**3.2. Application to confined environments modeled by arbitrary cross-section oversized waveguides: 2.5D TLM approach**

The analysis of electrically large structures like tunnels involves a high computation time by using full-wave methods like TLM. However, some astutenesses can be considered to efficiently model them, which translates in finding an approach to simplify the analysis of oversized waveguides. From the geometrical point of view, tunnels can have any arbitrary shape, and this constraint has to be considered. From the electromagnetic point of view, waves propagating in tunnels can be classified in modes. This problem is well known in electromagnetic theory. As we shall see, a full-wave model based on a modal decomposition of the waves for the analysis of the radio propagation in tunnels at high frequencies is presented. Besides, the concept is used to further reduce the complexity of this electromagnetic problem.

### *3.2.1. The TLM modal approach*

Modal approaches in frequency domain are based on the expansion of fields in terms of modes. In general, total fields can be represented as the sum of modes as shown in (5):

$$F\left(\mathbf{x}, \mathbf{y}, \mathbf{z}\right) = \sum\_{i=1}^{N} A\_i(\mathbf{x}, \mathbf{z}) e^{-(\mathbf{a}\_i + j\beta\_i)\mathbf{y}} \,, \tag{5}$$

where *N* is the total number of modes, *Ai*(*x*, *z*) the amplitude of the *i*-*th* mode at the coordinate (*x*, *z*) in the plane perpendicular to the propagation direction *y*, *α<sup>i</sup>* its attenuation, and *β<sup>i</sup>* its phase constant. The mode amplitude *Ai*(*x*, *z*) gives the electric or magnetic field distribution of the mode. The phase constant *β* represents the change in phase along the path traveled by the wave. Lastly, the attenuation constant is a measure of losses in the structure.

*3.1.1. Boundary conditions in TLM*

244 Advanced Electromagnetic Waves

accounted for in the TLM method [7].

this electromagnetic problem.

*3.2.1. The TLM modal approach*

**oversized waveguides: 2.5D TLM approach**

The boundary conditions link the electromagnetic fields through the tangential or normal field values. Since TLM is based on the equivalence between Maxwell's equations and equations for voltages and currents that travel in a mesh of interconnected transmission lines, a relationship of the involved voltages at the boundary can be found. The scattered voltages *V<sup>r</sup>* are always known values, and the incident voltages *V<sup>i</sup>* are unknown. Any resistive load at a boundary may be simulated by introducing a reflection coefficient Γ as shown in (3), [7].

*armj* (*x*, *<sup>y</sup>*, *<sup>z</sup>*) <sup>=</sup> <sup>Γ</sup>*kV<sup>r</sup>*

This formalism allows us to represent a variety of boundary conditions as long as a reflection coefficient Γ can be defined. For instance, for a perfectly electric conductor (PEC), boundary is simulated by choosing Γ = −1; a perfectly magnetic conductor (PMC) is implemented by choosing Γ = 1. The reflection coefficient for lossy boundaries [15], relating the incident and

where <sup>∆</sup>*<sup>y</sup>* and <sup>∆</sup>*<sup>z</sup>* are the cell dimensions; *Zzy* the impedance of the arms, equal to *<sup>Z</sup>*<sup>0</sup> <sup>=</sup> *<sup>µ</sup>*0/<sup>0</sup> for the standard SCN node; and *Zs* is the surface impedance (SI). In the case of a good conductor, an approximation by a real number for Γ is presented in [28]. However, Γ is in general complex and would alter the shape of the excitation pulses, which cannot be

**3.2. Application to confined environments modeled by arbitrary cross-section**

The analysis of electrically large structures like tunnels involves a high computation time by using full-wave methods like TLM. However, some astutenesses can be considered to efficiently model them, which translates in finding an approach to simplify the analysis of oversized waveguides. From the geometrical point of view, tunnels can have any arbitrary shape, and this constraint has to be considered. From the electromagnetic point of view, waves propagating in tunnels can be classified in modes. This problem is well known in electromagnetic theory. As we shall see, a full-wave model based on a modal decomposition of the waves for the analysis of the radio propagation in tunnels at high frequencies is presented. Besides, the concept is used to further reduce the complexity of

Modal approaches in frequency domain are based on the expansion of fields in terms of modes. In general, total fields can be represented as the sum of modes as shown in (5):

∆*yZs* (*s*) − *Zzy*∆*z*

∆*yZs* (*s*) + *Zzy*∆*z*

*armj* (*x*, *y*, *z*) (3)

(4)

*k*+1*V<sup>i</sup>*

reflected voltages, can be expressed in Laplace domain by the (4).

<sup>Γ</sup> (*s*) <sup>=</sup> *<sup>V</sup><sup>r</sup>* (*s*)

*<sup>V</sup><sup>i</sup>* (*s*) <sup>=</sup>

In the literature, the simplification of uniform guide problems in TLM was first introduced by [16] to obtain the dispersion properties. They proposed the reduction of the calculation region by introducing the field dependence solution in the propagation direction *y*. The longitudinal dependence can be described by exponential terms *e*−*jβy*, where *β* is the phase constant and *y* the longitudinal distance in the guide. Hence, for a specific mode, two points along the longitudinal distance of the guide have only a phase difference *β* (*y*<sup>2</sup> − *y*1). Then, it is possible to find a relationship between the reflected voltages of the node at the time *k*∆*t* and the incident ones at the time (*k* + 1)∆*t*. This approach allows one to reduce the computation domain from a 3D mesh of 3D nodes to a 2D one, avoiding to calculate the fields over all points along the propagation direction.

The concept of mode will be employed to find a more efficient formulation, to obtain a reduced 2.5-dimensional TLM modal approach for finding fields. The term 2.5D is used as the 3D computational domain is reduced to a 2D mesh in the guide cross section. However, cells are 3D ones that account for all 6 electromagnetic field components. The reduced model can be applied for a uniform (invariant cross section along *y*) tunnel of arbitrary cross section.
