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

2022年12月23日

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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

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