The iterative method is based on the concept of waves for the study of simple or multilayer planar structures and arbitrary shapes. The advantage of this method is its great ease of implementation and speed of execution mainly due to the systematic use of a fast Fourier transform in modes. So the behavior of a single or multilayer structure can be summarized by boundary conditions expressed at the global interface [1] [2] [3] [4]. These methods need to be improved and require updates as the evolution of these circuits is very fast. The manufacture of new circuits, non-stop and progressive, forces researchers to adapt their simulation tools, which requires the development of the numerical methods characterizing these devices.

The new formulation of the wave concept iterative method (FWCIP) [5] [6] [7] [8] extends it to the study of planar structure integrating flat and thick conductors. This method is based on the concept of waves for solving the problems of electromagnetic diffraction. Its principle is similar to that used in the TLM method (spatial mesh differential method). In this study, the circuit to be studied is placed in a case with metal walls. It makes it possible to define a basis for the decomposition of the electromagnetic field.

Model 1 Figure 1 proposed in this article, takes into account all faces of the metal conductor. Unlike model 2 Figure 2 was studied previously [9] [10].

Figure 3 shows the new electromagnetic model of the study structure. It consists of three layers of different materials. The first layer is filled with a dielectric material of relative permittivity ε_r1 and thickness h₁. It lies between the ground

plane of the study structure and the discontinuity plane Ω₁. The second layer is placed between the two planes of discontinuities Ω₁ and Ω₂. It consists of three environments. Medium 2-1 is a rectangular waveguide with metal walls. It is filled with air and of dimensions “a₀₁ × b × h₂”. The medium 2-2 is occupied by the metal conductor of dimensions “w₁ × L × h₂”. Inside this region the E_int field and the J_int current are void. According to the principle of the skin effect, microwave current circulates only on the periphery of the conductor. Medium 2-3 is a rectangular waveguide with metal walls. It is filled with air and of dimensions “a₀₂ × b × h₂”. Layer 3 is filled with air. It is located between the discontinuity plane Ω₂ and the top cover of the metal housing that surrounds the entire study structure.

Between the metal conductor (medium 2-2) and the walls of the housing are formed the two rectangular waveguides occupying the media (2-1 and 2-3). They are therefore made up of the three electric walls of the housing and one of the faces of the metal conductor parallel to the Oyz plane. The passage of waves from plane Ω₁ to plane Ω₂ Figure 4 (or from plane Ω₂ to plane Ω₁) is done through these two waveguides. The medium 2-2 occupied by the metal conductor will not be crossed by any waves (incident and reflected).

Study structure parameters Figure 5: a = 18.4 mm, b = 24 mm, c₁ = 0.75 mm, d₁ = 0.375 mm, w₁ = 2.3 mm, h₁ = 1.52 mm, h₂ = 10 µm, h₃ = 14.99 mm, ε_r1 = 4.32,ε_r2 = 1, ε_r3 = 1.

The diagram in Figure 6 summarizes the evolution of the multilayer iterative method, introducing into the calculations the effect of the two faces, from the metal conductor parallel to the Oyz plane. The Q₁ and Q₂ quadrupoles model these two waveguides that channel incident and reflected waves from layer 1 to layer 3.

Γ ^ Ω 1 et Γ ^ Ω 2 : Diffraction operators, giving the incident waves from the reflected waves that diffract at the level of the planes of discontinuities (Ω₁ et Ω₂). They are defined in the space domain.

Γ ^ k : Reflection operator ensuring the link between incident waves and reflected waves. It is defined in the spectral range kÎ {medium 1, medium 3}.

α: Te (Transverse Electrical), TM (Transverse Magnetic) mode indicator;

n: Number of iterations.

Γ ^ Q i : Reflection operator defined in the spectral domain. We find in this operator the nature of the walls which enclose the media 2-1 and 2-3, of layer n˚2 (Figure 4). We also find in this operator the nature of the dielectric filling these two medias. This operator connects the incident waves “ A → 21 , A → 22 ” reflected waves “ B → 21 , B → 22 ” when they move from the spatial domain to the spectral domain.

The expressions below relate the incident waves “ A → 1 , A → 21 , A → 22 et A → 3 ” reflected waves “ B → 1 , B → 21 , B → 22 and B → 3 ” when they pass from the spatial domain to the spectral domain.

( A 1 A 21 ) = Γ ^ Ω 1 ( B 1 B 21 ) + ( A 1 0 A 2 0 )

( B 21 B 22 ) = Γ ^ Q i ( A 21 A 22 )

( A 22 A 3 ) = Γ ^ Ω 2 ( B 22 B 3 )

( J → 21 J → 22 ) = ( Y ^ 11 Y ^ 12 Y ^ 21 Y ^ 22 ) ( E → 21 E → 22 )

According to the diagram of Figure 6, we can write:

{ J → 0 = J → 1 + J → 21 J → 2 = J → 22 + J → 3 J → 1 = Y ^ 1 ⋅ E → 1 J → 3 = Y ^ 3 ⋅ E → 3 E → 1 = E → 21 E → 22 = E → 3

FMT (Modal Fourier Transform) is a function for defining the amplitudes of the TE (Transverse Electrical) and TM (Transverse Magnetic) modes in the spectral domain. Its use in the iterative method significantly reduces computational time and accelerates the passage of incident waves from the spatial domain to the modal domain) [11] [12]:

[ [ A k T E ] [ A k T M ] ] = FMT [ [ A k , x ] [ A k , y ] ]

And also accelerates the passage of reflected waves from the modal domain to the spatial domain:

[ [ B k , x ] [ B k , y ] ] = FMT − 1 [ [ B k T E ] [ B k T M ] ]

The (FMT) and (FMT⁻¹) symbolize respectively the direct and inverse Fourier transform in modes.

FMT requires the discretization of the spatial and modal domains. The discretization of the first domain is achieved by meshing the different regions of the planes Ω₁ and Ω₂ into small “pixel” surfaces whose dimensions are linked to the dimensions of the passive components included in the considered plane. In this domain, the interaction between pixels is not considered in the formulation of the method. On the other hand, the use of the FMT introduces implicit pixel coupling in the modal domain. The electromagnetic quantities and the waves (incident and reflected) are represented by matrices whose dimensions depend on the density of the mesh chosen in the spatial domain.

Part of layer 2 of the study structure is filled with air (media 2-1 and 2-3) the other part is occupied by the metallic conductor (medium 2-2). To separate these two domains we use the operator H ^ which is a Heaviside echelon defined below. The operators H ^ i , H ^ S are defined as follows:

H ^ i = { 1 on the dielectric region 0elsewhere

H ^ S = { 1 on the excitation source region 0elsewhere

We can write the operator H ^ depending on the operators H ^ i et H ^ S :

H ^ = H ^ i + H ^ S = { 1 : in the medium filled with air 0 : in the medium occupied by the metallic conductor

Figure 7 shows the variation of the resonance frequency, the input impedance of the study structure, as a function of the thickness of the metal used. We find in this figure a comparison between the results of the two models, model 1 and model 2, with those calculated by Ansoft’s HFSS software. If the thickness of the metal used increases, the results of the model in Figure 1 get closer to the results calculated by Ansoft’s HFSS software. These results show that the model in Figure 1 is more rigorous. This model ensures an increase in the accuracy of the

analytical method used.

Figure 8 shows the relative error on the resonance frequency of the input impedance of the study structure. This depends on the thickness of the metal used. This figure shows a comparison of the relative error between the results calculated by Ansoft’s HFSS software and those calculated by the iterative method (models in Figure 1 and Figure 2). We note that the model 1 has a smaller error than the model 2. This is for a flat metal conductor whose thickness varies from 10 μm to 80 μm. These results confirm that the model in Figure 1 increases the accuracy of the analytical method used.

In this article we have shown the effectiveness of the correction made to the iterative method by the model 1 we have proposed. The latter ensures the modeling of planar structures integrating flat metal conductors with a significant thickness. The results found are compared with those calculated by Ansoft’s HFSS software showing the improvement made to the iterative method FWCIP.

The authors declare no conflicts of interest regarding the publication of this paper.

Mejri, R. and Aguili, T. (2022) Rigorous Electromagnetic Modeling of a Micro Strip Line Incorporating a Flat and Thick Metallic Conductor Based on the Skin Effect Phenomenon. Open Journal of Metal, 12, 1-9. https://doi.org/10.4236/ojmetal.2022.121001