An approach based on skin effect phenomenon for the modeling of a micro-strip line with thick copper conductor [3] - [5] . In fact at high frequency the skin effect phenomenon occurs and the current only flows on the periphery of the conductor. According to the law of Biot and Savart a moving current in a conductor generates a magnetic field H around it. If an alternating current flows through a conductive wire, fillers vibrate and the field H varies which creates an induced current loop that opposes to the change of current in the conductor. Therefore, the sum of the alternating current with that of the loop always lowers the center of the driver while the two currents are added to the surface (see Figure 1). The analytical method used is based on the Fast Wave Concept Iterative Process (FWCIP).

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 Figure 2). We modeled the thick metal tape by two metal strips without bulkiness placed one above the other at a distance. we have This allows us to neglect both thick coasts of conductive structure.

Figure 3 presents the variation of skin depth of copper conductor as function of frequency. The value of the skin thickness can be done by applying the following analytical equation:

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 “, , and” defined in spatial domain and the reflected waves “, , and” defined in the spectral domain. The passage of the spatial domain to the spectral domain is using modal Fourier transform (FMT). The passage of the spectral domain to the space domain is using the transform inverse Fourier modal (MTF⁻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 W₁. Figure 4 summarizes the iterative method for planar structures with three layers incorporating different mediums.

and: Diffraction operators, giving the incident waves from the reflected waves that diffract at the discontinuities plans (Ω₁ and Ω₂). They are defined in spatial domain and found in these image operators circuits

placed at plans (Ω₁ and Ω₂).

: Operator reflection ensuring the link between the incident waves and the reflected waves. It is defined in the spectral domain. It contains information on the housing walls and the relative permittivity of the different mediums of the structure, k Î {medium 1, medium 3}.

: Diffraction Operator at each interface (Ω₁ and Ω₂).

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 (Ω₁ and Ω₂) 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 “, , and”, to the reflected waves, , and” when they pass the space domain to the spectral domain:

The operators of diffraction and contains the images of circuit that being in the W₁ and W₂ plans. Figure 2(b) defines these two planes of discontinuities.

The flowchart in Figure 5 summarizes the evolution of the iterative method for a planar structure with three layers of different mediums.

・ 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 Ω₁ plane (metal region, source region of excitement, dielectric region):

where: H_s1 = 1 on the source and 0 elsewhere.

H_m1 = 1 on the metal and 0 elsewhere.

H_i1 = 1 on the dielectric and 0 elsewhere.

・ Diffraction Operator:

The overall diffraction operator is written from the diffraction operators in different regions of Ω₂ plane (metal region, dielectric region):

where: H_m2 = 1 on the metal and 0 elsewhere.

H_i2 = 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:

: Basic functions. It depends on the nature of the box.

: Impedance of the middle k Î {medium 1, medium 2},

: Vacuum impedance,

: Mode admittance reduced to the level of Ω plan.

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

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

: Mode admittance expressed by:

: Propagation constant

: Wave number in a vacuum.

: Speed of light (3 ´ 10⁸ 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}.

: Relative permittivity of the medium k Î {1, 2}.

: Vacuum permittivity (F/m).

µ₀: 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:

: Passing modal operator in the area expressed 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 “,” the reflected waves “,” they pass the space domain to the spectral domain. The quadruple Q ensures the passage of plan W₁ to plan W₂ and inversely (see Figure 2).

According to the diagram in Figure 6 we can write:

・ Parameters du quadruple Q:

The symmetry of the structure, allows us to write:

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

with

: Bases function of the box modes.

with

In this structure (Figure 9(a)) we have modeled the entire thick conductor. The upper and lower films of the conductor are modeled by two metal strips without thicknesses (d = 0), respectively placed in W₁ and W₂ 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 (Figure 9(b)). It is calculated from the complex relative permittivity of the thick conductor and the relative permittivity e_r2 = 1 of the air filling the remainder of the layer 2 of the study structure.

・ Calculus of the effective permittivity of the second layer:

Figure 10 shows the equivalent model of the medium 2 located between the two planes W₁ and W₂ of Figure 9(b).

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 W₁ and W₂ of the study structure (Figure 10). This medium consists of air e_r2 = 1 and copper calculated by the relation (21).

The effective permittivity “e_eff” is calculated by the relation

with:

V₁: Volume of air which occupies the second layer.

V₂: Volume of copper which occupies the second layer.

S₁: Surface of air which occupies the plan W₁.

S₂: Surface of the copper occupies the plan W₁.

Therefore:

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

The analysis of the two models is analogous, simply changer e_r2 = 1 by e_eff who becomes in complex form, so we obtain:

At the planes Ω₁ and Ω₂ diffractions operators are given by the following matrix:

(24)

And the diffraction operator at each interface (Ω₁ and Ω₂₎.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. Figure 11 shows that the real part of the structure of the input impedance, converges from 700 iterations and the imaginary part converges from 1500 iterations, for a frequency f = 1 GHz.

The simulation results (Figure 12) for a line with a thick conductor are in good agreement with those calculated by Ansoft HFSS software. For a micro-strip line with a driver without thickness the results obtained by the FWCIP method are shifted. This shows Indeed the use of exact model taking into account the thickness of the conductor seems necessary to improve the efficiency of the iterative method FWCIP.

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 Figure 13), requiring multilayer modeling structures.

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.