基于双区域增强体-面积分方程的有损互连结构的低频分析

张黎; 童美松

doi:10.12263/DZXB.20240834

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基于双区域增强体-面积分方程的有损互连结构的低频分析

学术论文 | 更新时间：2025-10-16

- 基于双区域增强体-面积分方程的有损互连结构的低频分析
- Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations
- 电子学报 2025年53卷第6期页码：1874-1884
- 作者机构：
  
  同济大学电子与信息工程学院，上海 201804
- 作者简介：
  
  [ "张黎女.同济大学电磁场与微波技术专业博士研究生，慕尼黑工业大学国家公派联合培养博士研究生.于2021年获得国家留学基金委奖学金.主要研究方向为无芯片RFID标签、分形天线和计算电磁学. E-mail: 1910692@tongji.edu.cn" ]
  [ "童美松男.华中科技大学学士、硕士，美国亚利桑那州立大学博士，美国伊利诺伊大学香槟分校博士后.现任同济大学微电子学院副院长、特聘教授、长聘教授、博士生导师.主要研究方向为电磁场理论、天线理论与设计、微波电路设计、芯片电子封装、电磁逆散射与微波成像、多物理电磁学、计算电磁学等.E-mail: mstong@tongji.edu.cn" ]
- 基金信息：
  
  国家自然科学基金(62071331);上海市科技委员会(21500714400)
- DOI：10.12263/DZXB.20240834
  中图分类号： TN4;
- 收稿：2024-09-11，
  
  修回：2025-03-10，
  
  纸质出版：2025-06-25
- 稿件说明：
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张黎, 童美松. 基于双区域增强体-面积分方程的有损互连结构的低频分析[J]. 电子学报, 2025, 53(06): 1874-1884.

ZHANG Li, TONG Mei-song. Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations[J]. Acta Electronica Sinica, 2025, 53(06): 1874-1884.
张黎, 童美松. 基于双区域增强体-面积分方程的有损互连结构的低频分析[J]. 电子学报, 2025, 53(06): 1874-1884. DOI：10.12263/DZXB.20240834

ZHANG Li, TONG Mei-song. Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations[J]. Acta Electronica Sinica, 2025, 53(06): 1874-1884. DOI：10.12263/DZXB.20240834

摘要

在实际应用中互连结构的导体是有损耗的，在低频时其趋肤深度会很大，电流会渗入导体内部并覆盖整个导体横截面，这时传统上采用完全电导体（Perfect Electric Conductor，PEC）近似或采用表面阻抗计入损耗的单一区域形式的积分方程可能不再有效，因此在积分方程法中需要采用双区域积分方程形式来描述.在低频时电场积分方程（Electric Feld Integral Equation，EFIE）容易出现崩溃现象，为此人们提出了采用增强电场积分方程（Augmented Electric Field Integral Equations，AEFIEs）来解决单一导体或介质结构中的低频崩溃问题.文章将有损导体视为可穿透的介质物体，提出了采用双区域增强混合场积分方程（Augmented Hybrid Field Integral Equati

ons，AHFIEs）来解决有耗导体互连结构的低频崩溃问题.混合场积分方程（Hybrid Field Integral Equations，HFIEs）由描述导体外部的EFIE和描述导体内部的磁场积分方程（Magnetic Field Integral Equation，MFIE）组成.由于磁荷密度出现在HFIEs的

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算子中，文章将磁荷密度作为新的独立未知函数并引入磁流密度连续性方程作为附加约束方程描述导体部分，对互连结构中任意可穿透的介质部分则采用体积分方程（Volume Integral Equations，VIEs）描述，将2类方程通过场耦合结合起来便建立了整个结构的双区域增强体-面积分方程（Augmented Volume-Surface Integral Equations，AVSIEs）.基于AEFIEs的传统方法只能求解包含PEC和各向同性及均匀介质衬底的封装结构，而文章提出的基于AVSIEs的方法可以求解包含有损耗的导体和任意性质的介质衬底的封装结构，因而大大增强了求解此类问题的能力. AVSIEs采用矩量法求解，其中RWG（Rao-Wilton-Glisson）基函数用来表示增强混合场积分方程（AHFIEs）中的表面电流密度和表面磁流密度，而SWG（Schaubert-Wilton-Glisson）基函数则用来表示体积分方程（VIEs）中的体电流密度或体磁流密度，脉冲基函数用来表示AHFIEs中的电荷密度和磁荷密度.文章通过数值算例验证了提出方法的有效性和优越性.

Abstract

Real conductors of interconnect structures are lossy and their skin depth becomes large at low frequencies. The traditional one-region formation with the approximation of perfect electric conductor (PEC) or surface impedances may not be valid anymore

and two-region integral equation formations are needed in the integral equation approach. Also

the electric feld integral equation (EFIE) tend to break down at low frequencies and augmented electric field integral equations (AEFIEs) have been proposed to remedy the problem. In this work

we treat lossy conductors as penetrable objects and propose two-region augmented hybrid field integral equations (AHFIEs) for low-frequency analysis. The hybrid field integral equations (HFIEs) consist of the EFIE of describing the exterior of a conductor

and the magnetic field integral equation (MFIE) of describing its interior. Since the magnetic current density appears in the

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operator in the HFIEs

we select the magnetic charge density as a new unknown function to be solved and introduce the continuity equation of magnetic current density as an extra equation. By incorporating the volume integral equations (VIEs) of describing the substrate with arbitrary penetrable media in the interconnect structures

the two-region augmented volume-surface integral equations (AVSIEs) are formulated for entire structures. The traditional method based on the AEFIEs can only be used for solving the problems including PEC interconnects and isotropic and homogeneous substrates while the proposed method based on the AVSIEs can applied to solve the problems with arbitrary materials so the capability of solving problems has been significantly enhanced. The AVSIEs are solved by the method of moments (MoM) where the Rao-Wlton-Glisson (RWG) and Schaubert-Wilton-Glisson (SWG) basis functions are used to represent the surface current densities of AHFIEs and volume current densities of VIEs

respectively

while a pulse basis function is employed to represent the charge densities of AHFIEs. Numerical examples are presented to illustrate the approach and good results have been obtained.

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references

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