^{1}

^{1}

In this work, we applied two electromagnetic models for the characterization of a planar structure including a flat, thick copper conductor. Indeed the first model is consisted by modeling two metal ribbons without bulkiness, placed one above the other at a distance of
h
_{2} equal to the thickness of the thick conductor. This approach has been implemented and tested by the iterative method. The results of simulations have been compared with those calculated by the Ansoft HFSS software, and they are in good concordance, validating the method of analysis used. The second model is based on the calculation of the effective permittivity of the medium containing the thick conductor. This medium consists of a metallic region of complex relative permittivity
, the rest of this medium is filled with air
e
_{r}
_{2}
= 1. The effective permittivity
e
_{eff}
calculated from these two relative permittivity
e
_{r}
_{2}
and
. Comparing the simulation results of this new formulation of the iterative method with those calculated by the software Ansoft HFSS shows that they are in good matching which validates the second model.

In Radio frequency, most devices are made in micro strip technology [

This method is well suited to the calculation of planar structures. In fact, TE and TM modes are used in the iterative method as digital basis of spectral domain in which the FFT. Subsequently, the concept of fast wave is introduced to reflect the boundary conditions and continuity of relationships in different parts of the interface W in terms of waves. The method involves determining an effective relationship to link the incident and reflected waves in different dielectric layers expressing thoughts in modal domain and the boundary conditions and continuity, expressed in terms of waves in spatial domain. The iterative process is then used to move from one field to another using the FMT thus accelerating the iterative process and then the convergence of the method. The use of the FMT requires the pixel description of the different regions of the dielectric interfaces [

An approach based on skin effect phenomenon for the modeling of a micro-strip line with thick copper conductor [

The studied structure is composed by a dielectric substrate of thickness H, on one of its two faces is deposited a metal strip of copper of thickness T, the other face is the ground plane of the structure. In fact the current in hyper frequency flows only on the surface of the conductor (see

d: Skin thickness in meters [m].

ω: pulsation [rad/s].

f: Frequency [Hz].

µ: magnetic permeability [H/m].

r: Resistivity [Ω∙m].

σ: Conductivity [S/m].

with r = (1/s).

The theoretical formulation for the iterative method is based on determining the relationship between the incident waves “^{−1}). These operations are done with repetitions until the convergence of the method. FMT and FMT^{−1} are used to speed up the computation time of the method. This evolution is initiated by the waves emitted by the excitation source on either side of the plane W_{1}.

_{1} and Ω_{2}). They are defined in spatial domain and found in these image operators circuits

placed at plans (Ω_{1} and Ω_{2}).

_{1} and Ω_{2}).

The evolution of iterations through the spectral domain to the space domain is done using the Fourier transform modal “FMT” which considerably reduces the calculation time. Modal Fourier transform requires the fragmentation of discontinuity planes (Ω_{1} and Ω_{2}) in pixels and this so that the electromagnetic behavior of the overall circuit will be summarized by writing the boundary conditions and continuity of the tangential fields on each pixel. The iterative process stops when it reaches the convergence of results.

The terms below link the incident waves “

The operators of diffraction _{1} and W_{2} plans.

The flowchart in

・ Diffraction Operator:

For a source of bilateral excitation polarized in (oy), the overall diffraction operator is written from the diffraction operators in different regions of Ω_{1} plane (metal region, source region of excitement, dielectric region):

where: H_{s}_{1} = 1 on the source and 0 elsewhere.

H_{m}_{1} = 1 on the metal and 0 elsewhere.

H_{i}_{1} = 1 on the dielectric and 0 elsewhere.

・ Diffraction Operator:

The overall diffraction operator is written from the diffraction operators in different regions of Ω_{2} plane (metal region, dielectric region):

where: H_{m}_{2} = 1 on the metal and 0 elsewhere.

H_{i}_{2} = 1 on the dielectric and 0 elsewhere.

・ Expression of the reflection operator:

It is defined in the spectral domain and contains information about the nature of the housing and the relative permittivity of the medium 1 and 3 of the structure. It is expressed by the following relationship:

For a top cover (or lower) placed at a distance h from Ω plan.

For an open circuit without top cover (or lower).

^{8} m/s).

m, n: Designating the index for modes Î {N}.

a: Mode indicator TE (Transverse Electric), TM (Transverse Magnetic).

k: Medium considered k Î {1, 2}.

µ_{0}: Magnetic vacuum permeability (H/m).

w: Angular pulsation equal to pulsation 2×Õ×f (rd/s).

・ Expression of the FMT

The Fourier transform in cosine and sine is defined by:

The Fourier mode transform (FMT) is defined by:

・ The reflection operator of the Quadruple:

The reflection operator of the Quadruple is defined in layer 2 of the structure to be studied. It links the incident waves “_{1} to plan W_{2} and inversely (see

According to the diagram in

・ Parameters

The symmetry of the structure, allows us to write:

After some mathematically manipulation, it is possible to determine the matrix:

with

with

This study begins by checking the convergence of results based on iterations. This is to optimize the calculation time and improve the accuracy of the method.

The simulation result (

In this structure (_{1} and W_{2} plans. The interior of the thick film conductor and both of the two vertical sides are modeled by effective permittivity e_{eff}_{ }characterizing the layer 2 of the structure (_{r}_{2} = 1 of the air filling the remainder of the layer 2 of the study structure.

・ Calculus of the effective permittivity of the second layer:

_{1} and W_{2} of

The permittivity of a metallic conductor is given by the following relationship:

σ: Conductivity [S/m].

f: Frequency [Hz].

The effective permittivity “e_{eff}” is calculated from the relative permittivity of the medium 2 (second layer located between the planes W_{1} and W_{2} of the study structure (_{r2} = 1 and copper

The effective permittivity “e_{eff}” is calculated by the relation

with:

V_{1}: Volume of air which occupies the second layer.

V_{2}: Volume of copper which occupies the second layer.

S_{1}: Surface of air which occupies the plan W_{1}.

S_{2}: Surface of the copper occupies the plan W_{1}.

Therefore:

Z_{02_eff}: Intrinsic impedance of the medium 2 (complex form).

The analysis of the two models is analogous, simply changer e_{r}_{2} = 1 by e_{eff} who becomes in complex form, so we obtain:

At the planes Ω_{1 }and Ω_{2} diffractions operators are given by the following matrix:

(24)

And the diffraction operator _{1} and Ω_{2)}.is given by the following matrix:

This study begins by checking the convergence of results based on iterations. This is to optimize the calculation time and improve the accuracy of the method.

The simulation results (

These models tested showed adapting iterative method considered in the calculation of complex structures. Indeed, through the new formulation of the method, we showed a net correction of the resonance frequency observed in the case of structures without thickness. These models provide almost the same results (see

We have shown the efficiency of the correction allocated to the iterative method by the two modals proposed for the modeling of the planar structures integrating the thick conductors. The results which were found and compared to those which were calculated by the HFSS software well demonstrate the improvement allocated to the iterative method.

This work allowed taking stock of two electromagnetic models with which we had characterized a planar structure including a flat, thick copper conductor. In fact, the first model based on the phenomenon of skin effect allowed us to model the latter with two metallic ribbons without thicknesses, placed one above the other with a distance h_{2} equal to the thickness of the thick conductor. Both sides of the conductor have summers neglected

since the width of the metal is strictly greater than its thickness. This is a simplifying assumption which has no effect on the results of the problem. This approach has been implemented and tested by the iterative method. Simulations results found were compared with those calculated by the software Ansoft HFFS, they were in good agreement, validating the method of analysis used. The second model is based on the calculation of the effective permittivity of the medium containing the thick conductor. This medium consists of a metallic region of complex relative permittivity, and the rest of this medium is filled with air e_{r}_{2} = 1. The effective permittivity e_{eff} calculated from these two relative permittivity e_{r}_{2} and

The different cases of the structures studied in this article allowed to highlight the potential of iterative FWCIP and suggest that it would be an essential tool in the global modeling of planar structures with thick conductors.

This work has been supported by The SYSCOM laboratory, National Engineering School of Tunis Tunis El Manar University.

Rafika Mejri,Taoufik Aguili, (2016) A New Approach Based on Iterative Method for the Characterization of a Micro-Strip Line with Thick Copper Conductor. Journal of Electromagnetic Analysis and Applications,08,95-108. doi: 10.4236/jemaa.2016.85010