The dispersion relation of the reltron modulation section has been derived using the lumped-element equivalent circuit model of a biperiodic side-coupled cavity resonator. Simultaneous equations involving voltage–current have been structured as eigenvalue problems and solved by the matrix method. The solutions, i.e., the eigenvalues and the corresponding eigenvectors, give the frequencies and electromagnetic (EM) field patterns of the three normal modes and, subsequently, the dispersion relation. The analysis shows that the frequency differences among the three modes are not equal, in general, because of the nonlinear dependence. A reltron modulation section has been designed in the S-band and modeled using MAGIC and CST Microwave Studio (MWS). The eigenvalues and the corresponding eigenvectors have been obtained at different coupling depths, and the corresponding coupling coefficients ( $k$ ) are calculated. The effect of the coupling depth on the device efficiency has also been studied through the particle-in-cell (PIC) solver.

中文翻译：

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Reltron调制段色散分析及耦合深度对器件性能影响的仿真研究

reltron 调制部分的色散关系已经使用双周期侧耦合腔谐振器的集总元件等效电路模型推导出来。涉及电压-电流的联立方程已被构造为特征值问题并通过矩阵方法求解。解，即特征值和相应的特征向量，给出了三个正常模式的频率和电磁 (EM) 场模式，以及随后的色散关系。分析表明，三种模式之间的频率差异通常不相等，原因是非线性相关。在 S 波段设计了一个 reltron 调制部分，并使用 MAGIC 和 CST Microwave Studio (MWS) 进行建模。得到了不同耦合深度下的特征值和对应的特征向量，并计算相应的耦合系数（ $k$ ）。耦合深度对器件效率的影响也已通过粒子内细胞 (PIC) 求解器进行了研究。