Analytical Analysis of Electromagnetic Properties of Shaped Metal Resonant Cavity in Spherical Coordinate System

LI De-gui; SUN Qi; MA Zi-yin; MEI Zhong-lei

doi:10.12263/DZXB.20250644

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Analytical Analysis of Electromagnetic Properties of Shaped Metal Resonant Cavity in Spherical Coordinate System

PAPERS | 更新时间：2026-04-24

- Analytical Analysis of Electromagnetic Properties of Shaped Metal Resonant Cavity in Spherical Coordinate System
- ACTA ELECTRONICA SINICA Vol. 53, Issue 12, Pages: 4288-4295(2025)
- 作者机构：
  
  兰州大学信息科学与工程学院，甘肃兰州 730000
- 作者简介：
- 基金信息：
  
  National Key Research and Development Program of China(2019YFA0405403)
- DOI：10.12263/DZXB.20250644
  CLC： TN815;O441.4
- Received：23 July 2025，
  
  Accepted：03 December 2025，
  
  Published：25 December 2025
- 稿件说明：
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李德贵, 孙琦, 马子寅, 等. 球坐标系下异形金属谐振腔电磁特性的解析分析[J]. 电子学报, 2025, 53(12): 4288-4295.

LI De-gui, SUN Qi, MA Zi-yin, et al. Analytical Analysis of Electromagnetic Properties of Shaped Metal Resonant Cavity in Spherical Coordinate System[J]. Acta Electronica Sinica, 2025, 53(12): 4288-4295.
李德贵, 孙琦, 马子寅, 等. 球坐标系下异形金属谐振腔电磁特性的解析分析[J]. 电子学报, 2025, 53(12): 4288-4295. DOI：10.12263/DZXB.20250644

LI De-gui, SUN Qi, MA Zi-yin, et al. Analytical Analysis of Electromagnetic Properties of Shaped Metal Resonant Cavity in Spherical Coordinate System[J]. Acta Electronica Sinica, 2025, 53(12): 4288-4295. DOI：10.12263/DZXB.20250644

摘要

分离变量法分析球形谐振腔的亥姆霍兹方程时，其通解只能取整数阶第一类联带勒让德函数，因此难以有效表征该结构派生得到的异形金属谐振腔的电磁特性.针对上述问题，本研究提出一种异形金属谐振腔电磁特性的普适性解析分析框架.通过引入广义超几何函数，构建了任意阶、次的第一类、第二类联带勒让德函数，获得亥姆霍兹方程的完备通解.在此基础上，基于伯格尼斯位函数法，推导得到异形金属谐振腔在横磁、横电模式下基模与高阶模电磁场的解析解，并通过有限元数值仿真进行验证.结果表明，解析解与数值解的谐振频率在基模、高阶模的相对误差分别为0.070%和0.069%；而且，两个解的归一化电磁场分布完全一致，其基模、高阶模的均方根误差仅为

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14.64733315

2.53999996

和

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14.98600006

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，验证了研究方法的准确性与可靠性.本研究不仅成功将经典球形腔结论推广至异形结构，也拓展了电磁场解析建模的应用范围，有助于新型微波、光学器件等的精确设计.

Abstract

As the general solution of the Helmholtz equations for spherical resonant cavities analyzed with the separated-variable method only takes the i

nteger-order first-type associated Legendre function

it is confined to characterize the electromagnetic properties of the shaped-metal resonant cavity derived from this structure invalidly. To address the aforementioned drawbacks

in this study

a universal analytical analytical framework of electromagnetic properties of shaped metal resonant cavity is proposed. The first and the second type of arbitrary order and degree of associated Legendre functions are constructed by introducing the generalized hypergeometric function

so as to complete the general solution of Helmholtz equations. Based on this

the analytical solutions of the electromagnetic fields of the fundamental and higher-order modes in the transverse magnetic and transverse electric modes are derived based on the Borgins’ method of potential function

as well as verified by finite element numerical simulations. The results indicated that the relative errors between the analytical and numerical solutions for the resonance frequencies of fundamental and higher-order modes are 0.070% and 0.069%

respectively. Furthermore

the normalized electromagnetic field distributions of both solutions are mutually consistent

with root-mean-square errors of only

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14.98600006

2.53999996

and

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14.98600006

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respectively

validating the accuracy and reliability of the research method. This study not only successfully extends the conclusion of classical spherical cavities to shaped structures

but also expands the application scope of analytical modeling of electromagnetic fields

which is instrumental to precise design of novel microwave and optical devices.

关键词

Keywords

references

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