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Temporal-spatial discretization and fractional latency techniques for wave propagation in heterogeneous.. De Rybel, Tom 2010-12-31

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Temporal-Spatial Discretization andFractional Latency Techniques forWave Propagation in HeterogeneousMediabyTom De RybelInd. Ing., Hogeschool Gent, 2002M.A.Sc., The University of British Columbia, 2005A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)February 2010c Tom De Rybel 2010AbstractThis thesis presents the development of a novel, transient wave propagation simulator usingtime-decoupled transmission line models. The models are based on the electro-magnetic tran-sient program (EMTP) power system transient analysis tools, extended to two dimensions. Thenew tool is targeted at acoustic wave propagation phenomena. The method, called TINA fortransient insular nodal analysis, uses temporal interpolation and fractional latency to main-tain synchronicity in heterogeneous media. The fractional latency method allows the modelcells to operate at a local simulation time step which can be a non-integer ratio of the globalsimulation time step. This simplifies synchronicity and saves computation time and memory.Th´evenin equivalents are used to interface the mesh cells and provide an abstraction of thecell content. Numerically, the method is of the transmission-line matrix (TLM) family. Inthe thesis, loss-less and distortion-less models are considered. The loss-less transmission linemodels are studied for their stability and numerical error, for which analytical expressions arederived based on the simulation parameters. A number of new relations were discovered anddiscussed. The TINA method is evaluated in 2D using acoustic experiments, and also a newmethod is proposed for obtaining impulse responses in time-domain simulation, based on aperiodic, band-limited impulse signals.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Scope of the Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Why Was This Research Done? . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Predicting the Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Modeling the Outcome or the System? . . . . . . . . . . . . . . . . . . . . . 41.5 Unification in Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Equivalence Between the Electro-Magnetic and Acoustic Wave Equation . . . 71.7 Relativity and Uncertainty in Simulation . . . . . . . . . . . . . . . . . . . . 82 The TINA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 The Transmission Line Modeling Method . . . . . . . . . . . . . . . . . . . . 122.1.1 TLM Circuit Equivalents in 1D and 2D . . . . . . . . . . . . . . . . . 132.1.2 Equivalence Between Network and Field . . . . . . . . . . . . . . . . 182.1.3 The Dispersion Relation of the Propagation Velocity in TLM . . . . . 222.1.4 The Use of Transmission Lines in TLM . . . . . . . . . . . . . . . . . 252.1.5 Synchronicity in TLM . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1.6 3D in TLM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.2 Transient Insular Nodal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 352.2.1 The TINA Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 372.2.2 The Interpolated Line Model in TINA . . . . . . . . . . . . . . . . . . 392.2.3 Error Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41iiiTable of Contents2.2.4 Domain Termination . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2.5 Validation of the TINA method . . . . . . . . . . . . . . . . . . . . . 442.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Convergence, Dissipation, and Dispersion in 1D Loss-Less Line Models . . . . . 503.1 Convergence of the Line Models . . . . . . . . . . . . . . . . . . . . . . . . . 503.1.1 The Ideal Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1.2 The Interpolated Line Model . . . . . . . . . . . . . . . . . . . . . . 573.1.3 A Physical Interpretation of Marginal Stability in the Line Models . . 643.2 Dissipation and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2.1 Influence of the Interpolation Factor . . . . . . . . . . . . . . . . . . . 693.2.2 Influence of the Reflection Coefficient and Line Length . . . . . . . . 763.3 Direct Comparison Between the Ideal and Interpolated Models . . . . . . . . . 773.4 EMTP 1/10 Rule of Thumb for Interpolated Lines . . . . . . . . . . . . . . . 873.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914 Fractional Sub-Area Latency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.1 The Need for Latency Techniques in TINA . . . . . . . . . . . . . . . . . . . 934.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2.1 Multi-Grid Methods in TLM and FDTD . . . . . . . . . . . . . . . . 964.2.2 Latency in EMTP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.3 Development of Fractional Latency . . . . . . . . . . . . . . . . . . . . . . . 1004.3.1 Equations and Parameters for Fractional Latency . . . . . . . . . . . . 1014.3.2 The Need for Extrapolation . . . . . . . . . . . . . . . . . . . . . . . 1044.3.3 Fractional Latency for Simulation Synchronization . . . . . . . . . . . 1064.3.4 Choice of the Latency Master Time Step . . . . . . . . . . . . . . . . 1064.3.5 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.3.6 Generating Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.3.7 The Accuracy and Stability of Fractional Latency . . . . . . . . . . . 1124.4 Implementations of Fractional Latency . . . . . . . . . . . . . . . . . . . . . 1124.4.1 Fractional Latency in EMTP . . . . . . . . . . . . . . . . . . . . . . . 1134.4.2 EMTP Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.4.3 Fractional Latency in TINA . . . . . . . . . . . . . . . . . . . . . . . 1184.4.4 Avoiding Extrapolation with Transmission-Line Decoupling . . . . . . 1184.4.5 Integration of the Fractional Latency Cells in the TINA Mesh . . . . . 1214.4.6 Simulation Parameters in TINA for Fractional Latency . . . . . . . . . 1234.4.7 TINA Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125ivTable of Contents4.4.8 Issues with Fractional Latency in TINA . . . . . . . . . . . . . . . . . 1364.4.9 Losses to Aid Stability . . . . . . . . . . . . . . . . . . . . . . . . . . 1404.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1435 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1455.1 Two-Dimensional Systems in a Three-Dimensional World . . . . . . . . . . . 1455.2 The Band-Limited Impulse Response Method . . . . . . . . . . . . . . . . . . 1465.3 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3.1 Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3.2 3D Vocal-tract model . . . . . . . . . . . . . . . . . . . . . . . . . . 1505.3.3 2D Room With Source at an Edge . . . . . . . . . . . . . . . . . . . . 1535.3.4 Comparison with measurements . . . . . . . . . . . . . . . . . . . . . 1545.3.5 2D Room With Central Source . . . . . . . . . . . . . . . . . . . . . 1585.3.6 Expansion Duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625.4 Magnitude Over-Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1666 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1676.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1686.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170AppendicesA Derivation of the 1D Transmission Line Models in Time and Frequency Domain 179A.1 The 1D Wave and Telegrapher’s Equations . . . . . . . . . . . . . . . . . . . 179A.2 The Ideal, Loss-Less Line Model . . . . . . . . . . . . . . . . . . . . . . . . 182A.2.1 Derivation in the Time Domain . . . . . . . . . . . . . . . . . . . . . 182A.2.2 Formulation of the EMTP time-domain CP-Line model . . . . . . . . 183A.2.3 Conversion into the Frequency Domain . . . . . . . . . . . . . . . . . 184A.3 The Interpolated Loss-Less Line Model . . . . . . . . . . . . . . . . . . . . . 185A.3.1 The Need for Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 185A.3.2 Derivation in the Time Domain . . . . . . . . . . . . . . . . . . . . . 187A.3.3 Conversion into the Frequency Domain . . . . . . . . . . . . . . . . . 188A.4 The Distortion-Less Line Model . . . . . . . . . . . . . . . . . . . . . . . . . 188vTable of ContentsA.5 Reducing the Computational Load of the Ideal and Interpolated Line Models . 190A.5.1 The Issue with Memory Access . . . . . . . . . . . . . . . . . . . . . 191A.5.2 Ideal Line Model in Voltage Form . . . . . . . . . . . . . . . . . . . . 191A.5.3 Interpolated Line Model in Voltage Form . . . . . . . . . . . . . . . . 193A.5.4 Data Structures for the History Term Storage . . . . . . . . . . . . . . 194B Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196B.1 Diagonalization on a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 196B.2 Finding the Modes of a System . . . . . . . . . . . . . . . . . . . . . . . . . 198B.3 Computing the Matrix Exponential . . . . . . . . . . . . . . . . . . . . . . . 199C Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202D List of Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208viList of Tables1.1 The Acoustical and Electro-Magnetic 1D Wave Equations . . . . . . . . . . . 82.1 TINA Error Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.2 Materials Used in the TINA Simulations . . . . . . . . . . . . . . . . . . . . . 474.1 Numerical Fractional Latency Simulation Event Table . . . . . . . . . . . . . . 1034.2 EMTP Case Simulation Parameters and Switch Times . . . . . . . . . . . . . . 1144.3 Materials Used in the TINA Simulations . . . . . . . . . . . . . . . . . . . . . 1284.4 TINA Simulation Results and Parameters Full Border . . . . . . . . . . . . . . 1294.5 TINA Simulation Results and Parameters Medium Border . . . . . . . . . . . 1314.6 TINA Simulation Results and Parameters Thin Border . . . . . . . . . . . . . 133A.1 Comparison Between the Ideal and Distortion-Less Line Models . . . . . . . . 189viiList of Figures2.1 One-Dimensional T-Circuit of a Differential Length of Line . . . . . . . . . . . 142.2 Two-Dimensional T-Circuit of a Differential, Loss-Less Area of Line . . . . . . 162.3 Velocity Dispersion in a 2D TLM Grid . . . . . . . . . . . . . . . . . . . . . . 232.4 Dispersion of the Velocity of Waves in the 2D, Loss-Less TLM Mesh . . . . . 252.5 Dispersion of the Velocity of Waves in the 2D, Loss-Less TLM Mesh . . . . . 262.6 Methods to Incorporate Different Media in TLM . . . . . . . . . . . . . . . . 302.7 1D TLM Node with Stub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.8 1D Node Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.9 Loss-Less, 1D TINA Node Cell . . . . . . . . . . . . . . . . . . . . . . . . . 362.10 1D TINA Node Cell Internals . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.11 Boundary Reflections in a Resistively Terminated 2D TINA Mesh . . . . . . . 432.12 Step Response in a 1D TINA Mesh . . . . . . . . . . . . . . . . . . . . . . . . 452.13 Expansion Duct Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.14 Magnitude Results for Various Mesh Sizes . . . . . . . . . . . . . . . . . . . . 472.15 Phase Results for Various Mesh Sizes . . . . . . . . . . . . . . . . . . . . . . 483.1 Lossless Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Pole Plots for Various Combinations of R and N for  1 . . . . . . . . . . . . . 623.3 Pole Plots for Various Combinations of R and N for  2 . . . . . . . . . . . . . 633.4 Loss-Less Line Model with Boundary Conditions in Frequency Domain . . . . 663.5 Voltage and Current Error Magnitude Plots for Various values of R,  t = 1s . . 703.6 Voltage and Current Error Phase Plots for Various values of R,  t = 1s . . . . . 713.7 Voltage and Current Error Magnitude Plots for Various values of R,  t = 0.2s . 723.8 Voltage and Current Error Phase Plots for Various values of R,  t = 0.2s . . . . 733.9 Voltage and Current Error Magnitude Plots for Various values of R,  t = 0.1s . 743.10 Voltage and Current Error Phase Plots for Various values of R,  t = 0.1s . . . . 753.11 Voltage Error Magnitude Plots for Various values of  and L = 1:5 . . . . . . . 783.12 Voltage Error Magnitude Plots for Various values of  and L = 2:5 . . . . . . . 793.13 Voltage Error Phase Plots for Various values of  and L = 1:5 . . . . . . . . . 80viiiList of Figures3.14 Voltage Error Phase Plots for Various values of  and L = 2:5 . . . . . . . . . 813.15 Current Error Magnitude Plots for Various values of  and L = 1:5 . . . . . . . 823.16 Current Error Magnitude Plots for Various values of  and L = 2:5 . . . . . . . 833.17 Current Error Phase Plots for Various values of  and L = 1:5 . . . . . . . . . 843.18 Current Error Phase Plots for Various values of  and L = 2:5 . . . . . . . . . 853.19 Frequency Response of Open Line for Various values of R and L = 1,  t = 1s 883.20 Frequency Response of Open Line for Various values of R and L = 5,  t = 0:2s 893.21 Frequency Response of Open Line for Various values of R and L = 10,  t =0:1s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.1 Mesh Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Two Connected Th´evenin Equivalents in Different Sub-Systems . . . . . . . . 984.3 Integer Latency Solution Time-Line . . . . . . . . . . . . . . . . . . . . . . . 984.4 Integer Latency History Interpolation . . . . . . . . . . . . . . . . . . . . . . . 994.5 Fractional Latency Bi-Directional Cell . . . . . . . . . . . . . . . . . . . . . . 1014.6 Fractional Latency Interpolation Interval . . . . . . . . . . . . . . . . . . . . . 1024.7 Fractional Latency Solution Issue . . . . . . . . . . . . . . . . . . . . . . . . 1024.8 Fractional Latency Solution Sequencing Examples . . . . . . . . . . . . . . . 1084.9 Fractional Latency Solution Sequencing Algorithm . . . . . . . . . . . . . . . 1114.10 Fractional Latency Domains in EMTP Solution Case . . . . . . . . . . . . . . 1134.11 Fractional Latency EMTP Solution Stable Case 1 . . . . . . . . . . . . . . . . 1154.12 Fractional Latency EMTP Solution Stable Case 2 . . . . . . . . . . . . . . . . 1154.13 Fractional Latency EMTP Solution Stable Case 3 . . . . . . . . . . . . . . . . 1164.14 Fractional Latency EMTP Solution Unstable Case . . . . . . . . . . . . . . . . 1164.15 Fractional Latency Applied to a Transmission Line Section . . . . . . . . . . . 1194.16 Fractional Latency Cells in a TINA Mesh . . . . . . . . . . . . . . . . . . . . 1224.17 Simulation Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.18 Expansion Duct Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.19 Full Border Time Domain Results at A . . . . . . . . . . . . . . . . . . . . . . 1294.20 Full Border Frequency Domain Transfer Function A ! G . . . . . . . . . . . . 1304.21 Medium Border Time Domain Results at A . . . . . . . . . . . . . . . . . . . 1314.22 Medium Border Frequency Domain Transfer Function A ! G . . . . . . . . . 1324.23 Thin Border Time Domain Results at A . . . . . . . . . . . . . . . . . . . . . 1334.24 Thin Border Frequency Domain Transfer Function A ! G . . . . . . . . . . . 1344.25 Fractional Latency Long-Term Stability . . . . . . . . . . . . . . . . . . . . . 1374.26 Thin Border Time Domain Results at A, Long-Term . . . . . . . . . . . . . . . 141ixList of Figures4.27 Thin Border Time Domain Results at A . . . . . . . . . . . . . . . . . . . . . 1414.28 Thin Border Frequency Domain Transfer Function A ! G . . . . . . . . . . . 1424.29 Nearly Under-Sampled Fractional Latency Case . . . . . . . . . . . . . . . . . 1445.1 Time Sequence of a Band-Limited Impulse Signal . . . . . . . . . . . . . . . . 1475.2 TINA Vocal-tract-model processing . . . . . . . . . . . . . . . . . . . . . . . 1515.3 Comparison of Measured and Simulated Acoustic Pressure for the Vocal Tract . 1525.4 Experimental Set-Up With Edge Source - Open Plane . . . . . . . . . . . . . . 1545.5 Experimental Set-Up With Edge Source (Cover Removed) - Duct . . . . . . . . 1555.6 Comparison of Measured and Simulated Acoustic Pressure (Point A!C) -Open Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.7 Comparison of Measured and Simulated Acoustic Pressure (Point A!C) - Duct 1575.8 Experimental Set-Up With Central Source - Open Plane . . . . . . . . . . . . . 1585.9 Experimental Set-Up With Central Source (Cover Removed) - Duct . . . . . . 1595.10 Comparison of Measured and Simulated Acoustic Pressure (Point C!A) -Open Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1605.11 Comparison of Measured and Simulated Acoustic Pressure (Point C!A) - Duct 1615.12 Expansion Duct Experimental Set-Up . . . . . . . . . . . . . . . . . . . . . . 1625.13 Expansion Duct Magnitude Plot . . . . . . . . . . . . . . . . . . . . . . . . . 1635.14 FFT Analyzer Screen Shots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645.15 Illustration of Loss-Less Model Errors . . . . . . . . . . . . . . . . . . . . . . 165A.1 Differential Length of Lossy Transmission Line . . . . . . . . . . . . . . . . . 180A.2 Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182A.3 CP Line Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183A.4 Non-Interpolated and Interpolated History Tables . . . . . . . . . . . . . . . . 186A.5 Interpolation Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187B.1 Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198B.2 Linear System with Eigenfunctions Applied . . . . . . . . . . . . . . . . . . . 199xAcknowledgementsIf I have seen farther than others, it is because I was standing on the shoulders ofgiants. Isaac NewtonWhen Newton made the famous statement that inevitably pops up in thesis acknowledgementsections world-wide, I believe that he had it only right in part. One must indeed stand on theshoulders of giants to look further, but one must also be able climb up there, understand thegiants strange language, and find food and shelter. There is an undeniable need to live andwork in an environment that allows one to spend so much time at lofty heights while the worldturns on below.Such environments, where curiosity is encouraged and the inevitable failures along the wayare accepted as the cost of business, where advise and support is readily available, be it of atechnical or personal nature, are rare and ultimately the creation of the efforts of many.Over the years, I havebeen supported in so many ways and by so many people that I cannot,by any stretch of imagination, call this thesis my work alone. Neither can I acknowledgeeveryone in these pages, for fear that this section would exceed the length of the actual text.Nor do I even know the names of all people involved.Thus, to those I missed: my deepest gratitude for your help. You are all heroes that madelife more livable and the world a nicer place to be in. And I sure as hell couldn’t have done itwithout you.When I came to Canada almost eight years ago, in the last week of August, 2002, I barelyknew the existence of three people and my understanding of the place was limited to rumorsthat the queen of England was somehow involved. And she only ended up getting tangled intothe whole mess through ways that needs the likes of Terry Pratchett to make it sound reason-able, if only for a non-dictionary definition of the word. Oh, and Monty Python’s “LumberJack” song taught me that British Columbia had many trees in it, and that the local woodsmenentertained rather strange vestimentary choices, given their line of work. South Park, the mo-tion picture, finally explained all I had to know about the political relations between Canadaand the United States. I had also consulted an atlas once, and noticed that large parts of thecountry looked like a sponge. Yes, one might say I definitely came prepared for the adventure.Of those three people who helped me settle-in, using actual facts, one is no longer withxiAcknowledgementsus. Eric Timothy “Tim” Baars, a friend who made me aware of the existence of the country inthe first place, and taught me much about the English language, passed away on October 23rd,2002, after a long fight with cancer. I still have his bicycle and helmet, a gift I use to explorethe beautiful Pacific Spirit park around the university.My first land lord, Andrew “Andy” Lai, a Malaysian ex-hippie who once toured SouthAmerica on a motor cycle, and a descendant of a failed Chinese revolutionary (his great grand-father, I think), taught me how to cook, live on less money than most spend on a single mealper week, and otherwise kept the bunch of international rabble in his house in line. It was therethat I also learned that Canada, and definitely Vancouver, tends to the multicultural side, withpeople from New Zealand, Poland, Germany, Belgium, and Malaysia under the same roof.Getting accepted in the university itself, for what was then a masters degree (little did Iknow where that was going to go, although I had plenty of misguided hopes) was eased by thepatience of Doris Metcalf, the department’s graduate secretary, who managed to hold things offhere, push things there, until I finally got all the paperwork in order. Also, when I discoveredthat the communications group I was accepted in wasn’t my thing (they do theory there!), shealso sent my file around again, I imagine with a fluorescent sticky that said: “Warning: severeallergy to theory”.And this is where things get going. My file was picked-up by Dr. Jos´e Mart´ı who, atthe time, was looking for someone who could hold a screwdriver and know which end goesforward (the pointy bit). In due course, the masters degree happened and, near the end of it,he offered me the opportunity to do a PhD with him. I, of course, said.... Let me think aboutit. After some consideration, I accepted, as this was by far the best offer I got, and stated, incapitals, that I Would Not Do Theory on this degree. It was to become a practical doctoralthesis. Upon hearing this, Dr. Marti gave me one of his patented little smiles. To put thatsmile in perspective, it is that smile that a fox makes when he knows he’s got the chicken downand fried-up with a nice sauce, far before the chicken even realizes that now would be a goodtime to start getting worried and book a ticket to the Bahamas. As to who won this particularEpic Struggle Against Theory (also capitalized), well, see the remainder of this text. Suffice tosay that my high school math teacher, who told my parents in no uncertain terms that I would,maybe, make a not altogether incompetent plumber, would have a fit. One could attribute thisrather monumental change in perspective to the wisdom and guidance of an experienced andinternationally acclaimed supervisor. Myself, I’d attribute it to being outfoxed in ways I’monly now starting to get a glimpse of. There are, indeed, reasons why Dr. Mart´ı is the professorand I’m the student.When it comes to the topic of this thesis, an acoustic transient simulator, it may seemincongruous with energy systems, the field I’m supposedly in. Like many long stories, itxiiAcknowledgementsstarted out with a question: how does the human ear work? To answer this ambitious question,a simulator was needed. This was to be based on power system analysis techniques, which usesome interesting tricks that are unique to the field. As, later on, I found it nearly impossibleto obtain good information on the human cochlea, and thus had no reliable way to actuallyverify any results, the topic shifted to the simulator itself. Of course, this acoustic simulatorwould also have to be verified somehow. Thus, I took a course in acoustics with Dr. MurrayHodgson. As seems to be a trend in my life, also this simple situation went completely out ofhand. Allow me to explain.When I started the whole “let’s use acoustics to validate the simulator, as it’ll be simpler towork with, with big wave lengths, inexpensive equipment, and easy to make a set-up becauseit can be big”, I proved to be a total, complete, and utter fool. After completing the first coupleexperiments using equipment that amounted to the value of a nice Porsche, I still couldn’t getany decent, let alone repeatable, results. And this is where Dr. Hodgson agreed to become myco-supervisor and teach me one or two thousand things about the realities of the field and howto get the universe to behave. This resulted in much gnashing of teeth, which in itself provedinteresting. I was unaware that the universe had them.Over the years, two degrees, and a countless number of “good ideas” aka “side projects”aka “scaring the bejezus out of the ethics committee, the health and safety board, as well as anybystander innocent or otherwise”, quite a few crazy contraptions resulted. I will save myselfthe embarrassment of listing them here, but nearly all of them involved building something outof parts that were never meant to be used in quite that way.Inordertopulloffsuchstunts, Ifrequentlyneededvariousmechanicalpartsmade. Luckily,our department has a good machine shop staffed with the kind of people who could literallybuild the space shuttle from a lipstick drawing on the back of a napkin that was subsequentlyburied in the bottom cruft of a handbag for fifty years. And it would work. On the first try. Inorder of harassed most to slightly less, I owe a lot to Donald “Don” Dawson, Neil Jackson, andDavid “Fletch” Fletcher. Although, one could argue that I managed to harass the latter thoughvarious indirect ways and this order is skewed enough to be circular.Then, there was the slight issue of getting parts in the first place. See, what one must knowis that, with few glorious exceptions, suppliers are a lying, conniving, and utterly morally de-prived bunch of puppy botherers. I write this in kindness, as the uncensored version would getme in trouble. Dealing with this lot of failed politicians requires a kind of calm perseverance, aparticular ability to not reach down the phone line and strangle the troglodyte on the other endwhen said waste of skin uses its sincere voice to explain that the part was shipped last weekand should arrive any day now. And of course they have it in stock. The person who handlesthese issues in our department is David “Chuch” Chu Chong, assisted by the now retired LancexiiiAcknowledgementsJones and his current incarnation, Kristie Henrikson. David has perfected the art of soundingabsolutely menacing and as unavoidable as a hovering mountain that is about to drop while be-ing absolutely pleasant and relaxed at the same time. This is why having friendly and helpfulassistants is necessary. Few cross the Purchaser and tell the tale. On the other hand, I’m happyto report that the lost art of getting parts yesterday is quite alive and well.On more ethereal planes of existence, those of software and networks, expensive coasters,paper weights and door stops, there is the extensive IT support that our department houses.Out of my own experience, our system works much better than that in the computer sciencedepartment, and people are far more flexible when it comes to strange, but not idiotic, requests.By the unwritten lore of IT, idiotic requests are, of course, met in much the same way starvedsaber tooth tigers meet bunnies. Anything else happens with an efficiency and effectivenessso unlike the general expectations from the industry that it becomes scary. My digital life hasbeen made a lot easier by, in alphabetic order, since I don’t know their MAC addresses: AndreyVlassov, Chris Dumont, Ken Madore, Luca Filipozzi, and Rob Ross.It might come as a surprise, but UBC has seen fit to allow me to mold the minds of the nextgeneration. Instrumentalinallowingmetodoso, andgettingpaidwellwhileatit, wasDarlaLaPierre, and her predecessor, Daria. I am very grateful for the continued teaching opportunities,as the extra income really helped keeping life fun. Being ever the scientist, I thus unleashednumerous didactic and otherwise diabolical experiments on what, by now, amounts to roughlyone thousand two hundred students. During these activities, mostly on EECE375/474, I joinedranks with numerous professors as a teaching assistant and had a generally good time doingso. Of special note is Dr. Paul Davies, who guided me through the terms I was hired to bethe instructor for the course, and prevented me from making many of the errors that result inpaperwork avalanches and other occupational hazards. Over the terms, I’ve had the pleasure ofworking with Peter Vautour, the assigned technician for the course, who has a similar outlookon students as I do. It boils down to the likeness between the majority of them and the Egyptianmythological critter of choice that made the sun move through the heavens. I’ve been told thefeeling is mutual. One thing I’m especially proud of is my earned nick name: Scary Tom.Actually leaving the institute of higher learning that is UBC proved about as convoluted asleaving those other esteemed places where those deemed too unsettling to let run free in societyare kept. The graduation process, which takes about nine months strangely enough, was madeless painful by Cathleen Holtvogt. Using a good sense of humor, she managed to coral thecommittee into attending, battled the Powers that Require Much Paper, and finally got my filesin order. It was all rather epic, if you ask me.Of course, all work and no play will land you in the actual loony bin before you can saycuckoo. It is, in fact, of note how close the UBC hospital psychiatric ward is to the departmentxivAcknowledgementsof electrical and computer engineering. Using the income from all the teaching assistant andinstructor work, I managed to fulfill my life-long desire of making the country unsafe on land,on water, and in the air. It started by the discovery that I couldn’t make it to the top floor ofthe building without being completely out of breath. This is when I picked-up this strangeVancouver fascination with hiking and joined the SFU hiking club, led by Andrew Haskell.Yes, that’s the competing university. The UBC version of it was just far too serious, filledwith the types who think of climbing Mount Everest as a day trip. Now, the fun in hiking is, Ibelieve, based on oxygen deprivation in the brain. By the time you make it to the view point,anything and everything looks spectacular though the hallucinatory daze. But it is fun. Withthis group, I also hiked the wold famous West Coast Trail, as well as its little brother, the Juande Fuca trail, both on Vancouver Island. The fame is well deserved. I also joined the UBCsailing club, got the White Sail I, II, and III certificates on dingies, and got roped into fixingthe boats for a year. Fiber glass and gel coat were no longer strangers after that. People hada habit of crashing boats and sailing a submarine. Yes, this means exactly what you think itdoes: sailing the boat back in with the deck below the water line.That leaves air. It was a long-standing dream of both my dad and me (and probably themajority of the human race) to learn how to fly. Unfortunately, when I was finally of age,the airport in my hometown was converted into a golf course. Three years ago, however, Idecided to give it a go in Canada, at Pitt Meadows Airport (CYPK) on a Cessna 172. Thegrin went ear to ear and stayed there for at least two hours after that first flight. The peopleat Pitt told me about Boundary Bay (CZBB), since the distances were too much to make itreasonable to do my training there. Then, after discovering the Vancouver Cooperative AutoNetwork and becoming a member (I am now the proud owner of about half a door handle), andsome shopping around the schools, I settled on Canadian Flight Center, and started off with JillSmith. I tended to be a bit low on the natural skills (choice sentence after pulling the mixtureto “off” for the second time in a row, delivered in a dead-pan, calm voice: “We need that tolive. Carry on.”). Eventually, when Jill got a real pilots job, I switched to Mike Stann, who alsogot a real job, then to Kwan Lam, who found a better paying instructor job in Malaysia, thento Amelia Kerr, who is looking for a real job. At this point, almost three years had gone by,the economy had tanked, and Jill was back again. With Amelia, I actually managed to finishthe license and am now ready to point that old Piper Cherokee (C-GOUP, the orange and whiterust bucket of love) into the sunset, with the landing light switched to “pulse” to make it mylighthouse in the sky. Kudos to all the good folks there who helped to make it happen!The search for, and maintenance of, sanity is also continued during the day job. First, I’dlike to tip my hat at the Lunch Team. This highly competitive and athletic endeavor at theLunching Hour has been upheld over the years. Members of this team were: Khosro KabirixvAcknowledgements(LordKhosroforfriends), Michael“Mike”Manarovici, AliDavoudi, Benjamin“Ben”Sch¨afer,Arvind Singh, Andrew Rutgers, Sina Chiniforoosh, and Mehmet Sucu. I’d furthermore like toextend my gratitude to my colleagues in general, and specifically to: Jorge Hollman, for help-ing me get going in research and support; Michael “Mike” Manarovici, for pleasant lunches,pop runs for the lab, and fun after work; Marcelo Tomim, for teaching me maths and thussucceeding where decades of teachers and tutors could not; Arvind Singh, for the ruthless ap-plication of common sense and dripping sarcasm to the furthering of research, enlivening ofthe Lunching Hour, and after work activities; and finally Michael “Mike” Wrinch, for sharingall his business and industry knowledge.Beyond work, there are those friends who don’t necessarily do electrical engineering for aliving. Living in Vancouver automatically makes this a colorful bunch from all over the planet,and my own little circle is no exception. I spend many great moments with this bunch. Weall know the details and what happened in Canada stays in Canada.... More or less in order ofappearance, warm wishes go out to: Eric Timothy “Tim” Baars, Jacek Kisynski, Alex Yuen,Linda Lay Tong, Allen Moore, Tihomir “Tiho” Tunchev, Daniel “Dan” Archambault, KeithMabbs, Donald “Don” Derrick, Laura Turner, Michael “Mike” Mueller, William “Bill” Drake,and Mathieu Sp´enard. Best of luck to the lot of you!Finally, there is the family clan. On the immediate offspring side, there is me and mybrother Bert. He’s the one who’s in plant genetics and somehow managed to get a PhD inthis. Something along the lines of side root development, I’ve understood. He’s said a greatdeal more about it, but beyond the bit of knowledge that the pink food dye is made of shieldlice, I can’t honestly say I’ve understood one iota. But that’s ok. That trick works both waysand it’s handy to have someone around who doesn’t have the curse of the black thumb. KarenBuysse, then, picked him up somewhere along the way. I’m very happy with a for all intentsand purposes sister in law like that. Extremely practical, ex scouts leader, and with the skillsto handle my bother. Quite a combination, if you ask me. She finished her PhD in humangenetics, on the subject of mental retardation in children. There is some irony there, I’m sure.Karen also spent half a year in Seattle, which is next door to Vancouver by Canadian standards.Our mutual visits were fun. Especially the bit where we managed to snap the customs officersinto matron mode. The indignant “Does your boyfriend know you’re visiting his brother?”, inthat tone of voice that suspected me and Karen were re-enacting the entire Kama Sutra behindmy brother’s back was particularly funny.And to round it all up with the cause of all of this: my parents. Without their constant loveand support, copious fire insurance, and emergency numbers on speed-dial, I would have nevermade it this far. Whether that is for better or worse, I leave up to the reader. I just know it’sbeen a blast so far and have no intentions whatsoever of quitting any time soon.xviChapter 1IntroductionThere are more things in heaven and Earth, Horatio, than are dreamt of in yourphilosophy. William Shakespeare, HamletIn this chapter, we will discuss the philosophical background behind the idea of simulating asystem, as well as the fundamental equivalences that exist between different fields of study. Itserves to introduce the concepts in the thesis, as well as explain why power systems simulationtechniques may be used to approach acoustical problems.1.1 Scope of the WorkThe objective of this thesis is the development of a new, one and two-dimensional, discrete-space, time-marching, transient wave propagation simulator based on time-decoupled trans-mission line models. These models are based on those used in transient power system analysistools, such as the EMTP.The newtool is targetedat acoustic and electro-magnetic applications,although the approach is universal to most wave propagation phenomena.The method, called TINA for transient insular nodal analysis, uses temporal interpolationand the newly developed fractional ratio latency technique to achieve synchronicity in het-erogeneous media. Th´evenin equivalents are used to interface the mesh cells and provide anabstraction of the cell content.Numerically, the method is of the transmission-line matrix family. For the purpose of thethesis, only loss-less models are considered, which means their poles are all on, or close to, theunit circle in the z-domain. The used transmission-line models are studied for their stabilityand numerical error, for which an expression is derived based on the simulation parameters.A number of new relations were discovered and discussed. The method is evaluated in 2Dusing acoustic experiments and a new method for obtaining impulse response in time-domainsimulations is shown and used.11.2. Why Was This Research Done?1.2 Why Was This Research Done?How would a new musical instrument sound? What would the acoustics of a new concert hallbe like? Will this expansion chamber filter work to block the machine noise in the ventilationduct? How effective is this new radio antenna design? Will this microwave waveguide designwork?Traditionally, one had little choice but to go out and build it, or at least a scale model,hoping that prior experience was enough to achieve a reasonable result. One could improve thechances of success by using approximate calculations, by using calculus on simplified shapes,if the problem was to be tractable by hand or early machine calculation. In fact, the abilityto try something out on paper, descriptive geometry as it was then called, was invented in the1760s by Gaspar Monge and kept a closely-guarded French military secret until 1799, whenMonge was allowed to publish his treatise on the subject [57]. Today, this art is referred to astechnical drawing; the ability of representing a three-dimensional object in two dimensions.Frequently, however, physicalmodelsofthesystemwerenotaccurateenough, ortheunder-standing of the system insufficient, to obtain a satisfactory result. Scale models often behavequite differently from the full-scale version. The latter is often the case with aircraft [82],where a design that was proven stable as a model ended up being dangerously uncontrollablewhen built full-scale. Another example can be found in the TRIUMF particle accelerator. Themagneticfield wasstudied by means of a scale model. In this model, half-inch thick steel plateswere used to construct the magnet geometry. Although great care was exercised, the ten inchthick steel plates used in the actual device had somewhat different metallurgical properties.As a result, the accelerator would not work until invasive modifications were made, involvinggrinding-off chunks of metal and welding bits to other places, so the required field distributioncould be obtained. Acoustically, many a concert hall sounded disastrous until major retro-fitting of acoustic panels, Helmholtz resonators, and other treatments took place.With the advent of sufficiently powerful electronic computers in the sixties, numericaldiscretized-space methods took flight, such as finite-differences and finite-element methods. Ingeneral, these methods discretized space and geometry in small, elementary cells from whichthe physical system could then be built in much the same way one may build something outof Lego bricks. This allowed many of the simplifications required for hand calculations to bedropped. Conceptually, these methods caused a complete change in the way complex systemscould be solved. One now only had to understand the behavior of the materials and how thesematerialsinteractwithoneanother. Thecomplexitiesofthegeometry, orthethingthematerialsform, are no longer important.Usingthese methods, it becamepossible to hear an instrumentnot evenbuilt, or testa shock21.3. Predicting the Futureabsorber that exists only in the engineer’s mind. And all one had to do was define the geometryand the materials with their interactions. Thus, after someone figured-out the details of theseelemental cells, anything could now be tried in the computer, with reasonable accuracy.This is, of course, too good to be true. It was soon found that the ten thousands to millionsof building blocks required for an accurate representation of a system present an enormouscomputational burden and thus take a long time to solve. The limitations on computationalspeed and memory meant that only coarse, highly simplified geometries could be studied. Thiswas still a leap beyond the approximations required for traditional hand-solutions, but in theend, the discretized-time methods have their weakness in the sheer number of little buildingblocks required to represent a system in detail.These issues are fundamental, but the applicability of the method has been steadily pushedforwardasmorepowerfulcomputersweredeveloped. However, therearetwowaysofpushing-forward the usability of the methods: one can try to improve the computer, or one can try toimprove the algorithm itself.This thesis attempts the latter and focuses on a family of algorithms known as transmission-line modeling and finite-difference time-domain methods. Improvements are made in the sensethat the presented methods allow one to trade some precision for significant numerical speedincreases. Using these methods, larger and more detailed systems may thus be studied inless time, or complex systems may be solved where, without these methods, the availablecomputers do not have enough memory to hold the problem or the computational time becomestoo unwieldy to be practical. Thus, the work in this thesis serves to improve the usability of afundamental tool used in many disciplines to give insight to complex systems and even allowsone to study problems that cannot – yet – be built, or are simply so expensive to realize thatone needs to be quite sure of the expected outcome before attempting construction. In the past,advancements to these methods have invariably lead to advances in many fields, and we hopethat the work presented here may once more allow the state of the art in many fields to bepushed a little further.1.3 Predicting the FutureThe ultimate purpose of scientific theory is nothing short of a flawless prediction of the future.For a theory to hold, it must be falsifiable through experiment. In other words, there must be away, through an experiment, by which it may be shown false, and thus the limits of its appli-cability discovered. And, most importantly, no theory can be Truth. They are approximations,their predictions limited to conditions that may take centuries to discover. And, when such alimiting case is found, the theory is to be rejected in favor of one that explains all that was31.4. Modeling the Outcome or the System?explained before, including the new discovery.The purpose of a scientific theory to the engineer is an understanding of nature detailedenough that the behavior of a complete system may be predicted to a sufficient degree ofaccuracy for the purpose at hand. For engineering, the requirements are usually not as stringentas those in science. Newtons laws of motions are still perfectly adequate for most applications.Automobiles do not operate at a significant fraction of the speed of light. Thus, significantapproximationsmaybemadeandagoodenoughresultobtainedinfarlesstimethanacompletesolution of the standard model of physics [61] would have required. In fact, determining theinteractions between the billions of atoms involved in a bicycle pump is, at the time of writing,not only impractical, but also impossible to calculate on any currently existing computer.In the end, no matter if the objective is science or engineering, the predictions of the modelmust be tested to reality, thus compared with measurements of a physical entity or effect.Maxwell, in his famous treatise on electricity and magnetism [54, 55], in fact starts the discus-sion with an exposition on what it means, to measure.Let us look at a classic high school example of such an engineering prediction. Supposewe take a ball, in vacuum, and we throw it. We are able to obtain the velocity of the ball asit leaves the thrower’s hand, as well as the angle at which it leaves. We may weigh the ballbeforehand, to ascertain its mass. From this, Newton’s laws of motion [86] will give us anaccurate prediction of where the ball will be at any time in the future. The more exact we knowthe initial conditions, thus, what the ball was doing when it left the thrower’s hand, the moreprecise the future location and trajectory, if it were to interact with structures, will be known.Of course, according to special relativity, the whole concept of time assumed by Newton ischanged [22] and, according to quantum mechanics, there are strict limits on the observable,though Heisenberg’s uncertainty principle [23] that imposes a trade-off between the accuracyof knowing either where something is and how fast it is going. Thus, the ultimate accuracywe may obtain in the prediction of this system is limited, even if we apply the relativisticcorrections. However, for this day-to-day engineering problem on a human scale, the throwingof a ball against a wall, these considerations have so little impact as to be imperceptible to eventhe most accurate of instruments and the fact remains that we may compute an accurate futureoutcome for the system of the ball being thrown and who it will hit in a classroom.1.4 Modeling the Outcome or the System?There are two fundamentally different ways in which a system may be modeled. We may eithermodel the outcome of the system, or we may model the system itself and see its outcome. Thedifference may be explained with an example.41.5. Unification in NatureConsider the current through a capacitor, given by the equation i(t) = C@v(t)@t . For an idealcapacitor, this equation is a model for the behavior of the device, relating the rate of change ofthe voltage over the terminals at a moment t to the current at a moment t. This equation thusmodels the capacitor, and is used in cases where the internal workings of the capacitor are notimportant, just its behavior. We must thus know the behavior of the system, the capacitor, inorder to derive a model of this type. Thus, this is a case of a model where the outcome of thesystem is modeled, but no insight into the actual mechanics in the box is shown, nor needed.A different approach is one where we obtain extensive knowledge about the behavior of,e.g.: aluminumandair. Throughexperiments, wemayfindamodeloftheirbehavior, muchlikeinthecaseforthewholecapacitorabove. However, whatisdonenowis, usingtheseconstituentmaterials, the geometric structure of the capacitor is represented in the simulator. From theknown, constituent materials, we may now see how the geometry of the capacitor gives riseto its function, how the fields and charges interact with the materials to store energy, how thebehavior of the whole device comes to be. Thus, in this case, we model the system itself, andsee the outcome. One could, theoretically, take this procedure to its logical extreme and modelthe individual constituent parts of the sub-atomic particles and the fundamental forces betweenthem. Although such an extreme case in not practical for macroscopic objects, these methodsdo allow one to find the behavior of complex systems without any prior knowledge but that ofthe constituent parts.The power in this method is thus that, if we know the materials well enough, that anygeometry, be it a capacitor, resistor, inductor, antenna, etc... may be modeled and the behaviorof the structure found. Even more, we gain insight in how the fields interact with the structureto give rise to the function. Compared with the other approach, we would only know how itbehaves, but not why.Both approaches have their place. When doing a typical circuit simulation, computing theinner workings of each capacitor is not only unpractical, but also unnecessary. On the otherhand, when we have a system and we do not know its behavior, this may be found through thesecond approach and captured in laws of the first kind for simplified computation later on. Bothmethods tend to be complementary. In this thesis, however, we will be focusing on methodsthat model the system to find its outcome.1.5 Unification in NatureIn this thesis, methods originating from electro-magnetics are used to compute acoustic phe-nomena. Although one could, at this point, simply give the one-dimensional, loss-less, acous-tic wave equation and show it to have exactly the same form as the one-dimensional, loss-less,51.5. Unification in Natureelectromagnetic wave equation, this does not truly show the deep equivalences on a fundamen-tal level.The great triumphs of the last century in physics were those of the unification of widelydifferent fields [61, 81] and, with that unification, the emergence of deep insights into thenature of the universe. In fact, the standard model [61] incorporates all that is currently knownin one, quantum mechanical equation, save for gravity. The unification of quantum mechanicsandgravityisstillunresolved,atthetimeofwriting,andthelastgrandquestionoflastcentury’sphysics.The situation, in the nineteenth century before the unifications took place, was that physicswas split in six different fields, namely [61]:Dynamics The laws of motion which, when combined with Newton’s laws of universal grav-itation, describe the motion of celestial bodies, as well as all day-to-day objects on ahuman scale.Thermodynamics The laws of temperature and heat energy, as well as the behavior of solids,liquids, and gases in bulk. It explains expansion, contraction, freezing, melting, andboiling.Waves The study of oscillations of continuous media.Optics The study of light.Electricity The study of static and dynamic charges.Magnetism The study of magnetic interactions.By the beginning of the twentieth century, these branches had been reduced to two. Theatomic hypothesis, strongly supported though the advances in chemistry, had shown that ther-modynamics, wave mechanics, and dynamics were the same. The theories of the electromag-netic field had shown that optics, electricity, and magnetism were the same. Thus, all in naturecould be explained by either particles, through the atomic theory, or waves, through the fieldtheories.As an aside, it is interesting, but disappointing, to note that these old territories are still verymuch alive. One of the issues that plagues the different traditional fields of science and engi-neering is that each method had its origins in some field, and the terminology and assumptionsof that field go with it. The unifications had little effect on this, and thus these obstructions tointerdisciplinary cross-fertilization remain to this day.The particle-wave duality remained a fundamental problem. How was one to reconcilethe relativity of motion with Maxwell’s laws of electricity and magnetism? Or: how can we61.6. Equivalence Between the Electro-Magnetic and Acoustic Wave Equationreconcile the theories of light with the existence of atoms? This issue, and with it the finalunification of classical physics, was performed by Einstein in 1905 [22], resulting in the well-known particle-wave duality.At the same time, the quantum revolution was also in full swing. With the work of Planck,who discovered the quantized nature of black-body radiation, Schr¨odinger, and Heisenberg,the old Newtonian dream of the predictable, clock-work universe was shattered. The universewas made of countable, discrete entities with properties that could only be observed to degreesof precision. Statistics took over from certainty.Eventually, also the strange quantum theories were reconciled with special relativity bythe generation of Freeman Dyson and Richard Feynman [23, 81]. By 1980, all then knownlong-lived particles and forces could be explained by the standard model of elementary particlephysics. All, except gravity. The quest for this final unification of what is known is still on.Although, in this thesis, electro-magnetic and acoustic phenomena are treated as identicalphenomena with different names for the governing parameters, from the standard model welearn that two different force carrying particles are involved. Electro-magnetic forces are trans-ferred through photons while the mechanical forces for acoustics in matter are transferred byphonons. Essential is that phonons can only work in matter, thus do not have the infinite reachthrough vacuum that photons have. Put differently, for photons, absolute vacuum may serve asa medium through which they propagate, for phonons vacuum, and thus the absence of matter,becomes a barrier to propagation.1.6 Equivalence Between the Electro-Magnetic andAcoustic Wave EquationAlthough the underlying phenomena are different, both the propagation of electro-magneticand acoustic energy, within certain approximations, are similar processes, when observedmacroscopically. For both phenomena, we may observe an energy wave traveling througha medium. For both, there is a propagation speed set by the medium, and a wave length relatedto that speed and the frequency of oscillation. For both, different media give rise to similareffects of scattering, transmission, and reflection at the boundary between the media.To illustrate these striking similarities, the one-dimensional wave equations for acoustics[13] and electro-magnetics [46] are shown in (1.1) and Table 1.1, for the loss-less case. Sincethe governing equations have the same form, it follows that a solution method devised for onewill also be applicable to the other. This is why a solution method for electro-magnetics mayalso be used to solve for acoustic phenomena, within certain limitations, e.g.: [67, 68].71.7. Relativity and Uncertainty in SimulationIt should be noted here that these equations appear in many different fields of study, asmany natural phenomena are, in essence, wave propagation problems. For example, the trans-mission line model used in this thesis as the basis of the actual numerical implementation ofthe solution to these equations, the Bergeron line model [6], finds its origins in hydraulics.More specifically, Bergeron studied the effects of both water hammer in pipes and lightningstrikes on transmission lines and found that both could, to a good degree of approximation, berepresented with the same model.@2p@z2 =  k@2p@t2@2v@z2 =   @2v@t2@2u@z2 =  k@2u@t2@2i@z2 =   @2i@t2(1.1)Acoustics Electro-Magneticsp = pressure v = voltageu = particle velocity i = current = density  = permeabilityk = compressibility  = permittivityt = time t = timez = distance z = distanceTable 1.1: The Acoustical and Electro-Magnetic 1D Wave Equations1.7 Relativity and Uncertainty in SimulationIt is interesting to note that two of the underlying main concepts of modern physics, relativityand quantum-mechanical uncertainty appear naturally in the discretized-space time-domainmethods this thesis talks about. Although, relativity is partly violated in the typical use of thesemethods, as the simulation is usually observed at all spatial locations instantaneously.Letusfirstconsidertheuncertaintyprinciple. Heisenberg’srelationisnotuniquetophysics.It is a fundamental property of any measurement in a discrete, thus quantized, system. For ex-ample, in information theory, the concept is known as the Nyquist limit. In a discretized-spacesimulation, where each constituent element in the simulation has a finite spatial extent, also aspatial Nyquist limit appears in addition to the temporal one.Indeed, if a computation happens at discrete times, and each value is represented by abinary number, thus has a finite precision, and the simulation space is divided into finite cells,81.7. Relativity and Uncertainty in Simulationwhat that really says is that the spatial, temporal, and quantity indicators are all quantized. Thedifference with the universe is that it operates at the Planck units [65] and a typical computationoccurs at much coarser devisions. In fact, from the Planck units, one may compute the Nyquistfrequency of the universe. Or: one may determine fixed limits to what may be observed inour universe, how small something may be, how fast something may move, so that we maystill observe it. If something were to exceed those limits, one would get aliasing effects, thusthe faster thing would appear slower, the smaller thing would appear larger, and so on. Forexample, when viewing a movie of a stationary wheel that accelerates, one first sees it turnforward, then become stationary again, then turn backward, become stationary, and eventuallyturn forward again, with this cycle repeating as the wheel turns faster and faster. By making amovie of the wheel, it is observed at twenty five frames per second, thus discretized in time,and aliasing effects occur as the wheel’s rotational speed exceeds the Nyquist frequency of thecamera.From the discretization of the simulation, we may now express uncertainty relations, as isdone in quantum mechanics. For example, if the spatial grid is discretized to one centimetercells, we cannot locate an event with better precision than that. We may say it is happening ina particular cell, but we cannot be more precise. Same with time. We may say that somethinghappened between two discrete time events, but we cannot say when precisely, as no detailedinformation is available between those discrete time events, just as there are no finer spatialdevisions between the cells.Thus now the subtlety: to know more precisely when something happens, we may reducethe simulation time step, and get more temporal precision. The problem is that, for an equalspatial discretization size, the acoustic wave still travels at the same speed. Thus, if before withthe coarser time step, it would travel further between time steps, and thus encounter more cells,with the finer time step, it cannot travel as far, as there was less time to travel in. Thus, weknow better when something happened, but the wave, having traveled only a little, may justas well still be in the same cell, and thus we will get less knowledge about the location of thewave between time steps, as it has less chance to interact with different cells.As for relativity, these discrete-space methods, when operated in the time domain, aretime-marching solutions where all phenomena may only propagate at a certain rate, set by therespectivemedium. DuetotheNyquistlimits,anyphenomenathatattemptstomovefasterendsup being aliased, and thus appears slower again. Thus, the temporal and spatial discretizationsplace an upper limit to the highest velocity, and frequency, possible in the computation. Moreimportantly, each medium will be configured for a finite propagation speed. Thus, since allphenomena may only travel at the speed of the medium, there is causality and we may alsoobserve relativistic effects, as phenomena are habitually traveling at the maximum speed for91.7. Relativity and Uncertainty in Simulationthe phenomena in the medium and, unless the simulation is multi-physics, there are no other,potentially faster, means of conveying information. E.g.: in an acoustics simulation, electro-magnetic effects are usually not present, thus, for that simulation, the acoustic wave can beconsidered to have similar properties as a light wave would in the real world, and its speed isthe absolute limit to how fast information may propagate in that medium.However, this relativistic notion, and its complications, are usually circumvented, as thewhole system is typically observed as a whole, not taking into account the time it takes thephenomena in the system to reach an observer, as it would in the real world. Thus, althoughthe various waves operate at the maximum speeds in the medium, and thus relativistic effectswould be expected, by using an observation technique that allows everything that happens tobe seen at the same time, relativistic effects are mostly removed. By using this technique,there is a preferred observer who watches at a preferred, absolute time, which is coupled to thesimulation time step. This is, however, not a requirement, and a non-privileged observer usingrelative time is easily possible.10Chapter 2The TINA MethodWe do not try to model the outcome of the system. We try to model the system andsee its outcome. Andr´e LaMotheInthischapter,wewillpresentadifferentformulationofthetransmission-linemodeling(TLM)method [31], called transient insular nodal analysis (TINA). The numerical solution used be-comes a hybrid between TLM and circuit analysis approaches used in Power Systems, suchas the electro-magnetic transients program (EMTP) [90]. We will show how the TLM methodcan be modified to enable time-decoupled, EMTP-style solutions and a temporal interpolation-based approach to synchronicity. The advantage of the TINA formulation lies in the extensiveuse of Th´evenin equivalents, which enable the transparent incorporation of the various trans-mission line models developed for EMTP simulations to be used in a TLM-style method, po-tentially allowing complex and non-linear media to be modeled in a straight-forward way. Nomatrix formulations are used for numerical efficiency on normal PCs, as the basic linear al-gebra sub-programs (BLAS) performance/loading and unloading of small matrices in memoryresults in significant and detrimental performance penalties. Although the TINA formulation,in practice, is more computationally, and significantly more memory, intensive than the basicTLM formulation, the simplicity and flexibility of direct Th´evenin equivalents offers a distinctpractical advantage when combining different types of models in a simulation. Also, trivialparallelization of the method is possible due to the full time-decoupled nature of each meshcell and latency techniques can readily be used to increase the numerical performance of themethod, as is shown in Chapter 4. The ability to easily integrate widely dissimilar materials,withdifferentpropertiesandunderlying models, in the samesimulationenvironmentand main-tain transient-free synchronicity is another advantage. A link-resistor approach with dissimilarcell impedances is used in the TINA formulation, as opposed to the traditional link-resistor orlink-line approaches in TLM, where all impedances are kept constant and stub lines are usedto modify the base material parameters and achieve synchronism. Also, the TINA mediumuses absolute parameters for all materials, as opposed to the relative ones used in TLM, wheretypically the base material is locally modified using stubs.The contributions presented in this chapter lie in the introduction of the TINA approach,which presents a new way of incorporating and maintaining synchronism between dissimilar112.1. The Transmission Line Modeling Methodmaterials in a simulation without the transients due to impedance discontinuities (and thussmall internal reflections) that come from stub-based approaches. Additionally, the TINA for-malism offers an automatic irregular grid in time that is still sampled in a regular pattern, dueto the use of variable-length transmission lines (which could be used with stubs as well), whichresultinaMinkowskyspace-time, whichisCartesianinspace, butnon-regularintime. Theuseof interpolation to affect sub-time step synchronicity adjustments in combination with variableline length for adjustments that exceed one time step, is another novelty in TLM-style simula-tions. The effect of interpolation on the transmission-line models will be studied in a furtherchapter.2.1 The Transmission Line Modeling MethodIntroducedbyPeterJohnsin1971, basedonanideabyRaymondBeurle[31], thetransmission-line matrix method, also known as the transmission-line method, has seen continued interestin many disciplines as diverse as diffusion processes in food and heat flow in jet turbines toelectro-magnetic wave propagation in wave guides.The time-domain method is conceptually different from finite-element and finite-differencemethods in that it does not attempt to discretize and solve the electro-magnetic field equations,or other equations that govern the flow of energy in the system, directly. Rather, it uses a cir-cuit equivalent approach where the physical phenomenon of energy propagation and reflectionthrough a mesh is simulated though an electrical network.This approach, using circuit equivalents to model field problems, has in itself a long his-tory, dating back to the pioneering work by Kirchhoff and Helmholtz. A particularly strikingexample was shown by Gabriel Kron in his papers on network equivalents for Maxwell’s fieldequations [36, 37]. In fact, when observing Kron’s work carefully, we may note the constituentcells of his equivalent networks are lumped  and T transmission line segments, which areelectrically equivalent and may be converted from one form to the other. Such equivalent net-works, as shown in Figure 2.1, are capable of representing elliptic (only R and G, Poisson’sequation), parabolic (either R and C or G and L, diffusion equation), and hyperbolic (only Land C, loss-less wave equation (Helmholtz equation)) partial-differential equations [73].In the end, the TLM method performs a field to circuit mapping. It allows the solutionof field problems through the simpler problem of solving a representative electrical network.Such an approach is possible when the circuit components each govern a spatial extent muchsmaller than the length of the physical wave to be modeled in the medium. In other words,the coarseness of the mesh components must be much smaller than the wave being modeled,similarly to the grain in a photograph, which must be much smaller than the object shown122.1. The Transmission Line Modeling Methodin the image for the grain to be no longer noticeable. This is an expression of a spatial dis-cretization criterion, like a temporal discretization criterion is given by the Nyquist frequency[63]. Similarly, spatial aliasing effects and numerical errors occur when the wave is under sam-pled in space, as the continuous fields are necessarily represented by an equivalent circuit meshwith finite and discrete spatial resolution. These considerations equally apply to finite-element,boundary-element, and discretized-space methods in general.What is different in TLM, compared to, for example, Kron’s work, is that it does not relyon lumped circuit equivalents. It uses a mesh of continuous two-wire transmission lines. Thismakes it a more general formulation, and allows it to perform better at higher frequencies,where the transmission and reflection properties can no longer be considered lumped [47].Second, the use of transmission lines allows for an elegant implementation in a computer,as, in the loss-less case, the system can be abstracted though a scattering process at the cellboundaries and propagation in the cells. The actual details of the line need not to be taken intoaccount explicitly [16]. However, one may still represent the system as an electrical network ofdiscrete transmission lines and solve it directly, which is what the TINA method will be basedon.Although the following discussion is centered on the TLM method, many of the proposedconcepts also apply to finite-difference time-domain methods (FDTD) [96] which, in certaincases, can be shown to be equivalent formulations [9, 11].2.1.1 TLM Circuit Equivalents in 1D and 2DThe derivation of TLM theory is usually done from a lumped circuit perspective. Then, it isshown that the lumped circuit equations, when the lumped elements are made infinitely small,is equal to the wave equation in one or more dimensions. Here, we will show how the T-circuityields the Telegrapher’s equation and how this circuit may be extended to a two-dimensionalsystem. In the next section, the field to circuit mapping will be shown. In both cases, thederivation is based on [73].One-Dimensional Wave EquationStarting from the model in Figure 2.1, we have an elemental portion of transmission-line oflength  l, with R, G, L, and C the resistivity, conductance, inductance, and capacitance perunit length, respectively. The wave is assumed, without loss of generality, to propagate in the+z direction, from generator to load.132.1. The Transmission Line Modeling Methodz+∆lzR∆l/2 L∆l/2 R∆l/2L∆l/2V (z,t) V (z+∆l,t)+−+−V (z+∆l/2,t)+−G∆lC∆lI(z,t) I(z+∆l,t)∆IFigure 2.1: One-Dimensional T-Circuit of a Differential Length of LineApplying Kirchoff’s voltage law to the left loop, we obtain: V (z +  l=2;t) V (z;t) l=2 = RI (z;t) + L@I@t (z;t) (2.1)Applying Kirchoff’s current law to the main node, we obtain: I (z +  l=2;t) I (z;t) l = GV (z +  l=2;t) + C@V@t (z +  l=2;t) (2.2)Taking the limit of both above equations, as  l ! 0, we can find continuous forms of thelumped circuit: @V@z = RI + L@I@t (2.3a) @I@z = GV + C@V@t (2.3b)Differentiating the above equations with respect to z and t, respectively, we find: @2V@z2 = R@I@z + L@2I@z@t (2.4a) @2I@t@z = G@V@t + C@2V@t2 (2.4b)Substituting the above equations into one another yields the wave equation, or more pre-142.1. The Transmission Line Modeling Methodcisely, Telegrapher’s equation in either voltage or current form [46]:@2V@z2 = LC@2V@t2 + (RC + GL)@V@t + RGV (2.5a)@2I@z2 = LC@2I@t2 + (RC + GL)@I@t + RGI (2.5b)This equation is in fact very generic in form and appears in a wide variety of problems. Infact, by setting the appropriate parameters to zero, the equation reduces to the classic partialdifferential equations that govern Poisson, diffusion, and (loss-less) wave propagation prob-lems.Thus, from the above, in the limit where the spatial extent of the lumped components tendsto zero, we find that the circuit equivalent of Figure 2.1 yields the lossy wave equation for one-dimensional waves. In TLM, if we construct a cascade of these circuit sections, we may modela one-dimensional problem where each section of line represents a small part of the physicalsystem, e.g.: a small section of air in a narrow duct. If we take the spatial discretization  lsmall enough with respect to the wave length and the geometry of the system, we find that wemay model one-dimensional wave propagation using these electric circuits, as they implementthe physical behavior of the electro-magnetic wave.Later on in this chapter, the losses will be neglected in the further derivation of the TLMmethod, as well as the field to circuit comparison. This is done since in this thesis only loss-lesswave propagation problems are studied and the inclusion of the losses significantly complicatesthe models and reduces clarity in the derivation of the methods.In the TLM method, when applied to wave propagation problems, the losses are addedback in later, as external to the wave propagation model itself. By doing so, the model re-mains simple and basic scattering techniques may be used to solve the propagation aspect, andmodification to the cell central node are used to add non-ideal behavior [16, 18].In the EMTP also, the losses are added to the ideal line model later on. Doing so keeps thecore model, the Bergeron line model, computationally effective. The derivation of this modelfrom the Telegrapher’s equation is shown in Appendix A.Keepingthelossesinthemodelcomplicatesthederivation. Thelineparametersofimpedanceand wave speed become frequency dependent [46], which precludes their direct solution ina time-domain computation, such as TLM or EMTP. One could solve them using a seriesexpansion, convert to the frequency domain, or curve-fit the response using, for example,the frequency-dependent (FD) line model [51]. All of this adds computation. However, theEMTP line models, by keeping the Bergeron model at the core, are modular and allow differ-ent types of losses to be added around the basic line model. This approach gave rise to the152.1. The Transmission Line Modeling Methodconstant-parameter (CP, frequency-independent losses), distortion-less (Heaviside condition),and frequency-dependent line models (FD, based on curve-fitting the desired response to themodel using LC sections). All of these are computationally effective and used extensively inEMTP power system simulations [20].Thus, we can keep the derivations sufficiently general by assuming the loss-less case andtreat the losses separate from the propagation model later on. In fact, the TINA method wasespecially designed to accommodate the EMTP line models in a TLM-style computation.Two-Dimensional Wave EquationL∆l/2L∆l/22C∆lIz (z−∆l/2) Iz (z +∆l/2)∆l/2∆l/2VyO L∆l/2L∆l/2Ix (x−∆l/2)Ix (x+∆l/2)yzxFigure 2.2: Two-Dimensional T-Circuit of a Differential, Loss-Less Area of LineFor the 2D case, the losses were neglected from the onset (R = G = 0). This was done tomaintain clarity in the derivation, since only loss-less wave propagation is considered in thisthesis. The result is the elemental area of 2D line, shown in Figure 2.2. The derivation followsalong the same lines as for the 1D case above.Note the presence of double the capacitance (2C l as opposed to C l) in the 2D model.162.1. The Transmission Line Modeling MethodThis is due to an artifact of the 2D spatial discretization of space in Cartesian coordinates,where the diagonal propagation in a cell takes longer than the propagation along the main axis(diagonal of a square is longer than the sides of the square). The TLM method corrects for thisin the model by including twice the permittivity  in the medium, which increases the wavespeed by 1p2 to make up for the apparent slow-down of the wave propagation in the model.This issue is discussed in detail in Section 2.1.3 of this chapter. The correction results in thedifference between the circuit model and Maxwell’s field equations, as is demonstrated later inSection 2.1.2. One could choose not to include this capacitance now and correct later, and havethe circuit to field mapping have the same parameters. However, in TLM, it appears that theconsensus [16, 18] is to modify the 2D cell model from the onset and build-in the correction.Thus, in this thesis, this convention was adopted.Applying Kirchoff’s voltage law around the loop in the x-y plane, we find:Vy (x  l=2) Vy (x +  l=2) l =L2@Ix (x  l=2)@t +L2@Ix (x +  l=2)@t (2.6)Applying Kirchoff’s voltage law around the loop in the y-z plane, we find:Vy (x  l=2) Vy (x +  l=2) l =L2@Iz (z   l=2)@t +L2@Iz (z +  l=2)@t (2.7)Applying Kirchoff’s current law at the central node, we find:Ix (x  l=2) Ix (x +  l=2) l +Iz (z   l=2) Iz (z +  l=2) l = 2C@Vy@t (2.8)Taking the limit  l ! 0 of the above equations, we may find the continuous forms of thelumped circuit: @Iz@z  @Ix@x = 2C@Vy@t (2.9a)@Vy@x =  L@Ix@t (2.9b)@Vy@z =  L@Iz@t (2.9c)172.1. The Transmission Line Modeling MethodDifferentiating the above equations with respect to t, x, and z, respectively, we find: @2Iz@z@t  @2Ix@x@t = 2C@2Vy@t2 (2.10a)@2Vy@x2 =  L@2Ix@t@x (2.10b)@2Vy@z2 =  L@2Iz@t@z (2.10c)Substituting the voltage equations in the first current equation above yields the 2D loss-lesswave equation, or more precisely, the 2D Helmholtz equation:@2Vy@x2 +@2Vy@z2 = 2LC@2Vy@t2 (2.11)2.1.2 Equivalence Between Network and FieldA TLM simulation is made-up of a large number of building blocks shown in Figures 2.1 and2.2, organized in a 1D or 2D regular, orthogonal mesh. Non-orthogonal meshes have, however,been studied. For example, cylindrical and spherical [1, 18], as well as un-structured meshes[16] have been successfully used with TLM.In order to show the field to circuit mapping, we start from Maxwell’s field equations [54,55] for the electric and magnetic fields in differential form for uniform plane wave propagationin one direction for the time-varying case. We assume a simple medium with constant, scalarpermittivity and permeability, and in which no free charges ( = 0) and currents (J = 0) arepresent [71].Maxwell’s equations for such a case reduce to:5 D = 0 (2.12a)5 B = 0 (2.12b)5 E =   @H@t (2.12c)5 H =  @E@t (2.12d)182.1. The Transmission Line Modeling MethodExpansion of the curl equations for E and H, in a rectangular coordinate system, yields:@Ez@y  @Ey@z =   @Hx@t (2.13a)@Ex@z  @Ez@x =   @Hy@t (2.13b)@Ey@x  @Ex@y =   @Hz@t (2.13c)@Hz@y  @Hy@z =  @Ex@t (2.13d)@Hx@z  @Hz@x =  @Ey@t (2.13e)@Hy@x  @Hx@y =  @Ez@t (2.13f)These equations will now be adapted for 1D and 2D fields below.Network and Field Equivalence in 1DFrom [71], for uniform, one-dimensional plane waves, we assume variation in only one di-rection. We take this as the z-direction of a Cartesian coordinate system. Doing so sets thederivatives with respect to x and y to zero, thus @@x = 0 and @@x = 0. Applying this to (2.13)yields: @Ey@z =   @Hx@t (2.14a)@Ex@z =   @Hy@t (2.14b)0 =   @Hz@t (2.14c) @Hy@z =  @Ex@t (2.14d)@Hx@z =  @Ey@t (2.14e)0 =  @Ez@t (2.14f)Since the time-varying parts of Hz and Ez are zero, the fields of the wave are entirelytransverse to the direction of propagation, as is expected of a plane wave in 1D. The remainingequations break into two independent sets, one relating Ey and Hx, and the second set relatingEx and Hy. These sets are merely an expression of the orthogonal electric and magnetic fieldsin the orthogonal coordinate system, perpendicular to the direction of propagation, z. Sinceboth an electric or a magnetic field can exist in the x-y plane, e.g. in the y direction, its192.1. The Transmission Line Modeling Methodcorresponding magnetic or electric field, respectively, must necessarily exist in the x direction.The wave propagation behavior may be illustrated with either set. We choose the pair Eyand Hx. Partially differentiating Ey with respect to z and Hx to t, we find:@2Ey@z2 =  @2Hx@t@z (2.15a)@2Hx@t@z =  @2Ey@t2 (2.15b)Substituting the second equation into the first yields:@2Ey@z2 =   @2Ey@t2 (2.16)Thus, we find the one-dimensional loss-less wave equation, with the wave propagating inthe z-direction. When we compare this result with the wave equation found from the circuitequivalent (2.5a), and set the losses to zero, thusR =G= 0, we find that Telegrapher’s equationin voltage reduces to:@2V@z2 = LC@2V@t2 (2.17)By comparing the equations (2.16) with (2.17), as well as (2.4) with (2.15), we find that thecircuit and field representations of the wave propagation have the same form, in the loss-lesscase. We may now express the following equivalences:Ey = V (2.18a)Hx = I (2.18b) = L (2.18c) = C (2.18d)Thus, in the equivalent circuit: The voltage at a node is Ey The current in the z-direction is Hx The inductance per unit length represents the permeability of the medium The capacitance per unit length represents the permittivity of the mediumFor completeness, in Appendix A.1, a similar derivation is shown, including the losses.202.1. The Transmission Line Modeling MethodNetwork and Field Equivalence in 2DReturning to [73], the two-dimensional case is studied for transverse electric (TE) wave propa-gation. This implies that Ex = Ez = Hy = 0 and that all derivatives with respect to y are zero,thus @@y = 0. As such, there is an electric field only in the y-direction, with its correspondingmagnetic fields in the x and z directions. The magnetic field in the z-direction is now possible,as this is a 2D case, and thus both the x and z are perpendicular to the y direction, in which theelectric field propagates. Thus, this is a plane wave in 2D, with the wave propagation possiblein any direction in the x-z plane, where only those fields with an electric component in the ydirection are considered for clarity.Maxwell’s equations of (2.13) thus reduce to:@Ey@z =  @Hx@t (2.19a)@Ey@x =   @Hz@t (2.19b)@Hx@z  @Hz@x =  @Ey@t (2.19c)Partially differentiating Ey with respect to x and z, and Hx to t, we find:@2Ey@z2 =  @Hx@z@t (2.20a)@2Ey@x2 =   @Hz@x@t (2.20b)@Hx@z@t  @Hz@x@t =  @2Ey@t2 (2.20c)Substituting the first and second equation in the third equation above yields:@2Ey@x2 +@2Ey@z2 =   @2Ey@t2 (2.21)This is a Helmholtz, or loss-less wave equation, in two dimensions, for transverse-electricpropagation in the x-z plane. We compare this result with the wave equation found from the2D loss-less circuit equivalent (2.11), repeated below:@2Vy@x2 +@2Vy@z2 = 2LC@2Vy@t2 (2.22)By comparing the equations (2.21) with (2.22), as well as (2.10) with (2.20), we find thatthecircuitandfieldrepresentationsofthewavepropagationhavethesameform, intheloss-less212.1. The Transmission Line Modeling Methodcase. We may now express the following equivalences:Ey = Vy (2.23a)Hx =  Iz (2.23b)Hz = Ix (2.23c) = L (2.23d) = 2C (2.23e)Thus, in the equivalent circuit: The voltage at a node is Ey The current in the z-direction is  Hx The current in the x-direction is Hz The inductance per unit length represents the permeability of the medium Twice the capacitance per unit length represents the permittivity of the mediumIt is important to note here that the circuit capacitance per unit length is twice the per-mittivity of the medium in the 2D case. This is critical when converting the input parametersfor a medium in the simulation, typically given as the field parameters  and  , to the circuitparameters for the lines. Failing to adjust the capacitance will result in incorrect wave speedand line impedance in the computed results. The reason for this correction factor will now bediscussed.2.1.3 The Dispersion Relation of the Propagation Velocity in TLMWhen propagating a wave on a TLM grid from a point source, it is noticed that the wave frontdoes not travel in a perfect spherical front. In fact, as can be seen in Figure 2.3(a), in somedirections it appears to go slower and there is significant fine detail in the wave where onewould only expect perfect concentric symmetry. Also, the overall wave propagation seemsslower than expected and for higher frequencies, the propagation does not seem to happen atall. These issues will now be discussed.Wave Speed in the 2D TLM MeshTo give an intuitive insight into this phenomenon, let us consider the grid in Figure 2.3(b). Ifa plane wave front were to propagate either perfectly left $ right or up $ down, the wave222.1. The Transmission Line Modeling Method(a) Point Source in a Grid11√2(b) Grid DistancesFigure 2.3: Velocity Dispersion in a 2D TLM Gridwill, as it propagates, always see a distance of one grid division between the cell center nodes,which are taken on the intersections of the grid lines. However, if a wave were to propagateat a 45 degree angle with the grid, the distance between grid nodes becomes p2. Thus, theperceived wave speed at these angles becomes less than that along the main grid lines. This isan issue common to most discretized-space methodsIf the grid is made fine enough, this error eventually blends out and a clean wave frontpropagating at a coherent speed is observed some reasonable distance from the source. Thiseffect can already be seen to happen in Figure 2.3(a). Near the source in the center, the wavefront appears nearly diamond-shaped, and it quickly rounds-out the corners to appear moreand more circular as the distance from the sources increases. It is generally taken that, if l=  1=10, the variation in velocity is acceptably small [18]. This is demonstrated by thefigure where, close to the sources, insufficient cells are available, compared to the size of thewave. As the wave propagates out, the wave becomes properly sampled.It may be mentioned here that this 1=10 rule for spatial discretization is a similar rule ofthumb as used in interpolated EMTP line models [20]. This ratio is an expression of a spatialsampling criterion, much like the temporal sampling criterion given by the Nyquist frequency,which in practice must also be taken so that the sampling rate is five to ten times the highestfrequency in the system.The remaining problem, that the overall wave front travels too slow in 2D TLM, can beexplained by looking at the equation for the wave speed on the transmission lines in TLM andthe expression for the propagation constant in the 2D TLM mesh.232.1. The Transmission Line Modeling MethodThe wave speed in a transmission line is given by [16], whereCd andLd are the capacitanceand inductance per unit length, for a 1D transmission line, respectively:a = 1pLdCd= 1p r 0 r 0(2.24)We may already observe the issue from the above equation, when comparing the field tocircuit equivalences, given by (2.23): Cd 6= C. Thus, if we define the 2D medium by usingthe field parameters  and  , the resulting wave speed and line impedance will be wrong, asthe circuit equivalent parameters do not have a 1:1 correspondence. In fact, the resulting meshwill have a propagation speed equal to 1p2 of the desired value. The solution is to change the parameter according to the field to circuit equivalences, thus divide the required  by twointernal to the model.Dispersion in the 2D TLM MeshDispersion means that the wave propagation is frequency dependent. For the loss-less 1D line,this is studied in detail in Chapter 3. Here, we will give the expression for 2D meshes, as shownin the literature. Intuitively, the issue may be thought of as follows: the TLM mesh consists of alarge number of interconnected transmission lines. A transmission line functions as a resonatorwhen the length equals specific fractions of the wave length, e.g.:  =2; =4, and for these wavelengths functions as an open and short circuit, respectively. Thus, it is not unreasonable toexpect that the TINA mesh will have similar cut-off effects at these wave lengths, due to theimpedance discontinuities on the the mesh. These discontinuities arise from the nodes wherefour transmission lines connect. Thus, between two nodes, there is a transmission line withlength equal to the spatial discretization  l. The line is split in two half-lines, one half in eachcell. When the impedances of these lines are equal, the result is a single line of length  l, onwhich the resonator effects may happen.From [73], the dispersion relation for a 2D TLM, uniform mesh is given by:sin( ra l )= p2 sin(  l )(2.25)where rn = un=a is the normalized velocity, here the ratio of the velocity of the waves onthe mesh relative to the desired wave speed. Solving this equation numerically for different val-ues of  l= results in Figure 2.4. From this, we do indeed observe that the mesh blocks prop-agation at the quarter wavelength point. Thus, the TLM mesh can only represent Maxwell’sequations over a range of frequencies from DC to the finest network cut-off frequency. Thisoccurs at  l= = 1=4. However, there is a significant error close to this frequency, and the242.1. The Transmission Line Modeling Method0.50.550.60.650.70 0.05 0.1 0.15 0.2 0.25Normalized Propagation Velocity rNormalized Frequency ∆l/λDispersion of the Wave Velocity in the 2D TLM MeshFigure 2.4: Dispersion of the Velocity of Waves in the 2D, Loss-Less TLM Meshusable range is thus lower. The rule of thumb,  l=  1=10 finds its origin here. Also notethat the wave speed in the mesh indeed levels off at DC to p2 of the expected velocity.2.1.4 The Use of Transmission Lines in TLMThe derivation of the loss-less TLM method has shown that the 1D TLM mesh is essentially acascade of transmission lines, each obeying the loss-less wave equation. The loss-less 2D meshconsists of cells that are a combination of four loss-less 1D lines, joined at the central node [16]in either a series or shunt configuration. Thus, we may conclude that the 1D transmission lineis the basic building block of 1D and 2D TLM (except for the correction factor applied to thecapacitance per unit length or the permittivity  , in the 2D case, depending on the particularformulation of the line model).Now, in Appendix A.2, we showed that an exact solution to loss-less, one-dimensionalwave equation is the Bergeron line model. In this model, no discretization is used, and thesystem is essentially two Th´evenin equivalents, where the source is a time-delayed version ofthe currents and voltages on the other side of the model. This makes the model very easy andeffective to implement in a time-domain solution, such as TLM.252.1. The Transmission Line Modeling Methodt t + ∆t12 4314321V − 12 V12 V12 V12 VFigure 2.5: Dispersion of the Velocity of Waves in the 2D, Loss-Less TLM MeshHowever, the model is not directly implemented in TLM like this. Using the model directlywould require a circuit solution at each cell’s central node, as well as at each line connectionbetween cells. Although these are small circuits which may be solved efficiently, in TLM thesolution is based on scattering.When looking at regular TLM, we may observe that all lines have the same propagationtime, usually set to the base medium in use, and are loss-less. Losses and material variationsare implemented on the cell’s central node by means of stubs and resistive losses [16]. Thismeans that a direct circuit solution is not required. Note that this will be a point where the newmethod presented later in this chapter diverges.In TLM, the solution of the system is based on Huygens principle, thus scattering of thewave at impedance discontinuities. This solution is efficient when all lines have the same,typically one time step, time delay and materials are defined relative to a base material bymeans of matching stubs. In such a case, the system may be easily expressed in terms of ascattering matrix, which also includes any matching stubs. The scattering process describesthe incident and reflected pulses, and forms the basis of the TLM algorithm, e.g.: [31, 73].To illustrate this, let us evaluate the process when a wave enters a cell. The cell is connectedto four identical neighbors, with normalized characteristic impedance Zc = 1, though portsnumbered 1-4, as shown in Figure 2.5. The traveling time of each line is normalized so that =  t. At time = t, a Dirac pulse of unit energy (Si) enters the central cell from cell 1. Onetime step later, the pulse is scattered to the connecting cells.Since line 1 has three other lines connected to it in the cell, its effective terminal impedanceis 1/3. Since we know the relevant impedances, we also know the reflection coefficient for both262.1. The Transmission Line Modeling Methodthe reflected and transmitted voltage [71]. It is given by: = ZL  ZcZL + Zc=13  113 + 1=  12 (2.26)Thus, the reflected and transmitted energy are, where the subscripts i, r, and t are incident,reflected, and transmitted, respectively:Sr = Si 2 = 14 (2.27a)St = Si(1  2) = 34 (2.27b)Thus, a voltage impulse of  12 is reflected back in terminal 1, while a voltage pulse of12 =√3=43 will be launched into each of the three other terminals.Generalizing the above case for four impulses incident on a node, we may apply superposi-tion by four single pulses, as the mesh is linear. Thus, if at a time t, voltage pulses tV i1, tV i2, tV i3and tV i4, are incident on lines 1-4, at any junction node, the combined voltage reflected alongline 1 at time t +  t will be given by:t+ tV r1 =12htVi2 + tVi3 + tVi4  tVi1i (2.28)In general, the total voltage impulse reflected along line n at time t +  t will be:t+ tV rn =12[ 4∑m=1tV im] tV in; n = 1;2;3;4 (2.29)This may be described in matrix form, relating the reflected voltages at time t +  t to theincident voltages at the previous time step t:V1V2V3V4rt+ t= 12 1 1 1 11  1 1 11 1  1 11 1 1  1V1V2V3V4it(2.30)Also, an impulse leaving a node at position (z,x) in the mesh (thus, a reflected impulse),272.1. The Transmission Line Modeling Methodbecomes automatically an incident pulse at the neighboring node. Thus:t+ tV 11 (z;x +  l) = t+ tV r3 (z;x) (2.31a)t+ tV 12 (z +  l;x) = t+ tV r4 (z;x) (2.31b)t+ tV 13 (z;x  l) = t+ tV r1 (z;x) (2.31c)t+ tV 14 (z   l;x) = t+ tV r2 (z;x) (2.31d)Thus, by applying the above equations (2.30) and (2.31), the magnitudes, positions, anddirections of all impulses at time t+ t can be obtained at each node in the network, providedthat the corresponding values at the previous time step were known. Thus, it is possible tosolve the system through scattering without doing a formal network analysis at each node inthe system. Also, note that the time steps are fully decoupled in time. Thus, it is possible tosolve any part of the system independent from another, making parallelization trivial.2.1.5 Synchronicity in TLMThus far, the discussion on TLM has focused on simulations with only one, homogeneousmaterial in the system. To do any interesting work with the method, different material typesmust be integrated in the simulation. This may be easily done by locally changing the mediumparameters  and  . However, this brings with it a new problem.When changing the material parameters, it follows that both the impedance and wave speedof the material, and thus line segments, must change. The central node solution remains un-changedfortheimpedancechanges; however, theconnectionsolutionnowmusttakeadditionalscattering at the impedance discontinuity between two neighboring areas into account.There are two main methods used in TLM to address these issues. One is based on chang-ing the impedance of the constituent transmission lines, and modifies the connection equations.This method, however, does not resolve the issues due to different wave speeds in the simula-tion. The second method uses loading stubs and addresses both. The latter method will now bediscussed.Connection Solution in the Presence of an Impedance DiscontinuityIn general, a pulse moving from a source s to a neighboring node a experiences a reflection asit passes the impedance discontinuity between the area with impedance Zs and the area with282.1. The Transmission Line Modeling Methodimpedance Za. These reflection coefficients are given by: s!a = Za  ZsZa + Zs(2.32a) a!s = Zs  ZaZs + Za(2.32b)Now, the connection process, expressed in (2.31), must be changed accordingly. Eachconnection will now get a portion of the neighbor’s contribution superimposed by a portion ofits own signal reflected back from the boundary. The terms in the first equation, for example,must be substituted as follows, where  is the reflection and  the transmission coefficient,respectively:t+ tV 11 (z;x +  l) =  a!st+ tV 11 (z;x +  l) +  a!st+ tV r3 (z;x) (2.33a)t+ tV r3 (z;x) =  s!at+ tV r3 (z;x) +  s!at+ tV 11 (z;x +  l) (2.33b)In essence, the connection process now completes a circuit solution based on incident andreflected waves. Also, it must be noted that this extended solution is only necessary at theboundary between two materials, thus in practice, the numerical cost of this is usually not veryhigh, as the material boundaries tend to only make up a small percentage of the simulation.What this method does not resolve is the issue caused by different traveling times in dif-ferent media. When two, or more, media in a simulation have traveling times that are not aninteger multiple of one another, only one of these can match the simulation time step. As aresult, the other areas cannot operate normally as the discrete transmission line is a time delayand thus can only be a multiple of the simulation time step, as each past value corresponds to adiscrete memory location. There is no simulation information for times in between these timesthat are multiples of the simulation time step.This is the problem of synchronicity. One could change the size of the lines in the othermedia, to insure they match the simulation time step. However, that introduces the issue thatthe physical cells themselves now no longer match with one another and partial overlaps mayhappen. Thus, doing so moves the synchronicity issue from a temporal to a spatial expression,but does not solve the underlying issue. Later in this chapter, we will show an approach thatovercomes these issues with this method.The Use of StubsThere is another way of handling the impedance differences, which also solves the issue ofdifferent traveling times in the computation. In this solution, all transmission lines in the mesh292.1. The Transmission Line Modeling MethodRegion A Region B(a) Scattering ConnectionRegion A Region B(b) Stub MatchingFigure 2.6: Methods to Incorporate Different Media in TLMare kept at the same impedance and, at the cell nodes, a transmission line stub of length  l=2 isconnected to adjust the material parameters [16, 18]. The result of this approach is a modifiedscattering matrix, and the connection process can be left as it was, since the line impedanceat the connections is now unchanged. The difference in concept between these methods isillustrated in Figure 2.6.The concept of a stub comes from microwave engineering, where, due to the very shortwave lengths involved, lumped components are unusable due to excessive parasitics. In thisfield, short lengths of transmission line are used to emulate capacitive and inductive behavior.In TLM, we may use the concept to modify the behavior of a medium by adding such a stubto the central node in a cell. If we have two materials, A and B, we may write the followingequivalence for the capacitive component of the line models in the mesh, where both materialsare referred to vacuum, as is customary in TLM for electro-magnetics [18]:CA =  AC0 CB =  BC0 (2.34)Thus, we may say that: B =  A +  S (2.35)There, S refers to the stub. Indeed, when looking at the circuit equivalent for a TLM node,302.1. The Transmission Line Modeling Methodadding a line adds to the capacitance of the node. Thus, we may write:CB = CA + CS (2.36)The impedance within region B is then:ZB =  tCA + CS(2.37)Thus, the medium with characteristic impedance ZB may be replaced by a parallel combi-nation of the medium with impedance ZA and a stub line. When    l, these stubs becomepart of the medium, in the same way the lumped line model components in the circuit equiva-lent did.This stub should be a storage element, so that all energy that enters it is eventually reflectedback into the mesh. Thus, open and closed stub lines must be used, depending on if we requireto add capacitance or inductance to the system. We may, in fact describe the stub impedanceas capacitive or inductive, depending upon the termination of the line. However, one can neverhave a line that has only capacitance without inductance, and vice versa. As a result, each ofthe above expressions is associated with a parasitic inductance and capacitance, respectively[16], for a standard stub with length  l=2:ZSC =  t2CSLerror = ( t)24CS (2.38a)ZSL = 2LS t Cerror = ( t)24LS (2.38b)Theseerrorsmaybereducedbymakingthesimulationtimestep tsmaller. This, however,will also make the simulation numerically much more expensive. Also, the error associatedwith the simulation depends on how much difference in material properties must be matchedbetween different regions. As the required L and C goes up, so does the associated error.InTLM,onewillusuallychooseabasemedium, andallothermediawillbedefinedrelativeto this by choosing appropriate stubs. Since the stubs allow us to influence both L and C, wemay thus calculate the stub so the material matches the required  and  , and thus match boththe desired impedance and wave speed, as both are related. When the errors associated with thematching stubs become too large, one may reduce the simulation time step to obtain sufficientaccuracy. This change will increase the ratio of the wave length to the mesh size, and thus pushthe mesh closer to the limiting condition of  l ! 0.312.1. The Transmission Line Modeling MethodWith the stub lines present in the system, the scattering matrix now becomes [18]:V1V2V3V4VSrt+ t= 14ZS + ZSV1V2V3V4VSit(2.39)withS = (Z + 2ZS) 2ZS 2ZS 2ZS 2Z2ZS  (Z + 2ZS) 2ZS 2ZS 2Z2ZS 2ZS  (Z + 2ZS) 2ZS 2Z2ZS 2ZS 2ZS  (Z + 2ZS) 2Z2ZS 2ZS 2ZS 2ZS (Z  4ZS)(2.40)In this solution, since all lines at the cell connections have the same impedance, the scat-tering process remains unchanged.Stubs and DispersionTheuseofstubsresultsinaneffectivereductionofthewavepropagationspeedinaTLMmesh.Since the stub has parasitic behavior, it is expected that this speed reduction will be frequencydependent. The dispersion of a computation is directly related to the wave speed. Thus, onemay expect that the dispersion relation for the 2D TLM mesh will change. From [18], the stubcapacitance may be expressed as:CS =  x y {1  t2  [ x2 +  y2 x2 y2]}(2.41)For a conventional, thus regular, TLM mesh with  x =  y, and with  r =  r = 1, the t = 1ap2, with a the wave speed. If  r and/or  r no longer equal to unity, we may find thefollowing relation for the cut-off frequency of the mesh:( x )cut off= 1 sin 1[1 xp r rpF](2.42)with:F =  t2 r r[ x2 +  y2 x2 y2](2.43)322.1. The Transmission Line Modeling MethodR1R=1 OhmR3R=1 OhmR2R=1 GOhmLine1Z=1 OhmL=10 mmLine3Z=1 OhmL=10 mmLine2Z=1 OhmL=10 mmV1U1=0 VU2=1 VT1=0T2=2 e-13 stransientsimulationTR1Type=linStart=0Stop=6 e-10 sNodeFigure 2.7: 1D TLM Node with StubThe cut-off frequency is the highest when no dissimilar materials are present in the simula-tion. WhenF increases, the cut-off frequency of the mesh decreases. Thus, the more dissimilarthe materials are, the larger the incurred error. The accuracy may be improved by reducing thesize of the mesh cells, which is directly related to changing the sampling rate of the system, asthe size of the cells is linked to the traveling time of the line segments, which is equal to aninteger multiple, usually unity, of the simulation time step.Stubs and Node ReflectionsThe use of stubs has a drawback in that the cell node exhibits an oscillatory transient response[18]. Indeed, the energy stored in the stub is reflected back to the node. Here, it encounters animpedance discontinuity, resulting in only a part of the energy in the line to be injected backinto the mesh. Thus, these reflections cause a transient behavior.Toillustratethis,considera1Dcellwithanopencircuitstuboflength l=2,asillustratedinFigure 2.7. An incident step of magnitude 1 is injected. The stub line has the same impedanceas the other lines, all set to unity. The lines are terminated in their characteristic impedance.When this system is solved, the voltage behavior on the node is shown in Figure 2.8. Thesolution was computed using QUCS v0.0.14 [70], an open-source circuit simulator. The resultsshow small spikes on the transitions. These are due to the use of an “infinite” rise-time, thuszero in one time step and unity in the next, pulse with the trapezoidal integration rule withoutcritical damping adjustment (CDA) [48], and may be ignored.The reflections due to the use of stubs are clearly visible. Although they die out after332.1. The Transmission Line Modeling Method0 5e-11 1e-10 1.5e-10 2e-10 2.5e-10 3e-10 3.5e-10 4e-10 4.5e-10 5e-10 5.5e-10 6e-10-0.0500. 2.8: 1D Node Reflectionsonly a few time steps, their magnitude is significant and will influence the transient behaviorof a computation. Thus, when using stubs, one must insure that the transient in the systemis much slower than the numerical transients caused by the stubs. The transient response ofthe interpolation-based method introduced later in this chapter, Figure 2.12, has an inherentlynon-oscillatory response.Note that, since the stub is half a time-step in length, the node would only be observed fromthe over-all simulation during the moments the wave-form is non-zero, as a reflection on thisline would require it to be traveled twice, and thus synchronize with the overall simulation timestep. The transient response is thus visible in the results.2.1.6 3D in TLMIn this thesis, we limit ourselves to 1D and 2D computations. 3D solutions are possible, butthe solution gains much complexity. For electro-magnetic fields, there is the phenomenon ofpolarization that does not exist in 1D and 2D fields. The acoustic equivalent would be theexistence of compression, shear, and torsional modes in a solid. In order to capture thesephenomena, the TLM cells must incorporate orthogonal field components on each connectingface. Where, in 2D, one only had to compute one voltage for one transmission line at eachconnection to the cell, here, two orthogonal transmission lines must connect, to form a plane inwhich the angle of the voltage vector may be described. Thus, the scattering matrix becomes12 X 12, as there are six sides to the unit cell [16] for the series condensed node (SCN). This342.2. Transient Insular Nodal Analysisis for a case where no stub lines are required.However, for acoustics in gasses and many liquids, there is no polarization (shear and tor-sionalforceswiththeirassociatedmodeconversion),andthe3Dcaseisfoundsimplybyaddingline segments in the third dimension and adjusting the various computations accordingly. Thus,the 3D acoustical node in these limited cases contains six line segments as opposed to four forthe 2D case [66, 67], for a case where no stub lines are required.When different media must be incorporated in a computation, the scattering matrix forsolutions beyond 2D incorporates more that one stub line. For example, the 3D SCN foracoustics [66] has one shunt and three series stubs in the system. This brings the total to tentransmission line segments per cell.The resulting circuit for the full model including polarization, incorporating stub lines, hasno relation to anything physical. For 3D, it merely becomes a formal representation of the fieldsolution at the centre of the node.2.2 Transient Insular Nodal AnalysisIn this thesis, a modified version of TLM was devised and used. The rationale behind this wasthe realization that EMTP power systems simulations have a host of very flexible and numer-ically efficient transmission line models available to them, varying from loss-less, delay-only,modelstofull-fledgedfrequency-dependentmodelswitharbitraryresponse. Thesemodelsthusshowed promise to allow complex materials to be modeled in a TLM context.Since the TLM method is, essentially, a circuit solution of a mesh of transmission lines, theEMTP models should be suitable for use in this context. In EMTP, temporal interpolation isused. Using this approach, stubs and their incurred node reflections, discussed above, may beprevented at a relatively modest increase in computation.Functionally, the EMTP line models are based on an ideal transmission line, thus a timedelay, with additional RGLC components to model losses and frequency-dependence, as re-quired. Thus, one of the main advantages of TLM, time decoupling, remains present and, withit, the possibility of effective parallelization.In order to make the TLM method suitable for direct insertion of EMTP transmission linemodels, we start from the observation that the majority of these models, in the time domain,may be represented by a Th´evenin equivalent and interface more easily with the connectingcircuit also represented by such an equivalent.Thus, if we modify the TLM method so that its mesh cells present a two or four port, in 1Dand 2D, respectively, and each port is a Th´evenin equivalent, we may obtain building blocksfrom which a simulation case may be build. The simulator then only needs to provide the352.2. Transient Insular Nodal Analysisuninw ineEastWestGridLoc∆tColorSolve historiesSolve Th´eveninsFigure 2.9: Loss-Less, 1D TINA Node Cellconnectivity between such cells, and manage the solution to instruct each cell to solve itself.Lastly, the simulator also manages the output processing, where the desired results from thecells are stored as required.From this, the name of the new simulator, TINA, may be explained: Transient InsularNodal Analysis. TINA is a transient simulator that exhibits full time-decoupling between itsmesh cells allowing them to be solved independent from one another, thus makes them insular.The solution is based on a localized nodal analysis, only incorporating the components perti-nent to the node under solution. Each insular cell is a fully self-contained model that exportsa universal interface to the simulator: Th´evenin equivalents for information exchange and astandardized API for the simulator to give instructions to the cell and obtain solution data fromthe cell. Such a cell, for one dimension, is shown in Figure 2.9. The internal details are notimportant to the TINA simulator, as long as the cell exposes suitable Th´evenin equivalents,variables, operators, and central node state data.From the figure, we may also observe that a link resistor formulation is used [18] to inter-face with the neighboring cells. In general, there are two choices: link-line and link-resistor.The first connects individual cells though a shared transmission line. The issue with this iswhen a material discontinuity is present. The line models used in this thesis cannot be readilysplit in such a way, where each half has different parameters, unless two separate line segmentsare used. In that case, one automatically has a link resistor formulation, where two neighboringcells are joined through a resistor. In TLM, however, a link-line formulation is possible withdifferent materials when stub-matching is used at the nodes, and all the transmission lines inthe mesh are thus identical. However, since TINA uses temporal interpolation, and thus line ofdifferent parameters for different media. Thus, a link resistor formulation is the more flexibleand logical choice.Also, TINA uses a single-wire formulation. This implies a ground, thus absolute reference,362.2. Transient Insular Nodal Analysisthroughout the simulation. This change from the two-wire formulation common in TLM wasmade to simplify the solution. The second wire is implied, as the ground serves as a mirror intransmission-line theory [7]. Since the TINA formulation uses a direct circuit formulation andsolution, a two-wire formulation was not required and a solution to ground sufficient, as it isfunctionally equivalent.InTINA,eachcellisacompletematerialmodelthatincorporatesthetransmissionlinesandall their associated components, as well as the node solution for that cell. This approach allowsfor model-specific optimizations, as well as potentially allow for any special functions to beintegrated into the cells, such as non-linearities and various dependencies, such as temperature.For the TINA implementation in this thesis, this segregation of the local and global solutionwas driven to its logical extreme, with each mesh cell a C++ object [44]. This implementationallowed a variety of cell models to be implemented and tested using the same solver code.Also, different solver synchronicity techniques, such as interpolation and fractional latency,could be used with the same, well-tested cell models. This proved convenient for developmentand presented a clean programming abstraction. However, the overhead incurred by literallyhundred of thousands to millions of objects did result in a significant performance penalty.Future versions will likely be based on non-object approaches for the actual solution of thecells.Another difference is the extensive use of interpolation in the TINA method for synchronic-ity, where TLM commonly uses stub lines. Although the TLM method can use either approach,its scattering matrix formulation favors stub lines as these can be implemented in a clean andefficient way, maintaining single time-step transmission lines.TINA, with its self-contained cell solutions, can also be used with either, but since goodtransient response was a priority in the development of the method, the use of stubs was foundunsatisfactory and temporal interpolation methods were chosen.We will now show how the TINA model is obtained from TLM, and that the method isessentially identical, but for the formulation and solution approach. The changes allow fora more flexible simulation, allowing the direct use of EMTP line models, as well the newfractional latency technique, introduced later in Chapter The TINA FormulationTo obtain the TINA formulation, we must first return to the development of TLM. AlthoughTLM uses a scattering solution, this is really just an implicit circuit solution of the cell. ForTINA, we will use an explicit circuit solution for the cell and its boundaries.The second observation is that the lumped element model used in TLM may be substituted372.2. Transient Insular Nodal AnalysisFigure 2.10: 1D TINA Node Cell Internalsby a transmission-line segment. This is, in fact, done in TLM though the scattering matrixformulation, which works with line parameters of impedance and the wave speed is implicitto one simulation time step. In TINA, the transmission line segments may be any temporallength and their actual models, which are derived in Appendix A, are solved explicitly. Thischoice allows the transmission line model to have any form, as long as it maintains time-stepdecoupling. It also allows the transmission line segments to have time histories that are deeperthan one time step. This requirement is discussed in the next section of this chapter, where thetemporal interpolation solution for synchronicity is introduced.Thus, TINAisatime-decoupled, explicitcircuitsolutionversionofTLMthatusestemporalinterpolation for synchronicity. The latter is made directly possible due to the explicit useof the transmission-line models, which can have any size of history buffers. This is not asstraight-forward in the common scattering TLM formulation, where a single time step historyis expected.By packaging all the lines and their associated components in cell models, each of which istime-decoupled from one-another, the basic TINA solution becomes trivial. All the solver hasto do is tell the cells, in any order desired, to first solve their internal histories. When that isdone, it tells the cells again, in any order desired, to update their external Th´evenin equivalents.The solver will also query each cell for its internal node state information, which is the solutionof the system. This sequence of events is identical to that used in EMTP circuit simulations[90]. Hence, one may think of TINA as a higher-dimensional extension to the EMTP powersystem simulator.The actual circuit solutions in each cell are made through basic nodal analysis [79] of thecircuit. Since the circuit is small and time-decoupled from the neighbors, this solution is trivial.Thesolutionisfurthersimplified, aseachneighborpresentsaTh´eveninequivalent, whichisfedthrough a length of time-decoupling transmission line that, in itself, also automatically presentsa Th´evenin equivalent, in the loss-less case. Thus, the node solution becomes a simple R-onlycircuit with two or four sources, for 1D and 2D, respectively. This situation is schematically382.2. Transient Insular Nodal Analysisillustrated in Figure 2.10, for a 1D case. Each pair of sources and resistors forms a section ofloss-less CP line model, as discussed in detail in the following chapters and Appendix A. Thecircuit shown in this figure is the contents of a typical TINA building block. When comparedto the abstract cell of Figure 2.9, we may see how directly this fits in the general TINA conceptof cell abstraction.Since nearly all EMTP transmission line models have a similar circuit formulation [20],this elemental cell can be readily adapted to accommodate such models and add losses to thesimulation. Also, other components may be integrated in the solution to add non-linearitiesto a simulation e.g.: [89]. None of this requires modification to the basic simulator algorithmitself, making the solve inherently modular.A direct, algebraic solution of the circuit in the cell was chosen over a matrix formulation.The problem with the matrix formulation for such a small system was that constructing thematrix and solving it, even using machine specific, optimized mathematics libraries, such asGotoBLAS [24], proved to perform at roughly half the speed of the direct solution. The directsolution, when the equations of same form are grouped, and using a suitable, optimizing com-piler, actually allows for some degree of parallelism in vector operations which modern CPUsoffer in their mathematical units.2.2.2 The Interpolated Line Model in TINAThe fundamental issue with the transmission line method, and also TINA, when different ma-terials are incorporated in the simulation, is that the traveling time of the wave in each lengthof transmission line must be an integer multiple of the simulation time step to maintain syn-chronicity. Clearly, with different materials, this will not always be the case.The reason why integer multiples are required has to do with the nature of the transmission-line models used. They are time-delays, thus the only values that exist in their memory, calledhistory sources, are values at the simulation time steps. When the traveling time does not matchwith the simulation time step, data points in between the available information is required. Thisis discussed in Appendix A.In TLM, stub-matching is usually employed to avoid this synchronicity issue. By choosingthe base medium so that each length of line is an integer multiple, usually unity, of the simula-tion time step, and then adjusting the medium with a stub line where required, the desired localwave speed and medium impedance is obtained. In TINA, the other method available, temporalinterpolation, is used, as this is the approach has a clean transient response, as is desired for atransient simulator. This is also the method used in the EMTP.The concept behind the interpolation process and its requirements is detailed in Appendix392.2. Transient Insular Nodal AnalysisA.3, and the behavior of the interpolation process, with respect to stability and accuracy, isstudied in Chapter 3. All of these studies are for the one-dimensional line, which is the funda-mental building block of all TINA models.From those chapters, it is found that the interpolation process, when applied to the loss-less line is stable. It is even more stable than the ideal line model itself, as it adds someamount of dampening to the system due to the losses incurred by the low-pass filter effect ofthe interpolation. The low-pass filter effect is also present in the stub matching approach [18].The TINA method, being nothing but a circuit solution, is in itself conserving and linear, as thecircuit solutions are conserving through Kirchoff’s laws. Thus, the whole system must also beconserving.When stubs are not used, and different materials are present, the transmission line segmentsmust now also have different lengths, or rather, history depths. This means that the temporalinterpolation method to synchronicity will require more memory than the stub-matched ap-proach.To illustrate the issue, consider a material with wave speed 1.03 m/s, and one with wavespeed 10 m/s. For a simulation time step, which would have to be linked to the faster material,there are two ways to incorporate the slower material.Using stubs, the slower material would be composed out of the same transmission lines asthefastmaterial, andsuitablestublineswouldbeaddedtobring-downthespeedofthe materialto the desired one. Thus, since all lines operate at the same time step, synchronicity is obtainedand all lines may be chosen so that their traveling time is precisely equal to the simulation timestep. In TLM is usually chosen so that the traveling time equals the simulation time step. Thus,the memory required to implement the lines is only one memory location for the one step deephistory buffer.Using temporal interpolation, the fast material would be chosen in the same way, but theslow material would be composed out of transmission lines that are much longer, temporally.Since the simulation operates at a rate roughly ten times faster than the traveling time of theline, it needs to store a data point for each value in the past in the buffer of the line. Thus,roughly simulation time steps. As such, the history buffers associated with the lines must beabout ten steps deep. The temporal interpolation method thus requires about ten times morestorage.In addition, the interpolation of the history sources requires more computation than the stubline requires.As such, temporal interpolation, while allowing more flexibility and direct compatibilitywith the EMTP transmission line models, does add a significant overhead to the computation,bothinnumberofoperationsandmemoryuse. Thatsaid, itdoeshaveacleantransientresponse402.2. Transient Insular Nodal Analysisdue to the absence of stubs.Much of the lost performance may, however, be re-gained through the use of fractionallatency, introduced in Chapter 4, which only requires interpolation at the borders of a material,as opposed to throughout its entire volume, and further allows a suitable local time step so thateach line in the simulation may have history buffers one step deep. Using this, the numericalperformance my be better than using stubs, as the overhead of calculating the stubs is removed,as they are no longer required.2.2.3 Error CriterionAs with any discretized method, there are limits to how coarse a computation may be for adesired precision. Here, we discretize in both time and space. Both are intimately related, aswill be seen below.First, consider the spatial discretization. To prevent spatial aliasing, two cells per wave-length are a minimum. However, in practice, about five to ten cells per wavelength are typicallyused for accuracy [18]. From this, it follows that the shortest wavelength in the computationwill determine the maximum allowed grid size in a material. This will thus be the highestoccurring frequency. However, and this is the subtlety, it is the highest occurring frequency inthe material with the slowest wave speed that will have the shortest wave length.For the temporal discretization, this is given by the traveling time of the line segments ineach grid cell. If each material has traveling times that are integer multiples of one another,and thus no interpolation is needed for synchronicity, the simulation time step simply becomesthe shortest traveling time in the system. This will be the fastest medium in the system, wherethe size of that cell, and thus the line lengths in it, is set by the slowest medium.If, however, interpolation is used for synchronicity, the simulation is no longer based onideal line segments. The interpolation adds additional accuracy requirements, as discussed inChapter 3. As is custom in EMTP [20], and shown in that chapter, the used simulation timestep must be five to ten times smaller than the traveling time of the line segments.Thus, two error criteria may be stated for the TINA method: one for when interpolation isused, and one for when it isn’t. These are summarized in Table 2.1, where amax is the wavespeed of the fastest material and fmax is the highest frequency in the simulation.  lmax and tmax are the maximum spatial and temporal discretization sizes allowed for these parame-ters, taking a ten times spatial oversampling into account. This automatically results in a tentimestemporal oversampling. When interplation is used, an additional ten times temporal over-sampling for the fastest line segments is required, resulting in a total hundred times temporaloversampling.412.2. Transient Insular Nodal AnalysisDiscretization Ideal Lines Interpolated LinesSpatial  lmax = amin10fmax  lmax = amin10fmaxTemporal  tmax = 110fmax  tmax = 1100fmaxTable 2.1: TINA Error CriteriaThus, the use of interpolation requires ten times more time steps than the non-interpolatedcomputation. However, it must be noted that the use of stubs in TLM also requires a reducedtime step to maintain accuracy [18], as it also exhibits a low-pass filtering effect.2.2.4 Domain TerminationA problem common to nearly all discretized-space methods is that of domain termination.In the computer, only a finite expanse of space can be represented, meaning that for mostcases, the outer bounds of the simulation space must somehow be terminated in a known way.Usually, one would prefer the medium to be unbounded, so that any wave incident on theedges of the simulation space continues as if the space were infinite. The two other commonboundary conditions are that of the open circuit or short [16].The termination into infinity for a 1D mesh is easy to obtain, and is simply the terminationof the transmission line into a resistance of equal impedance to the line [46, 52].Unfortunately, thisterminationintoinfinityis hardtoobtainbeyond1D,andmuchhasbeenwritten about this subject. The difficulty lies in the fact that one must terminate the medium,not the transmission lines that make-up the medium. Since it was already established in thischapter that the TLM, and by extention also TINA, mesh is anisotropic, with a different prop-agation speed depending on the direction of propagation. It follows that the effective mediumcharacteristics are than also different in different directions of propagation. This poses an issuewith the boundary, which has no direct information on the direction of wave propagation.Lately, the consensus is that the split-field perfectly matched layer (PML), proposed byBerenger in 1994 [4], and its more recent generalization, stretched-coordinate PML [12, 85],perform best. This first technique is based on decomposing the electro-magnetic field intotwo un-physical fields in the terminating region and applying complex termination materials tothese. In essence, the wave is de-composed in orthogonal components, which are now propa-gation direction in-sensitive and can thus be easily terminated in a suitable complex impedanceto match the frequency-dependant (dispersion) behaviour of the mesh.A newer formulation is based on a Cartesian to cylindrical coordinate system transforma-422.2. Transient Insular Nodal AnalysisFigure 2.11: Boundary Reflections in a Resistively Terminated 2D TINA Meshtion, where one or more coordinates are mapped to complex numbers. It represents an analyticcontinuation of the wave equation into complex coordinates, replacing propagating, thus oscil-lating, waves by exponentially decaying waves.In this thesis, a simple, resistive termination was used to terminate both the 1D and 2Dgrids [16]. However, this resulted in poor attenuation of the incident wave on the boundary for2D, and significant reflections occurred. This is why the precision experiments in this thesiswere done on closed ducts, so little energy would leave the system under investigation andhave a chance to interact with this boundary. Also, since all systems measured were 2D, thetermination of the 2D experimental setup into 3D free-space could not be modeled accurately,and thus some degree of inaccuracy would be incurred in those physical experiments. Chapter5 shows those results.Figure 2.11 show what happens when a 2D mesh is resistively terminated, using boundaryconditions matched to the medium [16]. The errors were somewhat amplified in this figureto make the effect more pronounced. But, as will be clear in Chapter 5, for the open-planeexperiments, the influence is quite significant.It must, however, be noted that the matched boundary conditions work reasonable well ifthe wave is incident at right angles [16]. The incident at right angle limitation is reminiscentof the dispersion of the mesh, with different propagation velocity depending on the angle the432.2. Transient Insular Nodal Analysiswave travels to the mesh coordinates. This is why a simple, resistive, termination does notwork. The mesh has different properties depending on the angle of incidence, thus a simple,real resistance is insufficient. Spit-field PML works around this be decomposing the field inorthogonal components and presents suitable complex terminations to these.For the purpose of the thesis, matched boundary conditions provided a sufficient, and com-putationally efficient, solution. Further work on TINA, however, would include the implemen-tation of PML-style boundaries.2.2.5 Validation of the TINA methodTo evaluate both the correctness of the implementation, and the method itself, three types ofvalidation were performed.First, the TINA solver itself was compared with EMTP, for identical small-scale systems.This validates the numerical implementation of the solver and its models. Also an interpolatedversion of the models was tested this way.Second, a time-domain evaluation of the step response for the interpolated model was per-formed to verify that the transient response is, indeed, clean and the use of interpolation justi-fied when the transient response is important.Third, the interpolated solution was compared with the non-interpolated version for largersimulations, with parameters chosen so the results should be nearly identical. This was toexperiment with different simulation parameters and partially validate the error criteria.Last, the TINA solutions were compared with measurements. No direct comparisons witha TLM solver were made, as the TINA method is a only a re-formulation of TLM. The experi-mental validation is detailed in Chapter 5.Validation with EMTPUsing Mictrotran [56], the EMTP version developed at UBC, a number of 1D and 2D caseswere constructed and compared to the TINA solutions. The 1D cases had three cells in them,the 2D cases were three by three grids. For each system, computations with and withoutinterpolation were performed. Also, different materials were included in the tests.It was found that all results were identical to the EMTP reference solutions, proving thatthe TINA solver and models are a correct implementation of both the circuit solution and theconstituent loss-less transmission-line models.442.2. Transient Insular Nodal Analysis 0 0.2 0.4 0.6 0.8 1 1.2 0  0.0002  0.0004  0.0006  0.0008  0.001  0.0012  0.0014Pressure (Pa)Time (s)SourceNode 1Node 2Figure 2.12: Step Response in a 1D TINA MeshTime-Domain Step Response of 1D TINA MeshSince one of the main reasons behind the use of interpolation for synchronicity was the cleantransient response, compared to stubs in Figure 2.8, also this was evaluated. The experimentwas performed on 1D grid of three cells in length. The simulation time step was 56  s, thewave speed 2000 m/s, and the medium impedance 100  . For this combination of parameters,interpolation is required. The mesh was excited with a step.In Figure 2.12, the source data was taken at the ideal source node, before its matchingresistor. Hence, the amplitude is twice that of the rest of the mesh. Also, the source wasimplemented so that it had no traveling time associated with it. Thus, the distance from thesource to node 1 was  l=2, while the distance between the grid nodes is  l. Hence, thepropagation time from the source node to node 1 is half that of the normal propagation time inthe mesh, as would be the case between node 1 and node 2.Two main effects are immediately apparent: the step response has no transient behavior, aswould be the case with stub matching. Second, the low-pass filter effect of the interpolationremoves some of the higher frequency components, causing the slope edges to roll-off, ashigher frequency components are more and more attenuated as the signal passes through the452.2. Transient Insular Nodal Analysis20 205405003002004090140190A GFigure 2.13: Expansion Duct Set-Upmesh. Hence the need for increased oversampling when interpolation or stubs are used, so thehigher-frequency performance may be retained, if required.Performance Comparison of Interpolation in TINAThree different spatial discretizations will be used for this comparison, namely 0.5 mm, 1 mm,and 2 mm square cells. The system under investigation is the expansion duct case, as describedin Chapter 5. The normal simulation uses slightly adjusted parameters for the aluminum toinsure no interpolation is used in the solution. The system is shown in Figure 2.13. The utmostleft yellow dot is the “A” microphone position and the utmost right yellow dot is the “G”microphone position. Transfer functions between those two physical locations were calculatedand compared.The physical size of the microphone itself was also taken into account, which is a sizableportion of the wave length for the higher frequencies evaluated. Thus, in the simulation, theacoustic pressure over an area comparable to that of the physical microphone was averaged toobtain the result for that location.The differences between the precise and adjusted materials are shown in Table 4.3, where is the compressibility, k the compressibility, a the wave speed, and z the impedance. Theseparameters are acoustic ones, but relate directly to their electrical equivalents of permeabilityand permittivity.From Figures 2.14 and 2.15, we see that the normal and interpolated methods are self-consistent and overlap. For the phase, the unwrapping algorithm made an number of incorrectunwraps due to small differences in the phase of the input signal. Thus, the error between thetraces is not representative of the actual phase error incurred. The different spatial discretiza-tion, and thus accuracy by which the system may be represented, results in some deviationsbetween the cases, as the physical size is somewhat different. The system has to be discretized462.2. Transient Insular Nodal AnalysisAir Al (correct) Al (adjusted) 1:1198 103 gm3 2:6989 106 gm3 2:6989 106 gm3k 6:99746 10 9ms2g 1:45138 10 14ms2g 1:584727 10 14ms2ga 3:45383 102ms 5:05261 103ms 4:83537 103msz 4:13769 105 gsm2 1:36365 1010 gsm2 1:30502 1010 gsm2Table 2.2: Materials Used in the TINA Simulations 0 1 2 3 4 5 500  1000  1500  2000  2500  3000  3500  4000  4500  5000MagnitudeFrequency (Hz)Transfer Function A->G MagnitudeNormal 0.5 mmInterpolated 0.5 mmNormal 1 mmInterpolated 1 mmNormal 2 mmInterpolated 2 mmFigure 2.14: Magnitude Results for Various Mesh Sizesto 1 mm to be able to accurately represent the dimensions. Also, for higher frequencies, the er-rors increase faster with coarser spatial discretization, mostly due to the spatial error becomingan increasingly more significant portion of the acoustic wavelength in the medium for thosefrequencies.Attempts to cause insufficient temporal discretization or spatial oversampling proved prob-lematic, as the bandwidth obtained from the required spatial discretization to represent thephysical system and provide sufficiently accurate microphone positions was in the megahertzrange, far beyond what could be verified through measurement with the available equipment.Thus, any conclusions drawn from such simulations would not be falsifiable through the avail-able experiments. This also implies that, for the auditory acoustic range usually of interest,472.3. Conclusions-1000-800-600-400-200 0 200 400 600 500  1000  1500  2000  2500  3000  3500  4000  4500  5000Phase (degrees)Frequency (Hz)Transfer Function A->G PhaseNormal 0.5 mmInterpolated 0.5 mmNormal 1 mmInterpolated 1 mmNormal 2 mmInterpolated 2 mmFigure 2.15: Phase Results for Various Mesh Sizeswhen simulating small cases that require high, millimeter-scale precision, the available band-widths are so high that a lot of computation is wasted. This is a problem of scale, and isdiscussed in Chapter 4.More detailed comparisons between these cases may be found in Chapter 4, where therelative memory use and computation times for 1 mm cell size cases is discussed. These arelong simulations, over seven hours compared to the less than half an hour required for the 2mm cell sizes. Thus, any cumulative errors will be more likely to show there. The experimentalvalidation of the method is discussed in Chapter 5.2.3 ConclusionsIn this section, we discussed the TLM method, and how TINA is a direct descendant of it.The TINA formulation, which is closely inspired by EMTP approaches, has the advantage ofdirect compatibility with EMTP transmission-line models, and allows complex material cellsto be inserted into its mesh, as long as these cells obey the API and provide suitable Th´eveninequivalents to the solver. The mesh cells are conceived as self-contained units that may becalled upon to solve themselves, based on the EMTP solution algorithm.482.3. ConclusionsThe use of temporal interpolation allows for clean transient responses, at the cost of in-creased computation and memory consumption, when compared to stub matching techniques.However, the ability to have materials with arbitrary history buffer depths makes the methodmore flexible than the scattering matrix formulation, which is based on history buffers only onestep deep. Also, using fractional latency, as discussed in Chapter 4 in this thesis, much of thelost performance may be re-gained.The TINA method was evaluated for numerical correctness and consistency with itself andEMTP. The interpolation was shown to yield good results. Further experimental evaluationis performed in Chapter 5 and further numerical validation is performed in Chapter 4 of thisthesis.Future work would include the optimization of the TINA solver to avoid the use of objectsin the actual network solution, as well as the implementation of PML mesh termination. Also,a detailed study of the distortions introduced by the mesh, inspired by prior work on TLM,would be of interest.49Chapter 3Convergence, Dissipation, and Dispersionin 1D Loss-Less Line ModelsA small error in the former will produce an enormous error in the latter.Henri Poincar´eIn this chapter, the properties of the one-dimensional line models themselves are studied indetail. Their numerical convergence, dissipation, and dispersion is evaluated analytically forthe loss-less ideal and interpolated cases. Although the studied line models have been in wideuse in EMTP power system simulations since the sixties, e.g.:[90], this analysis has not beenperformed before in the literature. Error expressions are derived that allow the incurred errorsof an interpolated line under a given set of operating conditions to be determined. This formsa contribution of this thesis.In previous chapters, we showed that, as in the TLM method, the mesh in the TINA methodis constructed from a spatial combination of one-dimensional transmission lines. The deriva-tion of these one-dimensional building-block line models is shown in Appendix A.The computation of the TINA mesh itself is entirely based on the application of circuitanalysis, and thus is stable and conserves energy. As such, the problem of evaluating thesimulation stability is reduced to analyzing the stability of the one-dimensional line modelsused to construct the mesh. However, the numerical accuracy of the TINA mesh, in termsof dissipation and dispersion error, is determined by both the spatial discretization and theinherent accuracy of the line models used to construct the mesh.From the analysis, a commonly used “rule of thumb”, the necessity for choosing the sim-ulation time step  t at least ten times smaller than the line traveling time  [20, 90], can beobtained. Also, the incurred error is quantified, making it possible to relax the requirement incertain cases.3.1 Convergence of the Line ModelsIn order for the line models to be useful, the numerical solution must be stable. However,stability in itself is not a sufficient condition. Numerical methods can be stable, but still yield503.1. Convergence of the Line Modelsthe wrong results when a solution converges to a wrong value. Thus, a better criterion must beused.What must be proven of a numerical scheme is convergence [83], which proves that theeventual solution is bound and yields the correct final answer. This is frequently non-trivial toshow in a direct manner. The difficulties in showing convergence arise directly from the factthat the numerical method is defined by a recurrence relation, and the differential equation isdefinedbyadifferentiablefunction. However, convergencecanbeproveninanindirectfashionby showing both consistency and stability. Together, these two tests provide necessary andsufficient conditions for convergence, according to the Lax-Richtmyer equivalence theorem[40]. The theorem applies to TLM and TINA as well, as these methods can be show to beequivalent [9, 11].For finite-difference schemes for initial-value problems, the Lax-Richtmyer equiv-alence theorem states that a consistent finite-difference scheme for a partial differ-ential equation, for which the initial value problem is well-posed, is convergent ifand only if it is stable [83].Thus, the problem of proving convergence is replaced by the equivalent and simpler toverify conditions of consistency and stability. The theorem can also be applied to higher-orderschemes [72, 83].ConsistencyConsistency, when applied to a discretized model, is a test that insures that the solution tendsto the correct answer. The method described by Strikwerda [83] is used. The test is doneby establishing a point-wise convergence on each grid point or, in other words, by letting thespatial and temporal discretization intervals tend to zero. The discretized model should thenapproach the non-discretized one. Thus, the error in the numerical solution, when compared tothe continuous model, should tend to zero.StabilityStability is established by testing if the numerical solution is bounded, or in other words, pro-duces a finite output for a finite input. In this thesis, eigenvalue analysis was used to find themodes of the system. These modes give direct insight in the dynamics of the system, e.g.:[3, 38, 62, 83, 91]. This method is further detailed in Appendix B. It is then only a matter ofinsuring that each of these modes remains bounded to insure stability.513.1. Convergence of the Line ModelsWell-posednessMathematical models of physical phenomena should have a unique solution that depends onthe data in a continuous way, and is insensitive to small variations in the input data or modelparameters. Such a model is referred to as well-posed. Or, in other words: does it even makesense to attempt to discretize the problem in the first place and can a well-behaving discretemodel be obtained?The study of well-posedness, pioneered by Hadamard in his seminal 1921 lecture [25], isa statement of the sensitivity of a problem to small changes in the input data. When we saythat an equation is well-posed, it is meant that the initial value problem for the equation iswell-posed.In well-posed problems, small changes yield small differences in the solution. This helpsthe system to converge to a stable and correct value. A well-posed problem, when solved witha stable algorithm, has thus a good chance of reaching a correct and stable result. Ill-posedproblems, by comparison, may yield wildly different results for small changes in the input. Inthe presence of noise, interpolation, modeling approximations, and numerical roundoff errors,such problems are typically difficult to solve. Thus, well-posedness is a test to ensure that anattempt to discretize a continuous-time system makes sense.Another consideration is that a partial differential equation derived to model a physicalsystem should be unaffected by the addition of, or changes in, the lower order terms, and bysufficiently small changes in the coefficients. This quality is referred to as robustness. It isimportant, as most models are derived by making assumptions and neglecting various effectsdeemed not important through an understanding of the system. Thus, these approximationschange the model equations. The resulting model should be insensitive to the small perturba-tions caused by these omissions and still produce results that allow correct conclusions to bedrawn from the computations.Totestforwell-posedness,thesystemofgoverningequationsmustbeevaluated. Suchtests,based on finding the modes in frequency domain though eigenvalue analysis, were describedby Kreiss [34]. Strikwerda, in his book [83], provides a treatment of the subject in the Englishlanguage following the methods proposed by Kreiss.The principle of the analysis, for equations of first order with one time derivative and con-stant coefficients, is to show that the real parts of the eigenvalues of the continuous-time systemin frequency domain are always less or equal to a constant for all !. In other words, the so-lutions, each of an exponential form, have exponents that ensure that each mode is bound foreach frequency. This is a necessary and sufficient condition for well-posedness.More exactly, the situation for the ideal line model, which is a system of two PDE of thegeneral form:523.1. Convergence of the Line Models_ut = Q(!)_u (3.1)In frequency domain, this system has a general solution of the form:_u(t;!) = eQ(!)t_uo(!) (3.2)The well-posedness condition for this system is now:The necessary and sufficient condition for the system (3.2) to be well-posed [83]is that for each non-negative t, there is a constant Ct such that, for all ! 2RN:keQ(!)tk Ct (3.3)A necessary condition for the above to hold is that, for each eigenvalue q(!) ofQ(!), there is some constant q, such that, for all values of !:Re q(!)  q (3.4)It can also be seen that the eigenvalues of Q(!) govern the exponentials that make-up thesolution of this model. Thus, the eigenvalues define the modes of the system, and their studywill give the required information: whether or not the system remains bound to some constantvalue.An exponential to the power of a general matrix, required for the first part of the well-posedness condition (3.3), is not trivial to compute. There is, however, a standard solution tothis problem [62], outlined in Appendix B.3. The second part of the condition (3.4) is mucheasier, and necessary, but not sufficient.The first part can be calculated if the Q(!) matrix is diagonalized. Hence the applicationof eigenvector and eigenvalue analysis. Aside from diagonalizing the system, the procedurealso yields the individual eigenvalues that are needed to perform the second part of the well-posedness condition.From the approach outlined in Appendix B.3, assuming that the eigenvalues are distinct, itis found that:eQt = Ve 1t 0...0 e ntV 1 (3.5)Thus, the first part of the well-posedness condition (3.3) can be evaluated quite easily using533.1. Convergence of the Line Modelsthe above solution. Using the individual eigenvalues themselves, the second part of the well-posedness condition (3.4) can also be readily performed.Now, applying all the above, starting from the transmission-line equations for the loss-lesscase [46], as derived in Appendix A.2:{ @v(x;t)@x =  l@i(x;t)@t@i(x;t)@x =  c@v(x;t)@t(3.6)In order to find the modes from this system, needed for the well-posedness criteria, weneed to solve it. This implies conversion to either Laplace or Fourier domain, as that reducesthe problem from a system of PDE to a system of ODE. Diagonalization will then yield therequired eigenvectors and eigenvalues.Applying the Fourier transform to space, we thus find:{j!V (!) =  l@I(!)@tj!I(!) =  c@V(!)@t (3.7)Re-arranging: {@I(!)@t =  j!l V (!)@V(!)@t =  j!c I(!)(3.8)Writing in matrix form suitable for diagonalization though eigenvalue analysis:@@t[V (!)I(!)]=  [0 j!cj!l 0][V (!)I(!)](3.9)Computing the eigenvalues of (3.9):det(Q  I) = 0 =        j!c j!l        = 0 (3.10) 2  (j!)2lc = 0 ,  2 = (j!)2lc ,  =  j!plc (3.11)Computing the eigenvectors, using these eigenvalues:v1 =[1 √cl]v2 =[1√cl](3.12)543.1. Convergence of the Line ModelsThus, V,  , and V 1 are:V =[1 1 √cl √cl] =[j!plc 00  j!plc]V 1 = 121  1pcl1 1pcl (3.13)Where all the above matrices are in Fourier (!) domain. We find that both the eigenvec-tors and eigenvalues have multiplicity one, and are thus distinct. The general solution for anexponential raised to the power of a matrix, outlined in Appendix B.3, can thus be used for thewell-posedness analysis.To calculate the norm of the matrix required for the first part of the well-posedness test(3.3), the expressions of (B.14) and (B.15) are required. Also, the following definitions regard-ing matrix norms are used [62]:kABk  kAkkBk (3.14a)kAk1 = max1 i nm∑j=1jaijj (3.14b)Applying the above to the first well-posedness test, for the infinite norm [62], yields:keQ(!)tk1  Ct (3.15a),kV(!)e (!)tV(!) 1k1  Ct (3.15b),kV(!)k1ke (!)tk1kV(!) 1k1  Ct (3.15c), 2 je j!plctj(12 +12√cl) Ct (3.15d)Since l and c are constant for a given case, only the exponential term requires considerationas it has variables. After simplification, the entire expression becomes a sum of two exponen-tials with purely imaginary exponents. It is thus bounded for any value of t and !. Thus, theexpression is changing, but always  a constant value given by the line parameters, l and c.From the eigenvalues, we can now apply the second part of the well-posedness test, asshown in (3.4), directly. Testing for the inequality, thus verifying if the real part of each eigen-value is  a constant:Re( j!plc)= 0  C (3.16)Thus, we find that the equations governing the ideal line model are well-posed.553.1. Convergence of the Line ModelsZc Zcvk(t) emh(t)ekh(t)im(t)ik(t)vm(t)Figure 3.1: Lossless Line Model3.1.1 The Ideal Line ModelConsistencyDerived in Appendix A.2, this line model is found as an exact solution to Maxwell’s equationsfor the one-dimensional traverse electro-magnetic plane wave in the loss-less, non-dispersive,homogeneous case. It is thus not a discretized form of a continuous-time equation. As such,the consistency check for this model can be omitted, as the model is the solution itself, and theconsistency test would thus be a test with itself.StabilityThe proof of stability requires a little more work. In order to do the analysis, eigenvalues areused to find the modes of the system, as outlined in Appendix B. It is the behavior of thesemodes that governs stability. We require all modes to be bounded if the solution is not to growinfinite for a finite input. To diagonalize the system, and thus find the eigenvalues, the modelmust first be written in terms of a single type of variables, e.g.. voltage. In case of the linemodels, we can write the equations in terms of only the voltage history sources ehk and ehmand applied external voltages vk and vm.From Appendix A, (A.30), the ideal line model in voltage only is:{ehk(t) =  ehm(t  ) + 2vm(t  )ehm(t) =  ehk(t  ) + 2vk(t  ) (3.17)Writing (3.17) in a matrix form suitable for eigenvalue analysis:[ehkehm]t=  [0 11 0][ehkehm]t  + 2[0 11 0][vkvm]t  (3.18)Now, to perform the eigenvalue analysis as outlined in Appendix B, we recognize that(3.18) has the formx(t) = Ax(t  )+Bu(t  ), whereuis the forcing function, which we563.1. Convergence of the Line Modelscan set to zero for the current analysis. Thus, the eigenvalues of A are the relevant modes, orthe poles, of the system described by the model. As such, setting the external sources vk andvm to zero we find the eigenvalues as follows:det(A  I) = 0 ,        1 1        = 0 (3.19)The eigenvector is then found as: 2  1 = 0 (3.20)This yields the eigenvalues: =  1 (3.21)For stability, thus to insure a bounded output for a bounded input, all modes of the system,and thus eigenvalues, must be so that j j 1. We find that they are equal to 1. This representsa special type of stability, where the solution is bound for a finite input, but does not increasenor decrease in value over time. Such a type of stability is described by Lyapunov [99]. Inz-domain analysis, this would indicate that all poles are exactly on the unit circle. The idealline model is thus marginally stable. This result is expected for the ideal model, as it has nolosses of any kind and all energy in the model must thus be preserved. Further discussion ofthe stability conditions follows in Section 3.1.3, later in this chapter, where these results arecompared to the interpolated line model.Thus, since the ideal line model is both consistent and stable, as well as well-posed, we canconclude from the Lax-Richtmyer theorem that it is convergent.3.1.2 The Interpolated Line ModelDerived in Appendix A.3, this line model is found from the ideal model by manually adding-ina linear interpolation function in time between two know history values in order to obtain therequired history at a time that does not coincide with the simulation step. This addition ofinterpolation makes the solution no longer exact, and may thus introduce undesired behavior.The convergence aspect will be studied here.ConsistencyConsistency can be easily checked by comparison with the ideal model, by letting the temporaldiscretization (simulation time step)  t and the spatial discretization  x (grid size) tend tozero. The interpolated model should then become identical to the ideal model [83]. If that is573.1. Convergence of the Line Modelsthe case, the interpolated model will, for sufficiently small spatial and temporal discretizationintervals, tend to the correct answer and thus be consistent.To perform the proof, we first need the line equations in a single variable. Voltage is se-lected here, as it is convenient and the resulting expressions can be used for the followingstability proof. Also, the line equations in terms of voltage for the ideal model were derivedpreviously in Appendix A, (A.30). The same methodology was then used to obtain the inter-polated version.From Appendix A, (A.35), the interpolated line equations in voltage only are:ehk(t) = 2vm(t  int) 2R[vm(t  int) vm(t  int   t))] ehm(t  int) + R[ehm(t  int) ehm(t  int   t)]ehm(t) = 2vk(t  int) 2R[vk(t  int) vk(t  int   t))] ehk(t  int) + R[ehk(t  int) ehk(t  int   t)](3.22)Comparing (3.22) with the previous result from (3.17), it can be seen that both are similarin form, but in the interpolated case, there are extra terms that relate to an additional history,one  t further in the past. The R term expresses where in between both known values theinterpolated value is to be computed. The  int terms are the line length expressed as an integermultiple of simulation time steps  t, rounded down.To assess consistency, we must now let  t to zero and subtract the interpolated equationsfrom the ideal ones, which are an exact solution for the loss-less model under investigation.The result should be zero, thus no error in the limit for initially small temporal discretization,for the interpolated model to be consistent. From the equations, we also find that the spatialdiscretization is of no consequence to consistency in the 1D model. This is due to the factthat the interpolated model is based on an exact solution, modified with an interpolation thatoperates only in the temporal domain.Thus, with  t ! 0 it can be found though inspection of (A.15) and (A.16) that  int !  and R ! 0.Substituting these values in (3.22), it reverts to the expressions for the ideal case, as givenin (3.17). Subtracting this limit result for the interpolated model from the exact expressions forthe ideal line model (3.17), the remainder is found to be zero. The interpolated line model isthus consistent.StabilityThe stability analysis for the model will be performed along the same lines as in Section 3.1.1,at least, initially. Due to the multiple history depths, the equations cannot be written in a form583.1. Convergence of the Line Modelsthat is directly amenable to standard eigenvalue analysis to evaluate the modes as described inAppendix B.Conversion to the z domain is used from that point forward. The stability is then assessedby computing the system transfer function matrix H(z). The eigenvalues (poles) of this ex-pression, without a forcing function, are used to assess stability.However, the resulting closed-form polynomial expressions in z have variables as expo-nents, making it impossible to find closed-form roots in general. Specific cases can be evalu-ated, though. Numerical examples are given to illustrate the stability trends, and show that theinterpolated model tends to the ideal case as the number of time steps per line increases, andthe interpolation error becomes less and less.Alessrigorousstabilityproofisthenshown, basedonverificationofthelong-termoutcomeof the system (homogeneous part) as t ! 1. This proof is not as rigorous, as it assumes along-term outcome is possible.Thus, writing the interpolated line model equation in voltage only (3.22) in a matrix formsuitable for eigenvalue analysis:[ehkehm]t=[0 11 0]  (1 R)[ehkehm]t  int R[ehkehm]t  int  t+2(1 R)[vkvm]t  int+ 2R[vkvm]t  int  t(3.23)We see that (3.23) has the form:x(t) = A1x(t  int) +A2x(t  int   t) +B1u(t  int) +B2u(t  int   t) (3.24)The various A and B terms can be readily obtained from (3.23) as:A1 =[0  (1 R) (1 R) 0]A2 =[0  R R 0]B1 =[0 2(1 R)2(1 R) 0]B2 =[0 2R2R 0] (3.25)u is the forcing function, which is not required for the current analysis. This brings therelevant part of the equation entirely in the same terms of x.The issue is that the various RHS terms of x are in different time scales, making it im-possible to find the system modes through regular eigenvalue analysis as outlined in Appendix593.1. Convergence of the Line ModelsB. This is because the A matrices for each different time step cannot be combined in a singleexpression that can be diagonalized to directly evaluate the system behavior.To overcome this problem, the system is converted to z-domain using the following conver-sions of the indices to make them consistent with the implied concept of time in this discretedomain:  int ! N Where N is the length of the line as an integer number of simulation timesteps, rounded down, per definition of (A.15).  t ! 1 One time step in z-domain is a single delay. t ! n The total simulation time is implicit in z-domain. It is an integer number ofsimulation time steps.Using this notation, and setting the excitation to unity, the skeletal equation of (3.24) be-comes:x(n) = A1x(n N) +A2x(n (N + 1)) +u(n) (3.26)Converting (3.26) to z-domain:x(z) = A1z Nx(z) +A2z (N+1)x(z) +u(z) (3.27)Grouping terms we find:x(z) =[I A1z N  A2z (N+1)] 1u(z) (3.28)The system behavior is now described by a single matrix in Z-domain. This form is thussuitable for eigenvalue analysis. Since the system is in z-domain, the resulting eigenvalues arethe poles of the transfer function in z-domain, and must be within the unit circle for stability[63].Substituting the A terms back into the above equation we find:x(z) =[1 Rz (N+1) + (1 R)z NRz (N+1) + (1 R)z N 1] 1u(z) (3.29)Computing the eigenvalues using Maxima [53], we find: 1 = 1Rz (N+1) + (1 R)z N + 1  2 = 1 Rz (N+1)  (1 R)z N + 1 (3.30)To evaluate stability, we inspect the above expressions. It becomes clear that, to ensureboth eigenvalues are  1, we need to find the poles (roots) of polynomial-like expressions603.1. Convergence of the Line Modelswith exponents in N, where N is a positive integer and a variable. This form is not a standardpolynomial, and the roots cannot be found directly. Thus, a closed-form expression cannotbe found in the generic case. However, since N is constant for any given case, evaluation inthe specific is possible. It is thus possible to exhaustively work through each value of N, aspractical values will be limited to a couple hundred.When evaluating (3.30) numerically, the poles of these expressions in z-domain show anumber of definite trends, as illustrated graphically below for a number of cases for varying Rand N.From the plots (Figures 3.2 and 3.3), it can be seen that the poles follow a number ofdistinct trends. As the line becomes longer (N increases), the poles tend closer and closer tothe unit circle. Looking at (3.30), it can indeed be seen that, as N ! 1, the expressions forthe eigenvalues tends to:limN!1 1 = limN!11z N + 1 limN!1 2 = limN!11 z N + 1 (3.31)Thus, all poles, as N ! 1, tend toward the unit circle, but never exceed it. Also, theinfluence of the interpolation factor R decreases. This is due to the fact that the interpolationonly acts on one segment of the line. The effect of interpolation is thus reduced with increasinglength.A second observation is that the interpolation factor R affects the symmetry of the poleplacement in the z plane. For both small and large values of R (recalling that 0  R < 1)the effect of the interpolation is small. This is explained by the fact that, for such values ofR, the point to be interpolated is very close to a known value. Thus, the interpolation erroris significantly reduced. For R = 0:5, however, the interpolated point is in the middle of theinterval, and thus at the highest uncertainty. The error is thus larger.The effects of N and R will be studied in more detail in Section 3.2, later in this chapter.Fromthetrendsexhibitedbythepolesforvariousparameters, itcanthusbeinferredthattheinterpolatedlinemodelisstable. Asthemodelisalsoconsistent, andthegoverningcontinuous-time equations well-posed, from the Lax-Richtmyer theorem, the interpolated line model isconvergent.An additional observation is that the model tends toward Lunyapov stability [99] as it tendscloser to the ideal line model. When the interpolation is of significant influence, there are clearerrors associated with the model and the poles are further away from the unit circle. Thus,the model may have dissipative properties which make it more stable. This dissipation anddispersion are studied in Section 3.2.613.1. Convergence of the Line Models(a) R = 0:1, N = 1 (b) R = 0:1, N = 10 (c) R = 0:1, N = 100(d) R = 0:5, N = 1 (e) R = 0:5, N = 10 (f) R = 0:5, N = 100(g) R = 0:9, N = 1 (h) R = 0:9, N = 10 (i) R = 0:9, N = 100Figure 3.2: Pole Plots for Various Combinations of R and N for  1623.1. Convergence of the Line Models(a) R = 0:1, N = 1 (b) R = 0:1, N = 10 (c) R = 0:1, N = 100(d) R = 0:5, N = 1 (e) R = 0:5, N = 10 (f) R = 0:5, N = 100(g) R = 0:9, N = 1 (h) R = 0:9, N = 10 (i) R = 0:9, N = 100Figure 3.3: Pole Plots for Various Combinations of R and N for  2633.1. Convergence of the Line ModelsA different approach to stabilityAlthough the above proof is mathematically rigorous, the resulting eigenvalues cannot be eval-uated in the general case due to the variable in the exponent of the polynomials. Evaluation inthe specific case is, however, straight-forward.To strengthen the inferred conclusion of the above section regarding the stability of theinterpolated line model, a different approach will now be used to show stability. The proof isbased on finding the long-term solution (t !1) of the interpolated line model. This particularsolution must be zero or bound, in case there are no losses and no input or a bound input. Thisproof is not as rigorous, as a long-term solution is assumed to exist.Thus, starting from (3.24):x(t) = A1x(t  int) +A2x(t  int   t) +B1u(t  int) +B2u(t  int   t) (3.32)In the long run, when t ! 1, (t  int)  (t  int   t). Using this simplification, wefind:x(t)  (A1 +A2)x(t  int) + (B1 +B2)u(t  int) (3.33)Writing in matrix form, and substituting:[ehkehm]t  [0 11 0][ehkehm]t  int+ 2[0 11 0][vkvm]t  int(3.34)This expression is the same as the ideal line model (3.18). Setting the excitation to zero, itwas shown to be Lyapunov stable. Thus, the particular solution is bound. From this, we canconclude that the interpolated line model is also Lyapunov stable, when the simulation time islong enough for the particular solution to be reached.3.1.3 A Physical Interpretation of Marginal Stability in the Line ModelsFrom the preceding analysis, it is found that both the ideal and interpolated line models haveeigenvalues/poles in the z-domain that are unity/on the z-domain unit circle. Thus, the linemodels are/tend to marginally stable. This is a curious stability condition where the outcomemay be oscillatory, but is still bounded. Or, in different words, the energy in the model isconserved, but the response may be infinite in duration.For ideal transmission lines, this is the expected behavior. If we consider an ideal lineterminated in a short on both ends and we inject a pulse, this pulse will keep bouncing backand forward forever, as there are no losses and no dispersion. This means that the model, for a643.2. Dissipation and Dispersionfinite input, produces an infinite response (in duration) which is still bound (in the total energyin the system). Thus, although the response is oscillatory, the system is stable in the sensethat it does not gain energy. The interpolated model has some losses, to be investigated in thefollowing section, and thus only tends to this infinite response behavior, as some energy is lostover time. This can be confirmed from the z-domain analysis above for the interpolated model,where the poles are in fact only tending to the unit circle. The interpolated model is thus morestable than the ideal model, due to these losses.These types of stability are describe by Lyapunov [99]. It can be summarized as that fora dynamic system that starts out near an equilibrium point xe will stay near this equilibriumpoint forever. If that condition is true, then the system is Lyapunov stable. If a solution startsout near xe and converges closer toward it, the solution is asymptotically stable. Exponentialstability guarantees a minimal rate of decay, and thus an estimate of how fast the solution willconverge.3.2 Dissipation and DispersionIn a kinder world, the performance of the loss-less, interpolated line model would be identicalto the ideal one. In practice, this is not the case and some frequency-dependent behavior occursdue to the interpolation process. In fact, the interpolation results in a low-pass filter effect, aswill become apparent in this section.The errors incurred through interpolation are frequency-dependent losses which, in fre-quency domain, result in a magnitude and phase error. The magnitude error is the dissipationerror, while the phase error gives the dispersion error. Or, in different words, the magnitudeloss and phase shift for each frequency as it passes through the model.As opposed to the previous section, where only the internal behavior of the models wasconsidered, here the boundary conditions play a role as well. For example, reflections onthe terminals due to a load mis-match will cause part of the energy to be reflected back intothe model, where it will be affected by the non-ideal behavior once more. Thus, a directcomparison of the ideal and interpolated model by only evaluating their transfer functions willnot provide all the required information.Since, in this section, we must evaluate the models in the presence of boundary conditionsto measure their response, we must first define both the used boundary conditions, as well asthe performance metric used in the evaluation.To facilitate the analysis, a version of the line models converted to frequency domain wasused, since it is the magnitude and phase response of the model with respect to frequency thatis of interest. The conversion from time to frequency domain is illustrated in Appendix A,653.2. Dissipation and DispersionSections A.2.3 and A.3.3.¯Vk(ω) ¯Vm(ω)Zc Zc¯Ekh(ω) ¯Emh(ω)¯Ik(ω) ¯Im(ω)ZlFigure 3.4: Loss-Less Line Model with Boundary Conditions in Frequency DomainAsillustratedinFigure3.4, theboundaryconditionsare, onthek-portofthemodel, anidealvoltage source, and on the m-port, a variable impedance. This impedance will be adjusted tohighlight both typical and border cases.The performance metric is a normalised, relative measure that compares the performanceof the interpolated model with to the ideal one. Since the latter is an exact solution, it is chosenas the benchmark. As the used models are expressed in frequency domain, the ratio is one ofphasors:Error = Interpolated modelIdeal model  1 (3.35)Thus, the evaluation is simply a calculation of the line model in frequency domain, withidentical boundary conditions for both the ideal and interpolated model, and then dividingthe respective output phasors to find the error ratio (3.35) for each frequency. Plotting theresults in magnitude and phase will yield the dissipation and dispersion error for the case underinvestigation.Let us now derive a general error expression for a line terminated in a complex load, anddriven from an ideal source. Two such expressions must be derived, one in voltage and onein current, to accommodate shorted and open-circuit boundary conditions. From Appendix A,the ideal and interpolated line model in frequency domain are:Ideal{V k  ZcIk =(V m + ZcIm)e j! V m  ZcIm =(V k + ZcIk)e j! (3.36a)Interpolated{V k  ZcIk =(V m + ZcIm)e j! int[1 + R(e j! t  1)]V m  ZcIm =(V k + ZcIk)e j! int[1 + R(e j! t  1)] (3.36b)We observe that both are of the same general form:{V k  ZcIk =(V m + ZcIm)AV m  ZcIm =(V k + ZcIk)A (3.37)663.2. Dissipation and Dispersionwhich is a general expression of an input subjected to a complex phase shift A. Thisexpression is now used to find a general solution for the model, including boundary conditionsas shown in Figure 3.4. The appropriate terms will then be substituted for A after the solutionis found.The boundary conditions for voltage are thus:V k = V s Im =  V mZl(3.38)Using these boundary conditions, the system reverts to two equations in two unknowns,and can thus be solved. Substituting:{V s  ZcIk =(V m  ZcV mZl)AV m  ZcV mZl=(V s + ZcIk)A(3.39)Solving for voltage, we find:V mV s =2ZlA(Zl  Zc)A2 + Zl + Zc (3.40)Repeating the above for current, using a current injection Is instead of a voltage injectionV s, the following boundary conditions are applied:Ik = Is V m =  ImZl (3.41)Using these boundary conditions, the system reverts to two equations in two unknowns,and can thus be solved. Substituting:{V k  ZcIs =(ZcIm  ZlIm)A ZlIm  ZcIm =(V k + ZcIs)A (3.42)Solving for current, we find:ImIs =2ZcA(Zl  Zc)A2  Zl  Zc (3.43)Using the expressions for voltage (3.40) and current (3.43), the appropriate dissipation anddispersion errors, found from the ratio between the interpolated and ideal models, as definedin (3.35), can be computed. In order to obtain these voltage and current error ratios betweenthe interpolated and ideal models, we recognize that, if both models use the same injection,673.2. Dissipation and Dispersionthe respective V s and Is terms are the same and the ratios are found by simple division of theinterpolated by the ideal voltage or current expression, respectively:errorV = V minterpolatedV sidealVmidealV sinterpolated 1 = Ainterpolated[A2ideal +K+1K 1]Aideal[A2interpolated + K+1K 1] 1 (3.44a)errorI = IminterpolatedIsidealImidealIsinterpolated 1 = Ainterpolated[A2ideal  K+1K 1]Aideal[A2interpolated  K+1K 1] 1 (3.44b)where K = ZlZc.The A terms for the models are given by:AIdeal = e j! (3.45a)AInterpolated = e j! int + Re j! int(e j! t  1) (3.45b)where the  int factor can be substituted as  int =   R t (Appendix A.3.2) to write theexpressions in a single variable of  :AIdeal = e j! (3.46a)AInterpolated = e j! [ej!R t + Rej!R t(e j! t  1)] (3.46b)Substituting these A back with the appropriate terms for the ideal and interpolated linemodel:errorV = 1 + e2j! K+1K 1ej!R t[1 + R(e j! t  1) + e2j!(  R t)1+R(e j! t 1) K+1K 1] 1 (3.47a)errorI = 1 e2j! K+1K 1ej!R t[1 + R(e j! t  1) e2j!(  R t)1+R(e j! t 1) K+1K 1] 1 (3.47b)The above are the error expressions, in voltage and current, for the interpolated line model,compared to the ideal line model, with boundary conditions applied.In order to analyze the behavior of these expressions, the variables are now expressed indimensionless ratios to permit a more general evaluation of the results.R Interpolation factor R =(   t  b   tc). Relative to one  t and 0  R < 1 Reflection coefficient  =(K 1K+1), incorporates both Zl and Zc through K = ZlZcL Length of the line in expressed as # of simulation time steps L =(   t)(can be non-integer)683.2. Dissipation and DispersionUsing the dimensionless variables R,  , and L, we will now analyze the expressions (3.47)in terms of the above parameters for various cases.3.2.1 Influence of the Interpolation FactorThe interpolation factor R describes where in the last history interval the unknown value hasto be computed. Intuitively, the closer this value is to the known values on either side of theinterval, or the less difference between these known boundary values, the better the accuracyof the estimate will be. Half-way in the interval, R = 0:5, should then be the worst. Thus, thelocation of the interpolated value in the interval, as well as the simulation step size comparedto the highest frequency of interest, will have an influence on the accuracy of the simulation.In order to isolate the effect of the interpolation factor from the line length, and avoid in-fluences due to reflections from the model terminals, matched conditions,  = 0, were chosen.Using this condition, the line length  drops out of the equations as the terms that incorporateit are forced to zero. This is discussed in more detail in the next section, where the influence ofthe reflection coefficient  is studied. Finally, the simulation time step was evaluated at  t of1s, 0.5, and 0.1s.Using these conditions, the error expressions (3.47) are reduced to:errorV = 1ej!R t [1 + R(e j! t  1)]  1 (3.48a)errorI = 1ej!R t [1 + R(e j! t  1)]  1 (3.48b)Plotting these equations for varying R,  t and !, in voltage error only as both voltageerror and current error expressions are identical, Figures 3.5-3.10 result. It has to be noted thatby varying R, the line length is also changing, but this had no visible effect in the error plotsas the line length does not feature in the governing equations. Also, in the plots, only valuesof the interpolation factor up the R = 0:5 were shown due to this complete symmetry of theresults around this value, e.g.: the trace for R = 0:4 was identical to the trace for R = 0:6.This omission reduced visual clutter. In addition, this symmetry in the error also coincideswith the intuitive assessment of the influence of the location in the interpolation interval on theinterpolation error, given earlier in this section.From the plots and error equations, we can conclude, under matched conditions ( = 0): The position in the interpolated interval has an influence on the interpolation error. The worst-case interpolation error occurs at R = 0:5, half-way in the interpolated inter-val.693.2. Dissipation and Dispersion00.511.522.533.540 0.1 0.2 0.3 0.4 0.5Error MagnitudeFrequency (pu of dt)Interpolation Factor (R) Error Magnitude (dt = 1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(a) errorV and errorI Magnitude00. 0.02 0.04 0.06 0.08 0.1 0.12 0.14Error MagnitudeFrequency (pu of dt)Interpolation Factor (R) Error Magnitude (dt = 1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(b) errorV and errorI Magnitude (detail)Figure 3.5: Voltage and Current Error Magnitude Plots for Various values of R,  t = 1s703.2. Dissipation and Dispersion-80-70-60-50-40-30-20-1000 0.1 0.2 0.3 0.4 0.5Error Phase (degrees)Frequency (pu of dt)Interpolation Factor (R) Error Phase (dt = 1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(a) errorV and errorI Phase-14-12-10-8-6-4-200 0.02 0.04 0.06 0.08 0.1 0.12 0.14Error Phase (degrees)Frequency (pu of dt)Interpolation Factor (R) Error Phase (dt = 1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(b) errorV and errorI Phase (detail)Figure 3.6: Voltage and Current Error Phase Plots for Various values of R,  t = 1s713.2. Dissipation and Dispersion00.511.522.533.540 0.1 0.2 0.3 0.4 0.5Error MagnitudeFrequency (pu of dt)Interpolation Factor (R) Error Magnitude (dt = 0.2 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5 R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(a) errorV and errorI Magnitude00. 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16Error MagnitudeFrequency (pu of dt)Interpolation Factor (R) Error Magnitude (dt = 0.2 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(b) errorV and errorI Magnitude (detail)Figure 3.7: Voltage and Current Error Magnitude Plots for Various values of R,  t = 0.2s723.2. Dissipation and Dispersion-80-70-60-50-40-30-20-1000 0.1 0.2 0.3 0.4 0.5Error Phase (degrees)Frequency (pu of dt)Interpolation Factor (R) Error Phase (dt = 0.2 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(a) errorV and errorI Phase-14-12-10-8-6-4-200 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16Error Phase (degrees)Frequency (pu of dt)Interpolation Factor (R) Error Phase (dt = 0.2 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(b) errorV and errorI Phase (detail)Figure 3.8: Voltage and Current Error Phase Plots for Various values of R,  t = 0.2s733.2. Dissipation and Dispersion00.511.522.533.540 0.1 0.2 0.3 0.4 0.5Error MagnitudeFrequency (pu of dt)Interpolation Factor (R) Error Magnitude (dt = 0.1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5 R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(a) errorV and errorI Magnitude00. 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16Error MagnitudeFrequency (pu of dt)Interpolation Factor (R) Error Magnitude (dt = 0.1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(b) errorV and errorI Magnitude (detail)Figure 3.9: Voltage and Current Error Magnitude Plots for Various values of R,  t = 0.1s743.2. Dissipation and Dispersion-80-70-60-50-40-30-20-1000 0.1 0.2 0.3 0.4 0.5Error Phase (degrees)Frequency (pu of dt)Interpolation Factor (R) Error Phase (dt = 0.1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(a) errorV and errorI Phase-14-12-10-8-6-4-200 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16Error Phase (degrees)Frequency (pu of dt)Interpolation Factor (R) Error Phase (dt = 0.1 s) R=0 R=0.1 R=0.2 R=0.3 R=0.4 R=0.5(b) errorV and errorI Phase (detail)Figure 3.10: Voltage and Current Error Phase Plots for Various values of R,  t = 0.1s753.2. Dissipation and Dispersion The phase error is zero for R = 0, no interpolation, and R = 0:5. The length of the line, under matched conditions, has no impact on the interpolationerror. The error does depend on the line length under non-matched conditions. Interpolation results in a reduced usable bandwidth due to rapidly increasing errors. The shape of the error graphs remains unchanged with different simulation time steps t. There is a dependency on the simulation time step  t. The smaller the time step, thehigher the acceptable error cutoff frequency, thus the higher the usable bandwidth inabsolute terms. It remains, however, the same relative fraction of the total simulationbandwidth.3.2.2 Influence of the Reflection Coefficient and Line LengthThe matching conditions, given by the reflection coefficient  , have a significant influence onthemodelerrorduetoacumulativeeffectasreflectedenergypassesthroughthemodelmultipletimes. Also the line length, especially for boundary conditions close to shorted and open circuitconditions, results in difference in the error behavior, as the line becomes a resonator. In orderto evaluate these effects, the worst-case condition for the interpolation factor R = 0:5 waschosen. Again, for ease of interpretation of the plots, the simulation time step was chosen t = 1s.Choosing a fixed value for R does lock the possible line lengths  to multiples of half thesimulation time step  t, as the expressions R =   t  b   tc and L =   t must be satisfied.Thus, the allowable line lengths L for the plots are L = 1:5, 2:5, 3:5, ....The reflection coefficient  is bound between  1    1, thus between shorted circuit( =  1), over matched conditions ( = 0), to open circuit conditions ( = 1) [46, 71]. Assuch, a limit case will be encountered while evaluating the error expressions for matched con-ditions, as the expressions require the evaluation of 1 . However, this case is easily resolvedby inspecting (3.40) and (3.43), and observing that, under matched conditions where Zl = Zc,only the respective A terms remain, which are independent of the line and load impedances.Computing the error ratios from these terms, and comparing with (3.47), it is thus found that,under matched conditions: 1 = K+1K 1 = 0. Thus, the error equations reduce to (3.48) in thiscase.In general, however, the equations (3.47) cannot be further reduced. Thus, a mutual depen-dency of the line length, given by  , and the reflection coefficient, given by  exists, even with763.3. Direct Comparison Between the Ideal and Interpolated Modelswith a fixed value for R and  t. Plotting for varying line lengths L and matching conditions , Figures 3.11 - 3.18 result.The plots for voltage and current are nearly identical. The difference is the location ofthe various resonances, e.g.: the dip for  = 1 for the voltage error is identical to the dip for =  1 for the current error. This is expected behavior for transmission lines [7, 46].Also, the various resonances in the error plots correspond with the resonance frequenciesof the line sections. Indeed, a transmission line terminated in an open or short circuit (orconditionstendingtothesecases)willbehaveasaresonatoratthenominal  4 and  2 frequencies.At such a resonance, the magnitude of the input versus output tends to limit conditions in thelossless case, and small discrepancies, e.g.: due to model losses, quickly result in large relativeerrors. Still, the center frequencies at which the resonances occur is not affected by the modelsinaccuracies, as will be demonstrated in Section 3.3.From these figures and the error equations, under worst-case interpolation error conditionsR = 0:5, we can conclude that: The error due to the matching conditions, within a usable band sufficiently far fromthe Nyquist frequency, is bound, as the reflection coefficient itself is also bound to theinterval  1    1. For matched conditions, the error experiences a minimum. The line length has an influence on the error, especially for greater cases of mismatch, asresonances appear on the line. The line becomes a resonator near open and close-circuitboundary conditions and the many reflections accumulate error. Line length has no influence under matched boundary conditions. Interpolation results in a reduced usable bandwidth, which is further reduced due tocumulative error on reflected energy in case of mismatched boundary conditions.3.3 Direct Comparison Between the Ideal and InterpolatedModelsIn the previous sections, the normalised error ratios for the interpolated line model, comparedto the ideal model, were computed. During worst case conditions, with the interpolation factorhalfway in the time interval (R = 0:5), a simulation time step  t close to the traveling time ofthe line, and limit cases for the boundary conditions (open and shorted), the errors grew rapidlyinto usability.773.3. Direct Comparison Between the Ideal and Interpolated Models-1012340 0.1 0.2 0.3 0.4 0.5Voltage Error MagnitudeFrequency (pu of dt)Rho Voltage Error Magnitude (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorV Magnitude-0.3-0.25-0.2-0.15-0.1-0.0500.050.10 0.02 0.04 0.06 0.08 0.1 0.12 0.14Voltage Error MagnitudeFrequency (pu of dt)Rho Voltage Error Magnitude (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorV Magnitude (detail)Figure 3.11: Voltage Error Magnitude Plots for Various values of  and L = 1:5783.3. Direct Comparison Between the Ideal and Interpolated Models-1012340 0.1 0.2 0.3 0.4 0.5Voltage Error MagnitudeFrequency (pu of dt)Rho Voltage Error Magnitude (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorV Magnitude-1-0.8-0.6-0.4-0.200.20 0.02 0.04 0.06 0.08 0.1 0.12 0.14Voltage Error MagnitudeFrequency (pu of dt)Rho Voltage Error Magnitude (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorV Magnitude (detail)Figure 3.12: Voltage Error Magnitude Plots for Various values of  and L = 2:5793.3. Direct Comparison Between the Ideal and Interpolated Models-80-60-40-200204060800 0.1 0.2 0.3 0.4 0.5Voltage Error Phase (degrees)Frequency (pu of dt)Rho Voltage Error Phase (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorV Phase-35-30-25-20-15-10-500 0.02 0.04 0.06 0.08 0.1 0.12 0.14Error Phase (degrees)Frequency (pu of dt)Rho Voltage Error Phase (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorV Phase (detail)Figure 3.13: Voltage Error Phase Plots for Various values of  and L = 1:5803.3. Direct Comparison Between the Ideal and Interpolated Models-80-60-40-200204060800 0.1 0.2 0.3 0.4 0.5Voltage Error Phase (degrees)Frequency (pu of dt)Rho Voltage Error Phase (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorV Phase-80-60-40-200200 0.02 0.04 0.06 0.08 0.1 0.12 0.14Error Phase (degrees)Frequency (pu of dt)Rho Voltage Error Phase (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorV Phase (detail)Figure 3.14: Voltage Error Phase Plots for Various values of  and L = 2:5813.3. Direct Comparison Between the Ideal and Interpolated Models-1012340 0.1 0.2 0.3 0.4 0.5Current Error MagnitudeFrequency (pu of dt)Rho Current Error Magnitude (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorI Magnitude-0.3-0.25-0.2-0.15-0.1-0.0500.050.10 0.02 0.04 0.06 0.08 0.1 0.12 0.14Current Error MagnitudeFrequency (pu of dt)Rho Current Error Magnitude (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorI Magnitude (detail)Figure 3.15: Current Error Magnitude Plots for Various values of  and L = 1:5823.3. Direct Comparison Between the Ideal and Interpolated Models-1012340 0.1 0.2 0.3 0.4 0.5Current Error MagnitudeFrequency (pu of dt)Rho Current Error Magnitude (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorI Magnitude-1-0.8-0.6-0.4-0.200.20 0.02 0.04 0.06 0.08 0.1 0.12 0.14Current Error MagnitudeFrequency (pu of dt)Rho Current Error Magnitude (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorI Magnitude (detail)Figure 3.16: Current Error Magnitude Plots for Various values of  and L = 2:5833.3. Direct Comparison Between the Ideal and Interpolated Models-80-60-40-200204060800 0.1 0.2 0.3 0.4 0.5Current Error Phase (degrees)Frequency (pu of dt)Rho Current Error Phase (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorI Phase-35-30-25-20-15-10-500 0.02 0.04 0.06 0.08 0.1 0.12 0.14Current Error Phase (degrees)Frequency (pu of dt)Rho Current Error Phase (L = 1.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorI Phase (detail)Figure 3.17: Current Error Phase Plots for Various values of  and L = 1:5843.3. Direct Comparison Between the Ideal and Interpolated Models-80-60-40-200204060800 0.1 0.2 0.3 0.4 0.5Current Error Phase (degrees)Frequency (pu of dt)Rho Current Error Phase (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(a) errorI Phase-80-60-40-200200 0.02 0.04 0.06 0.08 0.1 0.12 0.14Current Error Phase (degrees)Frequency (pu of dt)Rho Current Error Phase (L = 2.5, R = 0.5, dt = 1 s) rho=-1 rho=-0.75 rho=-0.5 rho=-0.25 rho=0 rho=0.25 rho=0.5 rho=0.75 rho=1(b) errorI Phase (detail)Figure 3.18: Current Error Phase Plots for Various values of  and L = 2:5853.3. Direct Comparison Between the Ideal and Interpolated ModelsIt may now be of interest to compare the actual model responses under these adverse con-ditions directly in a graphical way as opposed to the numerical ratio of the responses [19]. Thisway, the nature and practical limitations of the inaccuracies can become more clear. In fact, ifthe peak value of the resonances is of no interest, the accuracy remains usable far longer thanthe previous results would let one to believe. Also, more suitable values for the simulation timestep  t will be shown for comparison.To perform the comparison in frequency domain, to obtain the frequency response of asection of line under limit conditions of open and shorted circuit, we first must derive theappropriate equations. We will derive input-output relations for the models. Here, these aretrue transfer functions, not weighted errors as used in the previous analysis.For the open line, it is more natural to compute the input to output voltage ratio, as thisavoid limit conditions due to the zero current flow at the open end of the line. Starting fromgeneral line equations in frequency domain (3.37) and applying boundary conditions Im = 0:V mV s =2AA2 + 1 (3.49)For the shorted line, the input to output current ratio is used, as this avoids the limit condi-tion due to the zero voltage at the shorted end of the line. Starting from general line equationsin frequency domain (3.37) and applying boundary conditions V m = 0:ImIs =  2AA2 + 1 (3.50)To find the specific equations for the ideal and interpolated line, we now substitute theappropriate A terms from (3.46) and, after some manipulation, find for the ideal line model,for open and shorted cases, respectively:Open V mVs= 2e j! e 2j! + 1 (3.51a)Shorted ImIs=  2e j! e 2j! + 1 (3.51b)For the interpolated line model the corresponding equations, for open and shorted cases,respectively, are:Open V mVs= 2e j! [ej!R t + Rej!R t(e j! t  1)]e 2j! [ej!R t + Rej!R t (e J! t  1)]2 + 1 (3.52a)Shorted ImIs=  2e j! [ej!R t + Rej!R t(e j! t  1)]e 2j! [ej!R t + Rej!R t (e J! t  1)]2 + 1 (3.52b)863.4. EMTP 1/10 Rule of Thumb for Interpolated LinesThese equations are now used to evaluate the frequency domain behavior of the terminatedline models. We will only evaluate the voltage equations, as the current equations have thesame response, with only a sign change.The influence of the interpolation factor R and the simulation time step  t is studied inFigures 3.19-3.21. The traveling time of the line in all cases was kept constant, within thetemporal resolution afforded by the simulation time step and the variation added to the linelength due to the interpolated sections.It can be seen that, when the simulation time step  t is close to the traveling time of theline  (L = 1, thus the line is only one history deep), the worst-case condition evaluated in theprevious section result in significant errors. However, in the plots, it also becomes clear thatthe phase change at the resonance of the open line section does happen at the right frequency,and so does the resonant peak, when compared with the ideal line, for all values of R. Thus,the errors are magnitude errors.It must also be noted that the various peaks for different R are shifted with respect to eachother. This is correct behavior. The influence of the interpolated interval is large, as in Figure3.19, where the line itself is only one time step deep and, when R = 0:5, the interpolatedinterval adds 50% to the line length of L = 1, resulting in a line that is in fact  = 1:5s, with t = 1s, as opposed to the nominal  = 1s. Longer line lengths with increasing R cause the 4 resonance to shift to lower frequencies.When the simulation time step becomes smaller, as in Figures 3.20 and 3.21, the accuracyof the model quickly increases, as the simulation bandwidth goes up by a factor 5 and 10,respectively. From the plots of previous section, it was seen that the usable bandwidth wasroughly 1/5th to 1/10th of the bandwidth. By increasing the bandwidth, the model now pro-duces good results. Also, since the interpolated interval is now much smaller compared to thelength of the line, the total traveling times for the various cases of R are now closer to eachother and all resonances are thus closer to each other as well.3.4 EMTP 1/10 Rule of Thumb for Interpolated LinesWhen interpolated transmission lines are used in EMTP simulations, it is customary to ensurethat the simulation time step  t is five to ten times smaller than the traveling time  of theline [20, 52]. This is in addition to the usual requirements on bandwidth, as given by theNyquist theorem [64]. In practice, it is advisable to use only one fifth to one tenth of theavailable bandwidth for accuracy, as opposed to half. Applying both considerations results ina generally acceptable error.From the preceding study, similar conclusions are reached. In fact, using the error expres-873.4. EMTP 1/10 Rule of Thumb for Interpolated Lines11.522.533.544.550 0.05 0.1 0.15 0.2 0.25 0.3MagnitudeFrequency (Hz)Open Line Voltage Ratio Magnitude (L = 1, dt = 1 s) ideal R=0 inter R=0 ideal R=0.125 inter R=0.125 ideal R=0.25 inter R=0.25 ideal R=0.375 inter R=0.375 ideal R=0.5 inter R=0.5 ideal R=0.625 inter R=0.625 ideal R=0.75 inter R=0.75 ideal R=0.875 inter R=0.875(a) Voltage Magnitude-150-100-5000 0.05 0.1 0.15 0.2 0.25 0.3Phase (degrees)Frequency (Hz)Open Line Voltage Ratio Phase (L = 1, dt = 1 s)ideal R=0inter R=0ideal R=0.125inter R=0.125ideal R=0.25inter R=0.25ideal R=0.375inter R=0.375ideal R=0.5inter R=0.5ideal R=0.625inter R=0.625ideal R=0.75inter R=0.75ideal R=0.875inter R=0.875(b) Voltage PhaseFigure 3.19: Frequency Response of Open Line for Various values of R and L = 1,  t = 1s883.4. EMTP 1/10 Rule of Thumb for Interpolated Lines11.522.533.544.550 0.05 0.1 0.15 0.2 0.25 0.3MagnitudeFrequency (Hz)Open Line Voltage Ratio Magnitude (L = 5, dt = 0.2 s) ideal R=0 inter R=0 ideal R=0.125 inter R=0.125 ideal R=0.25 inter R=0.25 ideal R=0.375 inter R=0.375 ideal R=0.5 inter R=0.5 ideal R=0.625 inter R=0.625 ideal R=0.75 inter R=0.75 ideal R=0.875 inter R=0.875(a) Voltage Magnitude-150-100-5000 0.05 0.1 0.15 0.2 0.25 0.3Phase (degrees)Frequency (Hz)Open Line Voltage Ratio Phase (L = 5, dt = 0.2 s)ideal R=0inter R=0ideal R=0.125inter R=0.125ideal R=0.25inter R=0.25ideal R=0.375inter R=0.375ideal R=0.5inter R=0.5ideal R=0.625inter R=0.625ideal R=0.75inter R=0.75ideal R=0.875inter R=0.875(b) Voltage PhaseFigure 3.20: Frequency Response of Open Line for Various values of R and L = 5,  t = 0:2s893.4. EMTP 1/10 Rule of Thumb for Interpolated Lines11.522.533.544.550 0.05 0.1 0.15 0.2 0.25 0.3MagnitudeFrequency (Hz)Open Line Voltage Ratio Magnitude (L = 10, dt = 0.1 s) ideal R=0 inter R=0 ideal R=0.125 inter R=0.125 ideal R=0.25 inter R=0.25 ideal R=0.375 inter R=0.375 ideal R=0.5 inter R=0.5 ideal R=0.625 inter R=0.625 ideal R=0.75 inter R=0.75 ideal R=0.875 inter R=0.875(a) Voltage Magnitude-150-100-5000 0.05 0.1 0.15 0.2 0.25 0.3Phase (degrees)Frequency (Hz)Open Line Voltage Ratio Phase (L = 10, dt = 0.1 s)ideal R=0inter R=0ideal R=0.125inter R=0.125ideal R=0.25inter R=0.25ideal R=0.375inter R=0.375ideal R=0.5inter R=0.5ideal R=0.625inter R=0.625ideal R=0.75inter R=0.75ideal R=0.875inter R=0.875(b) Voltage PhaseFigure3.21: FrequencyResponseofOpenLineforVariousvaluesofRandL = 10,  t = 0:1s903.5. Conclusionssions derived above, it is now possible (in the cases for which the expressions are valid), tocompute the error for a given case and adjust the simulation parameters to achieve an accept-able result. Also, depending on the value of the interpolation factor R, the ratio of  t to  maybe relaxed. This can be seen in Figures 3.5-3.10 and 3.19-3.21 of the preceding analysis.However, also the matching conditions play an important role, as they amplify the incurrederrors due to the cumulative effect of multiple reflections, and thus passes through the model.This is shown in Figures 3.11-3.18.Thus, the selection of the simulation time step, especially during severe mis-matched con-ditionsandsimulationparameterswheretheinterpolationfactorRiscloseto0.5, requiresgreatcare, as the bandwidth over which the simulation gives acceptable results may be severely re-stricted. Conversely, in simulations where the lines experience relatively matched conditionsand R is close to zero or unity, the computation may be sped-up as the larger allowable timestep  t results in less steps to be computed for a given simulation time, required bandwidthpermitting. Equations (3.47) can help in the selection process, at least for cases where the sec-tion of line is driven from an ideal source. The same procedure of derivation can, however, beused to derive a more generic equation for a source with an impedance.3.5 ConclusionsIn this chapter, the behavior of the transmission line models that make-up TINA and the EMTPwas studied. It was found that both the ideal and interpolated line models are stable. Theinterpolated model was shown to be dissipative, and the requirement for the EMTP 110 rule wasshown to maintain accuracy when interpolation is to be used.The resulting error expressions can, under applicable conditions, be used to estimate theerror incurred, and thus choose the simulation parameters for a suitable balance between per-formance and accuracy. It was found that the location of the interpolation in the interval, aswell as the ratio of the time step to the traveling time of the line, and the termination conditionshave a significant influence on the performance of the interpolated model. These relations, asapplicable to the interpolated line model, have not yet been discussed in the literature, and mayallow the simulation error constraints to be relaxed in certain cases.Future work would be the extension of the discussion of the interpolated line to generictermination conditions, so a source with an impedance can be considered, as well as a furthergeneralization of the stability proof for this model.91Chapter 4Fractional Sub-Area LatencyThe law of the constant velocity of light in empty space, which has been confirmedby the development of electro-dynamics and optics, and the equal legitimacy of allinertial systems (special principle of relativity), which was proved in a particularlyincisive manner by Michelson’s famous experiment, between them made it neces-sary, to begin with, that the concept of time should be made relative, each inertialsystem being given its own special time. ...According to the special theory of rela-tivity, spatial co-ordinates and time still have an absolute character in so far as theyare directly measurable by stationary clocks and bodies. But they are relative in sofar as they depend on the state of motion of the selected inertial system.Albert EinsteinIn this chapter, the concept of fractional, sub-area latency is introduced. It is an extension andgeneralization of previous work on latency and temporal multi-grid techniques, allowing theratio between the simulation time step and that of the area where latency is applied to be anon-integer multiple. Also, current use of latency and temporal multi-grid techniques in TLMand finite-difference methods has been limited to a single, integer, latency ratio for the wholesystem, or integer ratios for the sub-grids. Here, the use of multiple different, non-integer ra-tios for different parts of the simulation is developed, which allow both speed improvementsin the solution and offer a new way of maintaining synchronicity in the simulation. The is-sues with non-integer multi grids, the asynchronous nature of the simulation and the need forextrapolation, and thus prediction, has been solved within the framework of the TINA method.The contributions are thus the development of fractional latency, which requires a differentapproach compared to traditional latency and temporal multi-grid techniques, and the use ofthis technique, in compatible, time-decoupled solution methods such as TINA and TLM, toallow different parts of a simulation to operate at different local simulation time steps that neednot be integer multiples of each other. As a result, the various sub-areas operate asynchronousfrom one another. This yields important performance gains, depending on the particular simu-lation case. It is demonstrated how the asynchronous simulation can be re-synchronized and, intime-decoupledmethods, howtheextrapolationnecessaryforfractionallatencycanbeavoided.The fractional latency technique also offers a new way of maintaining synchronicity in a simu-924.1. The Need for Latency Techniques in TINAlation between the various sub-area in TLM-style methods, such as TINA, each with their ownsimulation time step, through its use of interpolation in time. This removes the need for localsynchronicity methods, such as stub matching and temporal interpolation in the transmissionlines themselves.4.1 The Need for Latency Techniques in TINALatency techniques allow different parts of a simulation to operate at a different, local simula-tiontimestep. This abilityisimportant, asinmanycases, portionsofacomputationmayberunslower without sacrificing accuracy, and thus important computational gains can be achieved.Therequirementtohavemultipletimestepsinasimulationfollowsnaturallywhendifferentcell sizes, and different materials, are used in a solution mesh. In electro-magnetics, mostmaterials have wave propagation speeds that are relatively close to each other, typically withinthe same order of magnitude. However, when the FDTD and TLM methods were being usedfor non-EM problems, order of magnitude variations in wave speed readily occur. Acoustics isone such field. Another issue is the ratio between the wave length and the structure in whichthe energy must propagate.Architecturalacousticsinparticularpresentsgreatdifficultiestodiscretized-spacemethods,due to both the wave propagation speed variation between air and building materials, whichcan easily span over an order of magnitude, and the large solution space, compared to the wavelength, resulting in lengthy computations.Other methods are then commonly used, such as ray-tracing. This is a method that orig-inated in computer graphics to compute three-dimensional scenes where virtual light beamsare reflected and scattered off obstacles and finally observed at a receiver. Although compu-tationally efficient for these applications, ray-tracing methods do not give phase information.However, recent work on a ray-tracing method that preserves phase information is under earlydevelopment [10]. Discretized-space methods based on wave propagation naturally yield phaseinformation and are thus of interest, and are actively developed, by the acoustics community,e.g.: [66, 67].To clarify the situation, the following example may be considered: suppose we wish tostudy an acoustical system composed out of air and metal. The wave speed for air is roughly340 m/s, while the wave speed for the metal can easily be 4000 m/s. Thus, an order of magni-tude difference. It is clear that, given a minimum requirement of ten mesh cells per wavelengthfor accuracy, the air portion of the simulation, which has a short wavelength compared to themetal, will require a much finer spatial discretization than the metal.If the same spatial discretization grid is used throughout the simulation, as is typically the934.1. The Need for Latency Techniques in TINAcase in less developed solvers, it becomes clear that the mesh cell size is determined by thearea requiring the finest discretization. This means that large areas of the simulation may beoversampled, and thus significant computational resources squandered for unneeded accuracy[15, 16, 98]. The situation becomes worse when the temporal discretization requirements ofover-sampled areas are considered. The result is a simulation with unreasonably high band-width for most areas, just so sufficient accuracy may be maintained in one part. Sections 2.2.3and 4.4.6 treat this in more detail as error criteria for the TINA method are discussed.However, if one could operate these computationally unnecessary expensive, over-sampledareas at a more suitable, slower simulation time step, a significant portion of the lost computa-tion can be recovered while sufficient accuracy can be maintained.Typically, multi-grid techniques [15, 39, 98] would then be used in FDTD and TLM meth-ods to allow for a suitable different spatial discretization within each simulation area and main-tain accuracy. Spatial multi-grid techniques, however, are complex to implement due to thegrid mis-match on the interface between the various different grid size areas. This is espe-cially the case when the grids sizes are not integer multiples of each other, and partial celloverlaps exist. Hence, most multi-grid methods are restricted to integer multiples so that nopartial overlaps exist, or otherwise unstructured grids are used, such as typically done in finiteelement methods, which adds significant computation to TLM-style methods [16]. Temporalmulti-grid techniques are typically easier to implement and mis-matches are treated using in-terpolation [98], but havenot been extendedto TLM, or fractional ratios due to synchronizationand extrapolation requirements, and thus stability and accuracy issues.In this chapter, we will treat fractional sub-area latency as a multi-grid technique in timeto be used within the time-decoupled mesh that makes-up the TINA method. By allowingfractional time-step ratios between sub-areas, each area in the TINA mesh can be operated at alocal time step which is an integer multiple of the local transmission line , and thus not requireinternal interpolation, as was shown earlier in this thesis to maintain synchronicity. The useof fractional time steps, in itself, results in significant performance gains as the interpolationwill only be on the interface between areas to maintain synchronicity, which will be explainedfurther in this chapter, and no longer in each cell in the latency area. Accuracy of the compu-tation will increase due to this, as each interpolated cell no longer forms a parasitic, cascadedlow-pass filter. Using fractional latency, the interpolation errors are only incurred within thelatency cells on an area boundary, thus only one instance of interpolation per boundary cell asopposed to thousands within an area, allowing more relaxed simulation parameters, and thusfurther reduced computational time.In a sense, multi-grid methods in space and time are an expression of the space/time dualityprinciple [22, 45]. We can either use an irregular grid in space, or an irregular “grid” in time.944.2. BackgroundMulti-grid techniques are an expression of the first, and latency is an expression of the latter. Inthe TINA computation, irregularity in time is used throughout and synchronicity is maintainedthough temporal interpolation. Temporal techniques are easier to handle in a digital computer,as this can be achieved directly by changing the depth of the history storage. In TINA, the spa-tial parameter is kept constant while the temporal one is allowed to be irregular. In most TLMand finite-difference methods, the converse applies. Ideally, a combination of both irregularityin space and time should be used to match a material area to the simulation with the desiresspatio-temporal discretization.The literature describing previous uses of multi-grid and latency methods will now be dis-cussed, followed by the development of fractional latency as an extension and generalization ofthe method. It will be shown that, for time-decoupled meshes, the prediction required in frac-tional latency may be avoided and replaced with a look-ahead in the transmission line historiesthat connect the various TINA cells. This approach effectively removes the prediction from thetechnique and replaces it by a know value. Also, a method will be introduced to re-synchronizethe now asynchronous, event-driven simulation when output must be generated.Finally, the implementation of the fractional latency solution in EMTP and TINA is dis-cussed and the simulation results compared to non-latency techniques.4.2 BackgroundWhen discussing the latency technique, it must first be noted that different fields of study havedeveloped different terminologies for what is, essentially, the same method. In electrical circuitsimulation, incorporated in programs such as EMTP for power systems, where the systemsare one-dimensional electrical circuits, or more correctly, graphs, the technique is know aslatency, and applies to either elements of the electrical circuit being modeled, or entire sub-areas. Latency is the term we will use throughout this thesis.In TLM and finite-difference time-domain methods, the technique is known as multi-gridin time, although sometimes the term sub-gridding is used as well, depending on the context.Throughout the discussion it must also be noted that TLM and FDTD formulations of wavepropagation are closely related, and thus have significant overlap. In fact, both methods havebeen proven to be equivalent formulations under certain conditions [9, 11].Due to the parallel, but mainly independent development, of the methods, the discussionwill be split in two halves. First, we will discuss the TLM and FDTD methods, which have alonger history regarding these methods. Then, EMTP latency will be discussed, as this formedthe basis for the development of the fractional sub-area latency method.954.2. Background4.2.1 Multi-Grid Methods in TLM and FDTDSoonafterthedevelopmentofthefinite-differencetime-domainmethodbyYee[96]forelectro-magnetics in 1966, it became clear that in cases where small features in large structures hadto be studied, such as a wire in a larger cavity, the computational requirements would quicklybecome excessive. This is due to the large difference of scale between the structures, and thuslarge, finely sub-divided simulation mesh over a large simulation space. Similar issues werefound in the transmission-line modeling method proposed by Johns [31] in 1971, and is in factan issue common to all regular-grid discretized-space methods.In FDTD and TLM, methods were consequently proposed to embed thin wires and narrowslots right into the cell description, or other means of obtaining local solutions for such featuressmaller than the spatial discretization size. The concept was first proposed in [28], then laterextended to arbitrarily oriented wires [21, 41] and eventually to multi-wire looms [5, 92]. Inparallel, methods to model the thin wires as an equivalent circuit, and take into account theinteraction with the field in the mesh allowed the wires to be placed between the cells [93],while another description placed them in the center of the cells [68].Although computationally efficient, as the mesh can be kept at the desired coarseness forfree-space, such techniques have many drawbacks, as the detail in the local features of thestructure, and the corresponding local field, is lost. Allowable shapes and sizes of the featuresare restricted. Also, the spatial location of the wire is restricted to the center of the cell, or tothe boundary between the cell. This places restrictions on the spatial accuracy of the resultingsimulation. [15, 68, 98].Consequently, work was undertaken to allow the simulation mesh to have different cellsizes so it could be locally matched to the required feature size. For example, a thin 1 mm wireinteracting with an electro-magnetic wave of several cm length could be locally surrounded bya fine, sub-mm mesh while free-space could be modeled with much larger cells. Such methodsare known as multi-grid, or variable mesh algorithms. In 1981, one of the earliest methods,a two-stage solution to allow part of the mesh to operate at a local, finer discretization wasdeveloped by Kunz and Simpson [39]. The method would first calculate the fields in the coarseregion, and would then use spatial and temporal interpolation to obtain the tangential fields onthe boundary with the finer grid. Using these results, the finer part of the grid could then becomputed. This was later extended to allow for variable simulation step sizes [98], allowingfor each region in the mesh to have a different, and appropriate simulation step. Both the spaceand time discretization in the mesh were integer multiples of each other.In TLM, similar developments took place and mesh refinement techniques, both local(multi-grid mesh) and global (variable or hybrid mesh) were developed to operate in spaceand/or time [69, 74, 94, 95]. Of special mention is the technique pioneered by Saguet and Pic964.2. Background(a) Multi Grid Mesh (b) Variable, or Hybrid, MeshFigure 4.1: Mesh Types[74], called variable of hybrid mesh. Illustrated in Figure 4.1(b), it does not use a purely localmesh refinement. The advantage is that a one-to-one spatial relationship is established betweenthe cells, avoiding much of the complexity of the multi grid methods where the energy distribu-tion on the boundaries between different mesh sizes must be handled [15, 26, 27], illustrated inFigure 4.1(a). Again, integer multiples for both spatial and temporal grid ratios are maintainedto aid synchronicity between the various regions [14].From the literature, we find that the shown spatial and temporal multi-grid techniques haveonly been achieved with integer multiples between time steps. In the development of the frac-tional latency method, which is a multi-grid technique in time, it will become clear that thedifficulty with non-integer temporal multiples is that the simulation become asynchronous andthat they require the use of extrapolation. This makes the method potentially unstable. Also,the potential need for iterative methods to improve the accuracy of the extrapolation predictionwill increase the numerical burden. However, due to the formulation of the TINA method, theextrapolation issues can be readily overcome by the proposed fractional latency technique, aswill be demonstrated. It will also be shown how the asynchronous nature of the simulation maybe maintained and synchronized to an observer clock so synchronized output can be generated.4.2.2 Latency in EMTPWithin the field of power systems analysis, there is no spatial discretization, as the compu-tations are circuit simulations. They take the form of graphs, not discretized space. Thus,multi-grid methods are necessarily limited to the time dimension. The use of different time974.2. BackgroundUthevaZtheva ZthevbUthevbSlowsub-system Fastsub-systemFigure 4.2: Two Connected Th´evenin Equivalents in Different Sub-Systemsk∆ta (k + 1)∆ tak∆ta + ∆tb k∆ta + 2∆tbFigure 4.3: Integer Latency Solution Time-Linesteps for different parts of a simulation is referred to as latency.The technique is based on earlier work on multi-rate simulation programs for very large-scale integrated circuits (VLSI), using the waveform relaxation method [42, 75, 97]. Later,the use of latency in power systems simulations, using EMTP, was studied [78] and initialwork on numerical stability and accuracy of the method was shown in [43]. The methodwas further developed in 2002 by Moreira [59, 60] in the context of the Multi-Area Th´eveninEquivalents (MATE) technique [49] that allows for efficient parallelization [87, 88] of powersystem simulations.Within the context of EMTP, the latency technique shows the interesting property that, forinteger multiples of the involved time steps, only interpolation of the slower time step needsto be used to find the intermediate fast ones. This is because we know the next step needed tointerpolate with, as in EMTP the history sources compute their next result using only past andcurrent information [20, 60].To illustrate the concept, let us represent two parts of a system, each by their own Th´eveninequivalent. This representation has the advantage that it is a universal description for anysystem at a given simulation time and used as the core sub-system interconnection concept inMATE and TINA, as well in the transmission line models used in this thesis. In Figure 4.2, the984.2. BackgroundUthev aTimek∆ta k∆ta + ∆tb k∆ta + 2∆ tb (k + 1)∆ taexactexactinterp olatedFigure 4.4: Integer Latency History Interpolationsituation is illustrated. The left-hand side of the figure is the slow sub-system, while the right-hand side is the fast sub-system. Thus, each sub-system can be allowed to run at a differentsimulation time step, where the slow sub-system runs at a larger local time step than the fastone, and maintain local accuracy. When we take the fast system three times faster than the slowsystem, the corresponding simulation time line is shown in Figure 4.3.Thus, as time advances through the simulation, we observe that the sub-systems are solvedin sequence. Both are solved at time k ta, then the fast sub-system is solved twice, followedby another joint solution. Thus, for the fast system, there are no corresponding values fromthe slow system to compute its internal results with and vice-versa. However, this is not so.If the simulation is arranged so that the slow system is solved first, either from zero or knowninitial conditions, an exact value for its solution can be found, as EMTP solutions use currentand present data to compute the next step. Assuming that the slow system varies little withineach step, which is a key requirement for latency techniques to work, the next value for theslow system can be found. Then, using interpolation, the intermediate values that are neededby the fast system can be estimated. This means that the interpolation happens between twoexact values, and no prediction is used. Also, the system remains in synchronism at the rate ofthe slowest sub-system. The situation is outlined in Figure 4.4. In EMTP latency techniques,linear interpolation is typically used.The temporal interpolation used in EMTP latency techniques is typical of other forms oftemporal interpolation for discrete-time, time-domain solutions discussed in the literature con-cerning TLM and FDTD techniques. It is based on integer multiples of the respective time994.3. Development of Fractional Latencysteps, which allow the use of interpolation between known values. Since the time steps areinteger multiples of each other, the time steps always align, meaning the fast system, after afixed number of solutions, coincides with the slow one. We will now show why only integermultiples have been used so far in the development of latency techniques.4.3 Development of Fractional LatencyTaking a step back from latency, in a practical simulation, the different materials do not havesimulation time steps that are clean integer multiples of one another. For this reason, the TINAmethod makes extensive use of temporal interpolation to achieve synchronicity to a commontime step throughout the simulation. That implies that entire areas of a certain medium requireinterpolation in all constituent cells to match them to this time step. Thus, a material areabecomes a system of cascaded filters, due to the low-pass effect of the interpolation in eachcell, and the distortion increases. This happens in the stub-matching method used in TLM aswell [18], which adds a matching transmission-line stub to each cell.If it were possible to allow each material area to operate at a local time step that is aninteger multiple of the transmission time  of the individual lines that make-up the materialcells, then the interpolation in the material cells could be omitted. Aside from speed-gains, thisalso removes an important and cumulative source of error.Returning to latency techniques, the classical, integer multiple latency cannot help to com-pletely remove the interpolation in the cells, but it can reduce the memory use and computa-tional burden as the time steps in the local areas can be chosen closer to optimum values. Thisis especially the case when a constant grid size is used and there are large variations in wavespeed in the simulation. If it were possible to use non-integer ratios, then also the interpolationwithin each cell could be removed as the local time step could be exactly matched to the localtransmission time of the lines.This is what fractional sub-area latency allows: each sub-area, or material, can operate ata convenient, local simulation time step that is a precise, integer multiple of the local trans-mission time  . There is still need for interpolation to implement the fractional latency itself,but only on the boundaries between the different material regions. Thus, fractional latencymaintains the advantages of integer latency while also allowing synchronicity to be maintainedat a much reduced numerical and memory cost, compared to interpolation in every cell. Thismay result in better accuracy, as the interpolation is now reduced to only the boundary cells asopposed to the entire material area. Thus, fractional latency combines the advantages of integerlatency and temporal interpolation as a means to synchronicity, while also mitigating some ofthe biggest drawbacks of interpolation for synchronicity.1004.3. Development of Fractional Latency∆tA ∆tBK MethA ethBZthA ZthBZlK ZlMelK elMFigure 4.5: Fractional Latency Bi-Directional Cell4.3.1 Equations and Parameters for Fractional LatencyFractional latency, as it is considered here, is conceived as a special cell that is introducedbetween two parts of a system. Since connectivity in TINA is entirely based on Th´eveninequivalents, this is also the concept that will be used for the latency cell, to allow for easyinsertion into the simulation grid.The advantage of using Th´evenin equivalents is that they offer a universal abstraction forany circuit in a time-domain, time-marching solution scheme. Thus, the bi-directional latencycell can be described, and connected, as shown in Figure 4.5. It is essentially two Th´eveninequivalents, where each represents a time-interpolated version of the circuit connected to theother side of the cell.The governing equations, which are temporal interpolations of the Th´evenin equivalentcircuit components are now given, for the current time step tA or tB as:elK(tA) = ethB(tB +  tB) + RK [ethB(tB) ethB(tB +  tB)] (4.1a)elM(tB) = ethA(tA +  tA) + RM [ethA(tA) ethA(tA +  tA)] (4.1b)ZlK(tA) = ZthB(tB +  tB) + RK [ZthB(tB) ZthB(tB +  tB)] (4.1c)ZlM(tB) = ZthA(tA +  tA) + RM [ZthA(tA) ZthA(tA +  tA)] (4.1d)The interpolation factors, illustrated in Figure 4.6, are computed as:RK = (tB +  tB) tA tB(4.2a)RM = (tA +  tA) tB tA(4.2b)In Figure 4.6, to compute tA +  tA in a sub-area operating at a local time step of  tA, we1014.3. Development of Fractional Latencyrequire a solution at time tA. We have the solution at tA   tA, but to compute the solutionat tA, we require information from the connecting sub-area, which operates at  tB. Since thissub-area does not have this result available, the neighbor values needs to be interpolated sothey can be used in the sub-area operating at  tA at time step tA to find tA+ tA. Using linearinterpolation, the interpolation factor RK is found. The gray dot in the figure represents theinterpolated value for the neighboring sub-area operating at  tB, interpolated in time to matchthe time step of the area operating at  tA.As opposed to the interpolated transmission lines, which were studied in Chapter 3 and Ap-tAtA−∆tA tBtA+∆tAtB +∆tBtHRHNeedto findthisUsingthisFigure 4.6: Fractional Latency Interpolation Intervalt0∆ t10∆ t21∆ t1 2∆ t1 3∆ t1 4∆ t1 5∆ t112 ms33 ms 1∆ t2 2∆ t2Figure 4.7: Fractional Latency Solution Issue1024.3. Development of Fractional LatencyEvent  t1 (12 ms)  t2 (33 ms)  t3 (44 ms)  t4 (100 ms)0 0 0 0 01 122 243 334 365 446 487 608 669 7210 8411 8812 9613 9914 10015 10816 12017 132 132 13218 14419 15620 16521 16822 17623 18024 19225 1982627 20028 204Table 4.1: Numerical Fractional Latency Simulation Event Table1034.3. Development of Fractional Latencypendix A.3, the interpolation factor R is no longer a constant factor, but varies. The reason forthis is illustrated in Figure 4.7 and Table 4.1, which shows a numerical example of simulationevents for two and four latency areas, respectively. It can be seen that the location of the valueto be interpolated in an interval changes with each time step. Thus, different interpolation fac-tors must be used for each time step. The whole simulation, using fractional latency, operatesasynchronous. Data is only synchronized on the boundaries between two areas using the la-tency cells, and for output processing, as discussed in Section 4.3.6. Temporal interpolation isused each time to achieve synchronicity.4.3.2 The Need for ExtrapolationFrom the above equations, Table 4.1, and Figure 4.7, the main issues with fractional latency,at least in a generic EMTP context, will become clear: there is a chicken and egg problem.Throughout the discussion, keep in mind that EMTP solutions are found in two steps. First,using current and past data, the next, future step value for the history sources is computed [20].During this step, the components in the system are not changed. When all the internal historiesare solved, the system itself is computed using these new histories to obtain the completesolution for the next time step. Thus, there is a difference between knowing the values of thehistory sources at the next time step and knowing the system solution at the next time step.The cases discussed below are based on a simplified version of the numerical example given inTable 4.1.In the following discussion, a simple case consisting of two sub-areas is considered. Thelocal time steps are  t1 = 12ms and  t2 = 22ms. The ratio between the time steps isthus:  t1 t2 = 12ms33ms = 0:6363:::. Fractional latency applies. Each sub-system has a number ofinternal components, with their associated history sources. However, that internal complexityis abstracted to a Th´evenin equivalent on each connection port [30]. Thus, the whole circuitis similar to that shown in Figure 4.5, where two sub-areas are joined using a universal, bi-directional fractional latency cell. As such, the overall circuit must still be solved together inorder to reach a solution, as there is no decoupling in the solution. This last point is at the coreof the issue with fractional latency ratios.Starting at t = 0, values for both the  t1 = 12ms and  t2 = 22ms sub-systems can becomputed through, for example, zero initial conditions. Since the entire system is defined, theEMTP solution algorithm allows us the compute the updated history sources for the next timestep based on the known boundary conditions presented to each sub-area through the Th´eveninequivalents. Using the updated histories, the whole system may be solved and new values forthe Th´evenin equivalents for the next time step found. Thus, at t = 0, we can compute system1044.3. Development of Fractional Latencysolutions for 1 t1 and 1 t2.Now, since the sub-system operating at  t2 hardly changes relative to the sub-system oper-ating at  t1, which is the main condition for latency, we may use latency to compute the nextfew values for the faster sub-system using interpolation of the Th´evenin equivalent presentedby the slower sub-system. Thus, a local system solution for 2 t1 can be found using the in-terpolated Th´evenin equivalent presented by the slower sub-system. Also, the Th´evenin equiv-alents of the faster sub-system can be updated, as the interpolated Th´evenins for the slowersub-system are available. Thus, computation is saved as during the computations the slowersub-system needs not be solved for, only its Th´evenin equivalents must be interpolated.The solution will now fail as the slow sub-system needs to be computed at t = 1 t2. Tocompute the new history values for this sub-area and solve the system, we require the Th´eveninequivalent obtained from the fast sub-system at this time step. However, the solution of thisfaster sub-system has only progressed until t = 2 t1. Although we could compute the nextvalue of the internal history sources of this faster sub-system, its Th´evenin equivalent cannotbe found, as this requires a full system solution including the slow sub-system. However, sincethe internal histories of the slower system could not be computed for want of the boundaryconditions presented through the Th´evenin equivalent associated with the faster sub-system,the solution stalls. To find the solution for the slow sub-system, we need the solution of thefast sub-system. To find the solution of the fast sub-system, we require the solution of the slowsub-system. There is thus a dead-lock, a chicken and egg problem.In classical latency, as implemented in EMTP [60], and all other temporal interpolationmethods we found in the literature, the dead-lock problem is avoided by only allowing integerratios between the time steps. Indeed, if the time steps at 3 t1 and 1 t2 overlap, the samesituation as at the zero time step is encountered: the whole system is defined and thus solvable.This is likely the reason why we could find no examples of fractional latency in the literature.At the core of the problem lies the fact that both sub-systems are not decoupled, and canthus not be solved wholly independent from each other. Such a solution is possible whentransmission lines are present on the connections between sub-systems, which offer such de-coupling in time, e.g.: [29]. This type of solution is discussed in Section 4.4.3. However, sincein a generic EMTP computation one cannot rely on the presence of transmission lines, wemay continue the solution if extrapolation is used. This allows us to guess one of the missingsolutions and break the dead-lock. This solution is discussed in Section 4.4.1. The linear ex-trapolation used has the same form as the interpolation equations, only the interpolation factorR is allowed to grow larger than unity. Again, the extrapolation will be avoided in the TINAcontext, where each cell is time-decoupled from the other.1054.3. Development of Fractional Latency4.3.3 Fractional Latency for Simulation SynchronizationThe main advantage of fractional latency over standard latency is that the ratios between sub-areas no longer need to be an integer ratio. Stepping back from this chapter, in other parts ofthis thesis, the use of interpolation within transmission line models was extensively studied,Chapter 3 and Appendix A, and used to construct the TINA mesh when dissimilar materialswere present and maintain synchronicity. Also in EMTP simulations, temporal interpolationin the line models is used to achieve synchronicity in the computations. The reason tempo-ral interpolation is used was in these cases is match the traveling time of the line  with thesimulation time step.Returning to the latency concept, if only integer ratios are allowed, interpolation in the linemodels will frequently be required. This introduces a double error: the error caused by theinterpolation at the border of a sub-area to match the different sub-area time steps, and theerror caused by the interpolation within the transmission lines that make up the sub-area tomatch them to the local sub-area time step.Using fractional latency, it is now possible to choose the time step of the sub-area as aclean, integer multiple of the transmission time  of the line. Doing so removes the needto interpolate the transmission line, and the exact line model can be used. Aside from accu-racy improvements, this also removes the computational burden of having to interpolate everytransmission line segment in a sub-area. In a TINA simulation, with easily ten to hundreds ofthousands of transmission line segments in an area, this results in significant speed gains.Thus, fractional latency allows us to take the interpolation out of the line models and placeit at the boundaries of the sub-area. As a result, the number of cells performing interpolationis reduced to only the sub-area border. The effect of cascading thousands of filters, as theinterpolation has a distinct and important frequency response, as studied in Chapter 3, is re-moved. In TLM methods, the need for stub-lines at the center of each mesh cell used to achievesynchronicity [16, 18] is thus also removed.Through the use of fractional latency, the errors and computational load usually associatedwith synchronicity methods could thus be significantly reduced. Using the technique, all theinterpolation needed for the volume associated with a sub-are can be moved to only the cellsassociated with the bounding shell of the sub-area.4.3.4 Choice of the Latency Master Time StepInitially, there are two ways fractional latency could be implemented. One could choose tomake the master time step, the base reference for the whole simulation to which all areasare eventually synchronized, the slowest or the fastest in the simulation. This decision has1064.3. Development of Fractional Latencyimplications on the details of the solution algorithm discussed in the next section, and thusmust be addressed first.Exploring the implications of this choice, questions of a philosophical nature appear and,in the end, the decision becomes one of personal preference, unless special technical consider-ations force the decision.When choosing the master time step the fastest one in the system (thus, the sub-areas are“local slow”), the slower areas have a reduced bandwidth. Therefore, any signal in the systemmust be limited by the bandwidth of the slower sections or temporal aliasing effects will bepresent. This adds the difficulty of insuring that no signals exceeding any of the local bandwidths exist in the simulation to comply with all the different Nyquist criteria. Thus, there isa requirement for band-limited signal sources and care must be taken when non-linearities areintroduced in the system. More important, the system output data, which is produced at eachinstance of the master time step, contains many redundant time steps due to excess bandwidthfrom over-sampled slow ares that cannot be used, as it is not valid for the whole simulationdomain. For 2D and 3D simulations, this excess bandwidth results in wasted storage andprocessing time to write the output. This is especially the case with fractional latency, whereeach output cell that is not an integer multiple of the simulation master time step must beinterpolated in time to obtain the result at the desired time step.When choosing the master time step the slowest one in the system (thus, the sub-areasare “local fast”), the faster areas have an increased bandwidth. Again, the issue with non-linearities occurs, but if the sources are embedded in space that runs at the master time step,they are automatically band-limited, as the source is locked to the bandwidth afforded by themaster time step. One could use a similar approach in the previous case and insure the sourcesare all in an area that runs at the slowest time step in the simulation. The advantage of thelocal fast approach is in the amount of output data that needs to be processed, and the naturallimitation on the output bandwidth, leaving no doubt as to the Nyquist limit of the output data.The disadvantage is that the spatial wave propagation in fast sub-areas may be temporallyunder-sampled and spatial aliasing in the output data may exist. The data is still correct ateach instance, but strobe-like visual effects do appear when a simulation is played-back asa movie. Compare, for example, the effect where in movies car wheels appear to suddenlyturn backwards as the car slows down in a scene. The local slow approach does not have thisphenomenon, as the output is saved at the rate of the fastest time step.In the TINA implementation of fractional latency, the local fast approach was chosen toreduce the amount of data to be processed. Since output interpolation must be done at eachstored time point, not having to do this at the rate of the faster system resulted in orders ofmagnitude less steps to be output-processed, interpolated, and stored.1074.3. Development of Fractional Latencyt0∆t10∆t21∆t1 2∆t1 3∆t1 4∆t1 5∆t112ms33ms 1∆t2 2∆t2(a) Sequencing with Two Time Stepst0∆t10∆t20∆t31∆t1 2∆t1 3∆t1 4∆t1 5∆t112ms33ms44ms 1∆t2 2∆t21∆t3(b) Sequencing with Three Time StepsFigure 4.8: Fractional Latency Solution Sequencing ExamplesThe spatial aliasing effects were visible in all our acoustic simulations involving metalboundaries, thus fast materials compared to the slow air. But if this wave behavior within thefast material must be studied, the master time step (by means of configuring the simulationbandwidth) may be reduced to a sufficiently short interval to remove those spatial aliasingeffects. Since the simulations experiments in this thesis were all studying the acoustic field inair, and not the bounding materials, the area of interest, being air and the slowest, did not sufferfrom spatial aliasing in the output data, and thus the computation and storage advantages of thelocal fast method could be obtained.It must be noted that the spatial aliasing only happens in the output processing of the sim-ulation, not in the simulation itself, as the fast areas are solved at an appropriate, local simula-tion time step within the computation. Due to the event-driven nature, bordering fast sub-areasexchange data at an appropriate rate. Thus no spatial nor temporal aliasing occurs in the simu-lation itself. It is only in the output processing that the effect occurs.4.3.5 Solution AlgorithmUsing the local fast approach, the main simulation and output processing is thus operated atthe slowest rate in the system. The second consideration is the order in which the sub-areas aresolved. This sequence is important, as otherwise the solution can dead-lock. In this thesis, anevent table-based solution sequencing system was implemented.First, consider the two simplified cases shown in Figure 4.8 for two and three sub-areas.The examples are based on Table 4.1. When looking at the figures and table, it is clear that thevarious sub-areas require solution at different absolute simulation times.Let us first establish the concept of absolute simulation time, which is key to the algorithm1084.3. Development of Fractional Latencydeveloped to manage the event table. At the core of the problem is that, using the fractionallatency concept, the simulation is essentially asynchronous. With normal latency, the varioussub-areas run at different rates, but still synchronize every so many steps. This is not the casewith fractional latency. Thus, the standard idea of time in EMTP, TLM, and finite-differencecomputations, which is simply that of time-marching and wholly implicit, cannot be used asthere is no clear relationship between the temporal conditions between sub-areas.Using fractional latency, we thus have to synchronize an asynchronous system to a regulartime step in order to generate valid output. Since there is no direct relation between the indi-vidual sub-areas, we still need to establish a relation of some type. There is an entire field ofstudy that investigates such synchronicity problems. Called the science of synchrony, it bringsmathematics, physics, and biology together to study how spontaneous order occurs naturally inthe universe. Steven H. Strogatz’s introduction to the topic [84] suggest a solution to the issueat hand.In essence, it comes down to the realization that by observing the system as a whole atregular intervals, like using a strobe light to find the rotational speed of machinery, the asyn-chronous system may be synchronized at the observational times.Inordertodefinetheverynotionof“atthesamemoment”inafractionallatencysimulation,the concept of absolute time must be introduced. Observing at regular intervals thus becomesa matter of obtaining values at regularly spaced absolute times and interpolating between ab-solute times if no solution for the required observational moment exists, an issue which will beinvestigated in Section 4.3.6.The concept of absolute time is readily implemented in a time-marching solution. Sinceeach sub-area is aware that it is being solved, by virtue of the solution code for that areabeing called by the simulator, and it is aware of its local simulation time step, it is a matterof counting the number of these steps and multiplying the count by the local simulation timestep. The result is the absolute time of that sub-area. By means of this time, each solution forthat sub-area can be supplied with an absolute time stamp, and stored in the correct order withrespect to the other sub-areas. The result is a table of time-stamped simulation events, eachassociated with solution data, similar as Table 4.1.To establish the solution order of the simulation using these absolute time stamps we returnto Figure 4.8 and Table 4.1. From these, we can observe how the solution sequencing algorithmmust work.First, at the zero time, all sub-systems must be solved. The order in which is important.When multiple areas are to be solved at the same absolute time, we must always solve thesub-area with the largest simulation time step first. Doing so insures the future values neededto interpolate boundary conditions for the faster sub-areas are available. The solution for the1094.3. Development of Fractional Latencysystem will now yield the future values, each at the local time step of 1 tx.The second rule in the algorithm is to always solve the area which has the smallest absolutetime stamp. In the example being studied, that would mean that all areas will be solved att = 0, the slowest sub-area first. When all sub-areas whose local absolute time was set tozero are solved, the next are to be solved will be the sub-area operating at  t1. It has, at thattime, the smallest absolute time and only that area needs to be solved at that absolute time. Itneeds interpolated boundary conditions from the connecting sub-areas for solution. In orderfor this interpolation to be possible, a solution for that neighboring area must exist at a laterabsolute time than the absolute time at which the solution for the sub-area is sought. Due tothe described algorithm, this is always the case.When the above two rules are applied, an event table may be calculated to control thesolution sequencing in the simulation. In practice, the event table is computed on the fly andhas as many entries as there are sub-areas in the system. An example solution sequencing withabsolute time stamps, using the above rules is shown in Table 4.1.Thus, the algorithm to determine the solution order may be captured in a flow chart. Figure4.9 illustrates the process, including the output processing step, as discussed in the sectionbelow. The box labeled “Solve the latency area” refers to the standard EMTP-style solutionusedthroughoutthisthesis, wherethelatencycellsprovidetheinformationfromwhatnormallywould be the boundary The´evenin equivalents from other latency areas so the local area maybe solved at its local time step.4.3.6 Generating OutputRegardless of the choice of the master time step, output generation will be required each timethe areas operating at the master time step, which is directly related to the required simulationbandwidth, are called. One could generate output at any time, if required, but by synchronizingto a latency area, significant computation can be saved, as the area the output generation issynchronized to does not require interpolation. In TINA and the EMTP example later in thechapter, output generation is synchronized to the slowest time step in the system, which servesas the master time step. The required step for output processing is reflected in the flowchartof Figure 4.9, where the output processors are called when the master time step areas arecomputed.Output generation is then a matter of obtaining readings from the asynchronous simulationthat corresponds to the absolute time the output is requested for. Thus, for the master timestep areas and those areas that, at the time have the same local absolute time, this is simply theresults computed at the current time step.1104.3. Development of Fractional LatencyIs smallestabsolute time < totalsimulation time?Find the latency area(s) with the smallestabsolute timeDo multiple areas withthe same simulation time step exist?Find the latency area with largestsimulation time stepSolve the latency areaStopStartNYNYSolve the latency cells for this area Run output processorsIs area absolutetime = outputgeneration time?NYIncrement the area local absolutetime by the local time stepFigure 4.9: Fractional Latency Solution Sequencing Algorithm1114.4. Implementations of Fractional LatencyFor the other areas, where no simulation data exists for the required time step (thus, wherefractional latency is applicable), interpolation over the future and past solution will be used.Due to the solution sequencing algorithm described in the previous section, it is ensured thateither the area is at the correct absolute time for output, or a solution for a future time, relativeto the current absolute time, exists and interpolation can be used to find the output for therequired absolute time.The process of generating the output data itself, at least in our implementation, is doneusing the same linear interpolation (4.1) used for the fractional latency process. It has to benoted that the errors associated with output interpolation do not influence the simulation itself.The output results do not participate in the actual computation, they are only stored to disk forpost-processing of the data.4.3.7 The Accuracy and Stability of Fractional LatencyThe fractional latency interpolation algorithm and the resulting latency cell has a non-flat fre-quency response due to the interpolation. One important difference, compared to the inter-polated line model studied in Chapter 3 is that the interpolation factor R, is not constant andpotentially varies over all values in the normal interval 0  R < 1. Thus, the interpolationerror is not constant either. By assuming worst-case conditions, R = 0:5, the error criteriaderived for the interpolated line model may be indicated initially, but will be explored furtherlater in this chapter, when the simulation parameters are discussed in Section 4.4.6. However,no formal assessment of the stability and accuracy of the fractional latency cells will be given.Qualitative comparisons between latency and standard cases in the TINA context indicatethat accuracy is comparable to what is to be expected from the interpolated transmission lines,with a much reduced computation time. Given the close relationship between the used equa-tions and methodology, this is not unexpected.InamoregeneralEMTPcontext, however, theextrapolationbringswithitahostofstabilityissues, as can be seen from the simulation results. A formal stability analysis would be quitesimilar to the methods used in Chapter 3 and a Courant condition [83], which places limitson the acceptable time step and spatial discretization size, should be derived. We leave thisanalysis for future work.4.4 Implementations of Fractional LatencyIn this thesis, two implementations of the universal fractional latency cell, as described earlierin this chapter, were made. One was an EMTP-style computation using extrapolation. This1124.4. Implementations of Fractional Latency1 2 3 4LatencyArea0LatencyArea1 LatencyArea23 Ω 50Ω350mH 100mH10nF 600nFR1 R2 L2L1C1 C2ES230cosωt kVω=377rad/sTs =8msFigure 4.10: Fractional Latency Domains in EMTP Solution Casetest case was used for the original latency work, and repeated here to compare the fractionallatency results, which require extrapolation in this context, to the normal EMTP solution. Thesecond implementation is in the TINA framework, where the extrapolation can be omitted asthe TINA solution is fully time-decoupled. Performance and accuracy comparisons betweenthe ideal solution, synchronicity through interpolation, and fractional latency techniques willbe made.4.4.1 Fractional Latency in EMTPThe example used for the latency implementation is that shown in Figure 4.10, as obtainedfrom [52]. It is a lumped system where each side, before and after the switch, has a differenttime constant. Thus, each side should be computable with a different local time step. As anadditional complexity, in our simulation both the source and the switch operate in the baselatency area, which is the one operating at the slowest rate. Thus, there are three latency areasin the system.To implement this system using the fractional latency cells described in this chapter, thevarious sub-areas need to be abstracted as Th´evenin equivalents and connected to the frac-tional latency cells from Figure 4.5 at each area boundary. This means, since this computationconsists of only a few lumped components and is thus not time-decoupled, that the equivalentsystem for the entire system behind a given Th´evenin equivalent must be found. This is ofcourse computationally not effective. Hence, the latency formulation commonly used only op-erates on the history sources internal to a latency area themselves, e.g.: [43, 60]. However, thefractional latency cell was only applied here to investigate the extrapolation effects.A more practical use in an EMTP context of the fractional latency cell method could likelybefoundthroughtheuseofMulti-AreaTh´eveninEquivalents(MATE)[49, 87]. Inthismethod,1134.4. Implementations of Fractional LatencyFigure EMTP  t Area 0  t Area 1  t Area 2  t EMTP ts Frac. Lat. ts4.11 35  s 35  s 3.51  s 3.54  s 0.015995 s 0.01589 s4.12 35  s 35  s 35.1  s 35.4  s 0.015995 s 0.01547 s4.13 35  s 100  s 100.1  s 35.4  s 0.015995 s 0.0154 s4.14 35  s 200  s 100.1  s 35.4  s 0.015995 s 0.0088 sTable 4.2: EMTP Case Simulation Parameters and Switch Timessub-areas may be abstracted, and connected through Th´evenin equivalents. It was shown [87]that this method is effective at parallelizing EMTP simulations and it could likely be expandedwiththefractionallatencytechniquetoallowfordifferent, non-integerlocaltimestepsbetweensub-areas, further increasing the performance.The circuit of Figure 4.10 was implemented in Octave (an open-source Matlab derivative)both using a standard EMTP solution and the fractional latency cells. Since the use of frac-tional latency for this small system is inherently inefficient due to the overhead associated withfinding the various Th´evenin equivalents, compared to the circuit size, no timing results willbe given.4.4.2 EMTP ResultsThe circuit, in the EMTP comparison case, was computed with a time step of 35  s whilethe fractional latency case was computed with various time steps, as given Table 4.2. Theswitching event was set at 8 ms, however, the switch will only open at the fist zero crossing ofthe current through the switch. This causes the switching moment to vary between simulationas a phase shift may cause the switch event to be delayed by a period of the natural frequency ofthe system, in addition to the limitations of the time step of the master area in which the switchresides. Since the magnitude of the resulting oscillation is strongly dependent on the voltageat the switching moment, this difference in switching time resulted in significant differences inamplitude.A problem with the system being studied is that the required time steps of all the sub-areasare relatively close to one another. For the fractional latency to work, a sufficient ratio in timesteps must exist. This is due to the interpolation used in the method, where it is assumed thatone system varies much slower than another, and is thus essentially a stiff system during theinterpolated interval. In Chapter 3, it was shown that for the interpolated line model, a ratio ofabout ten times is required.This is the case in Figure 4.11. The simulation operates both sub-areas at a much smallersimulation time step ration of about 110. The results are rather good, except of course the1144.4. Implementations of Fractional Latency-200000-150000-100000-500000500001000001500002000000 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(one)EMTPFracLat01000002000003000004000000 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(two,three)-1500-1000-500050010000 0.005 0.01 0.015 0.02Magnitude (A)Time (s)I(four,gnd)Figure 4.11: Fractional Latency EMTP Solution Stable Case 1-200000-150000-100000-500000500001000001500002000000 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(one)EMTPFracLat01000002000003000004000000 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(two,three)-1500-1000-500050010000 0.005 0.01 0.015 0.02Magnitude (A)Time (s)I(four,gnd)Figure 4.12: Fractional Latency EMTP Solution Stable Case 21154.4. Implementations of Fractional Latency-200000-150000-100000-500000500001000001500002000000 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(one)EMTPFracLat01000002000003000004000000 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(two,three)-1500-1000-500050010000 0.005 0.01 0.015 0.02Magnitude (A)Time (s)I(four,gnd)Figure 4.13: Fractional Latency EMTP Solution Stable Case 3-200000-150000-100000-500000500001000001500002000000 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(one)EMTPFracLat-2e+06-1e+0601e+062e+063e+060 0.005 0.01 0.015 0.02Magnitude (V)Time (s)V(two,three)-5000-4000-3000-2000-1000010002000300040000 0.005 0.01 0.015 0.02Magnitude (A)Time (s)I(four,gnd)Figure 4.14: Fractional Latency EMTP Solution Unstable Case1164.4. Implementations of Fractional Latencyincreased computational time. The studied case did not have enough natural difference in thenatural frequencies of each area, and thus resulting simulation time step, to make fractionallatency work in a reasonable way. The ratio was thus artificially increased.In the next cases, Figure 4.12, the base latency area and sub-ares were operated at about thesame rate as the EMTP solution, thus about 35  s. The results are not as good, but still followthe general shape of the expected result. The time step ratio between the base and sub-areaswas thus almost unity.In Figure 4.13, the base area and latency area 1 were slowed-down to roughly 13 of theEMTP case rate, while the fast oscillating area was left to operate at a faster rate. The resultsdegrade further, mainly due to the shifting of the switching event, but are otherwise comparableto the previous cases. Here, the fractional latency is being used as intended, having one partrun at a faster rate than the others.The last case, in Figure 4.14 operated with latency area and main area time steps that weretoo low for the system being studied and not only are the results wrong, they are also unstableas the extrapolation is pushed too far to maintain stability. With the used sampling parameters,the Nyquist bandwidth for the slowest area, the base area, is 2.5 kHz. This is sufficient forthe 60 Hz source embedded in it. The fast oscillation has a frequency of about 2.2 kHz, andtakes place in a sub-area with a Nyquist frequency of 14 kHz. Thus, the local time steps areaccording to the EMTP guidelines.The instability is likely due to the system being too close to the stability limit imposed bythe extrapolation. Since extrapolated solutions usually have stability limits, the Courant limits,much lower than the Nyquist limits [83], this is not unexpected. In practice, one would choosethe areas guided by an analysis of the eigenvalues, to insure that the local simulation time stepsare sufficient for the simulation [60], and then choose the time steps so the Courant conditionfor the method is met.Table 4.2 summarizes the used simulation parameters for all the figures, as well as theactual times at which the switch opens for each simulation.Allinall,theextrapolationperformsreasonablywell,buttheeffectsofthelatencymatchingcells, due to the lack of sufficient oversampling between sub-areas, shows as a significant phaseshift in the signals. In order to use the fractional latency cell in EMTP-style solutions, asdescribed in this chapter, a stability analysis of the method would need to be performed andthe Courant condition determined.However, from the above, it is clear that the fractional latency cell concept is not a goodone for EMTP simulations. The concept was not developed for it and the application to EMTPwas only done to explore the concept and illustrate the issues with extrapolation that happenwith the method in non-decoupled methods. Fractional latency could likely be used in EMTP,1174.4. Implementations of Fractional Latencybut it should be applied to the history sources themselves, by modifying the traditional latencyformulation. Alternatively, itcouldbeintegratedintheMATEconcept[49, 87], whereasystemis split in sub-areas, communicated through Th´evenin equivalents. We leave these issues asfuture work, as they are not the focus of this thesis.4.4.3 Fractional Latency in TINAThe fractional latency method was developed for use in TINA, or other time-decoupled meth-ods, not a regular EMTP. Hence, the choice of formulation as a universal, stand-alone cell thatcan be readily and transparently introduced in a discretized-space solution grid.The issues with extrapolation in the EMTP implementation of fractional latency can bereadily overcome within the TINA framework. As was stated earlier in this chapter, whentransmission lines are present, the ability to “look ahead” becomes available without extrapo-lation, and fractional latency can be implemented directly. Since each and every cell in TINAis connected through transmission lines, full time-decoupling is in effect and transmission line“look ahead” can be readily implemented. In the next section, the equations are derived for the1D case, which is the basic building block for the higher-dimensional meshes used in TINA.4.4.4 Avoiding Extrapolation with Transmission-Line DecouplingTo illustrate how the prediction in fractional latency may be avoided, we will now show howto add fractional latency to an ideal line segment. In TINA, every part of the system is coupledthough such transmission lines, which has the benefit of both time decoupling as well as areadily available Th´evenin equivalent to work with. The resulting expressions are similar to theinterpolated transmission line, except that they now feature prediction, which will be overcomethrough transmission-line look-ahead.Since any number of ideal line segments can be cascaded, resulting in a single segment oftransmission time equal to the whole, the following argument also holds for a one-dimensionalsection of TINA mesh connected to two neighboring areas, where both neighboring domainshave equal, but different local time step than the sub-area the lines are in.Conceptually, the ideal line consists of two Th´evenin equivalents, with the sources givenby the the model equations and the resistance equal to the line impedance. The fractionallatency cell is essentially a Th´evenin conversion system, that takes the equivalent on one endand presents an interpolated equivalent on the other. The resemblance between the circuits iscoincidental. Combining both systems yields the interpolated line. We will apply the fractionallatency conversion to the model, resulting in the circuit of Figure 4.15.1184.4. Implementations of Fractional Latency∆tA ∆tBK MZlK ZlMelK elM∆tA ↔ ∆tB Latency al CellZcekh∆tAemhZc∆tB ↔ ∆tA Latency al CellFigure 4.15: Fractional Latency Applied to a Transmission Line SectionNote that in this figure, only fractional latency half cells are shown. This was done toclarify the derivations. In the shown figure and derivation, only the matching of the line mod-els Th´evenins is performed. The second part of the system, matching the external systemTh´evenins to the line model latency area is not shown. These second halfs are required to findthe various voltages and currents at the terminals of the line model and solve the system. InFigure 4.16, the whole system, comprising both half-cells for each size is shown.Note that there is no coupling between both Th´evenin equivalents in the full fractionallatency cell (Figure 4.5). Rather, each Th´evenin uses the Th´evenin presented by the systemneighboring on one side to find the values for the Th´evenin it presents on the other side. Thefact that the full latency cell circuit looks similar to the transmission line model used in thisthesis is coincidence and not functional.Starting from the ideal line equations (A.9) for the Th´evenin history sources, from Ap-pendix A:ekh(t) = vm(t  ) + Zcim(t  ) (4.3a)emh(t) = vk(t  ) + Zcik(t  ) (4.3b)The relevant fractional latency equations (4.1) are:elK(tA) = ethB(tB +  tB) + RK [ethB(tB) ethB(tB +  tB)] (4.4a)elM(tB) = ethA(tA +  tA) + RM [ethA(tA) ethA(tA +  tA)] (4.4b)Now, applying the fractional latency equations (4.1) to both end of the line, with constant1194.4. Implementations of Fractional Latencyimpedances and using the A as the external area time step and B as the internal area time step:elK(tA) = ekh(tB +  tB) + RK [ekh(tB) ekh(tB +  tB)] (4.5a)elM(tA) = emh(tB +  tB) + RK [emh(tB) emh(tB +  tB)] (4.5b)Substituting the line equations and simplifying, with the local absolute time given by tB:elK(tA) = vm(tB   +  tB) (4.6a)+Zcim(tB   +  tB)+RK [vm(tB   ) vm(tB   +  tB)]+ZcRK [im(tB   ) im(tB   +  tb)]elM(tA) = vk(tB   +  tB) (4.6b)+Zcik(tB   +  tB)+RK [vk(tB   ) vk(tB   +  tB)]+ZcRK [ik(tB   ) ik(tB   +  tb)](4.6c)The resulting equations have the same form as the interpolated line model, except that theLHS part is shifted one step ahead compared to the interpolated line model. This is wherethe extrapolation comes in. Also, the  value used here is the actual length of the line, not arounded-up integer multiple of the local time step as with the interpolated lines. However, inpractice when using fractional latency, the local time step will usually be an integer multipleof the transmission time of the line. Thus, we can think of the transmission line with fractionallatency as an interpolated line with the required history values shifted one step in the future,thus requiring the current and future values to compute a solution, as opposed to current andpast values, as in the interpolated line model.This is where the extrapolation, the requirement for the future value, may be avoided. Thetransmission line history buffers, as the model is implemented in this thesis, Appendix A,already store the (tB  + tB) values, as these are the values to be used at the next time step,and are thus simply waiting for use in the buffer. These are exact values, computed by the idealline model. Since they are in transit, nothing else will happen to them, except the time delayin the model. We thus have access to exact, future values already in memory which may beused in the latency equations to match the different  t domains. This is the advantage of time-decoupled methods. Since each area is decoupled, they may be solved independently fromone another and we can obtain the area solution, not just the history sources, for the next step.1204.4. Implementations of Fractional LatencyThis solution may then be used with fractional latency to remove the need for extrapolation.Transmission-line decoupling is a classic example of a time-decoupled method, and at the basisof both the TLM and TINA methods. TINA has the advantage that the formulation is entirelybased on Th´evenin equivalents, which makes interfacing the line models and the fractionallatency cells trivial.4.4.5 Integration of the Fractional Latency Cells in the TINA MeshThe latency cell, as described in the EMTP example, is used again and split in two uni-directional half-cells. This time, however, they will gain the ability to request the next valuefrom the connecting line segments. In practice, this construction is shown in Figure 4.16,for a 2D example where the central cell is surrounded by four different latency domains, andmust thus use the uni-directional fractional latency cells to interface with those neighboringregions. Using the latency cells, which are modeled like any other cell in TINA and presentTh´evenin equivalents to the outside world, they can be inserted into the grid at will and alloweasy separation of a simulation into sub-areas.The choice of using half-cells was done for purely practical reasons regarding the way theindividual cells in TINA were implemented. Each cell in the simulation uses a pointer to theneighboring cell’s Th´evenin data. It uses this pointer to access that Th´evenin equivalent fromeach neighboring cell in order to solve itself. Since this neighbor sits in a different latency area,its results must be converted to the local required time. This is done through the latency cell.It is inserted in the simulation as a 1D block (in the drawing indicated by the W-E and N-Sblocks) and connected in the mesh.The insertion process consists of initializing a uni-directional latency cell, obtaining theaddress the cell to be solved was pointing to, connecting the latency cell to that address, andconfiguring the cell to be solved to point to a new address, that of the output side of the frac-tional latency cell. The half-cell for the other data flow direction will be inserted when theneighbor cell is processed.For a given latency area, the latency half-cells are solved before the mesh cells, and thusmay obtain properly interpolated Th´evenin equivalents from the neighbors that the connectedmesh cell may then use to compute itself. By splitting in half-cells, the uni-directional nature oftheTh´evenininformationexchangesisexploitedandasymmetricaldistributionofthehalfcellsacross an area barrier can be used. This simplifies the programming of the system. The latencycell requests its required data from the neighbor, interpolates it, and presents a new Th´eveninto the internal cell. Later, when the actual mesh cells are solved, it will pull its data from thelatency cell it is connected to. Thus, inserting the latency cells merely becomes a matter of1214.4. Implementations of Fractional LatencyNSW ENSW ENSW ENSW ENSW ENorthWest EastSouthX,Y X+1,YX,Y-1X,Y+1X-1,YW EW EW EW ESNSNSNSNFigure 4.16: Fractional Latency Cells in a TINA Mesh1224.4. Implementations of Fractional Latencymoving data connectivity pointers around and connecting the latency cell appropriately.4.4.6 Simulation Parameters in TINA for Fractional LatencyTo evaluate the error incurred by fractional latency, it would be of interest to lower the ratio ofthe time step between the latency areas and the master time step base area. This would increasethe error in the simulation due to the interpolation used in the latency, likely in a similar wayas studied in Chapter 3. Doing so, we could establish a qualitative limit for the interpolationtime step ratios between the base area running at the master time step and the other areas in thesystem.However, this proves impossible in the TINA implementation. We can increase the la-tency ratio, making the already huge sub-area bandwidths even larger compared to the usablesimulation bandwidth given by the slower areas. But, we cannot reduce it below the spatialdiscretization limit. Or: if the system is discretized with, e.g.: 10cells , the interpolation andfractional latency ratios cannot be lower than 10 without also adjusting the spatial discretiza-tion factor.The only way to reduce the fractional latency time step ratio is to also reduce the spatialdiscretization. This, however, also influences the accuracy of the simulation and makes itimpossible to study the parameters independent from one another.In the end, the situation may be explained from the fact that, when using fractional latency,we enforce that the length of each transmission-line segment must be an integer number ofthe traveling time  and that the local sub-area time step is an integer multiple N of  , thus: t =  N. Doing so avoids the use of interpolation for that line segment and increases bothprecision and execution speed. This is the main reason for using the more complex fractionallatency versus the regular latency. Thus, the shortest a line segment may be is one  t, thus onememory location (N = 1). This puts a hard limit on the lowest simulation time step possiblein a latency area, which is thus so that  tmax =  .The traveling time  itself is given by the line parameters, which yield the wave speed a,and the spatial discretization of the problem, which determine the size of the cell, and thusthe physical length of the line segments used therein. Thus:  =  l2a, noting that TINA useshalf-line segments with a node in the cell center.The spatial discretization  l is now chosen so that the physical extent of the wave is suf-ficiently sampled. It is a spatial Nyquist criterion, and given by the shortest wave length inthe latency sub-area. Typically, one samples so that  l =  SD, where SD is the spatial dis-cretization and typically chosen as 10 cells per wavelength. The shortest wave length  minitself is given by the wave speed in the local medium a, as well as the highest frequency in1234.4. Implementations of Fractional Latencythe simulation, BW as:  min = aBW . In TINA, the bandwidth BW is defined by the user forthis purpose, and is equal to the desired Nyquist frequency of the computation, and the highestoccurring frequency.Thus, combining the above considerations together, the maximum time step usable in alatency area,  tmax, may be expressed as: tmax =  l2a (4.7)The spatial discretization used in the simulation  lsimulation, then, is a single global param-eter for a regular grid in space that must be less than the following limitation, where the wavespeed is that of the area with the slowest wave speed in the simulation amin: lsimulation = aminSD BW (4.8)The spatial discretization is a constant for the whole simulation, as a regular grid in space isusedintheTINAimplementationdiscussedinthisthesis. Thismeansthatthisglobalparameteris determined by that sub-area with the slowest wave speed, thus shortest wave length for thegiven bandwidth. This will be the master time step area, or base area, since it must operate atthe lowest simulation time step as the output generation is synchronized to it. From the aboveequations, it follows that the area with the slowest wave speed is also that with the largestsimulation time step, thus is solved at the slowest rate, and has the longest traveling times forits line segments.In a TINA simulation, SD, BW, and  l are global parameters chosen by the user andverified by the program to be in compliance with all areas in the simulation. The program willalso use these parameters to put limitations on the simulation time steps of each latency area.Thus, to return to the original argument in this section, of why the latency sub-area ratios couldnot be reduced below the spatial discretization, we may now discover why.First, the simulation base area operates at the lowest simulation time step in the system.This was discussed in Section 4.3.4. From the above, it also follows that it is linked to thematerial with the slowest wave speed. When now regarding a latency sub-area, this must bydefinition operate at a faster rate and have a higher wave speed. The rate between the local timesteps in these areas is the parameter of interest.In the TINA implementation in the thesis, with the base area time step  tbase = 12BW ,the base area cannot have any transmission lines in it, as these will automatically be spatiallyunder sampled. We find, in fact, that if we were to define a medium in this area composed oftransmission lines with  =  tbase, the spatial sampling ratio is unity.Thus, the time step ratio between the base area and a sub-area which has lines must be1244.4. Implementations of Fractional Latencyat least the spatial discretization factor, further compounded by the requirements posed bythe traveling time of the lines. Mathematically, the criterion for the minimum time step ratiobetween the base area and a sub-area becomes: tlatency area tbase = SDalatency areaBW l (4.9)It must be noted that, in the TINA implementation, each spatial cell is split in two half-lines. Thus, the  l parameter used in this discussion has to be halved to find the values used inthe simulation. Also, keep in mind that the 2D and 3D grid require an adjustment factor for theline parameters, which is the same as for TLM [18], and a result of the field to circuit mapping,as discussed in Chapter 2.Intheaboveequation, thewavespeedoftheareaunderinvestigationalatency area maynotbeslower than the slowest wave speed in the simulationamin. Otherwise, the spatial discretizationneedstobeadjusted, andthustheindependenceofthesevariablesislost, makingtheevaluationof the time step ratio between the simulation base area and the latency areas difficult.When both wave speeds are equal, the above expression may be reduced to the followinglimiting condition: tlatency area tbase = SD2 (4.10)We may also see that the minimum time step ratio between the base area and the varioussub areas is similar, in practice, to that found in the error criterion when interpolation is usedin TINA, as discussed in Section 2.2.3. With an oversampling of 10 for both the spatial andtemporal factors, we found a time step ratio of 100. Here, a spatial discretization of 10 squaredalso yields 100 as a minimum ratio between the time steps as an accuracy condition.4.4.7 TINA ResultsTo evaluate the use of fractional latency in the TINA method, three cases will be studied forboth accuracy and numerical performance. These cases will be computed using highly over-sampled non-interpolated and interpolated lines, to serve as a base reference, as well as differ-ent time step ratios of fractional latency. The non-interpolated case is the reference, but has tooperate with slightly adjusted parameters for some of the used materials, to insure the varioustime steps match.The physical size of the microphone itself was also taken into account, which is a sizableportion of the wave length for the higher frequencies evaluated. Thus, in the simulation, theacoustic pressure over an area comparable to that of the physical microphone was averaged toobtain the result for that location.1254.4. Implementations of Fractional Latency(a) Full Boundary(b) Medium Boundary(c) Thin BoundaryFigure 4.17: Simulation Cases1264.4. Implementations of Fractional Latency20 205405003002004090140190A GFigure 4.18: Expansion Duct Set-UpThe used 2D configurations are shown in Figure 4.17 and the setup itself in Figure 4.18.The spatial discretization was 1 mm. These are comparable cases, but with different widths forthe duct border. This was done to show the change in efficiency of both the interpolated andthe fractional latency technique with different ratios of fast and slow materials in the system.The case is presented in full detail in Chapter 5, where a physical realization is compared tosimulation. As described there, the utmost left yellow dot is the “A” microphone position andthe utmost right yellow dot is the “G” microphone position.The adjustments were made in the dense boundaries (red material, aluminum). This leavesthe main medium (blue material, air), where the measurements are made, at the same param-eters across all simulations. The error incurred is thus limited, as the dense boundary has fiveorders of magnitude higher impedance compared to air, and is thus nearly an infinitely denseboundary (open circuit). The reflection coefficient between air and the boundary thus doesn’tchange much and little energy can actually enter the boundary material and propagate in it.Thus, the change in its parameters has negligible effect on the simulation results, which areobtained in the air medium only. This is confirmed in the plots, where the interpolated andnon-interpolated (normal) cases overlap, as well as in Chapter 2, where this is discussed inmore detail. The use of significant oversampling for these reference cases aids in this agree-ment. It must be noted that the used oversampling is a result of previously established errorcriteria, and a valid simulation bandwidth of 25 kHz was maintained for each case.The differences between the precise and adjusted materials are shown in Table 4.3, where1274.4. Implementations of Fractional LatencyAir Al (correct) Al (adjusted) 1:1198 103 gm3 2:6989 106 gm3 2:6989 106 gm3k 6:99746 10 9ms2g 1:45138 10 14ms2g 1:584727 10 14ms2ga 3:45383 102ms 5:05261 103ms 4:83537 103msz 4:13769 105 gsm2 1:36365 1010 gsm2 1:30502 1010 gsm2Table 4.3: Materials Used in the TINA Simulations is the compressibility, k the compressibility, a the wave speed, and z the impedance. Theseparameters are acoustic ones, but relate directly to their electrical equivalents of permeabilityand permittivity.Comparison of Simulation Parameters and ResultsThe three simulation cases, repeated for each used computation method, are summarized be-low. The normal case, which is the non-interpolated simulation using the adjusted materialparameters from Table 4.3, served as the base line for the comparisons. In all cases, the normalcase case was used as the comparison base line.The frequency-domain averaged error was computed as a basic error criterion to help assessthe simulation approaches. It is a percentile number, computed by obtaining the magnitudeerror at each frequency point, at microphone location A, and compared to the normal case,summing these together, and normalizing to the number of spectral components.Memory use was determined using the Linux “top” program, and the indication for datasize was recorded. This is the total memory that the program uses for data, thus non-executablecode, including the stack and any portion swapped to disk. This served as an indication of howmuch memory the simulator requires to solve a particular case.All cases were executed on an Intel Core 2 Duo E8400 3 GHz system, running a stockKbuntu 9.04 “Jaunty Jackalope” 64 bit Linux distribution on kernel 2.6.28-15. The solveroperated in a single thread and was confined to one core. The average core usage was over 99%, indicating nearly unique usage of that core for the TINA solver process.Full BorderIn this case, the border comprised roughly one half of the total simulation volume. This meantthat the latency, which mainly applies on the air portion of the simulation, could only act ona comparatively limited portion of the simulation area. A speed gain of roughly two timeswas obtained, and a reduction in memory use of about 15 %, compared to the normal case.Compared to the interpolated cases, the performance was better due to the absence of per-cell1284.4. Implementations of Fractional LatencyFull Border Normal Interpolated Latency Latencyratio 10 ratio 100# Mesh cells 88400 88400 88400 88400Air/Aluminum cells ratio 2/1 2/1 2/1 2/1Latency/Interpolation ratio N/A 10 10 100Spatial discretization air 14 Cells/ 14 Cells/ 14 Cells/ 14 Cells/ Spatial discretization alu. 193 Cells/ 202 Cells/ 202 Cells/ 202 Cells/  t air 7.31182e-8 s 6.99744e-8 s 1.02366e-6 s 1.70609e-7 s t aluminum 7.31182e-8 s 6.99744e-8 s 6.99744e-8 s 6.99744e-8 s t Simulation/Base 7.31182e-8 s 6.99744e-8 s 2e-5 s 2e-5 s# Time steps base 1312942 1371931 4800 4800Computation time 31732 s 36395 s 18714 s 24236 sEffective cells/s 3.7e6 3.3e6 6.5e6 eqv. 5.0e6 eqv.Memory use 59 MB 60 MB 50 MB 55 MB% FD avg. mag. err. vs nor. N/A 0.06 % 13 % 8.5 %Speed-up vs normal N/A 0.9 x 1.7 x 1.3 x% Memory use vs normal N/A + 2 % - 15 % - 7 %Table 4.4: TINA Simulation Results and Parameters Full Border-0.4-0.2 0 0.2 0.4 0.6 0.8 0  0.001  0.002  0.003  0.004  0.005MagnitudeTime (s)Waveform at A Full BorderNormalInterpolatedLatency 10Latency 100Figure 4.19: Full Border Time Domain Results at A1294.4. Implementations of Fractional Latency 0 2 4 6 8 10 12 0  1000  2000  3000  4000  5000MagnitudeFrequency (Hz)Transfer Function A->G Magnitude Full BorderNormalInterpolatedLatency 10Latency 100(a) Magnitude-1500-1000-500 0 500 1000 1500 0  1000  2000  3000  4000  5000Phase (degrees)Frequency (Hz)Transfer Function A->G Phase Full BorderNormalInterpolatedLatency 10Latency 100(b) PhaseFigure 4.20: Full Border Frequency Domain Transfer Function A ! G1304.4. Implementations of Fractional LatencyMedium Border Normal Interpolated Latency Latencyratio 10 ratio 100# Mesh cells 88400 88400 88400 88400Air/Aluminum cells ratio 6/1 6/1 6/1 6/1Latency/Interpolation ratio N/A 10 10 100Spatial discretization air 14 Cells/ 14 Cells/ 14 Cells/ 14 Cells/ Spatial discretization alu. 193 Cells/ 202 Cells/ 202 Cells/ 202 Cells/  t air 7.31182e-8 s 6.99744e-8 s 1.02366e-6 s 1.70609e-7 s t aluminum 7.31182e-8 s 6.99744e-8 s 6.99744e-8 s 6.99744e-8 s t Simulation/Base 7.31182e-8 s 6.99744e-8 s 2e-5 s 2e-5 s# Time steps base 1312942 1371931 4800 4800Computation time 35774 s 46628 s 6090 s 16947 sEffective cells/s 3.2e6 2.6e6 19.9e6 eqv. 7.2e6 eqv.Memory use 75 MB 77 MB 50 MB 62 MB% FD avg. mag. err. vs nor. N/A 0.06 % 7.0 % 1.5 %Speed-up vs normal N/A 0.8 x 5.9 x 2.1 xMemory use vs normal N/A + 3 % - 33 % - 17 %Table 4.5: TINA Simulation Results and Parameters Medium Border-0.4-0.2 0 0.2 0.4 0.6 0.8 0  0.001  0.002  0.003  0.004  0.005MagnitudeTime (s)Waveform at A Medium BorderNormalInterpolatedLatency 10Latency 100Figure 4.21: Medium Border Time Domain Results at A1314.4. Implementations of Fractional Latency 0 2 4 6 8 10 12 0  1000  2000  3000  4000  5000MagnitudeFrequency (Hz)Transfer Function A->G Magnitude Medium BorderNormalInterpolatedLatency 10Latency 100(a) Magnitude-1500-1000-500 0 500 1000 1500 0  1000  2000  3000  4000  5000Phase (degrees)Frequency (Hz)Transfer Function A->G Phase Medium BorderNormalInterpolatedLatency 10Latency 100(b) PhaseFigure 4.22: Medium Border Frequency Domain Transfer Function A ! G1324.4. Implementations of Fractional LatencyThin Border Normal Interpolated Latency Latencyratio 10 ratio 100# Mesh cells 88400 88400 88400 88400Air/Aluminum cells ratio 60/1 60/1 60/1 60/1Latency/Interpolation ratio N/A 10 10 100Spatial discretization air 14 Cells/ 14 Cells/ 14 Cells/ 14 Cells/ Spatial discretization alu. 193 Cells/ 202 Cells/ 202 Cells/ 202 Cells/  t air 1.02366e-6 s 1.02366e-6 s 1.02366e-6 s 1.70609e-7 s t aluminum 7.31182e-8 s 6.99744e-8 s 6.99744e-8 s 6.99744e-8 s t Simulation/Base 7.31182e-8 s 6.99744e-8 s 2e-5 s 2e-5 s# Time steps base 1312942 1371931 4800 4800Computation time 37872 s 49410 s 2379 s 14603 sEffective cells/s 3.1e6 2.5e6 51e6 eqv. 8.3e6 eqv.Memory use 79 MB 82 MB 50 MB 63 MB% FD avg. mag. err. vs nor. N/A 0.06 -435e6 - 1Speed-up vs normal N/A 0.8 x 16 x 2.6 xMemory use vs normal N/A + 4 % - 37 % - 20 %Table 4.6: TINA Simulation Results and Parameters Thin Border-0.4-0.2 0 0.2 0.4 0.6 0.8 0  0.001  0.002  0.003  0.004  0.005MagnitudeTime (s)Waveform at A thin BorderNormalInterpolatedLatency 10Latency 100Figure 4.23: Thin Border Time Domain Results at A1334.4. Implementations of Fractional Latency 0 2 4 6 8 10 12 0  1000  2000  3000  4000  5000MagnitudeFrequency (Hz)Transfer Function A->G Magnitude Thin BorderNormalInterpolatedLatency 10Latency 100(a) Magnitude-1500-1000-500 0 500 1000 1500 0  1000  2000  3000  4000  5000Phase (degrees)Frequency (Hz)Transfer Function A->G Phase Thin BorderNormalInterpolatedLatency 10Latency 100(b) PhaseFigure 4.24: Thin Border Frequency Domain Transfer Function A ! G1344.4. Implementations of Fractional Latencyinterpolation and reduced data storage operations. Table 4.4 summarizes the results. With lessborder cells, larger speed-gains are expected, as will be illustrated in the next two cases.Accuracy wise, the fractional latency consistently underestimated the amplitude of the sig-nal, in time domain, but closely matched the signal phase. This may be due to the more limitedmaster bandwidth of this case, which was that of the base area, and thus 25 kHz, while the testsignal had components up to 5 kHz. In EMTP, it is usually recommended to use one tenth toone fifth of the simulation bandwidth for accuracy. Thus, in this case, one fifth was used.In frequency domain, the results are relatively close, except for a strong, exaggerated res-onance around 1 kHz, where the latency case significantly over-estimates the amplitude. Thisresonance causes a shift in the unwrapped phase, which otherwise fairly closely parallels thereference results. The origin of this resonance may be explained by the somewhat differentsimulation environment that exists due to the interpolation boundaries that result from frac-tional latency, as opposed to distributed interpolation used before. These may result in a slightdiscontinuity which, for certain frequencies and angles of incidence, may appear as a differ-ent impedance. Thus, complex border-structure-material interactions may result, which couldgive rise to spurious resonances in the complex structure. In fact, the resonance at 1.2 kHz isunder-estimated. This lends some credibility to the above explanation, where the slight changein the system due to the border caused a shift in the predicted main resonance frequency. Still,the averaged error was about 13 % for the ratio 10 latency case.Of interest is that the ratio 100 latency case, which is supposed to be more accurate than theratio 10 case, consistently performed worse visually in the plots, but better for averaged error.Later in this chapter, it is found that the ration 100 case is also less stable. This is likely due tonumerical or implementation issues, due to the many repeated computations in the presence ofthese errors, as is explained later in this chapter.Medium BorderWith a thinner border, there are now three times more air cells in the simulation, which canbe solved using fractional latency, and thus operate at a slower rate. This is very noticeablein the computation time and memory use, with a nearly 6 times speed-up and 1/3 less mem-ory required. Visually, the error in the computation increased, although phase performanceremains good. The averaged frequency-domain errors are smaller than before, although onemay wonder how usable this simple metric is in evaluating these results.It is of interest that the thinner boundary to the volume did not result in a much differentsystem response. Any changes are likely due to the presence of a second interface, from alu-minum to air, where in the previous case the aluminum was terminated at infinity, thus a infinitealuminum boundary around the air cavity. As such, the propagation of energy in this boundary1354.4. Implementations of Fractional Latencyis different. However, due to the large impedance mis-match between the air and aluminum,only a small amount of the acoustic energy in the air ends up interacting with the boundary,thus is transmitted into from the air volume, and vice-versa.This observation allows simulations to be modified in a way that reduces the computationalload, but still maintains reasonable accuracy. When the material impedances are sufficientlydifferent, one could substitute large sections of computationally expensive materials by othertypes, and make better advantage of the speed-ups and memory use reductions possible withfractional latency. This is, in fact, the reasoning behind a common simplification in TD-FDand TLM methods, where sufficiently high or low impedance boundaries are replaced by opencircuitsorshorts. Theserequirevirtuallynocomputation, andmaythusimprovecomputationalperformance.There is, however, an advantage of leaving at least some of the original material in placewhen the behavior of the system right at the boundary itself is of interest. Doing so avoids thetheextremeconditionsofopenorshortcircuit conditionsatthelocationwheretheobservationsmust be made.Thin BorderAlthough thin borders have proven to work in many different simulations done in preparationfor this thesis, this particular case causes the simulation to quickly become unstable whenfractional latency is used. In Figure 4.23, the simulation can be seen to diverge much moresignificantly, compared to the other cases. Soon after the time window shown the computationbecame unstable and the results oscillated to increasingly higher values.Through further experiments, by allowing the previous computations to continue much be-yond the time window of interest, similar instabilities were found when more and more cellswere of the medium requiring faster computation, in this case air. The instability appearedsooner, the more air cells there were in the simulation. From this, it seems that the stabilityissues are related to solution iterations, or more precisely, to the number of iterations summedover all the latency areas in the computation. A further hint to this is the visually worse perfor-mance of the ration 100 latency cases, which include ten times more solution cycles for thoseareas, resulting in earlier onset of instability. The issue is discussed further in the followingsection, together with a possible solution which will also be demonstrated.4.4.8 Issues with Fractional Latency in TINAThe current implementation of fractional latency in TINA still has a number of issues, whichwill be discussed here. In essence, the main problem appears to be a long-term stability issue,1364.4. Implementations of Fractional Latency-1-0.5 0 0.5 1 0.07  0.075  0.08  0.085  0.09  0.095MagnitudeTime (s)Waveform at A Full BorderNormalInterpolatedLatency 10Latency 100Figure 4.25: Fractional Latency Long-Term Stabilityprobablyrelatedtonumericalrounding. Apossiblesolutionwillbedemonstrated. Asecondaryissue is not a problem of fractional latency or TINA, but a potential user error that needs to behighlighted.Long-Term Stability IssuesWhen the prior simulations are continued for a long time, in most cases the fractional latencycomputations became unstable and unbounded. This happened repeatably with the main caseunder study in this thesis, although many other test configurations used did not show the prob-lem. Those cases were smaller, and contained far fewer cells in the mesh. The total numbercells and latency areas solved appear to have a significant influence on the long-term stabilityof the computation, as illustrated in the cases above.From Chapter 3, we find that the more a line is over-sampled, thus the more points perwave length, the closer the poles move to the unit circle. Small errors can thus push them intoinstability more easily than less sampled cases, where a poles are further away from the unitcircle, and there are thus more losses in the system. This is consistent with the observations,where smaller, coarser cases were never found unstable and larger, more detailed cases became1374.4. Implementations of Fractional Latencymuch more readily unstable.A typical example is shown in Figure 4.25, which is the same case as used before, withthe full border. All studied latency cases eventually became unstable, but the ratio 100 casebecame unstable much sooner than the ratio 10 case, at about 0.04 s versus 0.08 s, respectively,in the full border case.At this time, the origins of this behavior can only be speculated upon until a full stabilityanalysis of the fractional latency method is completed. However, a number of educated guessescan be made. The TINA implementation used float precision for the variables and computations inorder to conserve memory. Thus, as the energy in the system decays, numerical round-off errors between very small numbers can easily result in sizable errors when divisionsand multiplications are made. Cumulative errors in the computations, caused by insufficient accuracy in the interpola-tion, may result in run-away. This is possible, as the ratio 100 case performs many moreoperations than the ratio 10 case. The ratio 10 case, however, has a lower bandwidth, andthus introduces more dampening for high-frequency oscillations. The loss-less simulation does not allow the energy in the errors to be dissipated. Addingsome losses could be beneficial. This approach is persued later in the next section. There is an implementation error in the algorithm. It is quite easy to, somewhere in themanyinterpolationsandhistorymanipulations, beonetimestepoffinthehistorysourcesand thus compute with the wrong values, or have an interpolation factor computed be-tween incorrect time stamps. This will, over time, cause significant error. The border structure in the thin border case was only one cell wide. Although each cellin the simulation was chosen to be of a size that insures the desired spatial discretization,e.g. ten cells per wavelength, if the structure has only one cell in it, there effectively aspatial under sampling. In the medium border case, the border is ten cells wide, and wasmore stable. However, the normal and interpolated cases were stable in all cases, even inthe presence of thin borders. The fractional latency algorithm itself is unstable. This is rather unlikely, as interpola-tion was shown to be dissipative. However, the look-ahead, although using known andunchanging values, is essentially a violation of causality.It is, however, the author’s opinion that the instability is a result from a combination ofcumulative interpolation error, due to numerical round-off caused by insufficient precision of1384.4. Implementations of Fractional Latencythe “float” variable type, combined with the use of a loss-less, thus non-dissipative, mediumwithin the latency areas. This may push the effective latency area’s transfer function polesbeyond the unity circle.To offer a qualitative basis for this idea, from Figures 3.2 and 3.3, we find that the more in-terpolated line models are oversampled, the closer their poles approach the unit circle. Ideally,for the loss-less line model, they would have to precisely be on the unit circle. We also foundthat interpolation is dissipative, thus has a stabilizing effect.However, if round-off errors are present and accumulate, it is possible, when the case issufficiently over-sampled, and thus has its poles very close to the unit circle, that those errorscause some of the poles to exceed the stability limit. Thus, the more time steps and cellsinvolved in a computation, the more errors could accumulate, and the higher the likelihood ofinstability.When using interpolation in the line models for synchronicity, all cells have internal inter-polation, but also some dissipation. When using latency, the cells within a latency area are allimplemented using loss-less models, and interpolation only happens on the area borders. Thelatency interpolation in the borders naturally causes errors to and from the neighboring areas.These erroneous signals propagate through the loss-less medium within the latency areas and,at each interaction with the border, accumulate more error. Since there is only some dissipationin the borders, and the propagating energy only interacts with them as it reflects, further errorsare added with each reflection. There is no distributed dissipation, as in the interpolated case,which could aid with stability.In the simulations shown above, the wider the border, the less frequent the propagatingenergy in the area needs to interact with a border, and the less the errors accumulating eventoccurs, for each base-area time step. The thinner the border, the more frequent the interactionsoccur within the latency area for each base-area time step, and the quicker the errors accumu-late. The simulations show the stability and precision degrading with thinner borders. This alsohappens when more time steps within an area are used, which results in more events in the la-tency area per base-area time step, and thus more errors. The latter is illustrated by comparingthe ratio 10 and ratio 100 cases.A number of experiments could be performed to determine and confirm the cause of theinstabilities. First, increasingthenumericalprecision, bytransitioningto“double”asavariabletype, may give insight regarding the first two potential causes. If the computations are morestable using this type, numerical round-off is likely to be a factor. Second, the algorithm can berun an a small case and a couple of time steps computed by hand. The results could be verifiedto locate an implementation issue, and validate the third potential cause.The last cause is hard to validate, as it requires a complete, formal study of the algorithm1394.4. Implementations of Fractional Latencyand its stability. An abbreviated case could be composed, based on a single 1D transmissionline fitted with latency cells on both ends, and the study of Chapter 3 could be repeated. Thecase is, however, inherently more complex due to the variable interpolation factor and it maybe hard to find a closed-form solution.This reduced-complexity study should be sufficient to validate the fractional latency algo-rithm, as the TINA solution itself was not changed to implement fractional latency, and thesame circuit solutions are still used. The output interpolation does not affect the computationitself, so it does not need to be considered regard the stability of the method.It would also be interesting to implement a lossy line model in the simulation and use itin conjunction with fractional latency to introduce some dissipation in the latency areas andobserve stability in the presence of such losses.Finally, regarding the determination of locating any implementation errors is that the TINAalgorithm itself, it was found to be extremely robust. During the research, many serious errorswere discovered and fixed, however, they only had minor effects on the simulation results.Given the long computations required to trigger the instability in many cases (over seven hoursfor the full border case), and the literally millions of time steps over more than 88000 cells thatthis involves, it is hard to locate such issues, especially if they are caused by small round-offproblems or off-by-one issues in the history manipulations and interpolation functions.4.4.9 Losses to Aid StabilityIn order to help dissipate the energy associated with the errors in the fractional latency sim-ulation, a simple loss model was added to the TINA simulator, and used in both the normaland latency 10 thin border cases discussed earlier. Since the addition of losses makes the res-onances less steep, due to the decreased Q factor of the resonant structures, one must evaluatethis case on its own. Hence, a comparison with the normal case is made under the same losscondition.To implement the losses, the distortion-less line model of Appendix A.4 was used. Theattenuation function  was set to 10 3, which resulted in a loss multiplication factor of about0.9990005. Applyingtheselosses, thesimulationwasfoundtobeinagreementwiththenormalcase, and the instability disappeared completely. Under these conditions, the fractional latency,using the distortion-less line model, was eleven times faster than the normal case. In Figure4.26, we see that the simulation is now stable and in agreement with the normal case over thelong term. The negative going peak near the end is the beginning of the next impulse of theBLImp sequence. Figure 4.27 shows a detailed time-domain comparison between the normaland fractional latency case, while Figure 4.28 shows the magnitude and phase responses.1404.4. Implementations of Fractional Latency-0.1-0.08-0.06-0.04-0.02 0 0.07  0.075  0.08  0.085  0.09  0.095MagnitudeTime (s)Waveform at A thin BorderNormalLatency 10Figure 4.26: Thin Border Time Domain Results at A, Long-Term-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0  0.001  0.002  0.003  0.004  0.005MagnitudeTime (s)Waveform at A thin BorderNormalLatency 10Figure 4.27: Thin Border Time Domain Results at A1414.4. Implementations of Fractional Latency 0 0.2 0.4 0.6 0.8 1 0  1000  2000  3000  4000  5000MagnitudeFrequency (Hz)Transfer Function A->G Magnitude Thin BorderNormalLatency 10(a) Magnitude-3000-2500-2000-1500-1000-500 0 0  1000  2000  3000  4000  5000Phase (degrees)Frequency (Hz)Transfer Function A->G Phase Thin BorderNormalLatency 10(b) PhaseFigure 4.28: Thin Border Frequency Domain Transfer Function A ! G1424.5. ConclusionsFrom this, we may conclude that adding losses does suppress the oscillations and this lendscredibility to the notion that numerical rounding and its resulting error may be at the basis ofthe instabilities observed in the loss-less, and thus non-physical, cases.Under-Sampling the SolutionWhen setting the simulation band width, and thus the simulation time step for the base areaand output interpolation, what is really configured, with respect to the output generation, isthe rate at which the faster sub-systems are observed. It is thus very possible to under-samplethis output generation and encounter temporal and spatial aliasing issues while the simulationitself is internally correct. Visually, the time-domain output will not be able to follow fastervariations. The resulting samples themselves, however, are correct. This is illustrated in Figure4.29, which is an almost under sampled case.In this figure, the latency data points were shown, joined by a line. The normal and inter-polated cases were strongly oversampled to give precise results. It can be seen that the samplepointsobtained fromfractionallatencyarequite closeto thecorrect values. However, the band-width of the simulation wastoo low to catch the sharp peaks (high frequencycomponents). Thesignal in the plot had components up to 25 kHz, which is equal to the Nyquist bandwidth forthe simulation.Thus, when using fractional latency with a local fast implementation, one must take careto use a sufficiently fast sampling of the simulation to prevent under sampling the system. TheNyquist bandwidth is insufficient, as the fractional latency acts as a low-pass filter. In practice,oversampling by a factor of about five to ten gives good results, as was done in the rest ofthis chapter, where the signal bandwidth was limited to 5 kHz, compared to the simulationbandwidth of 25 kHz.4.5 ConclusionsIn this chapter, we introduced the concept of fractional latency and showed two implemen-tations thereof, with and without extrapolation. It was demonstrated how to use the methodto achieve synchronicity and achieve a computational performance gain over the previouslyshown interpolated line model approach by moving the interpolation out of the individualmodel cells and move it to the boundary of a latency area.Themethodoperatedthoughachievingsynchronisminanasynchronous,event-drivencom-putation and, in time-decoupled methods, allows operation without the need for an explicit ex-trapolation by means of transmission-line look-ahead. It is the novel combination of absolute1434.5. Conclusions-2-1 0 1 2 3 4 5 6 0  0.0002  0.0004  0.0006  0.0008  0.001MagnitudeTime (s)Waveform at A, Medium BorderNormalInterpolatedLatencyFigure 4.29: Nearly Under-Sampled Fractional Latency Casetime-stamp base synchronization and transmission-line look-ahead concepts make fractionallatency possible. These concepts allow the extension of traditional latency approaches, whichonly allow integer ratios between the time steps.Future work would focus on a detailed study of the stability and frequency-domain prop-erties of the algorithm, as well as improving the implementation to insure no algorithm errorsexist and investigate the long-term stability behavior by using higher precision variables in thesolver code. Also, the use of dissipative media and integration rule switching, should be inves-tigated to further aid long-term stability by introducing a level of dissipation in the simulation.144Chapter 5Experimental ResultsIn desperation I asked Fermi whether he was not impressed by the agreement be-tween our calculated numbers and his measured numbers. He replied, ”How manyarbitrary parameters did you use for your calculations?” I thought for a momentabout our cut-off procedures and said, ”Four.” He said, ”I remember my friendJohnny von Neumann used to say, with four parameters I can fit an elephant, andwith five I can make him wiggle his trunk.” With that, the conversation was over.Freeman Dyson ”A meeting with Enrico Fermi”, Nature #427, page 297)Inthischapter, varioustwo-dimensionalsystemswereconstructedandmeasuredtovalidatetheTINA method. The systems varied in size from roughly fifteen centimeters to over two meters.This allowed a reasonable range of different simulation parameter ranges to test the simula-tor with, and used cell sizes ranging from about one square centimeter down to one squaremillimeter. These cell sizes are consistent with the acoustic wavelengths in the media for theaudible range, as well as the spatial discretization accuracy required to faithfully represent thesystems in the simulation. Also, a novel method to compute impulse responses in simulationwill be shown, based on periodic, band-limited impulses computed in the time domain.5.1 Two-Dimensional Systems in a Three-DimensionalWorldThe TINA simulator, as presented in this thesis, can only compute one and two dimensionalsystems. This is not a fundamental limitation. However, we limited ourselves to two dimen-sions, as a result of the focus of the research. In order to perform experiments, we thus en-counter an interesting issue: how may one perform a two-dimensional experiment in a three-dimensional world?This is, of course, not possible, but a good approximation may be made if we limit onespatial dimension so that it becomes very small compared to the wave length. In practice,when using acoustics, this means that we can design a planar experiment if we constrain thegeometry with a sufficiently dense barrier in one of the spatial dimensions.1455.2. The Band-Limited Impulse Response MethodWhen this condition applies, two-dimensional plane-wave propagation will occur. Evenmore, the system becomes infinite in extent in the constrained dimension. A point sourcethus becomes an infinitely long line source due to the mirrored point sources introduced bythe boundaries. The original geometry of the system has become a slice projection. Thismeans that the usual 1=r acoustic pressure (amplitude) degradation, for a spherical wave frontemanating from a point source, is not applicable. It is replaced by 1=pr instead, which appliesfor cylindrical wave fronts from linear sources.The validity of this, that the 2D simulations exhibit cylindrical waves, was implicitly veri-fied with the vocal tract model lateron. This model was radially symmetric, and thus not a flat2D model. The simulation results for this model show poor agreement with the measurements,while the other models, being better 2D approximations, conformed reasonably well.5.2 The Band-Limited Impulse Response MethodTo reduce the computation time of the simulation to the strict minimum, the impulse responseof the systems was obtained, as this requires the simulation to take only as long as the inverseof the lowest frequency in the pulse. Since it is impossible, in an EMTP simulator, to exceedthe Nyquist frequency of the simulation when latency techniques are used, special care must betaken to ensure no excess spectrum exists, as temporal and spatial aliasing may occur, ruiningthe results. Also, if the impulse could be made truly periodic, a leakage-free spectral analysisof the time-domain signal would be possible by matching the FFT time window with the periodof the impulse.Traditionally, band-limited impulse responses are obtained by filtering a regular impulse,or by the use of Gaussian pulses. Both, however, result in signals with some spectral energy inthe tails. Although this amount may be low, it is non-zero.In this thesis, a novel method was used that produces perfect, band-limited impulses withany desired spectrum that can be matched precisely to the FFT parameters used for outputprocessing and obtaining the spectrum from the time-domain simulation.The signal required for this is little known, but is discussed by Moore in his book on com-puter music [58]. It is, in essence, the time-domain sum of all desired frequency components– stated differently: the conversion of a box-car spectrum into the time domain. The methodshown by Moore uses a closed-form, time-domain solution for this sum of sines that is very ef-ficient to compute in the time domain. From the expressions below, we may see it requires onlythree sine or cosine series development computations for any number of consecutive, summed1465.2. The Band-Limited Impulse Response Method-1.5-1-0.5 0 0.5 1 1.5 0  1  2  3  4  5  6  7  8  9  10MagnitudeTime (s)Band-Limited Impulse’blimp_sig.gpl’ using 1:2Figure 5.1: Time Sequence of a Band-Limited Impulse Signalfrequency components:n∑k=1sin(k ) = sin[(n + 1) 2]sin(n =2)sin( =2) (5.1a)n∑k=1cos(k ) = cos[(n + 1) 2]sin(n =2)sin( =2) (5.1b)Although the use of band-limited pulses for obtaining frequency responses is not new, wehave not seen the use of this type of perfect spectrum impulse, computed in the time domain,for impulse response measurements in simulation. Figure 5.1 show an example of a sequenceof these band-limited pulses.It must be noted that this is not an effective method to obtain impulse responses in practicalmeasurement, compared to, for example, a frequency sweep, as the full energy of the injectionis condensed in a narrow pulse. This results in relatively little energy that can be practicallyused in the experiments, and thus poor signal to noise ratio, unless many pulses are averaged.Also, practical amplifiers and loudspeakers frequently have problems faithfully reproducingsuch narrow impulses without significant harmonic distortion, which would again deteriorate1475.3. Experimentsthe experiment.However, when used in simulation, we have found this method to give very good results.Since each frequency component in the signal may be perfectly matched to the FFT frequencybins used in further processing, as the proposed band-limited impulse method is fully periodic,leakage-free signal processing is possible, which improves the observed signal-to-noise ratioin the computation.Although a time-domain computation was used to obtain the impulse signal, it can be de-rived from the frequency domain as well through analytical means using transform pairs, orthrough a suitable FFT transform.5.3 ExperimentsA number of experiments were performed to evaluate the TINA method. The first three setswere mainly exploratory, in order to learn how to design a suitable test configuration. It is thelast set-up, the expansion duct of Section 5.3.6 that was not only the best experiment, but alsoshowed the closest agreement with the simulation. Still, the earlier tests are included here toshow some of the pit falls and limitations of 2D systems and simulations and the care that mustbe taken when considering the termination of the 2D experiment into free-space.5.3.1 EquipmentThevariousexperimentswereperformedusingthesamesetofbasicequipment. Loudspeakers,microphones, and their associated pre-amplifiers varied, but due to the use of transfer functionsin all computations and comparisons, the influence of the signal chain was canceled out.FFT AnalyzerThe same FFT analyzer was used for all experiments, using the same settings. The device wasa SR770 Stanford FFT Network Analyzer, by Stanford Research Systems.The following FFT parameters were used: Span: 12.5 kHz Line width: 31.25 Hz Acquisition time: 32 ms Start frequency: 0 Hz1485.3. Experiments Center frequency: 6.25 kHz Averaging: 16Topreventtimealiasingduetothe finiteacquisitiontime, wemustverifythemaximumsizeof the environment that can be safely analyzed with the given time window. Such a criterion isgiven in [17], page 32. We quote:“For non-periodic signals time aliasing arises when the duration of each recordis similar to or less than the impulse response of the system under investigation,causing cross talk corruption in the signal processing.Time aliasing may be avoided by selecting the duration of each record to bemuch larger than the acoustic propagation times within the ... system, that ist  2xlc0t is the sample record length in seconds xl is the distance from the sample to thefurthest microphone c0is the sound velocity in meters per secondFor our system, with a spatial extent of two meters for some cases, we find that our timewindow t must be much larger then 11 ms. We meet this criterion, just barely, by a factor ofthree. Since all other systems are much smaller, we may consider the requirement met.”Microphones and amplifiersTwo different Br¨uel & Kjaer condenser microphones and amplifiers were used. For all experi-ments we used 1/2” types and the Nexus Conditioning Amplifier, except for the measurementsin Section 5.3.5 and 5.3.6, where we used a 1/4” model (type 4135) and the type 2609 micro-phone amplifier. For all measurements, the amplifiers were set to no weighting, and care wastaken that they were not driven in saturation.Power amplifier and loudspeakerThe amplifier used was an Alesis RA100. The loudspeaker was a Realistic midrange tweeter,cat. # 40-1288A fitted with a custom funnel to create a point source. For Section 5.3.6, aRadioSchack 4” midrange speaker cat. # 40-1197 was used on the source plate with a 1mmwidehole. OverdrivedistortionwasevaluatedbyFFTanalysisoftheoutputsignalandavoided.1495.3. ExperimentsAir humidityIn order to determine the speed of sound, we had to measure the air temperature and humidity.This was done with a wet and dry bulb thermometer, HB Psycro-Dyne, cat. # 23015. The fanwas allowed to run for one minute before the readings were taken. The results for the differentdays of measurement were found to be very similar, specifically: 22 degree Centigrade, 60 %relative air humidity. This yielded in an estimated speed of sound of 345 m/s. It was found,however, that the simulations in the most accurate system, the expansion duct of Section 5.3.6,showed much better agreement with a speed of sound of 330 m/s, which represents a 4 %deviation from the value found through the standard equation. This value was consequentlyused. The error could be in part due to inaccuracies in the experimental configurations, as wellas the measured environmental parameters.5.3.2 3D Vocal-tract modelInitially, we thought that the TINA method possessed circular symmetry in 2D. This wouldhave been a perfect match for the vocal-tract model, which is also circularly symmetric (thisis why it was studied). However, closer investigation revealed that TINA does not have suchsymmetry, but rather extends in an infinite plane. As such, the vocal-tract cross-section wouldappear as an infinitely wide “corrugated roof” as opposed to the desired profiled tube.We can deduce this from the assumptions made in the 2D model. In order for the acousticenergy not to spread above or below the plane, we either need perfect reflection, or equalacoustic pressure above and below the plane. As such, a point source becomes a line source.In the other experiments, this situation was emulated by using wooden boards as reflectors, sothat an infinite-plane approximation could be achieved and the sources in effect became linesources perpendicular to the plane.Experimental set-upThe physical measurement setup is described in Anderson [2]. In order to render the vocal-tract model in the computer, a special plug-in for the TINA simulator was written. It allowsany binary TIFF image to be drawn as a structure. Using this model, we used an optical scanfrom the physical model, processed it into a black and white binary image, and then renderedit in the simulator as aluminum. The process is illustrated in Figure 5.2. The top image is thescan. The second image is the binary outline, where black areas are rendered as a specifiedmaterial in TINA. The third image is the actual simulation structure as used by TINA. Thegreen border is the environment termination, the blue air, red and yellow aluminum, light-bluean ideal absorber, and off-white the acoustic source.1505.3. ExperimentsFigure 5.2: TINA Vocal-tract-model processingComparison with measurementsFirst, it must be noted that the measurements were done with a 31.25 Hz frequency step size.For the simulations, however, a 100 Hz interval was used for the vocal tract model. As such,dense clusters of resonances were blurred together in the simulation results.We do seem to cap-ture the general trend of the resonances (Figure 5.3). However, the magnitudes and frequenciesof the resonances are very different. These errors are mostly due to the incorrect symmetryassumptions, as well as reflections from the simulation boundary. The measurement point atthe end of the vocal tract is very close to the boundary, thus any reflections from it will havea significant effect on the simulation output in both magnitude and phase due to interferenceeffects. The phase plot is completely unusable, as the errors in resonant frequency also causelarge changes in the phase. With correct behavior, the simulated phase response would be akinto the envelope of the measured phase.1515.3. ExperimentsFigure 5.3: Comparison of Measured and Simulated Acoustic Pressure for the Vocal Tract1525.3. ExperimentsDue to the erroneous symmetry, the various resonances seen in the model do not agree well,as the simulated case results in wholly different structure than the physical model. In addition,the cells in the simulation were 1 mm by 1mm. There is an error of 1 cell in the drawingfunctions due to roundoff. Since some features in the model are only 3 mm in size, this wasa significant factor. However, compared to the wavelength, this error is still relatively small,and the response of the system should still be quite similar. The overall trend is still present,however.5.3.3 2D Room With Source at an EdgeIn order to emulate 2D plane waves as they appear in TINA, we need to construct a suitable2D experimental approximation. A practical method to such an approximation is to spatiallyconfine the acoustic waves in a narrow duct. If the constricting boundaries chosen are highlyreflective with respect to the acoustic medium, a 2D approximation is obtained, as long thefrequencies under investigation are below cutoff for all propagation modes of the duct.The reflective boundaries have the effect of giving the system infinite extent in the Z-axis.That is, a point source is converted into a line source in the Z axis, assuming the 2D plane is inthe X-Y directions, due to the reflections.In this first experiment, the source was placed at the edge of the structure. This was, inretrospect, not a good choice, as the discontinuity of the boundary to free space introducespartial reflections. The experiment was subsequently refined, the results of which are describedin section 5.3.5.For each of the two experiments, two configurations were measured: An open room: the waves were only restricted in the Z-direction. The X and Y dimen-sions were left unbounded. (The boundaries were the end of the experimental structure,where the waves terminate into free space. A tapered duct: the waves were now restricted in the plane by two more boundaries.A tapered shape was chosen to provoke resonances at different frequencies. The non-restricted dimensions of the duct terminated into free space and/or the 2D room approx-imation.Experimental set-upThe system was built out of plywood panels, spaced 3.8 cm apart using small wooden blocks,as illustrated in Figures 5.4 and 5.5 (cover board removed to show duct). This physical distanceallowed for the placement of the microphone and point source in-between the panels. However,1535.3. Experimentsthe distance also puts an effective upper limit on the frequency for which the two-dimensionalapproximation is valid. Furthermore, the lateral extent of the plywood was only 100 cm fromthe source on the Y-axis, and 221 cm on the X-axis. As such, the limiting dimensions are theZ-axis spacing of 3.8 cm and the Y-axis extent of 1 m. The waveguide cut-off frequency canbe found from equation 5.2. n and m are the propagation modes, while a and b are the spatialdimensions.fc = c2√(na)2+(mb)2(5.2)Investigating the fundamental modes (1,0) and (0,1), with an acoustic wave speed of 345m/s, a 3.8 cm by 100 cm wave guide has cut-off frequencies of about 4500 Hz (for 3.8 cm)and 170 Hz (for 1 m). Using these results, we choose 4000 Hz as an upper limit and 200 Hz alower limit for the two-dimensional approximation.FECBA D0. dimensions in metres1.0450.502.09Figure 5.4: Experimental Set-Up With Edge Source - Open Plane5.3.4 Comparison with measurementsThe transfer functions between point A and various other points in the system were computedfrom the raw measurement and simulation data. To reduce noise, the measurements were aver-1545.3. ExperimentsFECBA D0. dimensions in metres0.502.090.685 1.4051.0450.015Figure 5.5: Experimental Set-Up With Edge Source (Cover Removed) - Ductaged over 16 acquisitions. Also, the simulation was done with an infinite-plane approximation,while the measurements have impedance discontinuities at the boundaries that could not yetbe modeled, as this requires three-dimensional geometry to describe the end of the woodenpanels. However, the simulation boundary itself still exhibits some reflection, due to a match-ing problem. These reflections cause significant distortion in the simulation output, which isclearly visible in the phase plot. All these factors significantly distort both the measurementsand the simulation.For the C-location, the magnitude and phase responses were over-layed for direct compar-ison (Figures 5.6 and 5.7). The results were poor, given that an open plane should yield aperfectly flat response. However, the plane was of finite extent, and thus exhibited a plane tofree-space discontinuity, yielding reflections.For the duct, however, the situation improved somewhat, as the duct enclosed the source(Figure 5.5), effectively removing the 2D plane boundary discontinuity on that side from thephysical system, as well as the sides. However the other end of the duct still terminated in freespace. The situation is similar in the simulation. In the simulations, the duct spatial dimensionswere also truncated wrt the physical system.For higher frequencies, the duct resonances became so close together that they, for themeasured result, blended in a continuous bump. The simulation, due to the absence of losses,shows a strongly exaggerated magnitude response for these frequencies. Still, from the openplane simulations, we can see that the reflections in the simulation are in the order of 0.2 -0.4 relative magnitude. As such, the noise floor is so high that the majority of the duct data isessentially meaningless, except for the strong resonances at the higher frequencies. Thus, wemust conclude that the higher frequency bump, the only valid part in the simulation, exists in1555.3. Experimentsthe measurements, but the experiment can only be considered indicative at best.Figure 5.6: Comparison of Measured and Simulated Acoustic Pressure (Point A!C) - OpenPlane1565.3. ExperimentsFigure 5.7: Comparison of Measured and Simulated Acoustic Pressure (Point A!C) - Duct1575.3. Experiments5.3.5 2D Room With Central SourceIn order to prevent the problems associated with having the source close to the boundariesin the previous experiments, the source was next placed in the center to improve on the 2Dinfinite-plane approximation. Also, the source was now placed at the wide end of the duct toreduce the influence of the impedance discontinuity at the end. In the new configuration, thediscontinuity into free space is now reduced by the area exposed by the narrow end of the duct(Figure 5.9). Future improvements in the simulations should then yield results closer to themeasurements.Experimental Set-UpThe system was constructed in the same way as in Section 5.3.3. Figures 5.8 and 5.9 (coverboardremovedtoshowduct)illustratethedifferencesintheconfiguration. Thevalidfrequencyrange for the experiment is now somewhat different for the Y-dimension. It is now 120 cm,which puts the cut-off frequency slightly lower at 140 Hz. However, for consistency in thedata, we kept the previous 200Hz to 4000 Hz valid frequency range. A further addition was acustom frame to position the loud speaker in the setup. It allowed the source to be positionedvertically, and the depth of the funnel tip into the setup adjusted to the desired depth, whichwas midway in the clearance between the wooden panels.All dimensions in metres3.29S1.222.42 A J1.321.221.42(One point every 10 cm)B...IFigure 5.8: Experimental Set-Up With Central Source - Open Plane1585.3. ExperimentsAll dimensions in metres3.291.222.42 A1.321.221.42(One point every 10 cm)B...I JS5050112 132Figure 5.9: Experimental Set-Up With Central Source (Cover Removed) - DuctComparison With MeasurementsThe transfer functions were computed in the same way as in Section 5.3.3. Once more, bound-ary discontinuities in both the measurements and simulations existed. The measurements aver-ages agree reasonably well with the simulations. We attribute remaining errors to the boundaryreflections, bothinthephysicalset-upandthesimulation. Also, theresonancesinthesimulatedduct were not dampened, due to the absence of losses, which resulted in significant magnitudedeviations.The duct simulations do show most of the expected resonances, but the magnitudes areoff due to the undamped system. The phase response is in reasonable agreement with themeasurements. One has to, however, keep in mind that phase jumps occur in the plots. Thesewild flips are generally due to a small increase or decrease that causes the phase calculationsto wrap around. If one mentally unwraps the phase, the results are close.The boundary reflections are still a significant source of error, in the range of 0.2-0.3.However, due to the strong resonances, the signal to noise level in these simulations is muchgreater than before, and valid conclusions on the resonance frequencies of the system, as wellas phase response, could be obtained.1595.3. ExperimentsFigure 5.10: Comparison of Measured and Simulated Acoustic Pressure (Point C!A) - OpenPlane1605.3. ExperimentsFigure 5.11: Comparison of Measured and Simulated Acoustic Pressure (Point C!A) - Duct1615.3. Experiments5.3.6 Expansion DuctUsing the experience gained from the preceding experiments, a final, high-precision experi-ment was devised to validate the TINA method. Stiff boundaries from vocal tract combinedwith well-defined and simple, planar geometry of duct experiments, into a closed system, soboundary issues were avoided. The configuration gave good agreement with the simulateddata.Experimental Set-UpThe system, shown in Figure 5.12, consisted out of a duct, cut in a 2 mm thick plate of alu-minum clamped between a sandwich of two 2 mm thick aluminum plates, backed by plywoodsheets. The thin duct allows a much higher usable frequency range for the 2D approximation,while the stiffness of the metal, clamped by a multitude of bolts, and loaded with the mass ofthe plywood, prevented the strong spurious resonances that plagued the large setups describedearlier in this chapter.20 205405003002004090140190A GFigure 5.12: Expansion Duct Experimental Set-UpThe loudspeaker injected its signal through a 1 mm wide, 2 mm tall slit in a thick steelplate, from the edge of the configuration. The system is entirely closed, so no discontinuitiesto free-space, nor absorbing boundary condition issues arise in measurement and simulation.The microphones were inserted through holes in the top, while each unused hole was pluggedwith an aluminum insert to maintain continuity in the duct. Due to excess radiation from theback of the loudspeaker interfering with the measurements, it was fitted with a box made of1” thick particle board, sealed with foam against the speaker mounting plate. This reduced thespeaker interference to roughly 2 % of the average measured signal.Lastly, the physical size of the microphone itself was now taking into account, which is asizable portion of the wave length for the higher frequencies evaluated. Thus, in the simulation,1625.3. Experiments 0 1 2 3 4 5 500  1000  1500  2000  2500  3000  3500  4000  4500  5000MagnitudeFrequency (Hz)Transfer Function A->G MagnitudeMeasuredSimulationFigure 5.13: Expansion Duct Magnitude Plotthe acoustic pressure over an area comparable to that of the physical microphone was averagedto obtain the result for that location.Comparison With MeasurementsExcept for the magnitude overestimation, due to the absence of losses in the simulation, theresults from Figure 5.13 show excellent agreement. There were some issues with the accuracyof the duct itself, due to a problem with the water jet cutter, with the duct dimensions in error byroughly 1 mm. The simulation, however, was performed with a 2 mm cell size, thus relativelycoarse. As such, the errors in the physical dimensions still below the spatial discretizationresolution of the computation.The measurements themselves presented some issues due to the strong filter effect of thetiny speaker aperture, resulting in insufficient acoustic energy in parts of the spectrum. Thismay be observed from the FFT analyzer screen shots in Figure 5.14. Still, the simulationfollows the trends visible in this reduced signal.1635.3. ExperimentsFigure 5.14: FFT Analyzer Screen Shots1645.4. Magnitude Over-Estimation5.4 Magnitude Over-EstimationIn all simulations shown, the magnitude of the simulated spectra is significantly higher thanthose in the measurements. Also, for higher frequencies, an increasing shift in the systemresonances can be observed.Figure 5.15: Illustration of Loss-Less Model ErrorsThis behavior is typical of the use of loss-less models in the computation. This phenomenawas studied by Mart´ı [50, 51] during the development of the frequency-dependent line modelin the EMTP. A relevant plot from [50] illustrating this issue is reproduced, with permissionfrom the author, in Figure 5.15. In this figure, a solution for a 60 Hz power system withfrequency-independentparametersfor60Hz(P60)iscomparedwithacomputationusingexactparameters (C). It was found that the addition of correct, frequency-dependent losses resolvedthe mismatch.In the computations shown in this thesis, the use of the distortion-less line model did resultin a more accurate computation, but due to the frequency-independent nature of those losses,the improvement was limited. Frequency-dependent losses are important for acoustic phenom-ena in air, as the attenuation for 1 kHz is approximately 5dB/km, for 10 kHz approximately 95dB/km, and for 100 kHz approximately 4000 dB/km. Thus, these losses are both significant1655.5. Conclusionsand far from constant over frequency. Future work would definitely require the adaptation ofthe frequency-dependent transmission line model to the TINA simulator.5.5 ConclusionsThe TINA method works quite well, in that the simulation results give a reasonably closeagreement with the measured data, when those measured systems present a good 2D approxi-mation. The resonances are over-estimated, which is due to the absence of losses in the currentTINAmediamodels. Withoutlosses, resonancesexhibitatoohighQ-factor, andthusexcessivemagnitude. However, the predicted frequencies show good agreement.Future work would include the development of lossy media models and better techniquesof determining the acoustic parameters of the media in use. Also, the band-limited impulsemethod for transfer function computation requires further development, give the highly en-couraging experiences with this method throughout the thesis work in obtaining clean transferfunctions from the simulations.166Chapter 6Conclusions and Future WorkBefore I came here, I was confused about this subject. Having listened to yourlecture, I am still confused. But on a higher level. Enrico FermiIn this chapter, we will summarize the main contributions of the thesis, as well as out-line themost pertinent future work.6.1 ConclusionThe TINA method proved to work reasonably well. Although the method requires more com-putation than the basic TLM method it is directly related to, it offers increased flexibility and,when fractional latency is used, may even out-perform the basic TLM method in computationalspeed. Synchronicity through interpolation performs well, accuracy wise. The constituent, 1Dloss-less line models were characterized analytically and shown to be numerically stable. Sincethe TINA method is an EMTP-derived circuit solution, it is also conserves energy. From thosestudies, accuracy requirements were formulated.Through a number of experiments, the TINA approach was validated. Comparisons withEMTP showed the algorithm to be implemented correctly, while comparisons with variousmeasurementsshowedthe accuracy, butalsothelimitationsofa2Dsimulator. Also, toperformthe comparisons, the band-limited impulse method for transfer function calculation proved anexcellent approach, yielding clean spectra with a high signal to noise ratio.All in all, the loss-less TINA method performed as expected. Any resonances found wereover-predicted, due to lack of losses, but the frequencies were correct. The addition of losseswill resolve this issue. Also, the fractional latency technique worked quite well, except itshowed poor long-term stability. This is likely due to numerical issues, and further work willbe required to identify and fix these issues, as the method shows high promise, being ableto obtain good results at significant, order of magnitude, computational speed increases. Theaddition of losses did resolve the long-term stability issue1676.2. Contributions6.2 Contributions The development of TINA, a TLM-like method that allows the direct use of EMTP-styletransmission line models in the mesh, as well as any other circuit in its cells. Eachsuch cell is interfaced with the simulation though Th´evenin equivalents, and performs itsown internal solution. Through the pervasive use of transmission-lines, each such cellis time-decoupled from the next, allowing each cell to be solved independent from theother. The detailed, analytical study of the EMTP loss-less line model, both in its ideal andinterpolated form. Accuracy equations were derived, allowing the estimation of the nu-merical precision of the model under a given set of operating conditions. This study hasnot yet been reported in the literature and showed, amongst others, where the EMTP ruleof thumb for the interpolated line, that its traveling time must be five to ten times thesimulation time step, originates. The models were shown to be stable. The interpolatedmodel was shown to be dissipative, as a low-pass filter, with the error also dependingon where in the interpolation interval the result needs to be found. Also, the terminationconditions of the line were found to have a significant contribution to the total error. Latency techniques were expanded to fractional ratios. Use of the new fractional latencytechnique results in an event-drive, asynchronous computation that requires special tech-niques to not only solve, but also extract information from. The concepts of absolutetime and transmission-line look-ahead were introduced. The latter is an exact predictiontechnique based on the time-decoupling offered by the transmission lines. A new technique to obtain impulse responses in simulation, based on band-limited, pe-riodic impulses. By matching these pulses, and their spectral content, to the FFT usedto obtain the spectrum from the time-domain TINA solution, a leakage-free spectrummay be obtained with any desired spectral resolution. Also, since there is no spectralleakage, the signal to noise ratio is high. A computationally efficient method to computethe pulses in the time domain is also shown. The 2D TINA method, and thus by extension also the 2D TLM method, was verifiedfor a number acoustic experiments. It was found that good, 2D approximations requiremuch care to construct, but that the 2D computations could give good agreement withthem. Losses in the experimental configurations were relatively high, compared to theloss-less computations, resulting in over-estimation in the simulation transfer functions,compared to the measured ones. However, the predicted resonance frequencies werewell in agreement.1686.3. Future Work6.3 Future Work Theadditionoflossy,frequency-dependent,andnon-linearmaterialcellsbasedonEMTPline models. Extension of TINA to 3D. This will require parallelization of the method to maintainreasonable computation times. Parallelization of the TINA method. A parallel TINA solver core was written, usingOpenMP, but was not fully completed. Parallelization of the TINA method on a GPU. The self-contained, time-decoupled cellsolution appears an ideal fit for computation on a GPU, which has literally hundreds ofsmall arithmetic units that can perform simple operations on a small data set. These arean ideal fit for the cell computations, which are repetitive, simple operations to solve theinternal circuit and the line models and do not require much, if any, branching. Identification of methods to find the required model parameters for the lossy models,from real-world materials. Maybe impedance tube experiments could be used for thispurpose [17]. Further development of the error expressions for the interpolated line models under fullygeneralised termination conditions. Further study of the stability, dispersion, and dissipation of more advanced line models,such as the CP (constant-parameter) and FD (frequency-dependent) line models. Further study of the fractional latency technique to characterize it better and determinethe cause of the long-term stability issues. Investigate the use of integration rule switching to criticaly damp the long-term instabil-ities found in detailed fractional latency simulations. 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Boron.178Appendix ADerivation of the 1D Transmission LineModels in Time and Frequency DomainLife can only be understood backwards, but it must be lived forwards.Soren KierkegaardEMTP-style transmission line models were used and manipulated throughout this thesis. Giventhe pivotal role of these models in this research, their origins and derivation will now be dis-cussed.The EMTP models have a long history, starting with the work by Bergeron in the earlythirties on water hammer in hydraulics. He demonstrated that, to a certain degree of approx-imation, the one-dimensional hydrodynamics model he used has a similar form as the onedimensional wave equation used in electro-magnetics, with similar solutions. Bergeron notedthis fact and explored it in [6], where he demonstrated the similarities between water hammerin hydraulic systems and lightning strikes in electrical systems. Bergeron’s approach was con-sequently used by Dommel in his development of the EMTP [90] during the late sixties, dueto the elegance and simplicity of the model, when translated in numerical form. This modelbecame the standard model in EMTP, known as the constant parameter, or CP line model [20].The CP line model is a one-dimensional transmission line approximation that incorporatesthe distributed line impedance, lumped losses, and a time delay, all independent of frequency.When the losses are set to zero, only the impedance and time delay remain. This is the coreof the loss-less, energy propagation model used in this thesis. Since the EMTP line models,including those that have constant parameter and frequency-dependent behaviors [20, 51, 52],have this impedance and time delay model at the core, and the more complicated behavior canbe “bolted on” the basic model, only the basic model will be discussed in this appendix forclarity.A.1 The 1D Wave and Telegrapher’s EquationsThereareatleasttwowaysbywhichthepropagationelectro-magneticenergyinonedimensioncan be considered. The more general approach derives from Maxwell’s equations [54, 55] and179A.1. The 1D Wave and Telegrapher’s Equationsis a field approach, while a simpler and more intuitive approach derives the wave equations interms of a distributed-parameter circuit of a transmission line, where the behavior of voltageand current waves traveling along uniform conductors is studied. The latter is an equivalentcircuit approach. Both methods are well known and extensively treated in many references,e.g.: [7, 46, 52, 71, 80], and only the final results will be given here.L dz R dzC dz G dzFigure A.1: Differential Length of Lossy Transmission LineWhen solving for the one-dimensional, differential circuit equivalent from Figure A.1, weobtain the Telegrapher’s equation, which has the same form as the corresponding field solutionfrom Maxwell’s equations, but different parameters are involved. This brings up the issue offield to circuit mapping, and is at the core of some of the complexities that occur when com-puting a field solution using networks. Kron discussed this in his work on electrical networksto approximate the Maxwell field equations [36, 37], and later Johns [18, 31] with TLM.For the one-dimensional case, there is a direct one-to-one mapping, however, for higherdimensional cases, this is no longer the case and correction factors have to be introduced forsome of the parameters to assure correct results. In this thesis, the topic of field to circuitmapping is discussed in Chapter 2 for the proposed method in 1D and 2D, and the correctionparameters given where required.Of importance for our discussion here are the equivalences between circuit and field pa-rameters for the one-dimensional case. This is because the circuit parameters, expressed interms of distributed resistance, inductance, and capacitance are measurable for a structure, butare hard to define for a bulk material. Conversely, the field parameters of conductivity, perme-ability, and permittivity are measurable for a bulk material, but not easily found for an arbitrarystructure.Since the TINA simulator is conceived around constructing shapes of interest from bulkmaterials in order to find the emergent behavior of the structure, it uses field parameters asinput, but uses circuit equivalents internally for the component transmission lines that make-upthe simulation grid. It thus uses these conversions internally.180A.1. The 1D Wave and Telegrapher’s EquationsThus, both forms of the wave equation will be given here, as well as the conversion tablebetween the various parameters in the 1D case. From here on, the discussion will, however,concentrate on the Telegrapher’s equation form, as it relates more intuitively to the componenttransmission lines that are at the core of the TINA simulator, and are more familiar within thefield of power systems than the field equations are.From [46], the Telegrapher’s (wave) equations for the lossy, distributed circuit in the z-direction, with R is the resistivity, G the conductance, L the inductance, and C the capacitance,as illustrated in Figure A.1 are:@2v(z;t)@z2 = RGv(z;t) + (RC + LG)@v(z;t)@t + LC@2v(z;t)@t2 (A.1a)@2i(z;t)@z2 = RGi(z;t) + (RC + LG)@i(z;t)@t + LC@2i(z;t)@t2 (A.1b)while the wave equations from Maxwell’s theory, with  is the electric conductivity,  isthe magnetic susceptibility, and  is the dielectric permittivity are:@2Ex(z;t)@z2 =   @Ex(z;t)@t +   @2Ex(z;t)@t2 (A.2a)@2Hy(z;t)@z2 =   @Hy(z;t)@t +   @2Hy(z;t)@t2 (A.2b)We notice that there is no counterpart for the resistivity R in the field equations. Whenneglecting that term, we find that the Telegrapher’s equations take the same form as the waveequations and the following correspondences are found:Circuit Fieldv(z;t)  Ex(z;t)i(z;t)  Hy(z;t)G   L   C   When all losses, thus R and G, or the corresponding  , are neglected, we find the loss-less wave equation which forms the basis for the loss-less line models derived below. Theseloss-less, constant-parameter wave equations and resulting transmission-line model is quiteuniversal, and has the same form across many diverse disciplines. Within the this thesis, thisis taken advantage of to model acoustical phenomena, as the governing equations in 1D and2D, for the loss-less and non-dispersive case have exactly the same form as those for electro-magnetic fields, e.g.: [13, 33]. Note that this is no longer the case for 3D, as the acoustic waves181A.2. The Ideal, Loss-Less Line Modeldo not have polarization, and the model is just a simple extension of the 2D case, while theelectro-magnetic model becomes much more complex to accommodate polarization.A.2 The Ideal, Loss-Less Line ModelBy solving the Telegrapher’s equation (A.1) for the loss-less, constant-parameter case, theone-dimensional loss-less, constant-parameter transmission-line model can now be obtained[46, 52].The model is based upon the d’Alembert solution to the wave equation (A.1), and considersthe voltages and currents at the terminals of the line. Since the line is lossless, it effectivelyrepresents a time delay.The characteristic impedance of the line, in the loss-less case, is found as: Zc = √  andthe wave speed as a = 1p  A.2.1 Derivation in the Time DomainxK Mi(x,t)v(x,t)x=lx=0Figure A.2: Transmission LineThe d’Alembert’s solution, for a sinusoidal waves, which presumes two waves traveling inthe opposite direction.p = p+ cos(!t kz) (A.3a)p = p cos(!t + kz) (A.3b)We can write the voltages and currents at any point at the line as follows [52]:v(x;t) = vf(x;t) + vb(x;t) (A.4a)i(x;t) = if(x;t) + ib(x;t) (A.4b)With f the forward wave, b is the backward wave. Now, combine both the voltage andthe current together in one wave by relating the current to the voltage by means of the line182A.2. The Ideal, Loss-Less Line Modelimpedance. The wave speed is given by a.v(x;t) = vf(x at) + vb(x + at) (A.5a)i(x;t) = 1Zcvf(x at) 1Zcvb(x + at) (A.5b)Defining the traveling time  = la, with l the line length, we can add both equations (A.5),and write their total contribution for the forward perturbation wave for a point on the line (itsuffers no reflection regardless of termination):v(x;t) + Zci(x;t) = 2vf(x at) (A.6)From this, using the traveling time, we can express the terminating end of the line in func-tion of the sending end. With (t) values the present time values and the (t   ) values thehistory terms, we find:vm(t) + Zcim(t) = vk(t  ) + Zcik(t  ) (A.7)A.2.2 Formulation of the EMTP time-domain CP-Line modelZc Zcvk(t) emh(t)ekh(t)im(t)ik(t)vm(t)Figure A.3: CP Line ModelNotice that in (A.7), the current flows into one end and out the other, as shown in FigureA.2. This is not convenient in practical use, and the model will now be re-formulated to havethe current flow into the model from both k and m sides. The resulting pair of equations, onefor each side of the model corresponding to Figure A.3, are:vm(t) Zcim(t) = vk(t  ) + Zcik(t  ) (A.8a)vk(t) Zcik(t) = vm(t  ) + Zcim(t  ) (A.8b)183A.2. The Ideal, Loss-Less Line ModelRe-formulated in a circuit equivalent with history sources we find:ekh(t) = vm(t  ) + Zcim(t  ) emh(t) = vk(t  ) + Zcik(t  ) (A.9)This model has many advantages over a direct solution of Maxwell’s equations. It is notonly numerically simple, it also time-decouples nodes K and M. It is this time decoupling ofthe equations that allows the TINA model to function and give it its flexibility. The line modelis stable and exact, except for interpolation errors if  is not an integer multiple of  t, thesimulation time step, as shown in Chapter 3, and no integration rule was explicitly used in thederivation.A.2.3 Conversion into the Frequency DomainSince part of the stability analysis was performed in the frequency domain, we will now showhow to convert the time-domain form of the loss-less CP-line model into the frequency domain.Conceptually, this comes down to recognizing that the model is essentially a pure delay. Inthe frequency domain, the exponential function is the equivalent of the delay in time domainand does not change the signal in any other way.Thus, starting from (A.8), for valid for the circuit in Figure A.3, and using substitutions ofthe type v(t) = <fVej!tg, we find, for each end of the line:<fV kej!tg Zc<fIkej!tg = <fV mej!(t  )g+ Zc<fImej!(t  )g (A.10a)<fV mej!tg Zc<fImej!tg = <fV kej!(t  )g+ Zc<fIkej!(t  )g (A.10b)Grouping and re-arranging the terms, we find:(Vk  ZcIk)ej!t = (Vm + ZcIm)ej!te j! (A.11a)(Vm  ZcIm)ej!t = (Vk + ZcIk)ej!te j! (A.11b)After simplification we find the final frequency-domain form of the loss-less line:V k  ZcIk = (V m + ZcIm)e j! (A.12a)V m  ZcIm = (V k + ZcIk)e j! (A.12b)184A.3. The Interpolated Loss-Less Line ModelA.3 The Interpolated Loss-Less Line ModelLike most digital, time-marching systems, in EMTP and TINA time is implicit to the solu-tion. The system only knows of the progression of discrete steps and it is the user who assignsmeaning to those steps when processing the results. The above time-domain model computesa result at each simulation time step, using past values that were computed at prior simulationtimes. The amount of time in the past the model refers to is given by the length of the line  .When that line length is an integer multiple of the simulation time step, a suitable past valueexists in the history storage and a solution can be found. However, in a practical system, multi-ple materials are frequently required for an accurate representation of reality. These materialsrarely have wave speeds that work out to integer multiples of each other’s propagation time.As such, there is a problem of synchronicity, where different time steps would be required fordifferent material regions within a simulation.Thus, the models with non-matching transmission time would have to be forced into aninteger multiple of the shared simulation time step, which changes their effective transmissiontime, and thus properties, or the models have to be allowed to operate at their own, local timestep.One way by which the synchronicity between the simulation time step and the model trav-eling time can be restored is the use of time-step interpolation. This is the method used inEMTP simulations of power systems [20, 52]. This approach can also be used for elementsthat are not transmission lines, and the simplicity of implementation and computation make itsuitable for use in TINA.A.3.1 The Need for InterpolationWhen implemented on a digital computer, the expressions of (A.8) are programmed so thatthe past values in the equations, referred to by the indices (t   ), are available in programmemory in order for the results at the current time step (t) to be computed. For a given  ,which is defined by the length of the line l and its wave speed a, we can find the number ofhistory values that need to be stored in memory from the simulation time step  t: = al (A.13)history depth =   t (A.14)When the  of the line is an integer multiple of the simulation time step, this history depth willbe an integer number, and the value is available in memory since data is saved in the historystorage at each time step.185A.3. The Interpolated Loss-Less Line Model~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t(ms) Vk Ik Vm Im00. History Values in Memory,  t = 0.02 mst(ms) Vk Ik Vm Im00. 0.180.03 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~(b) History Values in Memory,  t = 0.03 msFigure A.4: Non-Interpolated and Interpolated History TablesHowever, when the  of the line is no longer an integer multiple of the time step, werequire access to history values that lie in-between exiting data in order to compute the nextcontribution at the current time step (t). The history memory depth is then rounded-up to thenext integer value to accommodate the time-step bound data and interpolation is used to obtainthe required history value.In Figure A.4(a), a simulation time step of  t = 0.02 ms and a traveling time of  = 0.10ms are used. It can be seen that, for a simulation time (t) of 0.18 ms, the required history valueof 0.08 ms is available in the table and the current value can be readily computed.In Figure A.4(b), a simulation time step of  t = 0.03 ms is used. This time, the requiredhistory value of 0.08 ms is not available in the table, as it lies in between the available historyvalues of 0.06 ms and 0.09 ms.When the difference between two such history values is relatively small, it can be seenthat the use of interpolation in time between the history values may be of use to find a goodapproximation for the variable at the required time. In EMTP, and TINA, linear interpolationis used for computational efficiency.The effects of the rate of change (and thus difference between two points in the historytable) on the accuracy of the linear interpolation is studied in detail in Chapter 3. It is foundto be stable in all cases and dissipative. As an aside, since the model is now no longer idealdue to the use of interpolation, which works as a low-pass filter, the label “loss-less” no longerstrictly applicable. However, with suitable simulation parameters, the errors can be kept smalland the model considered loss-less and constant-parameter for practical purposes.More complex interpolation schemes could be used, at the cost of numerical efficiency andpossible stability issues. However, linear interpolation has proven to be sufficient in most cases186A.3. The Interpolated Loss-Less Line Modeland, in those cases where it is not, the use of a smaller time step will alleviate the issues due toincreased usable bandwidth and accuracy.A.3.2 Derivation in the Time DomainTo use the linear interpolation [35], we must first establish the interval over which interpolationmust be performed. Since this is at most one time step, the following figure can be established:t(t − τint − ∆t) (t − τint )(t − τ)∆tR∆tFigure A.5: Interpolation IntervalIn this figure, two new variables are established: the interpolation factor R, which variesbetween 0  R < 1, and  int, which is the integer length of the transmission line in simulationtime steps, rounded-up. These rounded-up in time, and synchronous with the simulation timestep, history values are thus for a line that is longer than the desired one. The history value weactually need to find for the desired line length  is that at (t  )). Their expressions can bereadily derived from the figure: int =j   tk t (A.15)R =   t  j   tk(A.16)Applying linear interpolation to the line model from (A.8), the interpolated equation pair187A.4. The Distortion-Less Line Modelcan be found through inspection by linearly interpolating to the voltage and current terms:vk(t) Zcik(t) = vm(t  int) (A.17a)+R[vm(t  int   t) vm(t  int)]+Zcim(t  int)+ZcR[im(t  int   t) im(t  int)]vm(t) Zcim(t) = vk(t  int) (A.17b)+R[vk(t  int   t) vk(t  int)]+Zcik(t  int)+ZcR[ik(t  int   t) ik(t  int)]A.3.3 Conversion into the Frequency DomainThe conversion of this model into frequency domain can be done in a similar way as in SectionA.2.3 of this appendix. Starting from (A.17) we find:(Vk  ZcIk)ej!t = (Vm + ZcIm)ej!(t  int) (A.18a)+R(V m + ZcIm)ej!(t  int  t) R(V m + ZcIm)ej!(t  int)(Vm  ZcIm)ej!t = (Vk + ZcIk)ej!(t  int) (A.18b)+R(V k + ZcIk)ej!(t  int  t) R(V k + ZcIk)ej!(t  int)Re-ordering and simplifying, we find the final version of the frequency-domain form of theinterpolated, loss-less line:(Vk  ZcIk) = (Vm + ZcIm)e j! int[1 + R(e j! t  1)] (A.19a)(Vm  ZcIm) = (Vk + ZcIk)e j! int[1 + R(e j! t  1)] (A.19b)A.4 The Distortion-Less Line ModelAlthough the focus of this thesis is on loss-less models, there was a need for a simple lossmodel in the TINA simulator to allow the fractional latency simulations to be stable, as well asfind better agreement between the simulations and experiment. Ideally, this model would have188A.4. The Distortion-Less Line Modelfrequency-independent losses and be computationally efficient so its use does not degrade thespeed benefits from the fractional latency method.One such model, called the distortionless line model [20, 46], is of particular interest. Themodel is based on applying the Heaviside condition to the Telegrapher’s equation, and modifiestheideallinemodelbymultiplyingitwithasingle, constant, realattenuationfactor. Thismakesitcomputationallyefficient, asitaddsonlytwomultiplicationstoeachconstant-parameter(CP)line segment, resulting in a lossy, constant-parameter, thus frequency-independent, model. Inother words, these losses attenuate all frequencies equally in magnitude and have no influenceon the phase. Hence, the shape of the signal is not distorted, only reduced.The Heaviside condition is given by the following ratio between the line parameters for theTelegrapher’s equation (A.1):RL =GC (A.20)The propagation function  , of which the real part  is the attenuation function and theimaginary part  is the phase function, is given by: =√(R + j!l)(G + j!C) (A.21)Also, the phase velocity a and characteristic impedance Zc are given by:a = ! (A.22a)Zc =√R + j!LG + j!C (A.22b)Applying the Heaviside condition insures the phase velocity and characteristic impedanceof the line are the same as in the loss-less case, and the added losses are purely real andfrequency-independent. The solutions are summarized in Table A.1.Parameters Zc Phase Velocity a Attenuation Function  Loss-Less R = G = 0√LC1pLC 0Distortion-Less RL = GC√LC1pLC R√CLTable A.1: Comparison Between the Ideal and Distortion-Less Line ModelsApplying this to the ideal and interpolated lines, the following expressions result, for the189A.5. Reducing the Computational Load of the Ideal and Interpolated Line Modelsideal and interpolated models, respectively:vm(t) Zcim(t) = fvk(t  ) + Zcik(t  )ge  (A.23a)vk(t) Zcik(t) = fvm(t  ) + Zcim(t  )ge  (A.23b)vk(t) Zcik(t) = fvm(t  int) (A.24a)+R[vm(t  int   t) vm(t  int)]+Zcim(t  int)+ZcR[im(t  int   t) im(t  int)]ge  vm(t) Zcim(t) = fvk(t  int) (A.24b)+R[vk(t  int   t) vk(t  int)]+Zcik(t  int)+ZcR[ik(t  int   t) ik(t  int)]ge  Thus, the distortion-less models are the same as the ideal models, multiplied by a constantattenuation factor.A.5 Reducing the Computational Load of the Ideal andInterpolated Line ModelsIn TINA, where simulations can easily have ten thousands to hundreds of thousands of cells,each with two (1D), four (2D), or twelve (3D) transmission line segments. It is thus importantto achieve an efficient implementation of these models to maintain numerical performance asa naive implementation of the models quickly becomes ineffective as the computer’s memoryis depleted and swapping to hard disk becomes necessary. This greatly reduces the numeri-cal efficiency of the method. In this section, these optimizations are shown that significantlymitigate the issues.First, a more efficient form of the interpolated and ideal line models is shown, based on aformulation in voltage only, as done in the EMTP simulator. It reduces the memory use, andcomputational burden, compared to a naive implementation of the models. Second, the use ofC-style circular buffers for the line model history term storage, and their performance impact,is discussed.190A.5. Reducing the Computational Load of the Ideal and Interpolated Line ModelsA.5.1 The Issue with Memory AccessWhen implementing the ideal and interpolated line equations, (A.8) and (A.17), respectively,there are a multitude of terms to store and manipulate in the model in order to compute thehistory sources.Each of these variables must be stored to the history depth of the model. If the modelhas a  of, e.g., ten simulation time steps  t, this means that ten memory spaces per term inthe histories are required. More so, these many terms must be accessed from computer mainmemory, as the on-chip CPU cache is too small to hold the large data sets encountered in TINAsimulations.Given that modern CPUs are so fast that they spend much of their time waiting for data tocome from main memory when the required datum is not available in the limited on-chip cache[77], it becomes clear that all unnecessary access to main memory is to be avoided.The memory and computational load of the models can, however, be reduced by a differentformulation and implementation of the line equations. In EMTP, such a formulation is usedwhere the line models are expressed only in voltage terms. This formulation, together withexploiting the time-decoupling in the line model, makes it possible to, instead of storing all theindividual history terms, voltages, and currents at both ends of the lines, directly compute thehistory values themselves and only store these in the history sources.A.5.2 Ideal Line Model in Voltage FormThe ideal line equations (A.17) are:vk(t) Zcik(t) = vm(t  ) + Zcim(t  ) (A.25a)vm(t) Zcim(t) = vk(t  ) + Zcik(t  ) (A.25b)The left-hand side of the expressions are the history sources:ekh(t) = vk(t) Zcik(t) (A.26a)emh(t) = vm(t) Zcim(t) (A.26b)We now substitute the history sources on the left-hand side:ekh(t) = vm(t  ) + Zcim(t  ) (A.27a)emh(t) = vk(t  ) + Zcik(t  ) (A.27b)191A.5. Reducing the Computational Load of the Ideal and Interpolated Line ModelsApplying KVL on the model (Figure A.3) terminals, we find:vk(t) = Zcik(t) + ehk(t) (A.28a)vm(t) = Zcim(t) + ehm(t) (A.28b)Re-ordering the above expression:Zcik(t) = vk(t) ehk(t) (A.29a)Zcim(t) = vm(t) ehm(t) (A.29b)Substituting (A.29) in (A.27), we find the required expressions in voltage only:ehk(t) =  ehm(t  ) + 2vm(t  ) (A.30a)ehm(t) =  ehk(t  ) + 2vk(t  ) (A.30b)From this, it can be seen that this formulation requires less terms to be stored on the histo-ries, and also less calculations, compared to the naive form of the models.By computing the value to place on the sending end of the line at t, which is possible sinceat any time step, we can use the current values for the history source and node voltage andcompute the future history value at the other end of the line t +  . It is this only a matter ofcomputing this future result at the present time, and placing it on the history buffer. When t hasprogresses to t +  , this required value can then be read from the history buffers. The inherenttime-decoupling of the model allows for this optimization.As a result of these manipulations, only two history buffers are used for the entire model,as the node voltages vm and vk do not need to be stored. By comparison, the original modelrequired six in a naive implementation where the node voltages and currents needed to bestored, inadditiontothehistoriesthemselves. Thevoltage-onlyexpressionsallowforanumberof numerical optimizations in the implementation of the solutions, reducing the computationload.192A.5. Reducing the Computational Load of the Ideal and Interpolated Line ModelsA.5.3 Interpolated Line Model in Voltage FormUsing the same procedure as for the ideal line model, we start with the equations for the inter-polated model (A.17):vk(t) Zcik(t) = vm(t  int) (A.31a)+R[vm(t  int   t) vm(t  int)]+Zcim(t  int)+ZcR[im(t  int   t) im(t  int)]vm(t) Zcim(t) = vk(t  int) (A.31b)+R[vk(t  int   t) vk(t  int)]+Zcik(t  int)+ZcR[ik(t  int   t) ik(t  int)]We now substitute the history sources, (A.26), on the left-hand side:ekh(t) = vm(t  int) (A.32a)+R[vm(t  int   t) vm(t  int)]+Zcim(t  int)+ZcR[im(t  int   t) im(t  int)]emh(t) = vk(t  int) (A.32b)+R[vk(t  int   t) vk(t  int)]+Zcik(t  int)+ZcR[ik(t  int   t) ik(t  int)]Applying KVL on the model (Figure A.3) terminals, we find:vk(t) = Zcik(t) + ehk(t) (A.33a)vm(t) = Zcim(t) + ehm(t) (A.33b)Re-ordering the above expression:Zcik(t) = vk(t) ehk(t) (A.34a)Zcim(t) = vm(t) ehm(t) (A.34b)193A.5. Reducing the Computational Load of the Ideal and Interpolated Line ModelsSubstituting (A.34) in (A.32), we find the required expressions for the interpolated modelin voltage only:ehk(t) = 2vm(t  int) (A.35a) 2R[vm(t  int) vm(t  int   t))] ehm(t  int)+R[ehm(t  int) ehm(t  int   t)]ehm(t) = 2vk(t  int) (A.35b) 2R[vk(t  int) vk(t  int   t))] ehk(t  int)+R[ehk(t  int) ehk(t  int   t)]For the interpolated model, similar memory optimizations are achieved as for the interpo-lated model by calculating the history values at t for the future t +  . The added complicationis that two more storage places are needed to keep track of the t  int  t values of the nodevoltage, as well as history buffers that are one deeper. However, with careful implementationof the model, the extra storage for the node voltages vm and vk can be avoided.Asaresultofthesemanipulations, andmakinguseofthetimedecouplingofthelinemodel,once more only two history buffers are needed for the entire model, as opposed to the six in anaive implementation.A.5.4 Data Structures for the History Term StorageTheperformanceofthecomputationalsodependsstronglyonhowthehistorydataisphysicallystored in the computer memory. Although there are a number of convenient and suitable datastructures available in the C++ standard template library (STL) [32], we opted for the use of C-style circular buffers. In spite of required programming and extra computations to manage thedata pointers of the circular buffers, in practice they performed much better. This resulted in afar more efficient implementation of these history terms than STL or object-based approaches[44] due to a much lower overhead.A number of initial relative performance tests of these implementations, using the standardSTL libraries in the GCC 4.3.3 and our own implementation of C-style circular buffers, whencompared using TINA test cases, showed that the circular buffers were at least one order ofmagnitude faster and required far less memory, as they did not grow by themselves as the STLcontainers did. Given these initial test results, only the C-style circular buffers were used fromthen on in the code development.194A.5. Reducing the Computational Load of the Ideal and Interpolated Line ModelsThe main issue with the STL containers for storage proved to be that, as data was pushedon one end, and data pulled from the other, the memory required kept growing. This hap-pened even thought the data pulled from the container was no longer available, and shouldthus have been de-allocated. This behavior was further exasperated by the memory allocationmethods used in STL, where the memory pool used by a container is doubled whenever theavailable storage is exceeded. Although this makes the container faster in most cases, as lessre-allocation of memory is required, it did cause the machine to rapidly run out of resourcesin TINA, where histories are pushed on, and pulled off, at every time step. In addition, whenthe memory was not being depleted, the containers proved to have significant overhead in theiroperation.Circular buffers, as implemented in TINA, by comparison only allocate their storage onceand move the access pointers around instead. By using modulo operations, an increment ordecrement of an access pointer causes it to loop-round when the bounds of the memory storagewould be exceeded, causing previously used, but no longer required, storage to be over-writtenwith new data. As such, no memory allocation/deallocation has to occur during computationand memory fragmentation is avoided. The circular buffer does come at the cost of increasedcomputation to obtain the access pointers, but the reduction in memory bus operations and theavailability of a fast CPU proved to make this approach significantly more performant, giventhe easily hundreds of thousands of these buffers in a given simulation.195Appendix BEigenvalue AnalysisWe may note that, in these experiments, the sign “=” may stand for the words “isconfused with”. G. Spencer BrownThroughout the thesis, eigenvalue analysis is extensively used to obtain the modes, and thusstability information, of a system. They are also used to find system solutions, as eigenvalueanalysis is a way to diagonalize a matrix, and thus solve the system of equations governing thesystem.The flexibility of eigenvalue analysis has much to do with the nature of physical systems.The system response of linear, time-invariant systems is governed by complex coefficient ex-ponential functions. These encapsulate the sine and cosine behavior of physical systems which,when super-imposed for each mode, as exemplified by the Fourier theorem, can faithfully re-produce the system behavior. These exponential functions represent the relevant modes ofthe system and, though diagonalization, are encoded in the eigenvalues. Thus, it is no longerrequired to completely solve the system to evaluate stability, only the eigenvalues need beextracted, as illustrated below in Section B.2.B.1 Diagonalization on a MatrixThe subject of eigenvalues and their uses, such as matrix diagonalization, is treated by manyauthors. Here, we will adapt the treatment given by Boyce [8] to introduce the subject, aug-mented by other sources.Consider the equation:Qx = y (B.1)In a general sense, (B.1) can be seen as a linear transformation that maps a vector x into anew vectory, using the mapping encoded in matrixQ. This is a very common type of equationthat appears frequently in linear electrical network analysis, as such networks serve to map oneset of variables into another by means of a transformation encoded in the circuit. In manycases, such a circuit can be expressed mathematically, e.g.: through nodal analysis, as a matrix,resulting in a system described by the form (B.1).196B.1. Diagonalization on a MatrixTo find the vectors, we set y =  x, where  is a scalar proportionality factor. Now, weseek the solution to:Qx =  x (B.2a), (Q  I)x = 0 (B.2b)This has non-zero solutions if and only if  is chosen so that the polynomial p of  :p( ) = det(Q  I) =  n + c1 n 1 + + cn = 0 (B.3)Values of  that satisfy the characteristic polynomial (B.3) are called eigenvalues of thematrix Q. The solutions of (B.2) that are obtained using such a value of  are called theeigenvectors corresponding to that eigenvalue.Since (B.3) is a polynomial equation of degree n in  , there are n eigenvalues  1;:::; n,some of which may be repeated. If a given eigenvalue appears m times as a root of (B.3),then that eigenvalue is said to have multiplicity m. Each eigenvalue has at least one associatedeigenvector.If all eigenvalues of Q are simple (synonyms: multiplicity one, distinct), then all n eigen-vectors of Q are linearly independent. If there are repeated eigenvalues, this may not be thecase, asthenumberoflinearlyindependenteigenvectorsq foreacheigenvaluewithmultiplicitym is now 1  q  m.The preceding results can now be used to diagonalize a matrix, which has use in aidingsystem solution and finding the modes of a system.This is the diagonalization problem [3]. Given an nxn matrix Q, is there an invertiblematrix V so that V 1QV is a diagonal matrix? Or, can we find a matrix V that diagonalizesQ?Assuming Q has a full set of n linearly independent eigenvectors v1;:::;vn and corre-sponding eigenvalues  1;:::; n, the eigenvectors can be used to form a matrix V whosecolumns are the eigenvectors v1;:::;vn. Since the columns of V are linearly independentvectors, det(V) 6= 0. Thus, V is nonsingular and V 1 exists.The columns of QV are the vectors Qv1;:::;Qvn. Since Qvn =  kxn, it follows that:QV = 1v11     nvn1... ... 1v1n     nvnn= V (B.4)197B.2. Finding the Modes of a Systemwhere = 1 0...0  n (B.5)is a diagonal matrix whose diagonal elements are the eigenvalues of Q. From (B.4) itfollows:V 1QV =  (B.6)Thus, if the eigenvalues and eigenvectors of Q are known, Q can be transformed into adiagonal matrix. Note that, if Q has fewer than n linearly independent eigenvectors, there isno matrix V so that V 1QV =  . In this case, Q is not diagonalizable. Linear independenceand diagonalizability are, in fact, equivalent [3].B.2 Finding the Modes of a SystemThere is a direct relation between the modes of a linear system, described by an input-outputrelation or implicitly by a linear PDE, and the eigenvalues.To show this relation, let us consider the input-output description of a linear system [76].It is characterized by a linear operator H that operates on an input vector x to generate thecorresponding output vector y.x H yFigure B.1: Linear SystemThis system is described by the following relation:y = Hx (B.7)The modes of such a system are the special inputs that are unaltered (except for a multi-plicative constant) when passed through the system. For example:This system is now described by the following relation:Hxn =  nxn (B.8)where n is the number of the mode. The vector xn is an eigenvector of the system. The198B.3. Computing the Matrix ExponentialHxn λnxnFigure B.2: Linear System with Eigenfunctions Appliedmultiplication factor  n is an eigenvalue. The above expression is the eigenvalue problem forthis system and can be solved using the methods described earlier in this appendix.Considering a linear dynamic system described by N continuous variables constituting avector x(t). The behavior of any of the N variables of this vector is, in general, dependenton all N variables. By describing the system in a new coordinate system, the N variables canbe decomposed in N decoupled, one-dimensional systems. These decoupled variables are themodes of the system.This is where eigenvalue analysis is useful, as it performs this coordinate transformation bymeans of the transformation matrix V (and back again, using the inverse) of a diagonal matrix containing the eigenvalues, and thus modes, of the system, as follows: V V 1 = H. Thus,the modes of the system can be readily obtained from the eigenvalues alone.For a linear system described implicitly by a linear PDE that can be cast in the form (B.8),where H is a differential operator, the modes are the solutions of the differential equation, andthe eigenvalues are called eigenfunctions. These “indestructible functions”, which are onlyscaled by a multiplicative constant, will be of form: ej!t. The notion of input and output is notmeaningful in this context, as the modes are fully given by the solution of only the differentialoperator itself.B.3 Computing the Matrix ExponentialExponentials raised to the power of a matrix appear quite readily in the analysis of linear, time-invariant systems of differential equations. The main reason is found in the importance of thecomplex exponential as a governing function for the modes of oscillation of real systems. Toillustrate the issue, consider a common first order ODE system of equations and its knownsolution in time domain:_x = Qx) x(t) = CeQt (B.9)It is seen that the exponential function is raised to the power of a matrix that contains thesystem behavior. Such a structure cannot be solved directly. Assuming that the eigenvaluesare distinct, thus linearly independent as discussed in Section B.1, there is a standard solutionto this problem. The following proof is based on the general procedure as outlined in [62].199B.3. Computing the Matrix ExponentialAdditional steps were added to make the proof self-contained and easier to follow.First, the exponential function can be expressed as a series:eQt =1∑k=0tkk!Qk (B.10)From the diagonalization process of Q, we can obtain eigenvectors V and eigenvalues  .QV = V (B.11a), Q = V V 1 (B.11b)Substituting the above in (B.10):eQt =1∑k=0(tkk!)(V V 1)k (B.12)Since [3]:(V V 1)k = V V 1 V V 1 ::: V V 1 (B.13a)= V kV 1 (B.13b)Substituting this result in (B.12), and recognizing that the series on the RHS is still anexponential, as shown in (B.10):eQt =1∑k=0(tkk!)V kV 1 (B.14a)= V[ 1∑k=0(tkk!) k]V 1 (B.14b)= Ve tV 1 (B.14c)The  matrix is a diagonal matrix in Jordan canonical form, containing the eigenvalues ofthe system. The exponential of this matrix, when the eigenvectors are distinct, is [62]:e t =e 1t 0...0 e nt (B.15)200B.3. Computing the Matrix ExponentialUsing these results, the solution to the original problem of (B.9) can now be expressed as:x(t) = CeQt (B.16a)= CVe tV 1 (B.16b)= CVe 1t 0...0 e ntV 1 (B.16c)The solution is now described by two constant matrices and one square diagonal matrixcontainingexponentialstoapowerofafunction. Theinitialproblemofanexponentialfunctionto the power of a matrix is thus resolved.201Appendix CStatementsIn keeping with a long-established Dutch and Belgian academic tradition, this thesis is sup-plemented with a selection of statements. These may be humorous quips, serious thoughts, orgeneral observations regarding the state of the world.Nullius in verba(”On nobody’s authority” or: ”take nobody’s word for it”, “respect the facts”)The Royal Society’s motto from the enlightenment eraMuch about Vancouver can be explained by the fact that it is a rain forest. Halfwaythrough your first winter, the brain fungus sets in.Tom De RybelThere is a theory which states that if anyone ever discovers exactly what the Uni-verse if for and why it is here, it will instantly disappear and be replaced by some-thing even more bizarre and inexplicable. There is another theory which states thishas already happened.Douglas AdamsMost Americans considered Canada to be merely another state that figured out acute trick to avoid paying taxes to Washington, DC.Robert HorningPrediction is very difficult, especially of the future.Niels BohrMr. Madison, what you’ve just said is one of the most insanely idiotic things I haveever heard. At no point in your rambling, incoherent response were you even closeto anything that could be considered a rational thought. Everyone in this room isnow dumber for having listened to it. I award you no points, and may God havemercy on your soul.excerpt from Billy Madison202Appendix C. StatementsJust when you think you’ve seen it all, someone changes what “It” is.CmdrTacoA good scientist is a person with original ideas. A good engineer is a person whomakes a design that works with as few original ideas as possible. There are noprima donnas in engineering.Freeman DysonIf the aborigine drafted an IQ test, all of Western civilization would presumablyflunk it.Stanley GarnIf something comes with a lifetime warranty and it breaks, does that mean thatthere’ll be assassins on your doorstep?Tom De RybelInside every cynic there’s an idealist desperately yearning to be let out, and whenthey are let out they’re usually a real pain and cause all sorts of trouble.Chris BoucherThe most exciting phrase to hear in science, the one that heralds new discoveries,is not “Eureka!” (I found it!) but “That’s funny ...”Isaac AsimovMy take on all this is pretty simple: in a country where it is considered a normal,sane, andfunrecreationalactivitytostraptwogreasedstickstoyourfeetandthrowyourself down the side of a friggin’ mountain, nobody has the right to call *my*minor peccidillos “unsafe”.Nathan J. MehlAll wars are civil wars, because all men are brothers ... Each one owes infinitelymore to the human race than to the particular country in which he was born.Francois FenelonDistance doens’t make you any smaller, but it does make you part of a largerpicture.Yogi BerraEnjoy yourself while you are still old.Anonymous203Appendix C. StatementsHumanity has advanced, when it has advanced, not because it has been sober, re-sponsible, and cautious, but because it has been playful, rebellious, and immature.Tom RobbinsI can give you my word, but I know what it’s worth and you don’t.Nero Wolfe, “Over my dead body”A bus station is where a bus stops. A train station is where a train stops. On mydesk I have a workstation....Per HanssonThe aim of an argument or discussion should not be victory, but progress.Joseph JoubertI used to think that the brain was the most wonderful organ in my body. Then Irealized who was telling me this.Emo PhilipsI will follow the good side right to the fire, but not into it if I can help it.Michel Eyquem de MontaigneIf I had my life to live over, I’d try to make more mistakes next time. I would relax,I would limber up, I would be sillier than I have been this trip. I know of very fewthings I would take seriously. I would be crazier. I would climb more mountains,swim more rivers and watch more sunsets. I’d travel and see. I would have moreactual troubles and fewer imaginary ones. You see, I am one of those people wholives prophylactically and sensibly and sanely, hour after hour, day after day. Oh,I have had my moments and, if I had it to do over again, I’d have more of them.In fact, I’d try to have nothing else. Just moments, one after another, instead ofliving so many years ahead each day. I have been one of those people who nevergo anywhere without a thermometer, a hotwater bottle, a gargle, a raincoat and aparachute. If I had it to do over again, I would go places and do things and travellighter than I have. If I had my life to live over, I would start bare-footed earlier inthe spring and stay that way later in the fall. I would play hooky more. I probablywouldn’t make such good grades, but I’d learn more. I would ride on more merry-go-rounds. I’d pick more daisies.Anonymous204Appendix C. StatementsPerhaps a lunatic is simply a minority of one.George Orwell, “1984”Facts do not cease to exist because they are ignored.Aldous HuxleyMost things make sense when you look at them right. It’s just sometimes you haveto look really, really cockeyed.Florence Ambrose, FreefallHomophobia: The irrational fear that gays are going to invade and re-arrange yourfurniture against your will.Greywolf BladeReality is that which, when you stop believing in it, doesn’t go away.Philip K. DickWisdom is knowing what to do with what you know.J. Winter SmithYou are never given a wish without also being given the power to make it true. Youmay have to work for it, however.R. Bach, “Messiah’s Handbook: Reminders for the Advanced Soul”What we observe is not nature itself, but nature exposed to our method of ques-tioning.Werner HeisenbergClearly the secret of happiness, he reflects quite cheerfully, is a variation on thegeneral principle of banging your head against a wall, and then stopping.Stef Penny, The Tenderness of Wolves, P.250The Feynman Problem-Solving Algorithm:1. write down the problem2. think very hard3. write down the answerAttributed to Murray Gell-Mann205Appendix C. StatementsThe wind makes dust because it intends to blow, taking away our footprints.Bushmen folklore ( 1911)You take the old Goethe much too seriously, my young friend. You should not takeold people who are already dead seriously. It does them injustice. We immortalsdo not like things to be taken seriously. We like joking. Seriousness, young man,is an accident of time. It consists, I don’t mind telling you in confidence, in puttingtoo high a value on time. I, too, once put too high a value on time. For that reasonI wished to be a hundred years old. In eternity, however, there is no time, you see.Eternity is a mere moment, just long enough for a joke.Goethe (Hermann Hesse, Steppenwolf)Ad astra per alia porci (to the stars on the wings of a pig)John Steinbeck. A professor told him that he would be an author when pigs flew.Every book he wrote is printed with this insignia.Early to bed and early to rise makes a man stupid and blind in the eyes.Mazer Rackham (Orson Scott Card, Ender’s Game)For a long time it had seemed to me that life was about to begin – real life. Butthere was always some obstacle in the way. Something to be got through first,some unfinished business, time still to be served, a debt to be paid. Then lifewould begin. At last it dawned on me that these obstacles were my life.Alfred D’SouzaThat’s the problem with drinking, I thought as I poured myself a drink. If some-thing bad happens you drink in an attempt to forget; if something good happensyou drink in order to celebrate; and if nothing happens you drink to make some-thing happen.BukowskiThe superior man understands what is right; the inferior man understands whatwill sell.ConfuciusAt no time is freedom of speech more precious than when a man hits his thumbwith a hammer.Marshall Lumsden206Appendix C. StatementsYou can know the name of a bird in all the languages of the world, but when you’refinished, you’ll know absolutely nothing whatever about the bird... So let’s lookat the bird and see what it’s doing – that’s what counts. I learned very early thedifference between knowing the name of something and knowing something.Richard P. FeynmanI don’t have to know an answer. I don’t feel frightened by not knowing things;by being lost in a mysterious universe without any purpose – which is the way itreally is, as far as I can tell, possibly. It doesn’t frighten me.Richard P. FeynmanDr. Hoenikker used to say that any scientist who couldn’t explain to an eight-year-old what he was doing was a charlatan.Kurt Vonnegut, “Cat’s Cradle”There is one feature I notice that is generally missing in “cargo cult science”... It’sa kind of scientific integrity, a principle of scientific thought that corresponds to akind of utter honesty – a kind of leaning over backwards... For example, if you’redoing an experiment, you should report everything that you think might make itinvalid – not only what you think is right about it... Details that could throw doubton your interpretation must be given, if you know them.Richard P. FeynmanIt does not matter how beautiful your theory is, it does not matter how smart youare. If it does not agree with experiment, it is wrong.Richard P. FeynmanWhat I cannot create, I do not understand.Richard P. Feynman, On his blackboard at time of death in 1988; as quoted in TheUniverse in a Nutshell by Stephen Hawking.207Appendix DList of PublicationsJournal Marcelo A. Tomim, Tom De Rybel, and Jos´e R. Mart´ı. Multi-area Th´evenin equivalentsmethod applied to large power systems parallel computations. IEEE Transactions onPower Systems, 2009. Submitted. Tom De Rybel, Arvind Singh, Pak Phalmoniroth, and Jos´e R. Mart´ı. On-line signalinjection through a bus-referenced current transformer. IEEE Transactions on PowerDelivery, (TPWRD-00171-2009), 2009. Accepted. Tom De Rybel, Arvind Singh, John A. Vandermaar, May Wang, Jos´e R. Mart´ı, and K.D.Srivastava. Apparatus for on-line power transformer winding monitoring using bushingtap injection. IEEE Transactions on Power Delivery, 24(3):996–1003, July 2009. Ali Davoudi, Juri Jatskevich, and Tom De Rybel. Numerical state-space average-valuemodeling of PWM DC-DC converters operating in DCM and CCM. IEEE Transactionson Power Electronics, 21(4):1003–1012, July 2006.Invited Conference Tom De Rybel, Jos´e R. Mart´ı, and Murray Hodgson. Analytical validation of time-stepinterpolation in transient insular nodal analysis. In Acoustics’08 Paris, number ACOUS-TICS2008/177 in NS30-4aNSc, page 5, Paris, France, 29 June - 4 July 2008. Acous-tics’08.Conference Tom De Rybel, Arvind Singh, Pak Phalmoniroth, and Jos´e R. Mart´ı. Self-powered on-line signal injection based on a current transformer. In 8th IEEE Electric Power andEnergy Conference, number 1370, page 7, Vancouver, Canada, 6-7 October 2008. IEEE,EPEC’08.208Appendix D. List of Publications Tom De Rybel, Marcelo Tomim, Arvind Singh, and Jos´e R. Mart´ı. An introduction toopen-source linear algebra tools and parallelisation for power system applications. In 8thIEEE Electric Power and Energy Conference, number 1353, page 8, Vancouver, Canada,6-7 October 2008. IEEE, EPEC’08. Arvind Singh, Tom De Rybel, and Jos´e R. Mart´ı. FFT Tutor: A matlab-based instruc-tional tool for FFT parameter exploration. In Acoustics Week in Canada 6-8 October2008, volume 36, pages 82–83, Vancouver, Canada, September 2008. Canadian Acous-tical Association, Canadian Acoustics. Tom De Rybel, Jos´e R. Mart´ı, and Murray Hodgson. Modelling of acoustic wave propa-gation using transient insular nodal analysis (TINA). In 19th International Congress onAcoustics, number COM-06-004, page 6, Madrid, Spain, 2-7 September 2007. ICA’07. Arvind Singh, Tom De Rybel, Jos´e R. Mart´ı, and K.D. Srivastava. Field validation testsof the TLD box for online power transformer winding monitoring systems. In IEEECanadian Conference on Electrical and Computer Engineering, pages 141–144, Van-couver, Canada, 22-26 April 2007. CCECE’07. Tom De Rybel, Jorge A. Hollman, and Jos´e R. Mart´ı. OVNI-NET: A flexible clusterinterconnect for the new OVNI real-time simulator. In 15th Power Systems ComputationConference, page 6, Li`ege, Belgium, 22-26 August 2005. PSCC’05.Theses Tom De Rybel. OVNI-NET: A flexible cluster interconnect for the new OVNI real-timesimulator. Master’s thesis, The University of British Columbia, Vancouver, Canada,March 2005. Tom De Rybel. Elektrolysetoestel voor de aanmaak van vaste cyclotrondoelwitten. Mas-ter’s thesis, Hogeschool Gent, Gent, Belgium, 2002.209


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