Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations

ZHANG Li; TONG Mei-song

doi:10.12263/DZXB.20240834

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Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations

PAPERS | 更新时间：2025-10-16

- Low-Frequency Analysis of Lossy Interconnect Structures Based on Two-Region Augmented Volume-Surface Integral Equations
- ACTA ELECTRONICA SINICA Vol. 53, Issue 6, Pages: 1874-1884(2025)
- 作者机构：
  
  同济大学电子与信息工程学院，上海 201804
- 作者简介：
- 基金信息：
  
  National Natural Science Foundation of China(62071331);Shanghai Science and Technology Commission(21500714400)
- DOI：10.12263/DZXB.20240834
  CLC： TN4;
- Received：11 September 2024，
  
  Revised：2025-03-10，
  
  Published：25 June 2025
- 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

CHEW W C , TONG M S , HU B . Integral Equation Methods for Electromagnetic and Elastic Waves [M ] . San Rafael : Morgan Claypool Publishers , 2009 .

TONG M S . Efficient electromagnetic analysis for interconnect and packaging structures using dual basis function in the method of moments [J ] . IEEE Transactions on Components, Packaging and Manufacturing Technology , 2011 , 1 ( 7 ): 1089 - 1097 .

LI M K , CHEW W C . Multiscale simulation of complex structures using equivalence principle algorithm with high-order field point sampling scheme [J ] . IEEE Transactions on Antennas and Propagation , 2008 , 56 ( 8 ): 2389 - 2397 .

HARRINGTON R F . Field Computation by Moment Methods [M ] . London : Macmillan , 1968 .

RAO S , WILTON D , GLISSON A . Electromagnetic scattering by surfaces of arbitrary shape [J ] . IEEE Transactions on Antennas and Propagation , 1982 , 30 ( 3 ): 409 - 418 .

QIAN Z G , CHEW W C . Fast full-wave surface integral equation solver for multiscale structure modeling [J ] . IEEE Transactions on Antennas and Propagation , 2009 , 57 ( 11 ): 3594 - 3601 .

MORSEY J D , OKHMATOVSKI V I , CANGELLARIS A C . Finite-thickness conductor models for full-wave analysis of interconnects with a fast integral equation method [J ] . IEEE Transactions on Advanced Packaging , 2004 , 27 ( 1 ): 24 - 33 .

CHEN R , ULKU H A , ANDRIULLI F P , et al . On the low-frequency behavior of vector potential integral equations for perfect electrically conducting scatterers [J ] . IEEE Transactions on Antennas and Propagation , 2022 , 70 ( 12 ): 12411 - 12416 .

HENRY C , MERLINI A , RAHMOUNI L , et al . On a low-frequency and contrast-stabilized full-wave volume integral equation solver for lossy media [J ] . IEEE Transactions on Antennas and Propagation , 2023 , 71 ( 3 ): 2571 - 2586 .

RONG A S , CANGELLARIS A C , DONG L M . Comprehensive broadband electromagnetic modeling of on-chip interconnects with a surface discretization-based generalized PEEC model [C ] // Electrical Performance of Electrical Packaging . Piscataway : IEEE , 2003 : 367 - 370 .

DE ZUTTER D , ROGIER H , KNOCKAERT L , et al . Surface current modelling of the skin effect for on-chip interconnections [J ] . IEEE Transactions on Advanced Packaging , 2007 , 30 ( 2 ): 342 - 349 .

GLISSON A W . Electromagnetic scattering by arbitrarily shaped surfaces with impedance boundary conditions [J ] . Radio Science , 2016 , 27 ( 6 ): 935 - 943 .

HOPPE D J , RAHMAT-SAMII Y . Impedance Boundary Conditions in Electromagnetics [M ] . Bristol : Taylor Francis , 1995 .

SENIOR T B A , VOLAKIS J L . Approximate Boundary Conditions in Electromagnetics [M ] . Stevenage : IET , 1995 .

QIAN Z G , CHEW W C , SUAYA R . Generalized impedance boundary condition for conductor modeling in surface integral equation [J ] . IEEE Transactions on Microwave Theory and Techniques , 2007 , 55 ( 11 ): 2354 - 2364 .

SONG B , ZHU Z H , ROCKWAY J D , et al . A new surface integral formulation for wideband impedance extraction of 3-D structures [C ] //ICCAD-2003. International Conference on Computer Aided Design . Piscataway : IEEE , 2003 : 843 - 847 .

HUYNEN M , KAPUSUZ K Y , SUN X , et al . Entire domain basis function expansion of the differential surface admittance for efficient broadband characterization of lossy interconnects [J ] . IEEE Transactions on Microwave Theory and Techniques , 2020 , 68 ( 4 ): 1217 - 1233 .

SHARMA S , TRIVERIO P . SLIM: A well-conditioned single-source boundary element method for modeling lossy conductors in layered media [J ] . IEEE Antennas and Wireless Propagation Letters , 2020 , 19 ( 12 ): 2072 - 2076 .

SHEIKH H L F , MENSHOV A , GHOLAMI R , et al . Novel single-source surface integral equation for scattering problems by 3-D dielectric objects [J ] . IEEE Transactions on Antennas and Propagation , 2018 , 66 ( 2 ): 797 - 807 .

CHAI W W , JIAO D . Direct matrix solution of linear complexity for surface integral-equation-based impedance extraction of complicated 3-D structures [J ] . Proceedings of the IEEE , 2013 , 101 ( 2 ): 372 - 388 .

WANG Y , GOPE D , JANDHYALA V , et al . Generalized Kirchoff’s current and voltage law formulation for coupled circuit-electromagnetic simulation with surface Integral equations [J ] . IEEE Transactions on Microwave Theory and Techniques , 2004 , 52 ( 7 ): 1673 - 168 .

QIAN Z G , CHEW W C . An augmented electric field integral equation for high-speed interconnect analysis [J ] . Microwave and Optical Technology Letters , 2008 , 50 ( 10 ): 2658 - 2662 .

CHEN Y P , JIANG L J , QIAN Z G , et al . An augmented electric field integral equation for layered medium Green’s function [J ] . IEEE Transactions on Antennas and Propagation , 2011 , 59 ( 3 ): 960 - 968 .

LIU J , JIANG L J . Alternative AEFIE-EFIE method for broadband CEM simulations [J ] . Microwave and Optical Technology Letters , 2012 , 54 ( 7 ): 1737 - 1740 .

YOUNG J C , XU Y , ADAMS R J , et al . High-order nyström implementation of an augmented electric field integral equation [J ] . IEEE Antennas and Wireless Propagation Letters , 2012 , 11 : 846 - 849 .

TIAN X Z , XIAO G B . Time-domain augmented electric field integral equation for a robust marching on in time solver [J ] . IET Microwaves , Antennas Propagation, 2014 , 8 ( 9 ): 688 - 694 .

XIA T , GAN H , WEI M , et al . An integral equation modeling of lossy conductors with the enhanced augmented electric field integral equation [J ] . IEEE Transactions on Antennas and Propagation , 2017 , 65 ( 8 ): 4181 - 4190 .

GAN H H , XIA T , DAI Q I , et al . Augmented electric-field integral equation for inhomogeneous media [J ] . IEEE Antennas and Wireless Propagation Letters , 2017 , 16 : 2967 - 2970 .

XIA T , GAN H , WEI M , et al . An enhanced augmented electric-field integral equation formulation for dielectric objects [J ] . IEEE Transactions on Antennas and Propagation , 2016 , 64 ( 6 ): 2339 - 2347 .

CHEN W J , SUN S , LIU Y , et al . Improved A-EFIE system for electromagnetic simulation in low frequency regime [J ] . IEEE Antennas and Wireless Propagation Letters , 2022 , 21 ( 9 ): 1752 - 1756 .

CHIQOVANI G , DEMUROV A , EREMYAN D , et al . PEEC solution of EM-circuit problems for combined metalic and dielectric structures based on the augmented integral equations [C ] // 2023 International Symposium on Electromagnetic Compatibility-EMC Europe . Piscataway : IEEE , 2023 : 1 - 5 .

CHEN Q L . Electromagnetic Modeling of Three-Dimensional Piecewise Homogeneous Material Bodies of Arbitrary Composition and Geometry [D ] . Houston : University of Houston , 1990 .

RAO S M , WILTON D R . E-field, H-field, and combined field solution for arbitrarily shaped three-dimensional dielectric bodies [J ] . Electromagnetics , 1990 , 10 ( 4 ): 407 - 421 .

SHENG X Q , JIN J M , SONG J , et al . Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies [J ] . IEEE Transactions on Antennas and Propagation , 1998 , 46 ( 11 ): 1718 - 1726 .

HE Y , TONG M S . Accurate electromagnetic analysis of interconnect structures based on a mixed method [J ] . IEEE Microwave and Wireless Components Letters , 2020 , 30 ( 8 ): 733 - 736 .

TONG M S , YIN G Z , CHEN R P , et al . Electromagnetic modeling of packaging structures with lossy interconnects based on two-region surface integral equations [J ] . IEEE Transactions on Components, Packaging and Manufacturing Technology , 2014 , 4 ( 12 ): 1947 - 1955 .

CHU Y H , CHEW W C . A robust surface-integral-equation formulation for conductive media [J ] . Microwave and Optical Technology Letters , 2005 , 46 ( 2 ): 109 - 114 .

CHU Y H , CHEW W C , ZHAO J S , et al . A surface integral equation formulation for low-frequency scattering from a composite object [J ] . IEEE Transactions on Antennas and Propagation , 2003 , 51 ( 10 ): 2837 - 2844 .

SCHAUBERT D , WILTON D , GLISSON A . A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies [J ] . IEEE Transactions on Antennas and Propagation , 1984 , 32 ( 1 ): 77 - 85 .

YANG G M , XING X , DAIGLE A , et al . Electronically tunable miniaturized antennas on magnetoelectric substrates with enhanced performance [J ] . IEEE Transactions on Magnetics , 2008 , 44 ( 11 ): 3091 - 3094 .

TONG M S , CHEW W C . The Nystrom Method in Electromagnetics [M ] . Singapore : Wiley , 2020 .

CHAKRABORTY S , JANDHYALA V . Evaluation of Green’s function integrals in conducting media [C ] // IEEE Antennas and Propagation Society International Symposium Digest Held in Conjunction With: USNC/CNC/URSI North American Radio Sci Meeting . Piscataway : IEEE , 2003 : 320 - 323 .

SHENG W T , ZHU Z Y , YANG K , et al . Efficient evaluation of weakly singular integrals arising from electromagnetic surface integral equations [J ] . IEEE Transactions on Antennas and Propagation , 2013 , 61 ( 6 ): 3377 - 3381 .

TONG M S , CHEW W C . A novel approach for evaluating hypersingular and strongly singular surface integrals in electromagnetics [J ] . IEEE Transactions on Antennas and Propagation , 2010 , 58 ( 11 ): 3593 - 3601 .

KAMON M , TSUK M J , WHITE J K . FASTHENRY: A multipole-accelerated 3-D inductance extraction program [J ] . IEEE Transactions on Microwave Theory and Techniques , 1994 , 42 ( 9 ): 1750 - 1758 .

CHEW W C . Fast and Efficient Algorithms in Computational Electromagnetics [M ] . Boston : Artech House , 2001 .

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