# Numerical Methods

## Extending the Possibilities of the Linear Network Solver CONCIRC

Heinz-D. Brüns, Angela Freiberg, 01.10.18-31.01.19

In practice it is helpful in many cases to be able to display system responses not only as a function of frequency. Therefore the program has been extended such that it is now also possible to compute output quantities as a function of varying element values. In this context it has to be emphasized that all input quantities can be parameterized, i.e. in addition to the standard quantities R, L, C also the distributed inductance per unit length or capacitance per unit length, characteristic impedances or the control quantities of controlled current or voltage sources and so on.

The above figure shows a simple parallel resonant circuit fed by a current generator. It can be observed that the element values of R1, L1, and C1 are to be changed in certain intervals where in the example 10 steps for each element are assumed. As the frequency sampling comprises more
than 90 steps at the same time we already have more than 90000 data sets to be stored. This is accomplished in the h5 data format. The graphical output of arbitrary dependencies is possible in an easy way.

Furthermore the computation of S-parameters has been enabled based on the underlying MNA algorithm. For this purpose a special port element has been introduced. The circuit under investigation is allowed to contain all available elements of the program except for active voltage or current sources. As an example the computed S-parameters of a Wilkinson power divider are shown in the figure below. The divider is dimensioned for an operating frequency of 100 MHz.

Heinz-D. Brüns, Angela Freiberg, 20.09.17-20.12.17

After the release of the linear network solver CONCIRC which can be downloaded from the website of the institute numerous suggestions for extensions and improvements have reached the developers. Based on this the graphical output has been extended by

• Bode plots
• Nyquist plots

Numerous settings are provided for creating the curves to be plotted.
Two examples are demonstrated in the following figures:

Complex admittance of a circuit with frequency as parameter (Source: TET, TUHH).

Bode plot of the complex impedance of a circuit (magnitude and phase) (Source: TET, TUHH).

Furthermore the following features were introduced:

• Non-ideal transformer including the elements L1,L2, M12
• Extension of the current controlled voltage source for the modeling of magnetically coupled circuits
• Output of all computed quantities in dB
• Input of complex control data for all dependent sources from files
• Complex amplitudes of voltage and current generators from files
• Direct input of complex impedances

## Extension of the Contour Integral Method for Stochastic Modeling of Waveguiding Structures

Ph. D. Thesis Jan Birger Preibisch, 01.02.2013 – 30.06.2017

Electromagnetic fields confined in a printed circuit board (PCB) cavity, the field inside a substrate integrated waveguide (SIW), and the normal scattered fields of infinitely extended structures have in common that the wave vector lies in a plane. In this thesis, the Contour Integral Method (CIM) is extended to model practical problems and devices as PCBs and planar microwave devices with inhomogeneous substrates and optical devices in TE and TM polarization. Physics-based approaches are used to incorporate transitions to the planar
problem. For the handling of stochastic formulations of physics-based models, Polynomial Chaos Expansion (PCE) was used. Algorithms and methodologies are proposed to apply PCE efficiently and comprehensively. The proposed extensions and methods are validated using state of the art real world application examples.

Electric field strength in a ring-resonator embedded in a photonic crystal at a wavelength of 1562 nm calculated with CIM. (Source: TET, TUHH).

Electric field strength in a dielectric ring-resonator with a metal core cased on substrate integrated waveguide technology at 2.05 GHz calculated with CIM. (Source: TET, TUHH).

Relevant Publications:

Jan Birger Preibisch, Jayaprakash Balachandran, Torsten Reuschel, Katharina Scharff, Bidyut Sen, Christian Schuster
Exploring Efficient Variability-Aware Analysis Method for High-Speed Digital Link Design Using PCE
UBM DesignCon Conference, Santa Clara, CA, USA, 31. Januar 31 – 2. Februar, 2017

Jan Birger Preibisch, Christian Schuster
Extension of the Contour Integral Method for the Modeling of TE Scattering in Two-Dimensional Photonic Structures Using the Duality Principle
10th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics (Metamaterials‘), Crete, Greece, September 19-22, 2016

Jan Birger Preibisch, Piero Triverio, Christian Schuster
Design Space Exploration for Printed Circuit Board Vias Using Polynomial Chaos Expansion
IEEE Signal and Power Integrity Conference (SIPI 2016), Ottawa, ON, Canada, July 25-29, 2016

Jan Birger Preibisch, Torsten Reuschel, Katharina Scharff, Christian Schuster
Impact of Continuous Time Linear Equalizer Variability on Eye Opening of High-Speed Links
IEEE Workshop on Signal and Power Integrity (SPI), Turin, Italy, May 8-11, 2016

Jan Birger Preibisch, Piero Triverio, Christian Schuster
Efficient Stochastic Transmission Line Modeling Using Polynomial Chaos Expansion with Multiple Variables
IEEE MTT-S International Conference on Numerical Electromagnetic and Multiphysics Modeling and Optimization (NEMO), Ottawa, Canada, August 11-14, 2015

Jan Birger Preibisch, Andreas Hardock, Christian Schuster
Physics-Based Via and Waveguide Models for Efficient SIW Simulations in Multilayer Substrates
IEEE Transactions on Microwave Theory and Techniques,vol. 63, no. 6, June 2015

Jan Birger Preibisch, Piero Triverio, Christian Schuster
Sensitivity Analysis of Via Impedance using Polynomial Chaos Expansion
IEEE Workshop on Signal and Power Integrity (SPI), Berlin, Germany, May 10-13, 2015

Jan Birger Preibisch, Xiaomin Duan, Christian Schuster
An Efficient Analysis of Power/Ground Planes With Inhomogeneous Substrates Using the Contour Integral Method
IEEE Transactions on Electromagnetic Compatibility, vol. 56, no. 4, August 2014

Jan Birger Preibisch, Xiaomin Duan, Christian Schuster
Extensions to the Contour Integral Method for Efficient Modeling of TM Scattering in Two-Dimensional Photonic Crystals
7th International Congress on Advanced Electromagnetic Materials in Microwaves and Optics (Metamaterials’2013), Bordeaux, France, September 16-19, 2013

## Validation of Numerical Methods Using a Comparative Analysis of Canonical Electromagnetic Problems

Sangwook Park, M. Sc., (Ajou Universität, Süd Korea) 26.08.2015 – 11.08.2016

Nowadays 3D electromagnetic (EM) simulators based on different numerical techniques such as the finite element method (FEM), the method of moments (MoM), the finite-difference time- domain technique (FDTD), the transmission-line matrix method (TLM), or the finite integration technique (FIT) are available for the analysis of a wide variety of electromagnetic problems. However it is well known that the results provided by the codes depend on many factors such as a suitable discretization of the structure under investigation or a careful positioning of absorbing boundary conditions. In this context each 3D EM simulator has its own requirements concerning the boundary size, the port type, or the shape of mesh elements. All input quantities have to be specified carefully in order to obtain accurate results. In practice it is difficult to satisfy all conditions and, hence, the simulated results could be “inaccurate”. In this case the questions arises how to validate them.

Based on the IEEE standard 1597 “Standard for Validation of Computational Electromagnetic Computer Modeling and Simulation” various examples are investigated in this project. The goal is to find out how the different methods compare to each other when applied to different types of structures. Canonical validation problems such as dipole antenna, a loop antenna, a thick monopole antenna on a finite plate, and a rectangular cavity with apertures are considered using two commercial programs (based on FEM and FIT) and the code CONCEPT-II developed by the Institute of Electrromagnetic Theory (based on MOM). This project investigates the conditions necessary to obtain accurate results for all mentioned cases.

Far field pattern of thin dipole antenna of length 0.3m oriented along the z-axis simulated at 5-GHz using CONCEPT-II. (Source: TET, TUHH).

Numerical results for the input impedance of the dipole antenna using three different simulators. (Source: TET, TUHH).

## MoM-Based Computation of Multionductor Transmission Line Per-Unit-Length Parameters

Heinz-D. Brüns, Angela Freiberg. 01.07.2015 – 30.06.2016

The main task of the program CONMTL is the computation of the per-unit-length parameter matrices L’, C’, G’and R’ for multi-conductor transmission lines of arbitrary cross section. The program is based on a method of moments (MoM) technique by means of which a 2D integral equation is solved numerically for electrostatic problems. This integral equation is derived from the 2D Poisson’s equation of electrostatics. In a first step equivalent surface charge distributions are computed by discretizing both conductor and dielectric boundaries into appropriate numbers of straight segments. Over each segment a constant charge basis function is defined. Boundary conditions to be fulfilled are the constant potential on metal conductors and constant normal electric flux density on dielectric boundaries. In this way the integral equation can be converted into a linear system of equations. Solving this matrix equation provides an approximation of the overall charge distribution, both on the metal parts and on dielectric boundaries. This known charge distribution is necessary for the subsequent computation of the mentioned transmission line parameters. Comprehensive possibilities for visualizing the equivalent charge distribution on all cross sectional transmission line parts as well as the E field distribution exist for quickly checking the validity of the numerical solution. Given the length of a transmission line the corresponding Y-parameter matrix can easily be computed using well-known expressions from multi-conductor transmission line theory. Finally S-parameters are provided as a function of frequency.

Graphical user interface (Source: TET, TUHH)

## Implementation of a Linear Network Solver

Heinz-D. Brüns, Angela Freiberg. 01.07.2014 – 30.06.2015

The program CONCIRC is a simple network solver including all basic features necessary to support the studies of students attending the lecture „Electrical Engineering Fundamentals II’. They can use the program in order to get more insight into the specific behaviour of excercises. CONCEPT-II users can place lumped loads at ports for example. Y-parameter matrices can be read in various formats. The graphing utility „Gnuplot“ http://www.gnuplot.info is used for displaying voltages, currents or impedances as a function of frequency and needs to be installed additionally. Gnuplot is not part of CONCIRC.

Graphical user interface (Source: TET, TUHH)

## Fast Direct H-Matrix Solvers for Computational Electromagnetics

Ph. D. Thesis Alexander Vogt, 01.02.2014 – 30.06.2015

In this project, fast direct solvers based on hierarchical (H-) matrices are investigated and extended to cope with arbitrary boundary conditions. The goal of the project is a fast electromagnetic solver based on the method of moments (MoM) that is applicable to a wide range of problems.

These solvers are particularly interesting for highly resonant structures such as personal computer systems, as iterative solvers show a poor convergence. Furthermore, these systems comprise various components which leads to a high complexity of the simulation setup. Typical components include lossy dielectric and metallic materials, printed circuit boards, cables, and structural components. These are confined in the narrow space of the cavity and impact on the standing pattern. In this scenario, fast numerical methods based on multi-pole expansion of the far field are not feasible and algebraic methods such as the adaptive cross approximation need to be applied. The resulting H-matrices can be efficiently used to solve the MoM system matrix. The implemented solver is part of the Concept-II framework and shows an excellent performance for all frequencies and simulation scenarios a traditional (full) MoM solver can be applied to.

Electric fields calculated in a populated PC system: a metallic enclosure of dimensions (20x20x10 cm) contains a Mini-ITX mainboard with attached RAM modules and a CPU/heatsink configuration. In the depicted field distribution (8GHz), a monopole antenna is used to represent the RF excitation of the CPU. At these frequencies, 100000 unknowns are required to discretize the surface current distribution. The numerical calculation could be performed in less than an hour on four cores – including the evaluation of the electric fields at more than one million observation points. (Source: TET, TUHH).

Scaling behavior of the direct H-matrix solver at the example of a dielectric sphere: the asymptotic complexity of the matrix setup could be reduced from O(N^2) to O(N^(4/3) log⁡N ). The complexity of the solution stage is reduced from cubic to quadratic complexity. This illustrates the superiority of the H-matrix algorithm for solving large electromagnetic problems. (Source: TET, TUHH).

Related Publications:

Alexander Vogt, Torsten Reuschel, Heinz-D. Brüns, Sabine Le Borne, Christian Schuster
On the Treatment of Arbitrary Boundary Conditions Using a Fast Direct H-Matrix Solver in MoM
IEEE Transactions on Antennas and Propagation, Vol. 64, no.8, August 2016

Alexander Vogt, Heinz-D. Brüns, Christian Schuster
Auswirkung absorbierender Materialien auf die Verkopplung von Komponenten innerhalb eines PC-Gehäuses
Internationale Fachmesse und Kongress für Elektromagnetische Verträglichkeit (EMV Düsseldorf) , Düsseldorf, Germany, February 23-25, 2016

Alexander Vogt, Heinz-D. Brüns, Qi Wu, Frank Gronwald, Christian Schuster
A Measurement Setup for Quantification of Electromagnetic Interference in Metallic Casings
IEEE Transactions on Electromagnetic Compatibility, vol.57, no.6, December 2015

Alexander Vogt
Workshop: Solving highly resonant structures using a fast direct H-matrix solver in the method of moments
IEEE International Symposium on Electromagnetic Compatibility and EMC Europe, Dresden, Germany, August 16-22, 2015

Alexander Vogt, Heinz-D. Brüns
Workshop: Modeling using MoM – Solving challenging realworld problems
IEEE International Symposium on Electromagnetic Compatibility (EMC), Gothenburg, Sweden, September, 2014

Alexander Vogt, Heinz-D. Brüns, Christian Schuster
Einfluss von absorbierenden Materialien auf die elektromagnetische Strahlung von Leiterplatten
Internationale Fachmesse und Kongress für Elektromagnetische Verträglichkeit (EMV Düsseldorf) , Düsseldorf, Germany, March 11-13, 2014

Alexander Vogt, Heinz-Dietrich Brüns, Helge Fielitz, Christian Schuster
Modeling Absorbing Materials in Cavities with Apertures Using the Thin Sheet Approximation
Asia-Pacific EMC Symposium (APEMC), Melbourne, Australia, May 20-23, 2013

Alexander Vogt, Heinz Brüns, Samuel Connor, Bruce Archambeault, Christian Schuster
Applicability of the Thin Sheet Approximation to the Analysis of EM Emission from Coated PCBs
IEEE EMC International Symposium on Electromagnetic Compatibility, Pittsburgh, USA, August 05-10, 2012

## Extension of the Contour Integral Method for the Electrical Design of Planar Structures in Digital System

Ph. D. Thesis Xiaomin Duan. 01.09.2007 – 31.07.2013

The fast-paced evolution of the semiconductor industry is putting forth great challenges on electronic design automation tools.
This thesis focuses on the efficient and accurate modeling of planar structures and power planes in digital systems by the extension of a fast two-dimensional numerical technique, the contour integral method. The extension includes the derivation of analytical expressions for rapid analysis of cylindrical vias as well as the combination with other efficient approaches for handling of complex multilayer systems. The validation and evaluation of the method and its extension are thoroughly addressed by a variety of application examples.

TUHH Universitätsbibliothek. TUBDok Link:
http://doku.b.tu-harburg.de/volltexte/2012/1157/

(Source: TET, TUHH).

(Source: TET, TUHH).

Related Publications:

Xiaomin Duan
Extension of the Contour Integral Method for the Electrical Design of Planar Structures in Digital System.
Dissertation 2012. Shaker Verlag Aachen. 2012 (ISBN 978-3-8440-1059-6)

Xiaomin Duan, Renato Riomolo-Donadio, Heinz-D Brüns, Christian Schuster
Extension of the Contour Integral Method to Anisotropic Modes on Circular Ports.
IEEE Transactions on Component, Packaging, and Manufacturing Technology, vol. 2, No. 2, pp. 321-331, 2012.

Xiaomin Duan, Renato Riomolo-Donadio, Heinz-D Brüns, Christian Schuster
Circular Ports in Parallel-Plate Waveguide Analysis with Isotropic Excitations
IEEE Transaction of Electromagnetic Compatibility, vol. 54, No. 3, pp. 603-612, June 2012.

Xiaomin Duan, Renato Rimolo-Donadio, Sebastian Müller, Kijang Han, Xiaoxiong Gu, Young H. Kwark, Heinz-D. Brüns, Christian Schuster
Impact of Multiple Scattering on Passivity of Equivalent-Circuit Via Models
IEEE Electrical Design of Advanced Package & Systems Symposium, Hangzhou, China, December 12-14, 2011.

Xiaomin Duan, Renato Rimolo-Donadio, Heinz-Dietrich Brüns, Christian Schuster
A Hybrid CIM/MoM Approach for Power Plane Analysis Including Radiation Loss
Asia-Pacific EMC Symposium (APEMC), Jeju Island, Korea, May 16-19, 2011

Xiaomin Duan, Xiaoxiong Gu, Jonghyun Cho, Joungho Kim
A Through-Silicon-Via to Active Device Noise Coupling Study for CMOS SOI Technology
IEEE Electronic Components and Technology Conference, Florida, USA, May 31- June 3, 2011.

Xiaomin Duan, Renato Rimolo-Donadio, Heinz-D. Brüns, Christian Schuster
A Combined Method for Fast Analysis of Signal Propagation, Ground Noise, and Radiated Emission of Multilayer Printed Circuit Boards
IEEE Transactions on Electromagnetic Compatibility , vol. 52, no. 2, pp. 487-495, May 2010

Xiaomin Duan, Renato Rimolo-Donadio, Heinz-D. Brüns, Christian Schuster
Fast and Concurrent Simulations for SI, PI, and EMI Analysis of Multilayer Printed Circuit Boards (invited)
Asia-Pacific Symposium on EMC (APEMC), Beijing, China, April 12-16, 2010

Xiaomin Duan, Renato Rimolo-Donadio, Heinz-D. Brüns, Bruce Archambeault, Christian Schuster
Special Session on Power Integrity Techniques: Contour Integral Method for Rapid Computation of Power/Ground Plane Impedance
IEC DesignCon Conference, Santa Clara, USA, February 1-4, 2010

Xiaomin Duan, Bruce Archambeault, Heinz-D. Brüns, Christian Schuster
EM Emission of Differential Signals Across Connected Printed Circuit Boards in the GHz Range
IEEE International Symposium on Electromagnetic Compatibility, Austin, USA, August 17-21, 2009

## A Contribution to the PEEC-Method and its Hybridisation with the Method of Moments

Ph. D. Thesis Volker Vahrenholt. Completed in 2009

In order to inspect the elctromagnetic compatibility (EMC) behavior of complex electrical devices, numerical methods have been developed for current-simulation purposes. In this paper, the partial element equivalent circuit approach (PEEC) is treated and advanced for examining the current distribution on printed circuit boards (PCBs). Moreover, a hybridization of PEEC and the method of moments (MoM) is performed in order to combine the advantages of PEEC and MoM. The gain of the hybrid method is that the computation time is reduced significantly compared to a full-wave MoM-approach.

Related Publications:

Volker Vahrenholt
Ein Beitrag zur PEEC.Methode und deren Hybridiesierung mit der Momentenmethode
Dissertation 2009 (ISBN 978-3-8322-8181-6)

Volker Vahrenholt, Heinz-D. Brüns, Roland Krzikalla, Hermann Singer
Fast Analysis of Electromagnetic Field Coupling into PCBs based on the PEEC Method
IEEE Symposium EMC Europe, Hamburg, Germany, September 8-12, 2008

Volker Vahrenholt, Heinz-D. Brüns, Hermann Singer; Renato Rimolo-Donadio
Verkopplung einer schnellen PEEC-Methode mit der Momentenmethode bei gedruckten Schaltungen mittels der elektrischen Feldstärke
Internationale Fachmesse und Kongress für elektromagnetische Verträglichkeit (EMV), Düsseldorf, Deutschland, 18. -21. Februar 2008

Volker Vahrenholt, Heinz-D. Brüns, Hermann Singer
Razor Blade Fuctions in the PEEC method
IEEE International Symposium on EMC, Honolulu, Hawaii; USA, Jul. 8-13, 2007 WE-AM-1

Volker Vahrenholt, Heinz-D. Brüns, Hermann Singer
Reduction of Unknowns in PEEC Structures by Exploiting Connectivity of PEEC Cells
IEEE Transactions on EMC, vol. 29, no.2, pp. 412-418, May 2007

## Methods for Fast Solution of Large Systems of Equations in the Method of Moments

Ph. D. Thesis Miguel Astner. Completed 31.08.2009

The investigation of complex electrical systems with respect to electromagnetic compatibility (EMC) via the method of moments (MoM) usually leads to a large system of equations with dense matrices. Due to numerical complexity standard solution mehtods are not feasible anymore. In this work methods with much lower numerical complexity than in the classical MoM approach are introduced. Also the parallelization of one of these methods on multi-computer systems is addressed.

Related Publications:

Miguel Astner
Verfahren zur schnellen Lösung von großen Gleichungssystemen in der Momentenmethode
Dissertation 2009 (ISBN 978-3-8322-8568-5)

Miguel Astner, Tobias Pilsak, Heinz-D. Brüns, Jan-Luiken ter Haseborg, Hermann Singer
MoM-Based Analysis of the Immunity of Marine Equipment in a Control Cabinet of Cruise and Container Vessels against 2.4-GHz WLAN Application
IEEE Symposium EMC Europe, Hamburg, Germany, September 8-12, 2008

Miguel Astner, Heinz-D. Brüns, Hermann Singer
Frequenzabhängige Diskretisierung für Strukturelemente in CONCEPT II
Abschlussbereicht zu dem BWB-F&T-Projekt E/E590/1F037 “Lösung großer Systemmatrizen mit der Momentenmethode” im Rahmen des Vorhabens “Berechnung Komplexer Übertragungssysteme”, Projektzeitraum 01.03.2008 – 31.03.2008

Miguel Astner, Heinz-D. Brüns, Hermann Singer
Simple Load-Balancing in Binary-Tree Based Parallel Multilevel Low-Rank Compression Techniques
IEEE Int. Symposium on EMC, Detroit, USA, August 2008

Miguel Astner, Heinz-D. Brüns, Hermann Singer
Ein Verfahren zur schnellen MoM-Berechnung mittels Interpolation der Greenschen Funktion und FFT
Internationale Fachmesse und Kongress für elektromagnetische Verträglichkeit (EMV), Düsseldorf, Deutschland, 18.-21. Feburar 2008

Miguel Astner, Heinz-D. Brüns, Hermann Singer
Application of a Hierarchical SVD/ACA Compression Technique to Near-Field Calculations of Monopole Antennas
IEEE International Symposium on EMC, Honolulu, Hawaii; USA, Jul. 8-13, 2007 WE-AM-1