Rigorous Electromagnetic Modeling of a Micro Strip Line Incorporating a Flat and Thick Metallic Conductor Based on the Skin Effect Phenomenon

In most studies of microstrip circuits, the majority of researchers assume that the microstrip structures studied have flat metallic conductors of finite widths but without thickness. But in reality these types of structures integrate metallic copper conductors of different thicknesses. If we neglect this thickness we introduce error in the electrical parameters of the microstrip structure, which affects the effective permittivity, the characteristic impedance, the adaptation of the circuit, the resonance frequency, etc. Given the importance of this parameter (thickness of the metal of micro rubon structures), rigorous electro-magnetic modeling of the thick micro rubon line based on the skin effect phenomenon (In fact at high frequency the skin effect phenomenon occurs and the current only flows on the periphery of the conductor) has been proposed to improve the studied electric model and ensure the increase in the precision of the analysis method used: Wave concept iterative process. The good agreement between the simulated and published data justifies the improvement of the model.


Introduction
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 Open Journal of Metal 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].ˆQi Γ : Reflection operator defined in the spectral domain.We find in this oper- ator 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.

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

TM k y k
And also accelerates the passage of reflected waves from the modal domain to the spatial domain: 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 Ω 1 and Ω 2 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 Ĥ which is a Heaviside echelon defined below.The operators ˆi H , ˆS H are defined as follows: 1 on the dielectric region ˆ0 elsewhere     This 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.

Conclusion
In this article we have shown the effectiveness of the 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.

Figure 3 Figure 1 .
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 h 1 .It lies between the ground

Figure 6 .
Figure 6.Electrical diagram modeling the study structure.
Transform) is a function for defining the amplitudes of the TE (Transverse Electrical) and TM (Transverse Magnetic) modes in the Open Journal of Metal 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]:

1 :
We can write the operator Ĥ depending on the operators ˆi H et ˆS H : in the medium filled with air ˆˆˆ0: in the medium occupied by the metallic conductor

Figure 7
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

Figure 7 .
Figure 7. Variation of the resonance frequency of the input impedance of the study structure according to the thickness of the metal used.

Figure 8 .
Figure 8. Variation in relative error on the resonance frequency of Ze depending on the thickness of the metal used.

Figure 8
Figure8shows the relative error on the resonance frequency of the input impedance of the study structure.This depends on the thickness of the metal used.