease rapidly when one gets away from radiotelephone antenna. As a result the tissues around the ear are most exposed to electromagnetic fields.
The distribution of the local SAR values can be calculated directly from the electric field distribution, which results from the computer run. In figure 12 and 13 the absorption is greater in the skin of the head and the surface of the brain. Contrarily the skull absorbs radiation poorly, due to its low conductivity and the difference in the dielectric properties of the related tissues.
Figure11.The simulated SAR distribution of the dipole antenna in the homogeneous Head for z=00mm.
Figure12.The simulated SAR distribution of the dipole antenna in the heterogeneous Head for z=00mm.
Figure13. SAR variation as function of the transverse distance.
We can state that the maximum absorption occurs at the point where the phone is closer to the head. Using the relations
D =Îµ .E and , the ratio between the SAR values in these tissues under a uniform field distribution is around 0.35. In addition, the maximum SAR values are substantially higher for the heterogeneous model of head.
6. Near to far field transformation
While using FDTD method, the provided data are near fields. Therefore, these near fields are transformed to far fields. Then, the far fields are used to calculate the radiation pattern. First, we work within the frequency domain and assume that the analyzed antenna is surrounded by the enclosed surface S. Then, let's assume that this closed surface has the local unit outward normal vector n .
Thus, the electric and magnetic current densities can be written as follows:
r : the position of the observation point (x, y, z)
' r : the Position of the source point on S (x' , y' , z ' )
Ïˆ : the angle between r and ' r .
Figure 14. The virtual surface used for the nearto-far field transformation and the coordinate system used for its calculation.
E and H are electric and magnetic fields that propagate on the surface. Later, we can define the time harmonic vector potentials N and L [18-22].
j = âˆ’1 , k is the numerical wave number, r ,is the unit vector to the far field point and r' is the vector to the source point of integration. To obtain the far-field information from the
equivalent currents, it is necessary to integrate them over each of the six faces of the virtual box, here labelled surface S. That integration can be done by using the following pair of vector
potentials. These vectors' potentials are in the cartesian coordinate system. Next, these vectors potentials are converted into the spherical coordinate system. The Î¸ and Ï† components of
the vector potentials N and L are given by:
Finally the far electric field can be calculated by the following relation
the radiation pattern is given by
where Pe is the input power antenna, and =Î·0 is the free space .
Figure 15a, Î¸ =90deg, xy plane.
Figure 15b, Ï† =90deg, yz plane.
The radiating patterns were simulated in the XY plane (plan E) and in the YZ plane (H plane) for the dipole alone at the center frequency of the band (1.8GHz).
Figure 16a, Î¸ =90deg, xy plane.
Figure 16b, Ï† =90deg, yz plane.
Figure 16: Radiation patterns for linear dipole antenna radiating in the presence of homogeneous spherical head model. Figures 15 and 16 show the effect of the head model in the XY plane (plane E) and in the YZ plane (H plane). We can see that the head blocks the radiation patterns in the head direction. In the YZ plane the radiation in halfspace where the head is situated is affected seriously. So in the XY plane the radiation more decreased towards the head's direction.
7. The effect of frequency
To see the effect of the frequency on the distribution of the SAR, we have drawn the SAR's profile for the two used frequencies (900 MHz and 1800 MHz) and a homogeneous head.
Figure17. SAR normalized profile through a homogeneous spherical head for two different frequencies.
Figure 17 illustrates the profile of the local SAR across the homogeneous spherical head model. The distance was measured from the point of the source closest to the head model. The SAR
values were normalized to the maximum to show the effect of the frequency. We can observe that the SAR decreased faster in the
higher frequency range as expected due to the weak depth penetration.
As a conclusion, we can say that the important parameters affecting the energy absorbed in the human head exposed to radiation from radio is the distance between the head and the antenna. Although the power consumption in the case of the 1800 MHz frequency is lower than 900 MHz, the maximum values of SAR are more significant for the higher frequencies. The distribution of SAR in the spherical model illustrated in Figures
11, 12 and 13, indicates that large values of SAR are located in a volume close to the surface of the head. In other words, the absorption is greater in the skin of the head and the surface
of the brain. Contrarily, the skull absorbs radiations poorly due to its low conductivity and the difference in the dielectric properties of the related tissues. In addition, the maximum SAR values are substantially higher for the heterogeneous model of head. It can be inferred from the results that there is a good agreement
between the local SAR values of our study and the literature. In further works, we will study how to reduce the SAR in the human head emitted from cell phone radiation using the metamaterials