Assessment of the human exposure to transient and time-harmonic fields using the enhanced transmission line theory approach

ABSTRACT The paper deals with the assessment of human exposure to the transient electromagnetic fields and high-frequency (HF) radiation. The formulation of the problem is based on a simplified cylindrical representation of the human body. The analysis is based on the enhanced transmission line (TL) theory. For this purpose, in order to quantify the induced current inside the human body, we solve linear system equations, where the electromagnetic field excitation is represented by two equivalent current and voltage generators. Once the axial current is determined, it is possible to calculate the specific absorption rate (SAR). Some illustrative computational examples are presented in the paper.


Introduction
In the last 60 years, electricity requirements in the world have multiplied by twelve. Electricity has become an indispensable partner in our daily lives, our technological and economic development. Despite all the benefits associated with its use, as humans are constantly immersed in electromagnetic fields generated by different sources (power lines, mobile phones and broadcast antennas), there has been an increasing concern regarding the potential adverse health effects due to exposure to these fields. Consequently, there is a need to define safe exposure limits for both professional and general population.
To achieve this aim, it is especially crucial to provide researchers in Life Sciences objective evidence to quantify the electromagnetic phenomena occurring inside the human body and causing potential biological effects.
In addition to accurate, but at the same time computationally demanding, three-dimensional anatomical body models [1,2], some simplified models for rapid engineering estimation of the phenomena are also available, e.g. cylindrical body representation [3], involving numerical solution of Pocklington integrodifferential equation using the Galerkin-Bubnov boundary element method (GB-BEM) [3,4].
The present study extends the work reported in [5], by representing the body in terms of wire-junction models to account for the influence of arms. The formulation is based on the transmission line (TL) equations in the frequency domain [6] and valid for a range frequency of 50 Hz to 110 MHz. This paper is organized as follows: first, the axial current distribution induced along the body exposed to transient electromagnetic fields, taking into account the influence of the arms, is evaluated. Furthermore, a rapid estimation of specific absorption rate (SAR) in human body exposed to a high-frequency (HF) radiation is carried out.

Body models
Two representations of the human body are considered in this work: cylindrical body model and wire-junction model of the body. The human body assumed to be homogeneous is represented by a vertical cylinder of height L and radius a perpendicular to a ground plane with arms in contact with the body as shown in Figure 1, where the electrical properties of the human body such as conductivity s and relative permittivity e r are assumed the same value for all organs of the body (homogeneous body).
Furthermore, the body in a direct contact with the ground is considered, as shown in Figure 2. The arms are represented by a set of wires connected to the thick cylinder. The dimensions are given in Table 1.
The per unit length parameters of the horizontal segments are calculated using the formalism of Ametani et al. [7]. For vertical segments, we use the formalism of Rogers et al. [8].

Transmission line formulation
The TL theory allows the study of propagation phenomena along a slender support. The voltage and the CONTACT D. Poljak dpoljak@fesb.hr current distribution along the TL are governed by Telegraph equations. The set of equations accounting for the coupling between a TL (n conductors) and an external electromagnetic field is given as follows [9]: ½E e T ðz; vÞ: sources due to incident transverse electric field; ½E e L ðz; vÞ: sources due to incident longitudinal electric field. In our work, only the vertical incident electric field (longitudinal) to the human body is of interest. The vertical field E e z is the main cause of the exposition; the prevailing direction of induced currents and voltages is also vertical (axial).

Solution of the transmission line equations in the frequency domain
The telegraph equations (1) and (2) can be written in state variable forms as a coupled first-order, ordinary differential equations in matrix form as d dz Note that writing Equations (1) and (2) in the matrix form provides the solution of system (5) to be obtained directly by analogy with the formalism of state variables, whose demonstration is carried out in the work of Paul [6][7][8][9][10]. Thus, the solution of (5) is given by with XðzÞ ¼ UðzÞ IðzÞ ! , matrix of voltages and currents at any point z of the TL (the body in our case). The chain parameter matrix 'ðzÞ ½ is defined as Note that (6) provides the values of currents and voltages at an arbitrary point z of the line according to their value at the origin z 0 .   In the case studied in this work, values of voltages and currents at the extremities of the line (human body in our case) are of interest. Thus, (6) corresponding to a line that starts with z 0 = 0 and applied to the end z L = l becomes The n Â n submatrices of the chain parameter matrix are given by [6] where [T] is a matrix of size n Â n; [g] represents the diagonal matrix of propagation constants squared with (8) allows to show the terms Uð0Þ; Ið0Þ; UðlÞ and IðlÞ: which can be written as follows: where I 2n ½ is the identity matrix of order 2n (n: number of conductors by line).
And the total forcing functions are given in [6]: For the case where the body is exposed to an external electromagnetic field: ½U F ðtÞ ¼ ½E e z and ½I F ðtÞ ¼ ½0, Equations (12) and (13) become

Incorporation of boundary conditions
Obtaining the general solution of the TL coupling equations in terms of the chain parameter matrix and the external-field excitation, one then incorporates the boundary conditions to obtained corresponding voltages and currents for a given problem of interest. At both ends of the line, we have the following relationships: where Z 0 ½ and Z L ½ are the impedance matrices termination of the line.
In the present work, two specific ends representing the feet and head are considered. The contact of the human body with the ground can be taken into account via a capacitance C [3]: where C is the capacitance between the sole of the foot and its image in the ground, e 0 is the permittivity of free space, e r is the relative permittivity of the material constituting the sole of the shoe, A is the contact surface and d is the distance between the sole of the shoe and its image in the ground. The other end of the equivalent cylinder is connected by an impedance of great value, i.e. Z L ¼ 1 . Namely, cylinders 2, 3 and 4 in the case of the human body with the arms outstretched.

Extension to multiple wire model
The principle of the proposed concept is to transform the propagation equation (10) for all the conductors representing the body and the electric relations in the all nodes to a matrix: equation system: The matrix [A] is the topological matrix of a radial network. Note that the human is considered in our work as a radial network of TL. X ½ is the vector contains the unknown currents and voltages at each node; B ½ is the vector includes the external field effects.
Hence, one should transform the propagation relations (11) for all the lines representing the body and the relations in the all nodes (between different organs) to matrix equation (19).
The matrix A ½ is composed of two submatrices: The submatrix A 1 ½ takes into account the propagation in all line segments which form the human body (trunk and arms) and constructed from Equation (11).
The submatrix A 2 ½ takes into account the electrical relations (current and voltage) on all nodes of the human body topology [6]. Applying the Kirchhoff law in each mth node, the following equation is obtained [6]: where Z m i Â Ã and Y m i Â Ã are, respectively, the impedances and admittances matrices resulting from the application of Kirchhoff's laws in the mth node. They contain the numerical values 0, 1, ¡1, impedances and admittances values according to the human body topology.
Finally, solving the matrix system (19) yields the currents and voltages at all nodes of the human body topology.

Computational examples
The computational examples are related to the body exposure to transient and to HF radiation, respectively.

Cylindrical body exposed to transient electromagnetic pulse
The human body is represented by a cylinder of length L = 1.75 m and radius a = 0.14 m. The base and the top of the cylinder are terminated by impedances Z L and Z 0 , as depicted in Figure 3. In all calculations, the body is assumed to be well grounded, thus neglecting the capacitance between the soles of the feet and their image in the earth (Z 0 = 0 V), while the average value of the conductivity is chosen to be s ¼ 0:5 S=m (conductivity at 60 Hz).
The body is exposed to transient incident electric field in the form of double-exponential electromagnetic pulse (EMP) [11]: where E 0 = 1.05 V/m, a ¼ 4 Â 10 6 s À1 and b ¼ 4:76 Â 10 8 s À1 ; The incident field is vertically polarized to ensure maximum coupling with the body. It should be noted that the exposure duration is 100 ns. Figure 4(a,b) represents, respectively, the transient-induced current in the feet computed via the proposed TL approach compared to the direct time domain antenna theory approach based on the space-time Hallen integral equation solved via Galerkin-Bubnov indirect boundary element method (GB-IBEM) [11].
The results obtained via different approaches are found to be in a rather satisfactory agreement. In addition to plausible agreement in both the waveform and amplitude, the advantage of TL modelling is its mathematical simplicity and low computational cost. Figure 5 shows the spatial-temporal distribution of the induced current in the feet.

Human body with the arms outstretched
This subsection deals with the study of the human body with arms outstretched exposed to the bi-exponential pulse. Figures 6 and 7 represent the variation of transient current induced in the feet, the free end of the left and right arms, and the head due to exposure to the bi-exponential pulse. In this case, the current flows through the path of the least resistance [12]. Figure 6 shows a slight increase in the current induced in the feet compared to the result (Figure 4 (a)) obtained for the same excitation scenario but for the human body represented by a simple cylinder with arms in close contact with the body.
Furthermore, the human body is exposed to transient electric field in the form of Gaussian pulse [11]: where E 0 = 1 V/m, g ¼ 2 Â 10 9 s À1 and t 0 = 2 ns.

Human exposure to high-frequency (HF) radiation
The thermal effect of the human exposure to HF radiation is quantified in terms of the SAR.     The SAR (W/kg) at any point in the human head is defined as [13] SAR where s and r are electric conductivity and density of the biological tissue, respectively, jEj is the maximal value of the electric field induced in the human body. The current density induced in the body can be expressed in terms of the axial current I z as follows [13]: where J 0 and J 1 are the Bessel functions, k is the free space phase constant. The induced electrical field is given by [13]: The basic restrictions (SAR) of the ICNIRP guidelines for frequencies between 100 KHz and 100 MHz [14], given in Table 2, are established to account for uncertainties related to individual sensitivities, environmental conditions, and for the fact that the age and health status of members of the public varies.
In this case, a man with arms in contact of the sides standing on the earth, exposed to HF radiation where the value of the uniform electrical field is found to be 1 V/m at a frequency of 30 MHz, is represented by a cylinder of the entire length L and radius a. The average value of the conductivity and permittivity of the human body, respectively, is assumed to be s ¼ 0:6 S=m, e r ¼ 60 and r ¼ 1000 (kg/m 3 ) [15]. Figures 10 and 11 show, respectively, the induced current density and electric field calculated inside the human body due to HF radiation.
The maximum coupling conditions (uniform field along the body) are adopted in order to quantify in the most unfavourable conditions the maximum perturbation induced in the human body. Figure 12 shows the induced values of SAR inside the body. The maximum values of the induced current density, the induced electric field and SAR at the feet are presented in Table 3.
The results presented in Figures 10 and 11 show that the induced electric field decreases gradually as one approaches to the axis of the equivalent cylinder. Namely, at high frequencies, the induced current moves to the periphery of the cylinder due to skin effect. The outer layers of the body behaves like a "shielding".    The obtained results are compared to the exposure limits proposed by ICNIRP (SAR = 0.4 W/kg for General population, SAR = 0.08 W/kg for Occupational).
The SAR values do not exceed the basic restrictions (SAR max = 1.6693 £ 10 ¡5 W/Kg) and, as such, we conclude that heating effect is negligible.
Finally, the behaviour SAR versus frequency in the range from 5 to 40 MHz is analyzed.
The body model consists of a material with parameters simulating a muscle tissue: the frequency-dependent permittivity and electrical conductivity (see Table 4) [15].
The incident electric field amplitude is E z = 1 V/m, and the maximum values of the SAR at the feet obtained for Jðr ¼ a; 0Þ are shown in Table 5.
According to Table 5, we remark that at range frequency 5-40 MHz, the SAR inside the human body increases rapidly with frequency. The main conclusion is that the SAR max for different frequency never exceeds the limit of 0.08 W/kg defined by ICNIRP; moreover, the SAR depends on the frequency. Figure 13 shows the logarithm of frequency-dependent SAR: log10(SAR max ).
The obtained results stay far below the ICNIRP exposure limits.

Conclusion
In this study, the transient-induced current and SAR induced in the human body when exposed to transient and time-harmonic radiation, respectively, are evaluated by using the enhanced TL theory. The simplified representation of the human body is related to the thick cylinder with the thin wires attached to account for the influence of the arms, thus extending the previous study of the authors dealing with the single-cylinder body model. The external electromagnetic field that excited the body is represented by equivalent current and voltage generators.
The proposed TL approach, compared to other more sophisticated numerical methods, provides fairly conclusive results for the transient-induced current and SAR for different cases of human exposure. Therefore, this method (approach), considered reliable, represents a simplified tool for the assessment of the current induced in the human body due to both transient and time-harmonic electromagnetic field exposure, respectively, thus providing the rapid estimation of the phenomena and ensuring the useful results in an engineering sense.

Disclosure statement
No potential conflict of interest was reported by the authors.  Table 3. Maximum values of the induced current density, electric filed and specific absorption rate in the feet.   Figure 13. SAR max values as a function of frequency.