## 物理代写|电动力学代写electromagnetism代考|PHYSICS 2534

2022年7月15日

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## 物理代写|电动力学代写electromagnetism代考|Validation of the improved WCIP method

The FSS structure in Figure $3.16$ is analyzed by the method WCIP in which the perforated metal sheet has a thickness of $t=0.0175 \mathrm{~cm}$ and slits of dimensions $W=1.32 \mathrm{~cm}$ and $L=0.128 \mathrm{~cm}$. The dimensions of the unitary cell are $1.78 \mathrm{~cm} \times 1.78 \mathrm{~cm}$. The FSS structure is fed by a normally incident plane wave and polarized in the $y$ direction. The FSS structure to be analyzed may be used as a radome where a superior dielectric material (the so-called superstrate) may be added to insulate the antenna compartment [MON 07]. The resulting structure was made up of four dielectric layers in which the bottom layer and the top layer are both made up of air. The WCIP multilayer problem [TIT 08] should be applied to bring the TE and TM modes admittance of the air layers to the interfaces $\Omega_{1}$ and $\Omega_{2}$ through the dielectric slabs covering the perforated metal sheet which are the superstrate and the substrate, resorting to the model of the length of the guide. The interfaces $\Omega_{1}$ and $\Omega_{2}$ and the FSS unitary cell are each divided into $128 \times 128$ pixels and the iterative method is halted after 1,200 iterations.

Figure $3.20$ shows the power transmitted and the power reflected according to the thick FSS operating frequency based uniquely upon the perforated metal sheet with a thickness $t=0.175 \mathrm{~mm}$, without either a superstrate or a substrate and without a dielectric medium in the holes. The WCIP approach which is improved and adapted to thick FSS devices is validated in Figure 3.20. In filling the slits by an insulator with a dielectric constant $\varepsilon_{r}=4$, a reduction in the resonance frequency shown in Figure $3.21$ was observed comparative to the resonance frequency of the same FSS but without any dielectric in the slits, seen in Figure 3.20.

In Figures $3.22,3.23$ and $3.24$ the power transmitted relates to the perforated metallic sheet placed upon a substrate and covered with a superstrate both of the thickness $h$ and of the same dielectric constant $\varepsilon_{r}$. The resonance frequency is inversely proportional to the thickness $h$ of the substrate and that of the superstrate.

## 物理代写|电动力学代写electromagnetism代考|the definition of 

The modal scattering operator $\lceil 4.4\rceil$ is defined using the modes $F_{p q}$, which correspond to a circuit having the dimensions $\left(D_{x} \times D_{y}\right)$. The number of modes is taken as being equal to the number of cells within the structure:
$$\hat{\Gamma}=\sum_{p q}\left|F_{p q}\right\rangle \Gamma_{p q}\left\langle F_{p q}\right|,$$
with the equations:
$$\begin{gathered} F_{p q}=\frac{1}{\sqrt{\mathrm{D}{\mathbf{x}} \mathrm{D}{\mathrm{y}}}} \mathrm{e}^{\mathrm{j} \frac{2 \pi \mathrm{p}}{\mathrm{D}{\mathrm{x}}} \mathrm{x}} \mathrm{e}^{\mathrm{j} \frac{2 \pi q}{\mathrm{D}{\mathrm{y}}} \mathrm{y}} \ \Gamma_{p q}=\frac{Z_{p q}-Z_{0}}{Z_{p q}+Z_{o}} . \end{gathered}$$
The mode impedance is defined according to [4.7]. A convergence study in both $n$ and $m$ is made for each given $p q$ pair using the equation:
$$Z_{p q}=\sum_{m, n} \frac{j \omega \mu_{0}\left|\left\langle H \mid f_{p q, m n}\right\rangle\right|^{2}}{k_{n}^{2} \varepsilon_{r}-\alpha_{p, m}^{2}-\beta_{q, n}^{2}},$$
with $k_{0}$ being the wavenumber in free space; $\mu_{0}$ the substrate permeability; $\omega$ the pulsation; $H$ an indicator function for position, shape and size of the via in the cell; $f_{p q, m n}$ the generating function of the modal base of the elementary cell with dimensions $\left(d_{x} \times d_{y}\right)$ with a boundary consisting of periodic walls [4.8] using the following equations:
\begin{aligned} &f_{p q, m n}=\frac{1}{\sqrt{d_{x} d_{y}}} e^{j \alpha_{p, m} x} e^{j \beta_{q, n} y} \ &\alpha_{p, m}=\frac{2 \pi p}{D_{x}}+\frac{2 \pi m}{d_{x}} \ &\beta_{q, n}=\frac{2 \pi q}{D_{y}}+\frac{2 \pi n}{d_{y}} \end{aligned}
where $(\mathrm{p}, \mathrm{q}, \mathrm{n}, \mathrm{m}) \in \mathrm{Z}^{4}$ are the propagation constants, respectively in $x$ and in $y$.

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|the definition of 

$$\hat{\Gamma}=\sum_{p q}\left|F_{p q}\right\rangle \Gamma_{p q}\left\langle F_{p q}\right|$$

$$F_{p q}=\frac{1}{\sqrt{\mathrm{DxDy}}} \mathrm{e}^{\mathrm{j} \frac{2 \mathrm{~d}}{\mathrm{Dx}} \mathrm{x}} \mathrm{e}^{\mathrm{j} \frac{2 \pi q}{\mathrm{Dy}} \mathrm{y}} \Gamma_{p q}=\frac{Z_{p q}-Z_{0}}{Z_{p q}+Z_{o}} .$$

$$Z_{p q}=\sum_{m, n} \frac{j \omega \mu_{0}\left|\left\langle H \mid f_{p q, m n}\right\rangle\right|^{2}}{k_{n}^{2} \varepsilon_{r}-\alpha_{p, m}^{2}-\beta_{q, n}^{2}},$$

$$f_{p q, m n}=\frac{1}{\sqrt{d_{x} d_{y}}} e^{j \alpha_{p, m} x} e^{j \beta_{q, n} y} \quad \alpha_{p, m}=\frac{2 \pi p}{D_{x}}+\frac{2 \pi m}{d_{x}} \beta_{q, n}=\frac{2 \pi q}{D_{y}}+\frac{2 \pi n}{d_{y}}$$

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## MATLAB代写

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