The skin effect is an electromagnetic phenomenon that makes the current flows only on the surface of the conductors at high frequency. This article is based on the phenomenon to model a structure made in coplanar technology. In reality, these types of structures integrated metal layers of different thickness of copper (9 μm, 18 μm, 35 μm, 70 μm). The neglect of this parameter introduces errors, sometimes significant, in the numerical calculations. This is why an iterative method (FWCIP) based on the wave concept was restated. Validation of results was carried out by comparison with those calculated by Ansoft HFSS software and Agilent ADS Technology. They show a good matching.
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 Ω 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. Thus the electromagnetic behavior of a planar structure will be described by writing the boundary conditions and continuity of the tangential fields on each pixel containing the interface circuitry to study. This integral formulation retains the advantages well known iterative methods including ease of implementation and speed of execution.
The flowchart in
The theoretical formulation for the iterative method is based on determining the relationship between the incident waves
ing walls and the relative permittivity of the different mediums of the structure, k Î {medium1, medium 3}.
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 “
The operators of diffraction
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: Hs1 = 1 on the source and 0 elsewhere.
Hm1 = 1 on the metal and 0 elsewhere.
Hi1 = 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: Hm2 = 1 on the metal and 0 elsewhere.
Hi2 = 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).
m, n: Designating the index for modesÎ {N}.
a: mode indicator TE (Transverse Electric), TM (Transverse Magnetic).
k: Medium considered k Î {1, 2}.
µ0: Magnetic vacuum permeability (H/m).
w: Angular pulsation equal to pulsation 2Õf (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 “
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
To show the robustness of this new formulation of the iterative method we have applied to the study of two coupled micro strip lines, parallel, symmetrical and placed in the same plane. Metal foils which constitute them are copper and have a thickness which is important. The structure of these is presented below in
In our model both input and output voltage “VIN et VOUT” of the structure to study will be modeled by two field sources S1 {E1, J1} and S2 {E2, J2}.
Structural parameters:
The plan of discontinuity Ω1 contains the input and the output of the circuit and the two metal ribbons, without thickness, modeling the undersides of the two micro strip lines. The Ω2 plan only contains two metal ribbons, without thickness, modeling the upper faces of the two micro strip lines (
We present in
excitations considered S1 (E1, J1) et S2 (E2, J2). In this technique we apply the superposi- tion theorem that considers the study structure is alternately excited by the source S1 (E1, J1) then by the source S2 (E2, J2). This brings us to a problem with a single excitation source whose theoretical development is simple to prepare.
The study of structure (
1st Step: The localized source S1(E1, J1) is activated.
In this step we short circuit the excitation source n˚2 (E2 = 0), as shown in
The matrix representation (1) allows us to write:
J1 and J2 current densities are created by the excitation source {S1} respectively at the source {S1} and location of the source shorted {S2}. So that the parameters Y11 and Y21
are respectively the admittance seen by excitation source {S1} and the admittance viewpoints occupied by the source shorted {S2}.
2nd Step: The localized source S2(E2, J2) is activated
In this second step we short-circuit the source n˚1 (E1 = 0), as shown in
The matrix representation (1) allows us to write:
J1 and J2 current densities are created by the excitation source {S2} respectively at source level {S2} and the location of the source shorted {S1}. So that the parameters Y22 and Y12 are respectively the admittance seen by excitation source {S2} and the admittance viewpoints occupied by the source shorted {S1}.
The calculation of parameters Yij coupling matrix requires a convergence study of these items based on iterations, optimizing the computation time and increasing the accuracy of these results. By observing
Admittances Y11 and Y22, are identical, this is translated by the symmetry of the structure, same for the admittances Y12 and Y21.
The parameters Yij of the coupling matrix between the two circuit excitation sources have been found. It remains to calculate the characteristic impedance of the study structure is necessary for the calculation of Sij parameters.
We applied in a first step the empirical formulas of Hammerstad (based GARDIOL) to calculate the characteristic impedance of the study structure. These formulas we offer values approaching Zc. For more details on the characteristic impedance must use a numerical method for the calculation.
er is the relative permittivity of the dielectric which is the speeding of the electromagnetic wave. By cons in a microstrip line (
・ For the microstrip line such as:
・ For microstrip line such as:
With: λg: guided wavelength.
λ0: wavelength in free space.
eeff: effective permittivity of the micro strip line.
C = 2.99792458 ´ 108 m∙s−1: speed of light.
f: working frequency.
Zc: Characteristic impedance of the actual micro strip line.
Z0: Impedance characteristic of the imaginary line.
Adjusting for frequency:
It is possible to consider an approximate way, replacing eeff in formulas by eeff(f).
With:
Si
For our application, the study structure has a ratio between the width of the metal band “c1” and the dielectric thickness “h1” equal to 1.023622.
The relative error obtained with respect to the characteristic impedance calculated using the ADS Linecalc tool (which is equal to 47,618 W) is equal to 1.3% This is for equal work often 13.5 GHZ.
The Yij parameters of the structure study were calculated and so does the characteristic impedance Zc. This allows to deduce the Sij parameters of the two micro strip lines coplanar relationship defined below:
Calculating the parameters in decibel Sij is done by applying Equation (14) below:
We present in the following, the effect of the error obtained in the approximate calculation of the characteristic impedance of the parameters Sij.
Formulas Hammerstad:
LineCalc of ADS:
Error:
Error on Zc = 1.3%, Error on S11 = 2%, Error on S21 = 1.5%.
An error of 1.3% on the characteristic impedance results in an error of 2% on the coefficient of reflection S11 and an error of 1.5% on the transmission coefficient S21. This for equal work often 13.5 GHZ and 5000 iterations. We note that there is a slight increase of this error on the Sij parameters relative error on Zc.
By cons in the frequency band “22.4 GHZ - 28.8 GHZ” coupling is very small between the two sources of excitations. In this frequency band the coefficients of transmissions “S21 and S12” are strictly less than ?10 dB “S11 and S22” and coefficients of reflections are strictly greater than −1.2 dB. The circuit mismatched with the two inputs and it behaved like a notch filter.
The validation of these results was carried out by comparison with those calculated by Ansoft HFSS software and technology Agilent ADS software. This comparison shows that these results present the same variations and coincide in certain frequency bands.
We report in
These curves show that if we increase the thickness of the metal strips the transmission between the two sources of excitations increases regardless of the operating frequency.
We see in the results of 15 and 16 to a certain oscillation frequency 42.2 GHZ which is mainly due to the resonance of the housing. This oscillation is also observed in the results given by HFSS and ADS software. In fact to justify this we proceed to demonstrate.
We start with the ADS software. We compare the results of simulations of the structure with and without housing
We are interested in our method of analysis. We take a metal housing walls that we excited by two sources of localized fields, but without the presence of micro-strip lines. We keep the same physical parameters of the study structure (
The different results obtained by simulation of the coefficients of reflection and transmission (
This article allowed us to review an electromagnetic model with what we have characterized as a planar structure including a flat, thick copper conductor. Indeed this model which is based on the phenomenon of skin effect encouraged us to model the latter two metal ribbons without thickness, placed one above the other which has a h2 distance equal to the thickness T of conductor. Both sides, parallel to the plane Oyz, driver summers have been neglected because 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. The medium containing the thick conductor consists of a metal complex permittivity
This work has been supported by the SYSCOM laboratory, National Engineering School of Tunis Tunis El Manar University.
Mejri, R. and Aguili, T. (2016) New Formulation of the Mulilayer Iterative Method: Application to Coplanar Lines with Thick Conductor. Journal of Electromagnetic Analysis and Applications, 8, 197- 218. http://dx.doi.org/10.4236/jemaa.2016.89019