Power electronics use power semiconductors in switch mode. This operation gives the converter very high efficiency; on the other hand, it gives rise to numerous electromagnetic disturbances due mainly to the rapid switching of semiconductors. The disturbances propagate towards the converter’s power source and towards the load it supplies, a smaller part of this energy is radiated [22] .

The use of electronic equipment is increasing in all fields of activity, whether consumer, industrial or military [23] . The technologies used in the design and development of digital equipment systems are based on three parameters:

The use of electronic equipment is increasing in all fields of activity, whether consumer, industrial or military [23] . The technologies used in the design and development of digital equipment systems are based on three parameters:

Static converters are electronic devices that convert one form of electrical energy into another form of electrical energy [22] . These devices are often used in electrical power systems to provide stable voltage, current, or electrical power to a load [23] . These are devices with electronic components capable of modifying the voltage and/or frequency of the electrical wave.

There are two types of voltage sources:

DC voltage sources characterized by the value V of the voltage.

Alternative voltage sources defined by the values of the effective voltage V and the frequency f.

Then we list the energy modulators making it possible to manage the energy necessary for controlling electrical systems:

Figure 1 shows the table of all static converters as described above.

The principle of a converter consists of establishing then periodically interrupting the source-load connection using the power switch. The latter must be able to be closed or opened at will in order to have an adjustable DC output voltage [25] .

Disturbances in static converters can be caused by various factors, such as variations in supply voltage, variations in load, external electromagnetic disturbances, disturbances caused by transistor switching, etc. [25] .

These disturbances can lead to problems such as output voltage or current fluctuations, power losses, electromagnetic interference, component failures, etc.

To reduce disturbances in static converters, it is important to consider several design factors, such as component selection, component layout, PCB design, grounding, etc. [26] . Appropriate filtering techniques can also be used to reduce electromagnetic disturbances. Additionally, advanced control techniques can be used to minimize disturbances caused by transistor switching.

Finally, testing and verification must be carried out to ensure that the static converters meet the required specifications in terms of performance and electromagnetic compatibility.

Note that the main cause of producing disturbances is the rapid variation of voltages (dv/dt), currents (di/dt), magnetic fields (dH/dt) and electric fields (dE/dt). The most important consequence is the induction of parasitic voltages or currents in the devices (converters) under the influence of the disturbance. It can have the character of a source of voltage, current or energy depending on the nature of the physical phenomenon which produces it [27] . An electric wave is characterized by its frequency, voltage and intensity.

Use either SI (MKS) or CGS as primary units. (SI units are encouraged.) English units may be used as secondary units (in parentheses). An exception would be the use of English units as identifiers

The very principle of power electronics is to ensure the transformation of electrical energy by cutting at high frequencies (from 100 Hz to a few hundred kHz). Switching times are typically greater than 100 ns, while digital circuits dedicated to the signal have transition times often less than 5 ns. On the other hand, the voltages and currents involved which dictate the choice of power components lead to fairly similar voltage increases but to much higher current growth rates [28] - [32] .

The di/dt will create brief voltage pulses across the parasitic inductances or connections and subsequently, disrupt by so-called common impedance coupling the sensitive circuits using the same conductors called power bus.

The dv/dt will also create very brief current pulses in the various parasitic capacitances (card to chassis coupling for example) and cause, as previously, disturbances in the control-command cards [30] [33] [34] .

Depending on the frequencies there are 2 types of disturbances:

Low frequency disturbances the frequency range: 0 ≤ frequency < 1 to 5 MHz. Low frequency disturbances are found in installations mainly in conduit form (cables, etc.). Its duration is long (a few tens of ms). In some cases, the phenomenon can be permanent (harmonic). As for its conducted energy, it can be significant and results in the malfunction or even destruction of interconnected devices.

High frequency disturbances: frequency range: 30 MHz. High frequency disturbances are found in the installation in radiated form. Duration: the rise time of the pulse is less than 10 ns. The radiated energy is low and results in the malfunction of surrounding equipment.

Electronic energy conversion structures are ideally suited to illustrate the mechanisms of conducted emissions and provide the basic building blocks for modeling these phenomena.

The very nature of their operation is in contradiction with the common sense rules that we seek to apply to limit inter- and intra-system interaction. CEM tends to show that it is necessary to limit rapid variations in electrical quantities (voltage, current) and electromagnetic (fields) while switching structures generate sudden variations, at least electrical, to manage the transfer of energy desired with lower losses.

To talk about the sources of disturbance in power electronics we choose, for example, energy conversion: cutting is done by power switches with semiconductor components. There are switches with controlled switching (MOSFET, IGBT, JFET) which require control and others with natural switching. It consists of identifying the problem in three parts: the sources of disturbance, the propagation paths or channels and the victim [34] . The sources of disturbances are the switching of power switches. These switchings are the cause of sudden variations in voltage (dv/dt) and current (di/dt) which give rise to electromagnetic disturbances.

The sources of electromagnetic disturbance can be characterized by [35] - [39] :

To understand the origin of electromagnetic disturbances, let’s start by analyzing the spectral behavior of waveforms in power electronics.

Useful or parasitic electrical signals have two ways of propagating by conduction on a two-wire connection. We can therefore study the disturbances according to the two modes. Conducted and radiated disturbances have a common origin, they are linked to the spectral components of voltages and currents. The static converter, through its switching operation, produces sudden variations in voltage and current. These variations are the cause of parasites, called electromagnetic disturbances [35] - [39] .

In an electric field, disturbances are mainly radiated by conductors subject to variations in potential.

In a magnetic field, the disturbances are radiated by the loops, the switching of the voltage generates common mode HF currents which we find in the parasitic components, the switching of the current creates differential mode disturbances.

Any electromagnetic interference situation involves three different elements as shown in Figure 2 : a source of disturbance emission, a disturbance receiver (victim), and a coupling mechanism by which the disturbance reacts on the operation of the receiver.

Coupling is the phenomenon of propagation of disturbances that occurs between the source and the victim. The coupling modes can be classified according to the type of disturbance and according to the propagation medium, by conduction (characterized by currents and potential differences), or by radiation (characterized by electric and magnetic fields) [35] - [37] , this is illustrated by Figure 3 below:

Conductive coupling occurs when the source of disturbance and the victim are connected together by a conductor. It is done in two modes illustrated by Figure 4 and Figure 5 :

The common mode: in common mode, signals propagate in the same direction on all conductors. Common mode currents return through ground. This mode represents 10% of cases. It is also called “parallel”, “longitudinal”, or “asymmetric” mode.

Differential mode: differential mode is the ideal mode of signal transmission which represents 90% of cases. In this mode, all current that enters the receiver returns to the source through a return wire. It is also called “series”, “normal”, or “symmetric” mode.

We say coupling by radiation (field), if the propagation medium of a disturbance transmitted from the source to the receiver is space.

We know that the electromagnetic field radiated by a radiation source varies by the distance which separates the radiation source and the place where the field is observed, for this we distinguish two types of coupling by field [35] - [40] :

1) Near-field coupling

Near the source, the electromagnetic field depends on the characteristics of the source:

Coupling by electric field (capacitive):

The difference in potential between a conductor and its environment generates an electric field around the latter; the variation of this field injects a current into all nearby conductors. The cause of capacitive couplings is the parasitic capacitances formed by the presence of two conductors.

Coupling by magnetic field (inductive):

A current flowing in a wire generates a magnetic field. The variation of this field induces an electromotive force in the neighboring loops.

2) Far-field coupling

Beyond the near-field radiation zone, the so-called far-field radiation zone. In this region, the characteristics of the radiated electromagnetic field only depend on the properties of the medium in which the field propagates, and we say that we are dealing with coupling by electromagnetic field or by plane wave, so we must use the equations of Maxwell to calculate the amplitude of electromagnetic disturbances. Which means that the components of the electromagnetic field cannot be separated.

Figure 6 summarizes all the types of coupling.

Since the effects of electromagnetic fields in a biological cell are the focus of this article, we believe it is essential to describe the biological cell and the molecular structure of the plasma membrane. The biological cell is the structural and functional unit of all living beings, it is characterized by its nucleus, its cytoplasm and its plasma membrane [21] . The cellular plasma membrane plays an essential role in the life of the cell, it delimits the cell and separates the cytoplasm from the external environment. It surrounds the cytosol (i.e. the liquid phase in which the cytoplasmic organelles bathe) and forms a very thin protective layer composed of lipid and protein molecules. It thus has a heterogeneous molecular structure allowing it to play a dual role: the phospholipids which are the essential constituents of its basic material, make it insulating, while the protein molecules which are dispersed, ensure the exchange between the cytoplasm and the extracellular environment [22] .

The cell is the basic structural and functional unit of all living things. The cells are very small and very complex in organization. Knowledge of their structure, chemical composition and functioning (physiology) is very essential in biology and biomedical sciences. Studies on cellular functionality and behavior have been widely applied in many clinical and biomedical applications, such as diagnosing diseases and understanding their degree of progression, drug development, and cancer research [19] [23] .

This is why knowledge of their components and their characteristics is important for the further development of our research and in particular for the study and modeling of human biological tissues exposed to microwave electromagnetic waves.

In the case of biological environments, energy absorption phenomena in tissues can be relatively complex and depend on numerous factors. They are mainly linked to the type of coupling between the emission source and the biological environment. Galvanic coupling corresponds to the case of physical contact between the source and the medium. This contact causes the circulation of an ohmic type current in the person’s body. Depending on the intensity and frequency of the contact current, the consequences can lead to heating of the tissues, or even a burn [19] [23] .

Radiated coupling includes three fundamental mechanisms by which electric and/or magnetic fields, of variable frequency over time, interact with biological media [19] [21] [23] .

Coupling with low frequency electric fields: external electric fields induce, on the surface of the exposed body, a surface charge which causes, inside the body, the appearance of currents whose distribution depends on the conditions of exposure, body size and shape. The body as well as the position of the body in relation to the terrain [19] [21] [24] [25] .

Coupling with low frequency magnetic fields [26] [27] : the physical interaction between magnetic fields of variable frequency and the human body creates induced electric fields and causes the flow of electric currents. The magnitude of the fields and the density of the induced current are proportional to the intensity and frequency of the magnetic field B, the radius of the current loop in the body, and the electrical conductivity of the exposed tissues.

It should be noted that exposure of the human body to low-frequency electric or magnetic fields generally results in only negligible energy absorption and no measurable temperature rise. In contrast, exposure to electromagnetic fields with a frequency greater than 100 kHz can cause energy absorption and a significant rise in temperature [21] [28] - [30] . In general, exposure to electromagnetic fields results in highly inhomogeneous energy deposition and distribution in the body that must be assessed by dosimetry [31] - [34] . Finally, it remains to underline the indirect consequences caused by exposure to electromagnetic fields of people with active medical implants (cardiac pacemaker, cardiac defibrillator, insulin pump, etc.) which result in malfunctions of the implanted equipment.

These different phenomena can be analyzed mathematically in detail using Maxwell’s equations by defining each parameter and its role, some details of which are not given.

Then, we discussed the techniques most used to remedy the threats of two modes of conducted and radiated EM disturbances; filtering to protect against conducted disturbances and EM shielding against radiated.

Improving electromagnetic compatibility acts on the side of the sources by trying to reduce the disturbances they emit, and on the side of the victims by protecting them from external influences. We have seen that a source of disturbance reaches its victim through coupling. It is at this level that we must act. In order to protect equipment against conducted and radiated disturbances, we use EM filtering and shielding respectively.

From an electromagnetic point of view, biological media appear as materials at the same time [19] [21] [23] [41] [42] :

In general, biological tissues have a diamagnetic character. Certain substances such as ferritin, hemosiderin or methemoglobin with a paramagnetic nature are naturally present in the human body [41] [42] . However, the human body is still considered non-magnetic for the study of induced electromagnetic fields, and the magnetic permeability of biological tissues is therefore taken equal to that of a vacuum. Regarding electrical properties, given the chemical composition of biological tissues, the free charges capable of creating conduction currents are ions. These ions can move more or less freely under the effect of an electric field. They are subject to friction forces and stresses due to the structure of the tissues. Consequently, their mobility depends on the frequency of the source field. The presence of electric polar molecules of various sizes and also subject to friction, contributes to giving biological environments a lossy dielectric character. The human body therefore presents highly heterogeneous electrical properties at the microscopic (cellular structures) and macroscopic (organs) levels [41] [42] . The microscopic structure of a tissue can sometimes give it macroscopic anisotropic electrical properties: this is the case of muscles, for example, which are made up of cells that are very elongated in a single direction. In general, to characterize biological environments, we use the notions of conductivity/(σ) and relative permittivity ( ${\epsilon }_r$) such that the density of electric current induced by the $j$ pulsating electric field ω is $E$ [41] [42] :

$j=\left(\sigma +j\omega {\epsilon }_0{\epsilon }_r\right)E$(1)

where ${\epsilon }_0$ is the electrical permittivity of the vacuum.

These properties are often derived from macroscopic measurements on a given tissue considered homogeneous (and sometimes anisotropic) [41] - [43] [44] . The conductivity thus defined includes the static conductivity of the medium as well as the effect of dielectric losses. Sometimes, the notions of complex conductivity (σ) or complex relative permittivity ( ${\epsilon }_r$) are used. The current density and the electric field are then given by the relations:

$j=\sigma E=\left({\sigma }^{\prime }+j{\sigma }^{″}\right)E$(2)

$j=j\omega {\epsilon }_0{\epsilon }_{r}E=j\omega \left({{\epsilon }^{\prime }}_r+j{{\epsilon }^{″}}_r\right)E$(3)

For most tissues [43] [44] , it is not possible to carry out measurements allowing electrical characterization in vivo. It is often necessary to perform these in vitro measurements on tissue samples taken from deceased subjects. This very strong constraint poses the problem of conditioning the tissue to be studied. Indeed, the cellular structure can deteriorate rapidly after death, and the electrical properties can vary depending on many parameters that are difficult to control in vitro such as blood supply, hydration level or temperature.

This particular distribution of charges at the interfaces results in a very high impedance between the electrode and the biological environment for frequencies below a few kHz. The spectroscopic study of this interface impedance shows that it can be modeled in the form [43] - [45] :

$Z_i=K{\left(j\omega \right)}^{-\alpha }$ (4)

avec $0<\alpha <1$

The electrical characterization of biological media requires the use of a measuring device and a model allowing the extraction of conductivity and permittivity parameters. There are several measurement methods which differ depending on the frequencies studied [43] - [46] . For each method, there are different more or less complex models to represent the measuring device and the sample tested [46] .

Different empirical models can be used to approximate the frequency variations of the electrical properties of biological media.

Debye Model

The complex permittivity is expressed in the form [43] [46] [47] :

${\epsilon }_r={\epsilon }_{r\infty }-j\frac{\sigma }{\omega {\epsilon }_0}+{\displaystyle {\sum }_n\frac{\Delta {\epsilon }_{rn}}{1+j\frac{\omega }{{\omega }_n}}}$(5)

where ${\epsilon }_{r\infty }$ is the relative permittivity at infinite frequency, ${\omega }_n$ is the characteristic pulsation corresponding to relaxation n, $\sigma$ is the conductivity at zero frequency and $\Delta {\epsilon }_{rn}$ is the permittivity variation for relaxation n. This is the basic model for representing relaxation phenomena.

Cole Model Cole

This model introduces an additional parameter α_n characteristic of the frequency dispersion of each relaxation n [43] [46] - [49] :

${\epsilon }_r={\epsilon }_{r\infty }-j\frac{\sigma }{\omega {\epsilon }_0}+{\displaystyle {\sum }_n\frac{\Delta {\epsilon }_{rn}}{1+{\left(j\frac{\omega }{{\omega }_n}\right)}^{1-{\alpha }_n}}}$(6)

It is a simple model giving a good representation of the frequency behavior of the conductivity and permittivity of biological media, but it does not represent the physical phenomena at the origin of this behavior. It is used very frequently, notably by Gabriel. Generally, the Cole Cole model allows a better representation of the measured values than the Debye model [47] .

The universal dielectric response model represents the complex permittivity by a constant phase function of the form ${\left(j\omega \right)}^{n-1}$. The model combining the Debye and universal dielectric response models proposed by Raicu is of the type [50] [51] :

${\epsilon }_r={\epsilon }_{r\infty }-j\frac{\sigma }{\omega {\epsilon }_0}+\frac{\Delta {\epsilon }_{rn}}{{\left({\left(j\omega T\right)}^{\alpha }+{\left(j\omega T\right)}^{1-\beta }\right)}^{\gamma }}$ (8)

Regardless of the model used, the different parameters are adjusted using optimization algorithms to correspond as precisely as possible to the values resulting from the measurements.

Starting from the wave equations are obtained from the Maxwell equations described previously. For the electric and magnetic fields in which we are interested, the wave equations, at a point $r$ and at time t, are given respectively by [52] [53] - [55] :

$\nabla \times \nabla \times E\left(r,t\right)+{\mu }_0{\epsilon }_0\frac{{\partial }^2}{\partial t^2}E\left(r,t\right)={\mu }_0\frac{{\partial }^2}{\partial t^2}J\left(r,t\right)$(9)

$\nabla \times \nabla \times H\left(r,t\right)+{\mu }_0{\epsilon }_0\frac{{\partial }^2}{\partial t^2}H\left(r,t\right)={\mu }_0\nabla \times J\left(r,t\right)$(10)

where $E$ is the electric field, $H$ is the magnetic field and μ₀ and ε₀ are the magnetic permeability and electric permittivity of air (vacuum), respectively. The wave equations are written as follows [55] :

$\Delta E\left(r,t\right)-{\mu }_0{\epsilon }_0\frac{{\partial }^2}{\partial t^2}E\left(r,t\right)=\frac{1}{{\epsilon }_0}\nabla \rho \left(r,t\right)+{\mu }_0\frac{\partial }{\partial t}J\left(r,t\right)$(11)

$\nabla \times \nabla \times H\left(r,t\right)+{\mu }_0{\epsilon }_0\frac{{\partial }^2}{\partial t^2}H\left(r,t\right)=\nabla \times J\left(r,t\right)$(12)

The modeling of the contribution in radiated emissions from each discretization cell takes into account the currents in the structure, obtained by an appropriate method [55] . Firstly, a discretization cell is considered equivalent to a dipole. So, in this case, only one dimension, which is the length, is considered. To achieve this goal, two main approaches can be used for such a calculation: the quasi-steady state approximation and the infinitesimally small dipole approximation.

We know that the fields $E\left(r,t\right)$ and $H\left(r,t\right)$ can be written in terms of the vector potential $A$ and the scalar potential ϕ. The notion of potentials was used in order to simplify the resolution of Maxwell’s equations. Figure 7 below shows the structure of a discretized cell [55] .

We demonstrate that, for a cell crossed by a current and whose section is very small compared to the length, the radiation will be considered equivalent to that generated by an electric dipole. Thus, the vector potential is given by:

$A=\frac{{\mu }_0}{4\pi }I{\displaystyle {\int }_c\frac{{\text{e}}^{-jk\left|r-r^{\prime }\right|}}{\left|r-r^{\prime }\right|}\text{d}{l}^{\prime }}$ (13)

where I is the current passing through the cell and C is the length.

The radiated emissions are perfectly defined by the magnetic field and the electric field. Using the Lorentz gauge, we can write the electric field as a function of the vector potential alone.

$H=\frac{1}{{\mu }_0}\nabla \times A$ (14)

$E=\frac{1}{j\omega {\epsilon }_0{\mu }_0}\nabla \times \nabla \times A$(15)

We consider the discretization cell presented in Figure 9 . The vector potential is given by:

$A=\frac{\mu }{4\pi }{j}_{\gamma }{\displaystyle {\iiint }_{v^{\prime }}\frac{{\text{e}}^{-jk\left|r-r^{\prime }\right|}}{\left|r-r^{\prime }\right|}\text{d}{v}^{\prime }}$(16)

In our 1D case, we consider $\gamma =x$. The vector potential is written:

$A_x=\frac{\mu }{4\pi }{I}_x{\displaystyle {\int }_{-dx/2}^{dx/2}\frac{{\text{e}}^{-jk\left|r-r^{\prime }\right|}}{\left|r-r^{\prime }\right|}\text{d}{x}^{\prime }}$ (17)

By applying the quasi-steady state approximation to Equation (17), we therefore find:

$A_x=\frac{\mu }{4\pi }{I}_x{\text{e}}^{-jkr}{\displaystyle {\int }_{-dx/2}^{dx/2}\frac{1}{\sqrt{x^2+y^2+z^2}}\text{d}{x}^{\prime }}$ (18)

The calculation gives:

$A_x=\frac{\mu }{4\pi }{I}_x{\text{e}}^{-jk}\log \left(\frac{x-\frac{dx}{2}+\sqrt{{\left(x-\frac{dx}{2}\right)}^2+y^2+z^2}}{x+\frac{dx}{2}+\sqrt{{\left(x+\frac{dx}{2}\right)}^2+y^2+z^2}}\right)$ (19)

The infinitely small dipole approximation is widely used in electromagnetic modeling and especially in the field of antennas. In this case, the length of the dipole is infinitesimally small compared to the wavelength. Typically, it is less than a tenth. Note also that the distance of the observation point from the origin of the dipole is an important parameter in this approximation. The vector potential is written:

$A_x=\frac{\mu }{4\pi }{I}_x\frac{{\text{e}}^{-jkr}}{r}dx$(20)

In order to improve the calculation precision, we exploit the calculation approach based on the Maclaurin series. This approach is based on the fact that the length of the dipole is infinitely small compared to the wavelength. It resembles the infinitely small dipole approximation which is only a special case of it. Thus, we choose an order higher than the first order for calculation improvement.

By changing the variable ( $\alpha =\frac{x^{\prime }}{\lambda }$, $\eta =\frac{r}{\lambda }$ et $Q=\frac{x}{\lambda }$) in expression (17), we obtain the integral expression of the potential vector, considering Equation (21) below:

$\frac{{\text{e}}^{-jk\left|r-r^{\prime }\right|}}{\left|r-r^{\prime }\right|}=\frac{{\text{e}}^{-jk\sqrt{{\left(x-x^{\prime }\right)}^2+y^2+z^2}}}{\sqrt{{\left(x-x^{\prime }\right)}^2+y^2+z^2}}$ (21)

The new expression of the vector potential integrating the variable αis of the following form:

$A_x=\frac{\mu }{4\pi }{I}_x{\displaystyle {\int }_{-dx/2\lambda }^{dx/2\lambda }\frac{{\text{e}}^{-j2\pi \sqrt{{\eta }^2-2Q\alpha +{\alpha }^2}}}{\sqrt{{\eta }^2-2Q\alpha +{\alpha }^2}}\text{d}\alpha }$ (22)

Considering $\frac{{\text{e}}^{-j2\pi \sqrt{{\eta }^2-2Q\alpha +{\alpha }^2}}}{\sqrt{{\eta }^2-2Q\alpha +{\alpha }^2}}=f\left(\alpha \right)$, given that $\alpha \ll 1$ Taking into account the dimensions of discretization cells are very small compared to the wavelength, the development of the function $f\left(\alpha \right)$ in the form of a Maclaurin series is in the following polynomial form:

$f\left(\alpha \right)=f\left(0\right)+f^{\prime }\left(0\right)\alpha +\frac{1}{2}{f}^{″}\left(0\right){\alpha }^2+\frac{1}{6}{f}^{‴}\left(0\right){\alpha }^3$ (23)

Moreover, $f^{\prime }\left(0\right)=f^{‴}\left(0\right)=0$ because in the calculation of the integral of the polynomial equivalent to f between −dx/2λ and dx/2λ, terms of odd order, in particular those of the first and third order, are zero.

$A\left(x,y,z\right)=\frac{\mu }{4\pi }{I}_x\left(\frac{1}{2}f\left(0\right)\frac{dx}{\lambda }+\frac{1}{24}{f}^{″}\left(0\right)\frac{d{x}^3}{{\lambda }^3}\right)e_x$(24)

It is this last order which makes it possible to improve the precision. The component following ox of the vector potential is written, as in the case of the infinitely small dipole, as a function of the wavelength, of the length of the dipole. The expression of the vector potential is given by:

$A_x=\frac{\mu }{4\pi }{I}_x{\text{e}}^{-jkr}dx\left(\left(\frac{1}{r}+\frac{1}{24r^3}\left(x^2{\left(jkr\right)}^2+\left(3x^2-r^2\right)\left(1+jkr\right)\right)\right)d{x}^2\right)$(25)

Determining the radiated emissions of a cabling system involves two main steps: calculating the conducted emissions and deducting the radiated emissions. The first consists of determining for each discretization cell the current passing through it. Then, knowing both the geometry and the current value at each frequency, we use the analytical calculation approach to define the contribution of each of the discretization cells. The EM field at any point in space is the contribution of each cell and it is obtained by summing the different components of the magnetic and electric fields.

To study the consequences of magnetic fields on living beings using Maxwell’s equations, we can follow an approach based on the modeling of electromagnetic fields and their interaction with biological tissues. All electromagnetic phenomena can be described by Maxwell’s equations. These equations make it possible to link the electromagnetic field to the sources which gave rise to it. In fact, these four equations are split into two groups of two equations: the first group reflects the intrinsic properties of the field (independently of the sources) and the second really provides information on its dependence on the sources [21] .

To do this we have a few steps that we must follow to carry out this modeling:

The Maxwell equations to be implemented to model the coupling of electromagnetic fields-biological tissues are as follows:

$\nabla \cdot E=\frac{\rho }{{\epsilon }_0}$ (26)

$\nabla \cdot B=0$ (27)

$\nabla \cdot E=-\frac{\partial B}{\partial t}$ (28)

$\nabla \cdot B={\mu }_0\left(j+{\epsilon }_0\frac{\partial E}{\partial t}\right)$ (29)

Magnetic field-biological tissue model

Suppose we study human biological tissue exposed to a magnetic field radiated by industrial static converters located in a work environment. Solving equation 13 allows us to simplify the calculation of the magnetic field generated at a distance r illustrated in Figure 10 as follows:

$B_{\left(r\right)}=\frac{{\mu }_{0}I}{2\pi r}$ (30)

with:

${\mu }_0$: the magnetic permeability of the vacuum is 4π × 10⁻⁷ H/m;

I: Current intensity consumed by the converter in A;

r: the distance between the technician and the converter in m.

Figure 8 shows the interaction between the electromagnetic field and biological tissue.

The electromagnetic field generated around the converter induces a voltage in the biological tissue capable of raising its electrical potential in accordance with Faraday’s law:

$e=-\frac{\text{d}\varphi }{\text{d}t}$ (31)

with $\varphi$ is the magnetic flux through weber biological tissue (Wb).

If the magnetic field B varies with time, we can therefore write:

To fully understand, here is a global model to determine the e.m.f. in a cylinder (biological tissue):

$\frac{\text{d}\Phi }{\text{d}t}=2\pi f\Phi$ (32)

Gold:

$\Phi =B\cdot A$ (33)

From where:

$\frac{\text{d}\Phi }{\text{d}t}=2\pi fB\cdot A$ (34)

with: f: the frequency in Hz.

A: biological surface area in m².

Furthermore, surface area of biological A can be obtained via the following relation:

$A=\pi r^2$ (35)

Injecting Equations (30) and (35) into expression (34), we find:

$\frac{\text{d}\Phi }{\text{d}t}=2\pi f\cdot \frac{{\mu }_{0}I}{2\pi r}\cdot \pi r^2$(36)

Gold:

$e=\frac{\text{d}\Phi }{\text{d}t}$ (37)

Let us equalize Equations (36) and (37), we therefore find:

$e=\frac{\text{d}\Phi }{\text{d}t}=2\pi f\cdot \frac{{\mu }_{0}I}{2\pi r}\cdot \pi r^2$(38)

This model determines the voltage induced by the magnetic field in biological tissue.

To model and analyze the impact of electric fields on living beings, we made the hypothesis that human biological tissue is modeled as a continuous medium with specific dielectric and conductive properties.

We know well that, the continuity equation for electric charges in a conducting medium, the conservation of charge is described in the following form:

$\nabla \cdot j=-\frac{\partial \rho }{\partial t}$(39)

where: j is the current density in A/m³ and $\rho$ is the charge density in C/m³.

Furthermore, the relationship between the electric field and the current density is given by Ohm’s local law for a conducting medium is given by:

$j=\sigma \cdot E$(40)

where E is the radiated electric field in V/m.

We inject the expression (40) into (39), we obtain:

$\nabla \cdot \left(\sigma E\right)=-\frac{\partial \rho }{\partial t}$(41)

We know that the electric field derived from the scalar potential can be expressed by the following relation:

$E=-\nabla \cdot \Phi$(42)

Let us introduce the electric potential into the Maxwell-Gauss equations for a dielectric medium, we obtain the following Poisson equations:

$\nabla \cdot \left(\epsilon \cdot \nabla \cdot \varphi \right)=-\rho$(43)

By combining the Poisson equation with the relationship between the electric field and the current density, we arrive at the following final equations:

$\nabla \cdot \left({\epsilon }_r{\epsilon }_0\nabla \varphi \right)=-\rho +\nabla \cdot \left(\sigma \nabla \varphi \right)$ (44)

Equation (44) takes into account both the dielectric response and the conductivity of biological tissue. It also makes it possible to calculate the distribution of electric potential in tissues under the influence of an external electric field, which is crucial for assessing the biological impact of electromagnetic fields.

As a reminder, the greatest quantity of heat is evacuated from the biological environment by conduction. To implement the thermal model of biological tissue irradiated by the electric field, we consider that the biological environment is unidirectional and homogeneous. For this, the temperature propagation model in a biological environment is governed by the bio-heat equation.

$\langle P\rangle =\rho \cdot C_{th}\cdot \frac{\partial T\left(x,t\right)}{\partial t}$(45)

with $\langle P\rangle$ the power flow in the biological tissue in Watt; C_th: Thermal capacity of the tissues in [W·s/˚C] and $T\left(x,t\right)$: average increase in tissue temperature in ˚C.

Furthermore, the flow of incident power in biological tissue can be calculated using Poynting’s theorem, Poynting’s vector P is equal to the average power passing through the unit area of the wave plane, so the flux of P represents a power, in our case of a plane wave of direction O z, the vector P (z) has only one component P (z) because almost all of the energy received by the biological tissue is transformed into heat, which allows us to write:

$\langle P\rangle =\frac{1}{2}\left|E\times H\right|$ (46)

Let us replace Equation (46) in (45), we therefore find:

$\frac{1}{2}\left|E\times H\right|=\rho \cdot C_{th}\cdot \frac{\partial T\left(r,t\right)}{\partial t}$(47)

Let us multiply the expression (47) by $\frac{\partial r}{\partial t}$, we therefore find:

$\frac{1}{2}\left|E\times H\right|=\rho \cdot C_{th}\cdot \frac{\partial r}{\partial t}\frac{\partial T\left(r,t\right)}{\partial r}$ (48)

Now, $\frac{\partial r}{\partial t}$ represents the speed of temperature distribution in m/s and $r=\sqrt{x^2+y^2+z^2}$ the position or space in m.

$\gamma =\frac{\partial r}{\partial t}$ (49)

For this purpose, expression (48) therefore becomes:

$\frac{1}{2}\left|E\times H\right|=\rho \cdot C_{th}\cdot \gamma \cdot \frac{\partial T\left(r,t\right)}{\partial r}$ (50)

The absorbed power density D (W/m³) in the biological medium transformed into heat is given by the following relationship:

$D=-\frac{\text{d}P}{\text{d}r}=\frac{{\sigma }^2}{2}{E}^2{\text{e}}^{-2\alpha r}$ (51)

By introducing the density, we can also, from Equation (51), obtain the specific absorption rate (SAR) which is expressed in W/kg as follows:

$DAS=\frac{D}{\rho }=\frac{\sigma }{\rho }\cdot {\left|E\right|}^2$(52)

We can also write the following thermodynamic energy conservation equation:

$\frac{1}{2}\left|E\times H\right|\text{d}r=\rho \cdot C_{th}\cdot \gamma \cdot \text{d}\left[\Delta T\left(r,t\right)\right]$ (53)

We can infer the temperature variation in biological tissue as follows:

$\Delta T\left(r,t\right)=\frac{1}{\rho \cdot C_{th}\cdot \gamma }\cdot \left[\frac{1}{2}\left|E\times H\right|\right]\cdot r$ (53)

Poynting $\frac{1}{2}\left|E\times H\right|$ vector by the average power received by the biological tissue and the position r by $\sqrt{x^2+y^2+z^2}$:

$\Delta T\left(r,t\right)=\frac{P_{moy}\cdot \gamma }{\rho \cdot C_{th}\cdot \sqrt{x^2+y^2+z^2}}$ (54)

$\Delta T\left(r,t\right)$: Spatial distribution of temperature in ˚C.

$P_{moy}$: Average power received by the biological tissue in Watt.

When a material is exposed to an electromagnetic field, it is subject to a current density due to the movement of charges. Biological materials are not good conductors. Indeed, they conduct a current, however the losses can be significant, they cannot be described as lossless. This is due to the fact that the electromagnetic field only penetrates very superficially inside a conductor. The penetration depth is given by relation (55).

$\delta =\frac{1}{\omega \sqrt{\frac{\epsilon {\mu }_0}{2}\left[\sqrt{1+{\left(\frac{\sigma }{\epsilon \omega }\right)}^2}-1\right]}}$ (55)

Considering the electrical parameters of biological tissue presented in Table 1 below, the frequency range from 0.01 to 100 Gigas Hertz [Ghz] of propagation of electromagnetic waves radiated by in human biological tissue, the model of the tissue under the effect of the electromagnetic field and the equations going from (1) to (54), the simulation results are presented in Figures 9-12 . Figure 10 and Figure 13 the results obtained experimentally in the work of Rakotomananjara DF and Randriamitantsoa PA [56] . Then, Figure 11 , Figure 12 below respectively show the results of the spatial profile of the temperature and the current induced in the biological tissue under the influence of the electromagnetic fields radiated by the industrial static converters in a professional environment.