# 数学代写|数值分析代写numerical analysis代考|MATH3820

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## 数学代写|数值分析代写numerical analysis代考|Coupling Impedance

In order to compute the coupling impedance, just the longitudinal component of the electric field produced by the induced currents is required, since the electric field produced by the traveling charge does not contribute to the coupling impedance.

So, given Equation (24), the per-unit-length longitudinal coupling impedance (1) can be easily computed in a generic point in the transverse plane as
$$Z_{|}(r, \varphi, k)=\frac{j k \zeta_0}{2 \pi \varphi_a \beta^2} \sum_{n=0}^{\infty} \sigma_n(\kappa) b_n(\kappa, r, \varphi) .$$
It is worth noting that, since the matrix and the known term vector in Equation (15) are purely real, all the unknown terms $\sigma_n$ are real, too. So, the longitudinal coupling impedance is purely imaginary. This result is expected since there are not diffraction losses.

From Equation (26), by means of the definition (1) it is possible to find the expression of the transverse coupling impedance, that is
$$Z_{\perp}(r, \varphi, k)=\frac{j k \zeta_0}{2 \pi \varphi_a \beta^2} \sum_{n=0}^{\infty} \tilde{\sigma}n(\kappa)\left{\frac{\partial b_n(\kappa, r, \varphi)}{\partial r} \hat{r}+\frac{1}{r} \frac{\partial b_n(\kappa, r, \varphi)}{\partial \varphi} \hat{\varphi}\right} .$$ In order to compute the transverse coupling impedance in a practical way, it is worth recalling the addition theorem for the Hankel functions $$K_0(w R)=\sum{p=-\infty}^{+\infty}(-1)^p I_p\left(\rho^{\prime} w\right) K_p(\rho w) e^{j p\left(\varphi-\varphi^{\prime}\right)},$$
being $R=\sqrt{\rho^{\prime 2}+\rho^2-2 \rho^{\prime} \rho \cos \left(\phi^{\prime}-\phi\right)}$ and $\rho^{\prime} \leq \rho$.

## 数学代写|数值分析代写numerical analysis代考|Numerical Results

Some numerical results are presented in this section, in order to discuss the efficiency of the proposed method. In all the simulations, the shape of the angular slot is $a=1 \mathrm{~cm}, \varphi_a=60^{\circ}$. A Simpson rule with an adaptive spacing is adopted to compute the matrix coefficients (16), while a Gaussian quadrature algorithm is used for the coefficients (17). Since the kernel of the terms in Equation (16) exhibits a logarithmic singularity and gives rise to computational problems, proper numerical manipulations have to be introduced to navigate the problem. The adopted solution is discussed in the Appendix A. At first, the behavior of the coefficients $\sigma_n$ is shown for different values of the frequency and of the distance between the particle and the structure. In Figure $2 a$ the absolute values of expansion coefficients are shown for different frequencies. The particle is in the center of the axis, as that is the most realistic case in practice. At lower frequencies, the coefficients’ amplitudes quickly decrease, with few of them being enough to properly represent the current density: for $a \kappa=0.01$, the third coefficient is already four orders of magnitude lower than the first one. At higher frequencies, the amplitude of the coefficients decreases more slowly, so an higher number of coefficients is required, as expected.

In Figure $2 b$ the coefficients are shown in case of offset of the particle beam. As expected, the amplitude of the coefficients increases as the distance of the particle from the structure decreases. Additionally, while the odd coefficients vanish in case of centred particle, they grow proportionally to the particle offset.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Coupling Impedance

$$Z_{\mid}(r, \varphi, k)=\frac{j k \zeta_0}{2 \pi \varphi_a \beta^2} \sum_{n=0}^{\infty} \sigma_n(\kappa) b_n(\kappa, r, \varphi)$$

Z_{lperp}(r, Ivarphi, k)=|frac{j k \zeta_0}2 Ipi Ivarphi_a \beta^2} Isum_{n=

$$K_0(w R)=\sum p=-\infty^{+\infty}(-1)^p I_p\left(\rho^{\prime} w\right) K_p(\rho w) e^{j p\left(\varphi-\varphi^{\prime}\right)}$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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