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Free Space Propagation

The propagation inside a wireless channel such as air, space or sea water can be modeled as the electromagnetic waves undergo many effects including reflection, diffraction and scattering due to the presence of obstacles in the channel. These phenomena can lead to increase propagation losses and have to be taken into account for designing a system (transmitter, receiver or transceiver) with a link budget analysis.
The reflection effect occurs when electromagnetic waves strike an object, such as the ground, buildings or walls, which have a larger size than the wavelength of the propagating waves. Furthermore, if the radio wave falls on a propagating medium having different electrical properties, part of the energy is transmitted and part of it is reflected back. The diffraction consists of a phenomena occurring when a travelling wave interacts with a surface having sharp irregularities. This happens in the case of skyscrapers and creates a shadowed region where the received energy decreases as the user moves deeper. The Huygens’ principle can explain the diffraction effect and describes the field in the shadow region as a sum of multiple secondary sources producing a new wave front in direction of the receiver. Contrary to the reflection effect, the scattering manifests itself when the obstacle on the propagating channel is smaller than the wavelength of the waves. These various phenomena can lead to a multipath loss effect, increasing or decreasing the received power and must be included in the propagation equation.
In general case, antennas are used for transmitting signals and imply that the free space equations are mainly developed inside the far-field region, also called Fraunhofer region, where the propagating waves behave as plane waves and the power decays inversely with distance. Hence, this distance, df, is usually depicted as in (1.6) and depends on the largest dimension, D, of the antenna [2].
The most well-known equation which models the free space propagation is the Friis equation given in (1.7) for two antennas in direct sight without obstacles [2] as shown in Fig. 2.
where Pt is the transmitted power, Pr(d) is the received power, Gr and Gt are the receiver and transmitter antenna gain respectively, d is the distance between the antennas, ηrad is the radiation efficiency, and L is the coefficient taking into account all the perturbing effects. It has to be notified that the gain is related to the effective aperture, Ae, which depends on the physical size of each antenna as described in (1.8) [2].
Hence, it is also easy to represent the attenuation suffered by the propagating signal through a wireless channel. It is given by the difference of the transmitted and received power, called Path Loss, PL, and it is expressed in (1.9) [2].
To go further, some researchers had tried to find a better equation than this from Friis to match specific applications. Longley-Rice model, Durkin’s model, Okumura’s model, and Hata’s model are the most used for outdoor propagation model. As an example, Okumura’s model [3] is mainly used inside urban area from 100 MHz to 1900 MHz for a distance between 1 km and 100 km. This model has been essentially developed for mobile telecommunication. It is based on an empirical formula obtained from numerous measurements of signal attenuation between base stations to mobile phones. The empirical path-loss equation of Okumura is given by (1.10) [3] at a distance d.
where L(fc,d) is the free space path loss at a distance d and a carrier frequency fc, Amu(fc,d) is the median attenuation in addition to free space path loss, G(ht) and G(hr) are the base station and mobile antenna gain, and GAREA is the gain changing with the environment. Amu(fc,d) and GAREA are determined by the experiments. G(ht) and G(hr) are defined as follow:
(ℎ ) = 20 10 ( ℎ ) , 30 < ℎ <1000 (1.11) hr and ht are the height of the base station antenna and the mobile phone antenna respectively. This model has a derivation of 10-14 dB compared to empirical path loss and the path loss extracted from measurement.
Furthermore, contrary to Okumura’s model, it also exists indoor propagation models to take into account the specific environment related to the interior arrangement of our buildings. For example, the covered distances are smaller than outside and the composition of each wall influences the propagation channel. When the control of the propagating waves is required by the application and the shape of the channel, it is necessary to model the behavior of signals by solving the Maxwell’s equations.

Guided Waves [2]

In the case of harmonic conditions, it is assumed that the material used as propagation channel is isotropic, linear and homogeneous. Hence, the dielectric losses and the magnetic losses are defined as depicted in equations (1.14) and (1.15). Therefore, the Maxwell’s equations become as in (1.16).
In the case of guided waves inside uniform rectilinear transmission system, as depicted in Fig. 3, the wave’s equations can be applied and solved. For that, it is assumed that a planar section of the propagation material is uniform and the propagation is along z-axis in a Cartesian coordinate system. The other axis (x, y) are used for the transverse fields. Thus, by studying the wave’s equations in this material, the dispersion equation can be written as in (1.19) [4].
Γ is the propagation constant and is the same inside all the planar section, even if there is more than one material because of the continuity equations. Then, the general equations of propagation can be defined as the relation between the transverse and lengthwise components of the electromagnetic field and are defined in (1.20) and (1.21) [4].
Based on the previous conditions and on the equations (1.20) and (1.21), only three main modes can exist and propagate in such propagation channel. Transverse Magnetic (TM) and Transverse Electric (TE) mode are characterized by the fact that there is no propagation of the magnetic and electric lengthwise field respectively. They are depicted in (1.22) and (1.23) where ZTM and ZTE are the impedance wave of each mode [2].
The third main mode is the Transverse Electric Magnetic (TEM) mode, which means that there is any propagation of lengthwise electromagnetic fields. Hence, the general equations of propagation (1.20) and (1.21) cannot be used. It is required to combine equations (1.16) and the general expression of the electric (1.24) and the magnetic field (1.25) [4].

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Metallic Waveguides

The first wave guiding structure has been tested in 1894 based on the theory of Joseph John Thomson generalized by Lord Rayleigh [5]. He solved the boundary value problem of electromagnetic waves propagating in an arbitrary shaped waveguide. He has shown that the waves can travel with specific modes known as TE or TM modes, and a combination from them called hybrid mode. The cut-off wavelength was introduced to determine the section of a hollow tube and the frequency below which waves could no longer propagate. Therefore, it became possible to develop passive devices, as shown in Fig. 4, or systems using these technologies and transmit an electromagnetic signal from one point to another while controlling the direction of propagation.
There are different topologies of waveguide such as the dielectric-loaded waveguide, the circular or ridge waveguide [6] and the well-known rectangular metallic waveguide as depicted in Fig. 5. The behavior of the last one is very well-described by solving the equations (1.22) and (1.23) for the TM and TE mode respectively while the boundary conditions have to be applied. Then, the propagation equations of the electric and magnetic fields in each mode are obtained as described in [2]. The equations of each mode inside such waveguide are summed up in Table 1.
This structure presents different loss mechanisms that impact the performance of such waveguide. First, the dielectric loss is mainly due to the dissipation of the electromagnetic energy inside the dielectric material. It can be explained by the fact that the molecules are naturally polarized and when an electric field is applied, it changes this polarization. As the electric field alternatively changes its polarization, it switches the polarization of the molecules constituting the dielectric material. As a result, a dissipation of the energy appears and molecules evacuate this energy by heat. It becomes obvious that this phenomenon depends also on the frequency. The metallic walls create the second loss mechanism. Each metallic material is mainly composed of atomic ions. When a current created by the magnetic field go through a metallic surface, collisions between electrons and atoms are generated. The created energy is evacuated by heating the conductor and leads to the conductor loss. These losses can be described by (1.29) and (1.30) for the dielectric and ohmic loss respectively for a rectangular waveguide as depicted in [2]. Hence, the total loss can be extracted and is illustrated in Fig. 6 [7] based on previous equations for different standard rectangular waveguide.
Another important property of the rectangular waveguide is the power handling. This results in different phenomenon that can limit the maximum transmitted power. For example, it exists the corona and multipaction effect1 which take place generally for space and airspace applications. Here, the study is mainly focused on the average and peak power handling. The first one concerns the continuous power and the second one describes the maximum power that a wave-guiding structure can reach during a short time. Fig. 7 illustrates both of these parameters for several standard waveguides [8].

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