@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Electrical and Computer Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Zhou, Dan H."@en ; dcterms:issued "2008-09-15T18:09:42Z"@en, "1993"@en ; vivo:relatedDegree "Doctor of Philosophy - PhD"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """Signal lines such as microstrip lines have recently assumed increased importance in VLSI circuit design and computer package design. Their behavior has become a key factor in system performance. As the circuit implementation size or area reduces almost to its physical limits, the circuit speed is so high that devices can react in less than nanoseconds. Under these conditions, the trans-mission line characteristics of microstrip lines become dominant compared to their simple function for signal linkage. Many numerical methods have been introduced to simulate these signal lines. To employ these models, the transmission line characteristics must be determined first. With signal transients containing the frequency of interest, traditional formulae are inappropriate to calculate line parameters because of skin and proximity effects. A common strategy is to apply the subdivision principle — to subdivide the line conductors into smaller parts so that the traditional formulae may be used. Based on this strategy, new numerical methods have been developed in this thesis to determine microstrip line parameters including skin and proximity effects. The developed techniques include a Linear Current Subconductor Technique (LCST), and an advanced method of subareas (ANS). These methods are derived from the traditional subconductor method and the traditional method of subareas, respectively. LCST combines the simplicity of the traditional subconductor method with the accuracy of finite element methods. AMS avoids the procedures of optimization and recursion of traditional subareas methods. To generate the LCST, the conductors of microstrip lines are firstly divided into subconductors by a 26 rule (6 is the skin depth at the considered frequency). For the subconductors system, an impedance matrix is then built using Ohm's law and the concept of Geometric Mean Distances (GMD). Secondly, the current distributions in the subconductors are solved from the telegrapher's equations. Thirdly, a linear current distribution is evaluated from the results of the previous step. After substituting linear currents back into the telegrapher's equations, correction factors are obtained for the subconductor's impedance matrix. Finally, the corrected impedance matrix is reduced to an equivalent line impedance matrix by a bundling procedure. Another important application of the proposed sub conductor technique is in the calculation of the parameters of underground cables in electric power systems. Due to the irregular arrangement of the conductors in the cable no analytical formulas are available for these calculations. As an example, the LCST technique is ap-plied to pipe-type cables and the results are compared to those of previously published work using a finite element technique. In the case of ANS, after a similar subdivision procedure, the Maxwell coefficients matrix of subareas is setup from Green's functions. A bundling procedure is then used to convert the Maxwell coefficients matrix of the subareas into the Maxwell coefficients matrix of the line conductors. The inverse of the resultant matrix is the line capacitance matrix. In simulations of microstrip lines and pipe-type cables, the proposed LCST and ANS techniques proved to be efficient and accurate. Compared to the traditional uniform current density technique, the LCST results in savings of up to 99% and 98% in memory requirement and CPU cost, respectively, while the ANS technique results in savings of up to 80% as compared to the conventional uniform charge distribution technique. In comparisons with a finite element method and a traditional subconductor method, the results from LCST presented an average difference of about 4.0%. The resulting average difference was of 6.31% for the ANS technique as compared with two finite element methods and a software package from A. Djordjevic et al, ("Analysis Of Arbitrarily Oriented Microstrip Transmission Lines...", IEEE Trans. on MTT, vol.MTT-33,no.10, Oct.1985). Yet to be researched is the extension of LCST and ANS into otherline structures with open boundaries. Other potential applications are in telecommunications and computer high speed networking, as well as supercomputer packaging."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/1949?expand=metadata"@en ; dcterms:extent "5697097 bytes"@en ; dc:format "application/pdf"@en ; skos:note "NUMERICAL METHODS FOR FREQUENCY DEPENDENT LINE PARAMETERSWITH APPLICATIONS TO MICROSTRIP LINES AND PIPE-TYPE CABLESbyDAN HONG ZHOUB.Eng., The Beijing Institute of Technology, 1985M.Eng., The Beijing Institute of Technology, 1988A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF APPLIED SCIENCE(Department of Electrical Engineering)We accept this thesis as conformingTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993© Dan Hong Zhou, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Electrical EngineeringDepartment of ^The University of British ColumbiaVancouver, CanadaDate^September 23, 1993DE-6 (2/88)ABSTRACTSignal lines such as microstrip lines have recently assumedincreased importance in VLSI circuit design and computer packagedesign. Their behavior has become a key factor in system perfor-mance. As the circuit implementation size or area reduces almost toits physical limits, the circuit speed is so high that devices canreact in less than nanoseconds. Under these conditions, the trans-mission line characteristics of microstrip lines become dominantcompared to their simple function for signal linkage.Many numerical methods have been introduced to simulate thesesignal lines. To employ these models, the transmission linecharacteristics must be determined first. With signal transientscontaining the frequency of interest, traditional formulae areinappropriate to calculate line parameters because of skin andproximity effects. A common strategy is to apply the subdivisionprinciple — to subdivide the line conductors into smaller parts sothat the traditional formulae may be used. Based on this strategy,new numerical methods have been developed in this thesis to determinemicrostrip line parameters including skin and proximity effects.The developed techniques include a Linear Current SubconductorTechnique (LCST), and an advanced method of subareas (ANS). Thesemethods are derived from the traditional subconductor method andthe traditional method of subareas, respectively. LCST combinesthe simplicity of the traditional subconductor method with theaccuracy of finite element methods. AMS avoids the procedures ofoptimization and recursion of traditional subareas methods.To generate the LCST, the conductors of microstrip lines arefirstly divided into subconductors by a 26 rule (6 is the skin depthat the considered frequency). For the subconductors system, animpedance matrix is then built using Ohm's law and the concept ofGeometric Mean Distances (GMD). Secondly, the current distributionsin the subconductors are solved from the telegrapher's equations.Thirdly, a linear current distribution is evaluated from the resultsof the previous step. After substituting linear currents backinto the telegrapher's equations, correction factors are obtainedfor the subconductor's impedance matrix. Finally, the correctedimpedance matrix is reduced to an equivalent line impedance matrixby a bundling procedure.Another important application of the proposed subconductor tech-nique is in the calculation of the parameters of underground cablesin electric power systems. Due to the irregular arrangement ofthe conductors in the cable no analytical formulas are availablefor these calculations. As an example, the LCST technique is ap-plied to pipe-type cables and the results are compared to those ofpreviously published work using a finite element technique.In the case of ANS, after a similar subdivision procedure,the Maxwell coefficients matrix of subareas is setup from Green'sfunctions. A bundling procedure is then used to convert the Maxwellcoefficients matrix of the subareas into the Maxwell coefficientsmatrix of the line conductors. The inverse of the resultant matrixis the line capacitance matrix.In simulations of microstrip lines and pipe-type cables, theproposed LCST and ANS techniques proved to be efficient and accurate.Compared to the traditional uniform current density technique, theLCST results in savings of up to 99% and 98% in memory requirementand CPU cost, respectively, while the ANS technique results insavings of up to 80% as compared to the conventional uniformcharge distribution technique. In comparisons with a finiteelement method and a traditional subconductor method, the resultsfrom LCST presented an average difference of about 4.0%. Theresulting average difference was of 6.31% for the ANS technique ascompared with two finite element methods and a software packagefrom A. Djordjevic et al, (\"Analysis Of Arbitrarily OrientedMicrostrip Transmission Lines...\", IEEE Trans.on MTT, vol.MTT-33,no.10, Oct.1985).Yet to be researched is the extension of LCST and ANS into otherline structures with open boundaries. Other potential applicationsare in telecommunications and computer high speed networking, aswell as supercomputer packaging.Table of Contents ABSTRACT ^Table of Contents ^ ivList of Tables viiiList of Figures Chapter 1^Introduction ^ 11.1^SIGNAL LINES IN VLSI 11.2 SIGNIFICANCE OF THE THESIS WORK ^ 31.3^LINE MODELS ^ 5^1.3.1^Literature Review for Line Models ^ 51.3.2^Frequency Range of Transmission Line Model forMicrostrip Lines ^ 8Electromagnetic Wave along Microstrip Lines^ 8Transmission Line Equations For Microstrip Lines ^ 91.4^LINE CHARACTERISTIC PARAMETERS ^ 101.4.1^Literature Review for Line Parameters ^ 101.4.2^Problems with the Traditional Methods ^ 12Resistances and Inductances ^ 13Capacitances ^ 141.5^CONTRIBUTION OF THIS THESIS WORK ^ 151.6 ORGANIZATION OF THIS THESIS 16REFERENCES ^ 17Chapter 2^Linear-Current SubconductorTechnique for Matrices ofFrequency Dependent Resistanceand Inductance ^ 25^2.1^GENERAL DESCRIPTION 252.2 CONDUCTOR'S CROSS-SECTION SUBDIVISION^282.2.1^Subdivision of Microstrip Lines ^ 282.2.2^Pipe-Type Cables ^ 312.3^SUBCONDUCTOR RESISTANCE AND INDUCTANCE FROMTRADITIONAL FORMULAE 342.3.1^Microstrip Lines ^ 34Self Geometric Mean Distance (GMD) ^ 36GMDs between Subconductors in Microstrip Lines ^ 37iv^2.3.2^Pipe-Type Cables ^ 39Paralleled Curved Cells 39Non-paralleled Curved Cells 41Self Geometric Mean Distance ^ 41GMD for Circular Cells ^ 422.4^CURRENT DISTRIBUTION 432.5 APPROXIMATION OF LINEAR CURRENTDISTRIBUTION ACROSS SUBCONDUCTORS^ 452.5.1^Microstrip Lines ^ 462.5.2^Pipe-type Cables 472.6^LINEAR CURRENT DISTRIBUTION CORRECTIONFACTORS ^ 512.7^INDUCTANCE CORRECTION ^ 532.7.1^Inductance Formula • Option 1 ^ 54Microstrip Lines ^ 57Pipe-Type Cables 582.7.2^Inductance Formula • Option 2 ^ 592.8^BUNDLING PROCEDURE ^ 622.8.1^Voltage Drops along Subconductors ^ 622.8.2^Currents in the Subconductor System 652.8.3^Matrix Manipulation For Matrix Reduction . . . ^ 672.9^SUMMARY ^ 672.9.1^Summary of the Linear Current SubconductorTechnique 672.9.2^Discussion ^ 69REFERENCES ^ 72Chapter 3^Simulations Using LCST andOther Methods ^ 753.1^SOLUTIONS FOR MICROSTRIP LINES ^ 75^3.1.1^Current Distribution from Different Subdivisionsunder Uniform Distributed Current Assumption . 753.1.2^Inductance Correction Factors ^ 763.1.3^Solution and Discussion of Line Resistancesand Inductances — LCST vs. TraditionalSubconductor Method ^ 773.2^SOLUTIONS FOR PIPE-TYPE CABLES ^ 923.2.1^Inductance Correction Factors 923.2.2^Solution and Discussion for the ImpedanceMatrix — LCST vs.FEM, and LCST vs.TraditionalSubconductor Method ^ 933.3^CONCLUSIONS ^ 1033.4 SUMMARY 105REFERENCES ^ 106Chapter 4^Computation of the Capacitanceand Conductance Matrices ofMicrostrip Lines ^ 107^4.1^MICROSTRIP LINE CONFIGURATIONS ^107^4.2 FREQUENCY DEPENDENCE ^1084.2.1^Capacitance Formed by Line Conductor and GroundPlane (With Constant 0 ^109Solution from Maxwell equations ^109Solution from empirical formulae ^1114.2.2 Capacitance With Frequency Dependant E(f) . . . 1134.3 AMS VERSUS TRADITIONAL METHOD OF SUBAREAS 1144.4 SUBDIVISIONS ^ 1164.5 GREEN'S FUNCTIONS 1174.6 MAXWELL COEFFICIENTS OF SUBAREAS ^ 1184.7 CHARGE DISTRIBUTION ^ 1204.8 DISCUSSION^ 1224.9 CONDUCTANCE MATRIX ^ 1234.10 SUMMARY ^ 124REFERENCES ^ 126Chapter 5^Simulations and Comparisonsfor AMS ^ 1285.1^MICROSTRIP LINE CAPACITANCE WITH er=1. . 1285.2 MICROSTRIP LINE CAPACITANCE WITH sr==10 . 1305.3^COMMENTS ON THE AMS TECHNIQUE ^ 1305.4 LINE CONDUCTANCE MATRIX ^ 1315.5^CONCLUSIONS ^ 1315.6 SUMMARY 132REFERENCES ^ 134viChapter 6^Future Work ^ 135Chapter 7^Conclusions 136Appendix I Current Split in Subdivided andNon-Subdivided Conductors . .139Appendix II Symbols And Abbreviations . ^ 141SUMMARY OF REFERENCES ^ 150vi iList of Tables Table 3.1Table 3.1Table 3.2Table 3.2Table 3.3Table 3.3Table 3.4Table 3.•Table 3.6Table 3.7Table 3.7Table 3.8Table 3.8Impedance Correction Factors as Functions ofFrequency ^ 83Impedance Correction Factors as Functions ofFrequency ^ 84Correction Factors for Subconductors at f = 10-6Hertz ^ 85Correction Factors for Subconductors at f= 10-6Hertz ^ 86Correction Factors for Subconductors atf = 5 x 109 Hertz ^ 87Correction Factors for Subconductors atf = 5 x 109 Hertz ^ 88Comparison of LCST with Traditional SubconductorMethod Assuming Uniform Current Distribution,for the Case of Two Conductor Microstrip Lines .89Comparison of LCST with Traditional SubconductorMethod, for the Case of Three ConductorMicrostrip Lines ^ 90Comparison of LCST with Traditional SubconductorMethod, in Memory Requirement and CPU Time Cost.91Impedance Correction Factors as Function ofFrequency and Conductor Sequence ^ 96Impedance Correction Factors as Function ofFrequency and Conductor Sequence ^ 97Correction Factors for Subconductors at f = 60Hz 98Correction Factors for Subconductors at f = 60Hz 99Table 3.9^Correction Factors for Subconductors atf = 600, 000Hz ^ 100Table 3.9^Correction Factors for Subconductors atf = 600, 000Hz ^ 101Table 3.10^Comparisons of LCST with FEM: Resistances andInductances at 60Hz ^ 102viiiTable 3.11^Comparisons of LCST with FEM: Resistances andInductances at 600,000Hz ^ 104Table 5.1^Capacitance Values and CPU Time from ANS, FEMs,and A.D., for the Line Configuration of Fig.4.1(with Er =1) 129Table 5.2^Results of ANS and of A. D. for the Configurationsof Figs. 4.1 and 4.2 (with Er =10) 133Table 5.3Table 5.4Results of the Conductance Matrix with Er =10and at f = 106Hertz for the Configurations ofFigs. 4.1 and 4 2 133Results of the Conductance Matrix with cr =10and at f = 109Hertz for the Configurations ofFigs. 4.1 and 4.2 133ixList of FiguresFig. 1.1 Typical Configurations of Microstrip Lines.^. .^.^2Fig. 1.2 Transmission Line Models in the EMTP^ 6Fig. 1.3 Subdivision Strategy in the Subconductor andSubarea Methods. ^ 13Fig. 2.1 Flow Chart for the Linear Current SubconductorTechnique (LCST) ^26Fig. 2.2 Coordinate System for Microstrip Lines ^ 30Fig. 2.3 Pipe-Type Cable Configuration and Subdivisions. 33Fig. 2.4 Coordinate System for Pipe-Type Cables ^ 35Fig. 2.5 Cell Index and Fields Distribution in MicrostripLines ^ 37Fig. 2.6 Loop Definitions Related to GMD's forInductances 38Fig. 2.7 Schematic of Cells in Microstrip Lines.^. . 38Fig. 2.8 Two Kinds of Cells and Indices of Pipe-TypeCables ^ 40Fig. 2.9 Cells for GMD's in Pipe-Type Cables. ^ 42Fig. 2.10 Schematic of a Cell and its Current Densitiesfor Microstrip Lines. ^ 48Fig. 2.11 Schematic of Cells and Their Current Densitiesfor Pipe-Type Cables. ^ 49Fig. 2.12 The Voltage Drops Related to theSubconductors. ^ 63Fig. 3.1 Current Distributions with Different Number ofSubdivisions ^ 79Fig. 3.1 Current Distributions with Different Number ofSubdivisions. ^ 80Fig. 3.1 Current Distributions with Different Number ofSubdivisions. ^ 81Fig. 4.1 Typical Configuration of Microstrip Lines (fromFig.1.1).^Dimensions in pm ^ 108Fig. 4.2^Another Configuration of Microstrip Lines.Dimensions in pm ^ 108Fig. 4.3^Capacitance of a Single Conductor MicrostripLine with Ratio^= > 0.7 and DielectricConstant Er = 4.5. 111Fig. 4.4^Capacitance of a Single Conductor MicrostripLine with Ratio T = 0.5 and Dielectric ConstantEr = 4.5. 113Fig. 4.5^Linear Charge Density and a Schematic ofCoordinates Relationship in a Subarea ^ 121Fig. 1.1^A subconductor in forming two systems ^ 139xiACKNOWLEDGMENTI wish to state my gratitude to my parents, who haveinspired and supported my exploration into new worlds.I wish to express my sincere appreciation to mysupervisor, Dr.Jose Marti, for his excellentguidance, kindness, assistance and patience.I wish to acknowledge the Department of ElectricalEngineering at the University of British Columbia (UBC) forgiving me the financial support to accomplish this venture.I would like to thank Dr.Martin Wedepohl for sharing hiseigenvalue and eigenvector extracting routines, and offeringhelp during my thesis work. I would also like to thankDr.Yanan Yin for many inspiring discussions in the earlystage of my thesis work as well as for offering his resultsto validate the new numerical methods in this thesis, and toMr.David Michelson, Miss Ruth Harland, Miss Carly Wongand Miss Ellen Ho, for their helpful suggestions andEnglish polishes regarding the writing of this thesis.Last but not least, I have also appreciated the courtesy andassistance of the staff of the Department of ElectricalEngineering, UBC. Mr.Robert Ross and Mr.David Gagnehave been especially kind in patiently answering myquestions and facilitating my use of department computers.xi iChapter 1IntroductionThis thesis is concerned with the modelling and simulation ofmicrostrip lines for Very Large-Scale Integrated (VLSI) circuit andcomputer packaging design.In this chapter, the background to the thesis work will beintroduced. The significance of the work is covered along witha review of the literature in related fields, and an overviewof the development of two new numerical methods. Finally, theorganization of the thesis will be outlined.1 . 1 SIGNAL LINES IN VLSIIn VLSI, the signal lines can fall into two groups: intercon-nects of devices on a chip and interconnections between chips orcomponents on circuit boards. The interconnects and interconnec-tions are often implemented as microstrip lines.Three typical configurations of microstrip lines are shown inFig.1.1. Figure 1.1(a) shows the microstrip conductors coated in asubstrate or dielectric with a metal layer adhered to the other sideof the substrate, which functions as a ground plane. In Fig.1.1(b),the line conductors are sandwiched between two ground planes andsit on the homogeneous dielectric. Finally, in Fig.1.1(c), twobatches of paralleled lines are placed between two ground planes in1(c) Multilayers of microstrip lines.Fig. 1.1 Typical Configurations of Microstrip Lines.such a way that they are perpendicular to each other in homogenousdielectric. The structure displayed in Fig.1.1(a) is the mostcomplicated because of its open boundary condition. Among thesemicrostrip lines, the line-width, line-space and line-length of theinterconnects can be as small as 0.8pm — 1.2pm on a silicon chip[9,226, 61]. The dimensions of the interconnections between the chipscan range from lOpm — 200pm[47].1.2 SIGNIFICANCE OF THE THESIS WORKPrediction of the fast signal behavior or signal transientanalysis in VLSI has assumed increasing importance in recent years.Signal transients influence the lifetime of the devices, theirfunctioning and the performance of the whole system. Undesiredsignal transients are mainly caused by microstrip lines. Thedistributed nature of microstrip lines can induce harmful transientsin the other parts of the circuit, especially after power switchingand circuit logic transition. In the operation of high speedelectronic devices, the behavior of microstrip lines is criticalfor system performance.Besides introducing transients, microstrip lines are the mainchannels of electromagnetic interference and major sources of extrapower dissipation. In VLSI operations, the pulses on a microstripline suffer loss, distortion and dispersion[4, 15, 24, 28, 34, 35,40, 47, 52, 53] in addition to signal delays. Circuit performancemay be compromised by the decaying or coupling of signals. In highspeed computers the situation is even more critical. Hence, it isessential to have microstrip lines carefully designed.Current VLSI technology reduces the size or area occupied by adevice. This results both in a high circuit complexity on chips3and in an increase in the reacting speed of the device. There cannow be up to 500,000 gates and 250 I/O pins on a single siliconchip[9, 44, 61], such as the Intel i486 chip which contains 1.2million transistors[48]. While the ultimate clock rate of 200 -400 megahertz is found in current computers[7], gate transitions orgate delays of 250ps have become common in silicon, and the valuesof 100ps are observed in gallium arsenide[44, 51]. This greatcomplexity leads to an increase in the number of interconnects andinterconnections while the high operational speed makes it necessaryto model interconnects and interconnections as transmission lines.Microstrip line models must then be incorporated in VLSI circuitand computer packaging design procedures.From a practical point of view, electromagnetic compatibilityis very important in circuit and system design. Very often a\"theoretically\" correct circuit does not work as expected whenbuilt, or even worse, it may exhibit intermittent and unpredictableoperation, especially in VLSI[23, 44, 45, 51]. Experience hasalso shown that electromagnetic performance deficiencies or defectsare difficult, if not impossible, to correct after fabrication.Thus, accurate modelling and simulation of microstrip lines isan extremely important aspect of the complete electromagneticprediction of the high frequency behavior of high density chips,components, and circuit boards prior to fabrication.41.3 Ion= mamas1.3.1 Literature Review for Line ModelsThere is a large amount of literature related to microstrip linemodels. The proposed models can be roughly grouped as follows:• Wave-guide models[3, 15, 21, 42];• Full wave models (3-dimenssion)[16, 25, 54, 56,55]; and,• Transmission line models[13, 18, 36, 37, 41, 19].Although the first and second methods may accurately simulate theelectric and magnetic fields along the line, all models proposedin the literature have difficulties in their implementation intopractical circuit simulation packages. They need at least to solvePoisson or Laplace equations with the complicated boundary condi-tions of microstrip lines. Due to the open-boundary structure ofmicrostrip lines, these models have very large memory requirements.The transmission line approach has advantages in terms of theease of solving only linear equations, of dealing with open-boundaryproblems, of the direct access it provides to circuit variablessuch as voltages and currents, and of the almost mature and simplenumerical methods that have been developed for its description. Italso requires much less computer memory. The transmission linemodel has been well developed in the Electromagnetic TransientProgram (EMTP). For the purpose of the study of this thesis, thistransmission line model has been used for microstrip lines.51/1V1+AV1L1R1V2+AV2V3+ AV3node mVm(t)V2V3C3,G3(a) Transmission line parameters;Zeq(f)(b) Transmission line model.Fig. 1.2 Transmission Line Models in the EMTP.The transmission line model in the EMTP is illustrated in Fig.1.2.Figure 1.2 (a) shows the distributed parameters per unit length.Fig.1.2 (b) illustrates the line model. At one end of the line(e.g., node k) , the entire line is represented as a time-varyingvoltage source (grounding one terminal) connected to a frequency6dependent impedance.^The voltage source is determined by theparameters and history of the branch current linking node k andground.In Fig.1.2, two sub—models construct the microstrip line model:the parameter model (Fig.1.2(a)) and the line model (Fig.1.2(b)).The parameter model is the first step in simulating microstrip lines,it includes the distributed parameters of resistance (R), inductance(L), capacitance (C), and conductance (G). After obtaining theseparameters, the line model can be setup for node voltages andbranch currents.In comparison with the other two strategies, which focus onthe electromagnetic field around the microstrip lines, there isone alleged weak point in the application of the transmissionline model to microstrip lines. When the signal frequency risesto a high enough level, the transmission line theory loses itseffectiveness for microstrip lines. For example, beyond a certaincutoff frequency, microstrip lines will become leaky wave guidesrather than transmission lines[14, 15, 20, 64]. However, it can beproven that the transmission line model is still able to simulatethe behavior of microstrip lines with dimensions in the order ofmicrons within the frequencies of interest. It is also very likelyto have a further extended frequency range as VLSI technology isupgraded.7C • tan E rfTEIW1=N/2-rhVer -1(1.3.2)1.3.2 Frequency Range of TransmissionLine Model for Microstrip LinesElectromagnetic Wave along Microstrip LinesTransmission lines are defined as structures that allow guidedTransverse Electromagnetic waves (TEM) to travel from one point toanother[5, 14]. From this definition and from the reason that thewave on the lines is \"quasi-TEM\" type [15, 10, 11, 14, 20, 64], themicrostrip lines are lossy (distortion) and dispersive transmissionlines (signals on the lines will suffer delay, loss or distortion,dispersion, and introduce crosstalk in the circuits). They thussupport a hybrid wave mode containing TEM, TE, and TM modes[6, 14,15, 29]. The latter two modes contain the longitudinal componentsof electric and magnetic fields on the microstrip line and may causethe generation of eddy currents on the conductor's surface. Thecutoff frequency of the lowest TE mode (TED) is given byLutoff= 4h/Er -1'(1.3.1)while the frequency for strong coupling of TEM and TM modes[15]can be expressed asBelow both these frequencies, the TEM wave mode will dominate inthe propagation. TT° and TM() surface wave modes only hold a verysmall portion of power. Then the higher order modes are neglectedin the modelling and simulation of microstrip lines by transmissionline model. (A schematic of the electromagnetic field in a singleconductor microstrip line is shown in Fig.2.5(b) [15, 33, 54, 64].)8Therefore, transmission line theory will be valid for microstriplines as long as the signal frequencies are under the thresholdsindicated in eqns.(1.3.1) and (1.3.2).In eqns.(1.3.1) and (1.3.2), c is the travelling speed of thewave in vacuum or air; h is the height of the dielectric or thedistance between the line conductor and the ground plane; and Eris the relative permittivity or dielectric constant, as shown inFig.1.1(a).Transmission Line Equations For Microstrip LinesApplying transmission line theory to microstrip lines, thevoltages and currents along the lines satisfy the transmissionline equations (TEM mode) 1 az_az = yv----0L=zi-Tz-^ (1.3.3)where z is the direction of wave propagation along the line, and Zand Y are, respectively, impedance and admittance matrices formedby the line parameters. As shown in Fig.1.2(a), Z = R-FjcoL andY G jcvCEqn.(1.3.3) implies that on a cross-section of the line, per-pendicular to the z direction, the voltage drop is zero, i.e.,AVcross—section = 0.^ (1.3.4)This means that the cross-section of the line is an equipotentialplane. In another instance, if Across_ section <()q,, i.e., the cross-section area of the line is much less than the squared wavelength of9the signal on the line, the condition in eqn.(1.3.4) is satisfied,and the line is called electrically \"fine\"[35]. The cross-sectionthen forms an equipotential plane and the transmission line modelis valid.1.4 LINE CHARACTERISTIC PARAMETERSTo apply the transmission line model to microstrip lines, theline parameters must be calculated. In the past, RC networkshave been used to approximate microstrip lines in VLSI studies,but this is not regarded as acceptable[70]. For example, amongthe parameters in Fig.1.2(a), L becomes very important in highfrequencies.1.4.1 Literature Review for Line ParametersMany methods have been proposed to compute the frequency depen-dent parameters of microstrip lines. One method is based on theassumption of zero thickness of the conductors. In this approach,C can be obtained by using conformal mapping theory and the conceptof effective permittivity[2, 22, 30, 31, 32, 39, 43, 63, 62, 671.The resistance R can then be calculated by the method of GeometricMean Distances (GMD) and Ohm's law[13]. Finally, L and G can beobtained using the relationship between the four line parametersaccording to electromagnetic theory[5]. This approach is simpleand based on analytical formulae. The calculation for the line1 0inductance, however, is incorrect at low frequencies because theinternal inductance has not been taken into account. Also, althoughan adjustment for the thickness can be included, according to anempirical or asymptotic formula[49, 62, 63], the actual thicknesseffect has not been adequately modelled for the line capacitance.This is because the surface charge concentrates on the corners ofthe line conductors, and thickness becomes the dominant factor forthe capacitance.Besides the capacitance, the thickness assumption is also im-portant for the resistance and inductance parameters. Taking intoconsideration that the current will distribute more densely towardsthe corners and edges of the conductors, and will change the lineresistance and inductance as frequency increases, it is concludedthat the assumption of zero thickness is not appropriate for model-ling microstrip lines in the simulation of circuit transients. Itmay only be suitable to simulate situations such as the asymptoticbehavior in microwave circuits at single frequencies.Another general approach to calculate transmission line param-eters is the finite element method (FEM). The parameters R, L, G,and C can be calculated from electromagnetic field densities[66].However, for open-boundary structures such as those in Fig.1.1(a),the boundary conditions are very complicated and difficult to spec-ify for FEM techniques. In addition, there are intensive computermemory requirements for the initiation and solution process usingFEM techniques.11Another approach to determining the line's frequency-dependentparameters is the method of subconductors for R and L, and themethod of subareas or moments for C [1, 2, 8, 27, 38, 50, 58,59]. The parameter G can be obtained from C, the signal frequencyand the dielectric loss tangent. These methods share the samesubdivision strategy as FEM, but avoid the main difficulties ofthe FEM approach. The subconductor technique accommodates theinfluences of dynamic current distribution on the line when asignal has a wide frequency range. As frequency changes, thecurrent automatically redistributes itself among the subconductors.Hence, there is no need to assume zero line thickness or specificboundaries in the free space region surrounding the conductors.Similar considerations apply to the method of subareas and chargeredistribution. In addition to these advantages, the subconductorand subarea methods are simple and straight forward to implement.Figures 1.3(a) and (b) illustrate the subdivision strategiesadopted in the subconductor and subarea methods.1.4.2 Problems with the Traditional MethodsThe traditional methods for calculating the line resistance andinductance in microstrip lines face a common problem in their appli-cations — the large amount of memory storage required for accurateresults. In addition, the traditional methods for calculating theline capacitance in microstrip lines are based on recursion andoptimization procedures which are costly in CPU time. Following12#1^#Nline conductordielectricground plate (a)The method of subconductors;#1^#Nfr3 line conductordielectricground plane(b)The method of subareas.Fig. 1.3 Subdivision Strategy in the Subconductor and Subarea Methods.an investigation of these problems, the new proposed algorithmswill be introduced.Resistances and InductancesTo evaluate the frequency dependent R and L matrices in microstriplines by the subdivision strategy, most of the methods proposedin the literature (e.g.[4, 20, 44, 46, 47, 51, 59, 62, 63])assume a uniform current distribution across the subconductors'cross-section. For line signals with frequencies of interest forthe transient analysis, however, the uniform current distribution13assumption requires a very large number of subconductors to produceaccurate results.The linear current subconductor technique (LCST) proposed inthis work greatly reduces the number of subconductors required foraccurate results. Instead of assuming a uniform current distri-bution across the subconductors, the proposed technique assumes apiecewise linear current distribution[69]. Typically sixteen timesfewer subconductors are required with the new technique, resultingin savings of up to 98% to 99% in storage and CPU time requirements.CapacitancesIn most published material dealing with calculation of the Cmatrix (WO) of coupled microstrip lines[2, 12, 17, 20, 38, 51,57, 58, 60, 65, 66], the matrix elements Cri.7 are calculated byrepeatedly supplying a unit voltage on one line at a time untilall the lines have been excited, and determining the total chargeaccumulated on each line. The dipole method from [12] is used toobtain the accumulated charges by putting an electric dipole onall the surfaces (conductors and dielectric). The total charge iscalculated by integrating the field charge distributions inducedby the voltage source or dipole. This process involves numericalintegration, recursion and optimization; all are CPU time consuming.A new method for C calculation is proposed and derived fromthe traditional method of subareas 158]. The proposed method,which will be referred to here as the advanced method of subareas14(ANS), computes the Maxwell coefficients matrix and inverts itin order to obtain the C matrix. It is more efficient sinceit eliminates explicit calculations of charge distributions andtotal charge accumulations. Hence, the process of recursion andoptimization involved in the conventional application of the methodare avoided[68].The method of moments used in reference[38] is equivalent tothe method of subareas, but it requires the use of delta basisfunctions. Since the method of subareas uses Green's Functions,the underlying physical concept is simpler, and it is easier toimplement as a numerical algorithm.1.5 CONTRIBUTION OF THIS THESIS WORKThe main contribution of the work in this thesis is the devel-opment of numerical methods that allow more efficient and accuratecalculations of the parameters of microstrip lines. The linearcurrent subconductor technique (LCST) has been developed and im-plemented to compute the frequency dependent R and L of microstriplines. The technique has also been applied to pipe-type cablesfor underground power transmission. Another new algorithm, theadvanced method of subareas (ANS), has been proposed to calculatemicrostrip line capacitances.The methods developed in this thesis are validated by comparisonswith existing numerical techniques. These techniques include the15traditional subconductor method[59], the method of subareas[58],the finite element method[66], and the dipole method[l2] . Thenew methods result in savings between 38.8% and 99% in memoryrequirements, and between 50% and 99% in CPU time.Although the microstrip line structure in Fig.1.1(a) is used indeveloping the proposed numerical techniques, the techniques canbe extended to other configurations (for example, Figs.1.1(b) and(c)), or to other transmission line systems, with little difficulty.1 . 6 ORGANIZATION OF THIS THESISThis thesis is organized as follows. Chapters 2 and 3 presentthe theory and the proposed calculation techniques for microstripline resistances and inductances. Chapters 4 and 5 describe theproposed technique for the computation of the capacitances. Chapter6 makes suggestions for future research work and Chapter 7 outlinesthe conclusions.Finally, the appendices include division of currents in splitand non-split conductors, list the symbols and abbreviations usedin the thesis and provide a summarized reference list. 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The Propagation Of Signals Along A Three-LayeredRegion: Microstrip. IEEE Transactions on Microwave Theory andTechniques, vol.MTT-36, no.6, JUNE 1988.[30] M. KOBAYASHI. A Dispersion Formula Satisfying Recent Require-ment In Microstrip CAD. IEEE Transactions on Microwave Theoryand Techniques, vol.MTT-36, no.8, AUGUST 1988.19[31] M. KOBAYASHI and R. TERAKADO. Accurately Approximate FormulaOf Effective Filling Factor For MicrostripLine With IsotropicSubstrate And Its Application To The Case With AnisotropicSubstrate. IEEE Transactions on Microwave Theory andTechniques, vol.MTT-27, no.9, SEPTEMBER 1979.[32] H. KOBER. DICTIONARY OF CONFORMAL REPRESENTATIONS. New York:Dover Publications, 1957.[33] S. KOIKE, N. YOUSHIDA, and I. FUKAI. Transient Analysis OfMicrostrip Line On Anisotropic Substrate In Three-DimensionalSpace. IEEE Transactions on Microwave Theory and Techniques,vol.MTT-36, no.1, JANUARY 1988.[34] T. LEUNG and C. A. BALANIS. Attenuation Distortion Of TransientSignals In Microstrip. IEEE Transactions on Microwave Theoryand Techniques, vol.MTT-36, no.4, APRIL 1988.[35] K. K. LI, G. ARJAVALINGAM, A. DIENES, and J. R. WHINNERY.Propagation Of Picosecond Pulses On Microwave Striplines. IEEETransactions on Microwave Theory and Techniques, vol.MTT-30,no.8, AUGUST, 1982.[36] J. R. MART I. FREQUENCY DEPENDENCE OF TRANSMISSION LINES. PhDthesis, University of British Columbia, 1981.[37] L. MART I. SIMULATION OF ELECTROMAGNETIC TRANSIENTS IN UNDER-GROUND CABLES WITH FREQUENCY-DEPENDENT MODEL TRANSFORMATIONMATRICES. PhD thesis, University of British Columbia, 1986.[38] G. L. MATTHAEI, G. C. CHINN, C. H. PLOTT, and N. DAGLI. A SimpleMeans For Computation Of Interconnect Distributed CapacitancesAnd Inductances. IEEE Transactions on Computer Aided Design ofIntegrated Circuits and Systems, vol.CAD-11, no.4, APRIL 1992.[39] F. MEDINA, M. HORNO, and H. BAUDRAND. Generalized SpectralAnalysis Of Planar Lines On Layered Media Including UniaxialAnd Biaxial Dielectric Substrates. IEEE Transactions onMicrowave Theory and Techniques, vol.MTT-37, no.3, MARCH 1989.[40] G. METZGER. TRANSMISSION LINES WITH PULSE EXCITION. New YorkAcademic Press, 1969.20[41] W. S. MEYER and H. W. DOMMEL. Numerical Modelling Of Frequency-Dependent Transmission-Line Parameters In An ElectromagneticTransients Program. IEEE PES Winter Meeting, New York, N. Y., USA, JANUARY 27--FEBRUARY 1 1974.[42] J. R. MOSIG. Arbitrarily Shaped Microtrip Structures AndTheir Analysis With A Mixed Potential Integral Equation. IEEETransactions on Microwave Theory and Techniques, vol.MTT-36,no.2, FEBRUARY 1988.[43] H. B. PALMER. The Capacitance Of A Parallel-Plate CapacitorBy The Schwartz-Christoffel Transformation. Transactions ofthe American Institute of Electrical Engineers, vol.56, MARCH1937.[44] G. W. PAN, K. S. OLSON, and B. K. GILBERT. Improved AlgorithmicMethods For The Prediction Of Wavefront Propagation Behavior InMulticonductor Transmission Lines For High Frequency DigitalSignal Processors. IEEE Transactions on Computer Aided Designof Integrated Circuits and Systems, vol.CAD-8, no.6, June 1989.[45] E. A. PARR. LOGIC DESIGNER'S HANDBOOK. London: Granada, 1984.[46] C. R. PAUL. Modelling Electromagnetic Interference PropertiesOf Printed Circuit Boards. IBM Journal of Research andDevelopment, vol.33, no.1, JANUARY 1989.[47] R. F. PEASE and 0. K. KWON. Physical Limits To The UsefulPackaging Density Of Electronic Systems. IBM journal ofResearch and Development, vol.32, no.5, SEPTEMBER 1988.[48] T. S. PERRY. Intel's Secret Is Out. IEEE SPECTRUM, APRIL 1989.[49] S. Y. POH, W. C. CHEW, and J. A. KONG. Approximate FormulaeFor Line Capacitance And Characteristic Impedance Of MicrotripLine. IEEE Transactions on Microwave Theory and Techniques,vol.MTT-29, no.2, FEBRUARY 1981.[50] S. TALUKDAR R. LUCAS. Advances In Finite Element Techniques ForCalculation Cable Resistance And Inductance. IEEE Transactionson Power Apparatus and Systems, vol. PAS-97, no.3, MAY/JUNE1978.21[51] A. E. RUEHLI. Survey Of Computer-Aided Electrical Analysis OfIntegrated Circuit Interconnections. IBM journal of Researchand Development, vol.23, no. 6, NOVEMBER 1979.[52] R. A. SAINATI and T. J. MORAVEC. Estimating High Speed CircuitInterconnect Performance. IEEE Transactions on Circuits andSystems, vol.36, no.4, APRIL 1989.[53] K. C. SARASWAT and F. MOHAMMADI. Effect Of Scaling OfInterconnections On The Time Delay Of VLSI Circuits. IEEEJournal of Solid State Circuits, vol.SC-17, no.2, APRIL 1982.[54] T. SHIBATA, T. HAYASHI, and T. KIMURA. Analysis Of MicrostripCircuits Using Three-Dimensional Full-Wave ElectromagneticField Analysis In The Time Domain. IEEE Transactions onMicrowave Theory and Techniques, vol.MTT-36, no.6, JUNE 1988.[55] C. SHIH, R. B. WU, S. K. JENG, and C. H. CHEN. Frequency-Dependent Characteristics Of Open Microstrip Lines With FiniteStrip Thickness. IEEE Transactions on Microwave Theory andTechniques, vol.MTT-37, no.4, APRIL 1989.[56] C. SHIH, R. B. WU, S. K. JENG, and C. H. CHEN. A Full-Wave Analysis Of Microstrip Lines By Variational ConformalMapping Technique. IEEE Transactions on Microwave Theory andTechniques, vol.MTT-36, no.3, MARCH 1988.[57] J. VENKATARAMAN, S. M. RAO, A. T. DJORDJEVIO, T. K. SARKAR,and N. YANG. Analysis Of Arbitrarily Oriented MicrostripTransmission Lines In Arbitrarily Shaped Dielectric Media OverA Finite Ground Plane. IEEE Transactions on Microwave Theoryand Techniques, vol.MTT-33, no.10, OCTOBER 1985.[58] W. T. WEEKS. Calculation Of Coefficients Of CapacitanceOf Multiconductor Transmission Lines In The Presence Of ADielectric Interface . IEEE Transactions on Microwave Theoryand Techniques, vol.MTT-18, no.1, JANUARY 1970.[59] W. T. WEEKS, L. L. WU, M. F. MCALLISTER, and A. SINGH. Resis-tance And Inductance Skin Effect In Rectangular Conductors. IBMJournal of Research and Development, vol.23, no.6, NOVEMBER1979.[60] C. WEI, R. F. HARRINGTON, J. T. MAUTZ, and T. K. SARKAR.22Multiconductor Transmission Lines In Multilayered DielectricMedia. IEEE Transactions on Microwave Theory and Techniques,vol.MTT-32, no.4, APRIL 1984.[61] N. H. E. WESTE and K. ESHRAGHIAN. PRINCIPLES OF CMOS VLSIDESIGN: A System Perspective. Reading, Mass. : Addison-Wesley,1985.[62] H. A. WHEELER. Transmission Line Properties Of A Strip On ADielectric Sheet On A Plane. IEEE Transactions on MicrowaveTheory and Techniques, vol. MTT-25, no.8, AUGUST 1977.[63] H. A. WHEELER. Transmission Line Properties Of Parallel WideStrips By A Conformal Mapping Approximation. IEEE Transactionson Microwave Theory and Techniques, vol. MTT-12, no.5, MAY1964.[64] J. F. WHITTAKER, T. B. NORRIS, G. MOUROU, and T. Y. HSIANG.Pulse Dispersion And Shaping In Microstrip Lines. IEEETransactions on Microwave Theory and Techniques, vol.MTT-35,no.1, JANUARY 1987.[65] E. YAMASHITA. Variational Method For The Analysis OfMicrostrip-Like Transmission Lines. IEEE Transactions on Mi-crowave Theory and Techniques, vol.MTT-16, no.8, AUGUST 1968.[66] Y. YIN. CALCULATION OF FREQUENCY-DEPENDENT PARAMETERS OFUNDERGROUND POWER CABLES WITH FINITE ELEMENT METHOD. PhDthesis, University of British Columbia, JUNE 1990.[67] X. ZHANG, J. FANG, K. K. MEI, and Y. LIU. Calculations Of TheDispersive Characteristics Of Microstrips By The Time-DomainFinite Difference Method. IEEE Transactions on Microwave Theoryand Techniques, vol.MTT-36, no.2, FEBRUARY 1988.[68] D. ZHOU and J. R. MARTI. A New Algorithm For The CapacitanceMatrix Of Microstrip Lines In VLSI. BALLISTICS SIMULATION andSIMULATION WORK AND PROGRESS, Proceedings of the SCS SimulationMultiConference, New Orelean, USA, pages 60--65, APRIL 1991.[69] D. ZHOU and J. R. MARTI. A Linear Current Distribution TechniqueIn The Computation Of Resistance And Inductance Matrices OfMicrostrip Lines. MODELING AND SIMULATION, Proceedings of the23IASTED International Conference, Calgary, Canada, pages 47--49, JULY 1991.[70] D. ZHOU, F. P. PREPARATA, and S. M. KANG. InterconnectionDelay In Very High-Speed VLSI. IEEE Transactions on Circuitand Systems, vol.38, no.7, JULY 1991.24Chapter 2Linear-Current SubconductorTechnique for Matrices of FrequencyDependent Resistance and InductanceIn Chapter 1, we indicated that transmission line theory canbe used to model microstrip lines, and that to determine the linecharacteristic parameters (Fig.1.2(a)) is the first step towardssetting up the model. Existing techniques for microstrip lineparameter calculations share similar problems of inefficiency forpractical use. In this chapter, we derive and examine a newnumerical method — linear current subconductor technique (LCST) —to efficiently calculate the frequency dependent R and L matrices.2 . 1 GENERAL DESCRIPTIONFor signal frequencies of interest in microstrip lines, the val-ues of resistance and inductance are strongly frequency dependent.This is due to the skin effect, which limits the wave penetrationdepth in the conductors along the line, and to the proximity ef-fect, which is the current redistribution in the presence of othercurrents and is generated in wave propagation on coupled lines[9,8, 12, 13, 3]. As the frequency of the signal increases, thewave penetration depth decreases and the conductor's resistance Rincreases. In turn, the current distributed inside the conductorchanges and the inductance L varies accordingly. Traditional for-25Subdivision of line conductors+Rand L of subconductors+Current in subconductors:V =-..Z I+Modification of current withlinear current distribution+Correction ofRandL+Subconductors' bundling intoconductors, finalZ matrix Fig. 2.1 Flow Chart for the Linear Current Subconductor Technique (LCST).mulae for resistance and inductance of microstrip lines are basedon zero frequency signals and thus of very limited value.As introduced in Chapter 1, the proposed LCST technique is usedto compute the R and L parameters accurately and efficiently, with asubdivision strategy and a linear current distribution assumption.26A flow chart to illustrate the implementation of the LCST tech-nique is shown in Fig.2.1. To employ LCST, firstly the microstripline conductors, including the ground conductor, are subdivided intosmall segments (subconductors or cells) in their cross-sections ac-cording to a 28 rule. Here 5 represents the skin depth at a specificfrequency. Secondly, the segment's resistance and inductance arecalculated from traditional Ohm's law and Geometric Mean Distance(GMD). Thirdly, the current held in each cell is solved from theTelegrapher's equations defined by the initial resistances and in-ductances, as well as assumed initial voltages. Then new currentsunder assumption of linear current distribution across segment arecalculated from the currents obtained in the third step. Thesenew values of current distribution are subsequently used to modifythe resistances and inductances from the second step. Meanwhile,an inductance correction factor is incorporated into the modifi-cation. Finally, a bundling procedure is applied to the complexmatrix of subconductor impedances to obtain a reduced matrix withdimension /V — the number of equivalent conductors. The real partof the reduced matrix is the desired line resistance matrix, and itsimaginary part is the line admittance matrix. In this procedure,R and L are obtained simultaneously.In what follows, the theory of LCST will be derived. It includessubdivision, current distribution, correction factors, and bundlingprocedure. Microstrip lines and pipe-type cables are used asexamples to illustrate the technique.272 . 2 CONDUCTOR' S CROSS-SECTION SUBDIVISIONThe criterion for the conductor's cross-section subdivision isbased on the fact that the wave propagating along the line penetratesa small depth into the conductors when the signal frequenciesincrease to relatively high levels. (At zero frequency, the wavepenetrates the whole conductor which results in a uniform currentdistribution across the conductor's cross-section [5].) The skindepth is defined as_ ^ (2.2.1)A1ufawhere- Skin depth;y — Permeability of the conductor;a — Conductivity of the conductor; andf — Frequency of signals on the line.A sinusoidal distribution rule is used to determine the size ofthe subconductors, according to the line geometry and the skindepth[16]. The number of subconductors may change at differentfrequencies. The schematics of the subdivision for microstriplines and pipe-type cables are shown in Fig.1.3(a) and in Fig.2.3,respectively. The formulae to determine the subdivisions arederived as follows.2.2.1 Subdivision of Microstrip LinesIn order to take into account the nature of the current distribu-tion, the subdivisions are made denser near the conductor edges and28L2 arcsin (V 25^Thickness),7r (2.2.3)corners according to sinusoidal functions[7, 16]. The microstripline conductors are divided according to a 28 rule. The groundplate is considered as N pseudo conductors (with N the number ofline conductors in the system), as shown in Fig.2.2. An equalsegments rule is used for the subdivision of the ground plate.Let J be the number of subdivisions on the line width, and Lthe number of subdivisions on the line thickness. The values ofJ and L are calculated from7TJ= (2 . 2 . 2 )2 arcsin (V Width )andwhere Width and Thickness are shown in Fig.2.2. Equations (2.2.2)and (2.2.3) are derived from reference [16] using the 28 rule todetermine the size of the subdivisions. In the line conductors,the end points (x,y) enclosing a cell are calculated from (referto Fig.2.2)1x =^• [xv(i, 2) + xv(i, 1) — cos j, • (xv(i, 2) — xv(i, 1))]I^1y =^• [yv(i, 2) + yv(i,l) — cos jy • (Y.„(, 2) — yv(i, 1))ydi= 1,...,N (2.2.4)where29Yv(i,2)ConductorMI••^I^IDielectric—.^)c,(i,1)^)s,(i,2)Width:heightthickness•••Ground platePsuedo conductor(a) Co-ordinates with even number of conductors; Conductor^ Y(i,2)Widththickness•••:heightDielectric^MO)^)c,(i,2)^y,(1,1)Ground platePsuedo conductor(b) Co-ordinates with odd number of conductors.Fig. 2.2 Coordinate System for Microstrip Linescos j x = cos((j — 1) x 7r),j = 1 . . . J,^(2.2.5)andcos jv = cos j = 1 . . . L.^(2.2.6)30For the ground plate, the numbers J and L are used to equallysubdivide the pseudo conductors{x =^xv(i,2) — x v(i ,1)y yv(j,1) yv(i, — MO) i = N +1,...,2N.^(2.2.7)The ground plane is assumed to be very large compared to the areaoccupied by the line conductors. In the above equations, xv(i/i)and yv(i,j) ascribe the coordinates of the j-th vertex and the i-thconductor.2.2.2 Pipe-Type CablesPipe-type cables consist of different kinds of conductive mate-rials with an arrangement as shown in Fig.2.3. Notation A, B, andC in Fig.2.3 specifies Cable A, Cable B, and Cable C, respectively.Each cable is formed by a core and a sheath. A steel pipe isholding the cables.A similar subdivision method to that used for microstrip linesis employed to subdivide the core, sheath and pipe of pipe-typecables. To adapt to the circular geometry, arc-shaped cells orcurved cells are employed[1], as shown in Fig.2.3. The outside pipecorresponds to the ground plate in microstrip lines. Following theazimuth direction, the circles are divided into equal sectors. Thediameter or the annulus is sinusoidally divided along the radialdirection.31For the core, the number of subdivisions, NRi, isIrNRi =^2 arcsin (AP11-)for the sheath, Arli2,7rNR2 = ^2 arcsin (V-212—)and for the pipe, NR3,(2.2.8)(2.2.9)NR3^ir^ (2.2.10)2 arcsin^pr2 -priIn the above equations,81 — Skin depth in the core;82 — Skin depth in the sheath;63 — Skin depth in the pipe;ri — Radius of the core;r2 — Inner radius of the sheath;r3 — Outer radius of the sheath;pri — Inner radius of the pipe; andpr2 — Outer radius of the pipe.Along the azimuth direction, the number of subdivisions isN^27rri^2z. =^=168^86(2.2.11)with i, the sequence index of conductors, as shown in Fig.2.4. Herepri takes the place of ri in the pipe.The coordinates of the vertices of each cell are determined(refer to Fig.2.4) byI x = xo xi cos a — ysinc= Yo x sin a + cos a(2.2.12)32r1 = 24.25mmr2 = 40.25mmr3 = 42.25mmr4 = 44.25mm240mmPipe: a = 3907.7 S/mmpr =500Sheath: a = 4800 S/mmCore : a = 34060 S/mm[is = 1 for allFig. 2.3 Pipe-Type Cable Configuration and Subdivisions.in which coordinates (x,y) represent the position in the maincoordinate system with its origin as the center of pipe circle,,^,and coordinates (x , ,y , ) are measured with respect to three secondarycoordinate systems with their origins (x00/0 at the centers of thecables. In the above equations, a is the axis' angle between themain and secondary coordinate systems.33Additionally, in the secondary coordinate system, we have= p cos (^1 y^p sin 13^ 2.2.13)where p and 0 are the radius and azimuth of a cell. The cell'scircular size p in the core is calculated from(NRI—j rp^ri cos ^ 2)' j^N^(2.2.14)with index j starting from the centre of the cable.^For thesheath,— 1 7r)p = r2 (r3 — r2)sin ^NR2 2 )^j = 1, . , NR^(2.2.15)and for the pipe,Pi = pri (pr2 — pri) sin (i NR 21 7r^ • • •) — 1^N Rq3 (2.2.16)2.3 SUBCONDUCTOR RESISTANCE AND INDUCTANCEFROM TRADITIONAL FORMULAEAfter the line conductors have been subdivided into smallsegments, cells or subconductors, the initial R and L of thesesubconductors can be calculated using Ohm's law and the concept ofGeometric Mean Distances (GMD).2.3.1 Microstrip LinesIn Fig.2.5(a), a given segment is labeled (i,j,/), meaning row jand column 1 in the i-th conductor. (Fig.2.5(b) shows schematically34Fig. 2.4 Coordinate System for Pipe-Type Cables.the electromagnetic field in a microstrip line system when a staticsignal travels down the line[18, 15, 10, 6].) The resistanceper unit length for subconductor (i,j,/), assuming uniform currentdistribution in its cross-section, is = 1 (2.3.17)whereaiÄjil- Cross-section area of subconductor (i,j,/);— Resistance of subconductor (i,j,/); and35cri - Conductivity of conductor i.The inductance per unit length (see Fig.2.6) is[14, 11]42= (GadGbc)—1n.2r^GacGbd)^(Him)with(2.3.18)1 ^fIn (Gad) = AaAd J.]AA bint ditadAb (2.3.19)where a, b, c and d correspond to four long parallel subconductorsor filaments, ja = ,aoX jar is the permeability of the line conductor,r is the distance from a point in cell a and a point in d, and AL,,, isthe cross-section area of cell a. Constant Ito is the permeabilityof air or vacuum, and pr is the relative permeability of the lineconductors. Gad, for example, is the GMD between filaments a andd. As shown in Fig.2.6, loop 1 is formed by connecting the endsof filaments a and b, and loop 2 by connecting those of c and d.Then 42 is the mutual inductance between loop 1 and loop 2. Ifloop 1 and loop 2 are connected to become one loop, 42 becomes theself inductance 41 or 122.In the microstrip line system, the line permeability is assumedto be that of the air, i.e. Ito = 4r x 10-7 (Him) and /4=1.^Theshape of the subconductors cross-section (cells) can be rectangularor square. The GMDs are calculated from the geometric positionsof every two cells.Self Geometric Mean Distance (GMD)The self GMD for rectangular or squared cells is, with /4=1,Gpp = 0.2335 x (Aw + At)^ (2.3.20)36(a) A cell or subsection denoted as (i,j,I),meaning row], column I in conductor i.(b) Schematic of magnetic and electric field ina single conductor microstrip line.Fig. 2.5 Cell Index and Fields Distribution in Microstrip Lineswhere Alv and At represent the cell's width and thickness, respec-tively[1, 14].GMDs between Subconductors in Microstrip LinesMutual GMDs of retangular or squared cells in microstrip lines37loop 1^loop 2Fig. 2.6 Loop Definitions Related to GMD's for Inductances.(a) Paralleled cells;(b) Non-paralleled cells.Fig. 2.7 Schematic of Cells in Microstrip Lines.38(x — x')(y — y) {(3^x x ,)2 arctan Y — Y, + (y y')2 arctan^x — x^x — xy — y'l 1 y2 y, X2 X2,Y1 Y1 xi xi(2 3.21)are the integral of geometric distances between two cells [17],s425^(Y — y ) — 4(y — yi)2(x — xi)2 + (x — x')4^, 2^, 26 24^in [(x — x ) + (y — y ) 1in (Gad) 7---where (x,y) is located in cell a, (x',i) is in cell d andxi , x2, xi, x2 IYI,Y2,Y1,Y2 are the co-ordinates enclosing the cells.When two cells are merged, the mutual GMD between them becomethe self GMD.2.3.2 Pipe-Type CablesFor pipe-type cables, the cross-sections of subconductors (orelementals [14]) have different shapes from those of microstriplines. The permeability for the core and sheath is po while thatfor the pipe is go x,tir = 500p0. The subconductors resistance andinductance are still determined by eqns.(2.3.17) and (2.3.18). Thecorresponding GMDs, however, will follow different formulae.The index of a cell in pipe-type cables is given by (k,m,n),indicating the m-th angle in the azimuth and the n-th circle alongthe annulus in the k-th conductor (core, sheath or pipe), as shownin Fig.2.8.Paralleled Curved CellsAs shown in Fig.2.9(a), the cells in the conductors of thepipe-type cables are curved and close to each other. For these39(0,1)Fig. 2.8 Two Kinds of Cells and Indices of Pipe-Type Cables.shapes, the GMD can be calculated with the same formula as that ofmicrostrip lines, eqn.(2.3.19), except that the average arc lengthsof cells, ZVI and A102, are used instead of the widths Am4 and Atv2,respectively,in^GIN(.955)4-i.e.,1 — Y‹ +1)2 In + D2)+—2,41, /< — ><^Y< + D2 in (1q + v2)+2,6.1°1,42 2,41,422DX, 2DX^ arctan (N2)^, arctanAP1z-1192^D^APiAP21)2^ ln ()( +D2)P1,42 \\ + D2,41 — A P2 )(1 =2(2.3.22)with(2.3.23)and= Aroi AP2 X, 2(2.3.24)40Non-paralleled Curved CellsAS shown in Fig.2.9(b), when two cells are located relativelyfar apart, each cell is subdivided into four sectors such that thesectors enclosing one straight cell edge will have one-sixth ofthe arc's length, while the middle sectors will have one-third ofthe arc length. Let A be the spanned angle of a curved cell.Then pairs of points are symmetrically placed about the edges ofthe sectors, at an angle distance of 0.564x 4[14]. The mutualGMD between two cells is, therefore, the geometric mean of the 36distances between six points in one cell and six points in theother, as shown in Fig.2.9.If P is one of six points in cell p, and -0 is one of six pointsin cell q, the GMD between cells p and q will beGpq^36 II II (Pi-01i=1,6 j=1,6(2.3.25)where /5i and 01-i are coordinate variables of points P and -0,respectively.Self Geometric Mean DistanceThe self GMD for curved cells is given byG ^0.2315 x (Ap + At) x e1-4632(1-ur)^(2.3.26)41{ 3u,6 (Po —6Gpq(b) Narrow curved cells that are far apart.Fig. 2.9 Cells for GMD's in Pipe-Type Cables.where Ap and At are curved cell's arc length and thickness,respectively[14, 11].GMD for Circular CellsWhen cell p is a circle, eqn.(2.3.27) becomes(2.3.27)where P0 is the origin of circular cell, and Q3 ascribes thecoordinates of one of the six points located as in Fig.2.9(b).42The self GMD of a circular cell is:G = r x e-7\"-PPwhere r is the radius of the circle.(2.3.28)2 . 4 CURRENT DISTRIBUTIONIn a linear system such as microstrip lines or pipe-type cables,when the currents are assumed to be uniformly distributed acrosseach subconductor, their values can be obtained from solving thefollowing linear equations:[5,^4]VI- Z11^Z12^ Z1K - - Ii -172 Z21^Z22 Z2K 12(2.4.29)Zii^Zi2^ ZiK1/‹- zic2^ ZKKor in compact notation,-VKxi = ZKxKIKx1^ (2.4.30)wherei=j (2.4.31)rooZia =with K being the dimension of subconductor's impedance matrix. Formicrostrip lines,K=2xNx,IxL - 1^ (2.4.32)with Ar the number of line conductors, J the number of subdivisionson the width, and L the number of subdivisions on the thickness ofthe line conductors and pseudo conductors. For pipe-type cables,6K=^Nk X Nk' Npk X N 13'^3 x (Nki — 1) — 1,^(2.4.33)k=143with Ark the number of subdivisions on the radius, Aq the number ofsubdivisions along the azimuth for the conductors, and Arpk subdivi-sions on the radius and /NT the subdivisions along the azimuth in thepipe. Also, rij (i,j = 1^K)^is the subconductor's resistancefrom eqn.(2.3.17) and^(i,j = 1^K) the subconductor's mutualor self inductance, calculated from eqn.(2.3.18). In particular,1^roo =^ (2.4.34)o-A000is the resistance of a reference cell which is arbitrarily choseninside the ground plane or pipe.For a different subdivision criterion, e.g., 2(5 or S rule, therewill be different current distributions. Because of the assumptionof uniform current distribution across subconductors, the currentdensities will present \"jumps\" at the edges between cells. Theplots in Chapter 3 show these discontinuities in the solutions ofcurrent density.To minimize the error caused by the discontinuities betweensections, a relatively large number of subdivisions would berequired. This, in turn, would require large amounts of computermemory and resources, especially at high frequencies.By assuming linear current distribution across each subconduc-tor, the current density can be made continuous across line sec-tions. This results in much smaller computer memory and resourcesrequirements for a given accuracy of the results.442.5 APPROXIMATION OF LINEAR CURRENTDISTRIBUTION ACROSS SUBCONDUCTORSLinear current densities at the cell boundary can be evaluatedby averaging the values of the current densities in the cells aroundthat boundary. Those current densities in the cells around theboundary are calculated assuming uniform current distribution. Thelinear current held in the cell (subconductor) will be the integralof the assumed linear current density. The procedure implementedis as follows:Step 1. Calculate values of uniformly distributed cur-rent held in cells, by solving eqn.(2.4.30);Step 2. Approximate boundary values of linear currentdensities as the geometric mean of uniform cur-rent densities in the cells around the boundaryobtained from Step 1; andStep 3. Integrate the linear current density in each cellto obtain a new total current value in the fullconductor.Step 1 is the same for all cases. In Step 2, the values of currentdensity on the vertices or edges of a cell are approximated by thegeometric mean value of those current densities of the particularcells around the vertices or edges. They are, therefore, dependenton the line configuration. Microstrip lines and pipe-type cablesare used as examples to describe in detail this procedure.452.5.1 Microstrip LinesAfter the values of uniformly distributed current density areobtained from eqn.(2.4.31), the values of linearly distributedcurrent density at each vertex of cell (k,m,n), mn, (2\" =1,2,3,4),are estimated by the geometric mean of the uniformly distributedcurrent densities of the four cells around that vertex, i.e.JZmn =^Jkmn X jk(m±i)n X ikm(n±1) X jk(rn±1)(n±1) z =1..4^(2.5.35)wherejkmnikrrin 24krnn(2.5.36)is the value of uniformly distributed current density in cell(k,m,n). The operator \"±\" means either \"+\" or \"- If . Vertex indexz is shown in Fig.2.6 while cell index (k,m,n) is shown in Fig.2.8.For cells or subconductors located at the edges of the lineconductors, excluding the ones containing conductor vertex,z = 1,2 orJlzcrnn =^Jkmn X k(m±l)n,^z = 3,4^ (2.5.37)z = 1,2 orJlzcmn =^j knzn X Jkm(n±1))z = 3,4(2.5.38)For cells containing a vertex of line conductors,JZmn = j Icrnn,^z= 1, or 2, or 3, or 4.^(2.5.39)and46The new current in cell (k,m,n) is the integral of the assumedlinear current density on the rectangular or squared cells:x2Y2= lli(x, y)dx dyx2Y2[ii (x ,^+ in (x , y)] dx dyximx.,2y2= I f ((%' +QC' ) x^+ 93\") y (C'^)) dy(2.5.40)XiYi71^, 9 72^, 73^, 9 74= krnnl- kmn-r\" krnm. 6in which r. is the linearly distributed current carried by cell3(k,m,n), while 244mn is the area of that cell. The xi,x2,yi,y2 pointsare the four coordinates enclosing the cell (refer to Fig.2.10).The linear functioni(x ,^= 9ix + 93y^ (2.5.41)is assumed to be the linear current density distribution acrossthe subconductor, with JLin (z = 1,2,3,4) as the values at its fourvertices. 94,93 and t can be determined by three of four vertexcurrent densities. Then in each cell, two sets of 2t,93 and Q areobtained for the integration: { } and2.5.2 Pipe-type CablesThe same approach described above for microstrip lines is appliedto approximating a linear current density in each subdivision ofpipe-type cables. In these cables the subconductor's cross-sectionis shaped as a part-annulus or curved cell. Equation (2.4.31)is built for the resistance and inductance of these subconductors.Then from Step / above, uniformly distributed currents are obtained.47J[i(x,Y)]Fig. 2.10 Schematic of a Cell and its Current Densities for Microstrip Lines.The conductors in pipe-type cables are numbered in the sequence:cable 1, sheath 1, cable 2, sheath 2, ..., then pipe; from number1 to 7 as shown in Fig.2.4. Cell (k,m,n) refers to the k-thconductor, m-th layer along the radial direction and n-th angle inthe azimuth direction.In Step 2 of the solution procedure, for circular cells (k,1,1),the linear current densities at the edges are estimated by47nn = V ikll X ik2n X ik2(n±1) , (z = 1, 2, 3)^(2.5.42)with three randomly chosen points on the circle. For curved cellsadjacent to the circle, the linear current densities at the vertices48(b) A circled cell and its current densities.Fig. 2.11 Schematic of Cells and TheirCurrent Densities for Pipe-Type Cables.are evaluated fromT1 ,3^\\Idkmn =^ikll X ik2n X i k2(n±1) 1T2,4 =^/,:' kmn^4\\1 Jkmn X ik(m+i)n X ikm(n±1) X i k(m+1)(n±1) •(2.5.43)49For general curved cells, those densities are calculated from4mn = 4VIilonn X jk(rn±1 )7z X ikm(n±1) X k(m±1)(n±1)^(z^1,2,3,4).^(2.5.44)Figure 2.11 displays a schematic of current density planes on acurved and circled cells.With linear current density assumed in cells, one has i(x,y) =Qtx C, where = 94' + QC', 93 = 93' + 93\", and Q = C1+ c\". Theconstants 2(1, 931, C' or Qt\", 93\", C\" result from the following linearequationsx2x3Y^11[2111^[,ItY2^93 = JY3^Ci^47-nn(2 5 45in a plane supported by cell vertices (xi,Y1), (x2,Y2), and (x3,y3), andX4 Y4X2 Y2x3 y311[2t1:,1^[Jtcni1^= 47nnnmn(2.5.46)in the other plane formed by cell vertices (x4014)7 (x200, and (x3,y3),as shown in Fig.2.11. The quantity Z e 0,201 or z G {2,3,4},corresponds to the current density evaluated at cell vertices.Because four vertices of a cell make two linear current densityfunctions, the integration in Step 3 of the solution procedurehas to be calculated on two split planes enclosed by curved cellboundaries. Then the new current value in a cell under the linear50current distribution assumption is= Ki x intel1 + K2 X int el23where02p2= I (911 • p cos + 93' • p sin 0 + C') pc/000,11 3 3\\= — Pi) [QC (sin 02 — sin 00 — '(cos 02 — COS el)] +—2- — I9?) • (0and02 P2(2.5.47)(2.5.48)(2.5.49)intel2 = f (21\" • p cos 9 + 93\" • p sin0+ C\") pc/pc/08 P1=^— pT)[91\" (sin 02 — sin 01) — 93\" (cos 92 —CH 2^22 (P2 —Pi) • (02 —00.In the above equations, pu p, p, 01,02 and 0 are the correspondingpolar coordinates of the cell positions in the local (secondary)coordinate system, and will take the values from pl to p4 or fromal to a4 as illustrated in Fig.2.11. K1 and K2 are between 0 and1, K1,K2 E [0,11. Both are assumed to be 0.5 in the programs.Here intel2 follows the same formula as inteh's except forQC, 93\", and C\" instead of 911, 93', and C', respectively.2 . 6 LINEAR CuRREbrr DISTRIBUTION CORRECTION FACTORSIn this section, the linear current distribution approximationis employed to modify the subconductor impedances in eqn.(2.4.33).cos 01)] +51V1V2ViVK -zi,+(f;-14z2J+(f;+.6)zt,+(ft+ f,)^•Z1K+(.6.0Z2K+ZICK+ (fK+ fK)- (2.6.50)11;zic.,+(fK+ f.;)ti'^-^Y'Kx1^KxK Kxl (2.6.51)With the correction for linear current distribution, eqn. (2.4.31)is written as:Zi 1+ fl+ fl )^Z12+ (f1+ f2iZ21+ (f;+f; )^Z22+ (f2+ /2 )Z1+ (f:+ f;)^;2+ (f:+ f )(f1{-1- fi) ZK2+ fk+orwhere 4+ and fi+ fi are the linear current distribution correctionfactors for the impedance of cell (k,m,n), zij(m-1) x J n) , with=^x f.^- ^ziiand^„, ^zij= — x fj.(i, j= (k-1) xJ x L+(2.6.52)(2.6.53)By rearranging eqn. (2.6.52), the linear distribution correctionfactors, f (i= 1,...,K), can be found by solving the equation withthe supplied initial voltages, V, (i =1,...,K). That is,+^112z„ 2z=n-1-;^a2aLr.11 I^Z22 2Z22 -- V1 + CQ1 -V2 CQ2-ZKKzKK -laj.z j j 3•^ti +a,12– CQiv, – CQKzili_tZKK R1,,^(2.6.54)fK -52in whichCQi = E^(2.6.55)j=1and(2.6.56)• •j=i ZzzThe corrected value for the cell's impedance in eqn.(2.4.33) is2/,3 =+ (i;+) . (2.6.57)This quantity can also be written as= rii jxii 1(2.6.58)y ^jbij^(gii^j(bi3 Abii)•Here, r and^are the real and imaginary parts of 23 respectively,while Aqii and Abij are the increments introduced to gij or bij withthe linear current distribution.2.7 INDUCTANCE CORRECTIONIn the test cases using the linear distribution correction, itwas found that the percentage error in the final inductance valueswas higher than that of the resistance. This is because the conceptof GMDs is established under the assumption of an infinitely finecurrent loop (or uniform current distribution along the currentloop), regardless of the subdivision rules. There are two options53to improve the correction of the inductance for linear currentdistribution:1. Post-processing - Use of correction factors to compensate forthe error; or2. Pre-processing - Re-calculation of the inductance for linearcurrent density before finding the Zjj correction factors.2.7.1 Inductance Formula • Option 1The parameter inductance is derived from the flux linkages of thecurrents. From electromagnetic theory[14], the flux linkage at apoint on subconductor d due to a fine current Ja•dAa in subconductora is, in per unit length,r2^2Oad = g • Ja • dAa • ln [^+ 1) — ,\\/(—) + 1 + —Di^(2.7.59)27^ D 1with D being the distance between subconductor a and the point ind, and ja the current density in a.With the assumption (>1), 'Oa,/ becomesOad = -1±- • Ja^[ln () - 1].2r (2.7.60)Then the flux linkage of d with a, due to the whole current in a,will be the integrals on their cross-sections ALa and Ab:Wad =^if OadditadildAdA dA.• f f [111 (-2r) — 1] • a • ditad4d •2r AdAd A.(2.7.61)54Integrating by parts, the linear current distribution across sub-conductor a will be'F ad =^kJ. f (ja^[ha^— daa — f I [hi^— lidAadj a) dAd=^.^f (4, f [In (-)) - 1] 4a - f I [1 n^- 11 clAa • J:dAa)dAd2ir AdAd^Aa^ A.A.p 1 {^f [=^ in (i1_,) _ddAadAd—27r —AdAdA„- J: I j I [In (51,) - dAadAddAa}Ad^Aa AAA. Ad Anr>D p^1•^• {1- a • 1 1 [111 (-2i) - 11dAadAd- 27r Ad VAd A.—Jailif[(-2()-1]dAadAddAa}Aa Ad An= P 1 {J a 1 I [in (7721) - 11dAadAd27r Ad •Ad Aa(2.7.62)—5- 2(.4 I I r_() — A a + In 2[1 dAadAddAa},^2( Aa Adp127r • Ad{fa. 11 [in (_2() _ i]deladAdAdAa2[ • Jai [ln (-21) - 1 + Aa + 1 -11121PAadAddAa}Aa Ad127r • Ad •(Ja^2[ • Ja')•I .1 [in (52[) ddAadAd p 127r Ad 2 [• J:(Aa + 1 -11120 -AaAdAdA.=^Ad^+ 2 jai )^{In (^— dAad4dAdA.+ Li- • Aa 2[ • J: • Oa + 1- In 20.27rFrom definition of GMD,11nD • actadln (Gad) = AaAd I fAd A.then, it follows that(2.7.63)Wad = 2Pr • Aa • (Ja + 2L/a)• {ln (-2[ - 1} + Li • Aa • Ja • (Aa + 1 - ln 20. (2 .7 .64)Gad^27r55The mutual inductance between loop / and loop 2 in Fig.2.6 is,therefore,1,^1F12^Tac — Tad — (Tbc Ibd) 12^ 1-1 f Jad4aA.,a Aa(J, + 2[4)^[GadGbc 127r^f Jadlia^Ga,Gbd..1A.=^' 112 •(2.7.65)In the above equation, the inductance correction factor is definedaswhere= Aa (Ja + 2 Ucl) f JadA.A„(2.7.66) A.2= Ja(Aal) Ja(Aa2)21.1— 0.7 and Dielectric Constant Er = 4.5.Since microstrip line conductors contain a very small surfacearea compared to the ground plane, the dielectric constant for the111capacitance will not always be the exact physical value Er. Aneffective permittivity, eeff, can be defined for the structure [6,7]. As frequency increases, seff goes from an initial value upto cr. The frequency dependence of Eeff is one of the reasons forwave dispersion on microstrip lines and causes the capacitance tochange as frequency changes.However, within the frequency range of transient analysis andthe range of validity of the transmission line model, the variationof Eqf is very small for silicon. For example, with 1.2pm CMOStechnology in VLSI, the width to height ratio of microstrip linesWis generally about -- = 1.2m = 3388 [1]. Its 50% dispersiveh 3.5417gfrequency is then around 5.0x 1012 Hertz (Er =4.5), calculated fromthe empirical formulae with Eeff and the assumption of an infinitethin line conductor[6, 7]. Beyond the frequency of 5.0x 1012 Hertz,the capacitance will begin to eventually saturate to the asymptoticsolution calculated from the dielectric constant E. Below thisfrequency, we can use eeff as a constant. The capacitance can,therefore, be approximated as a constant value.Figures 4.3 and 4.4 show two curves of capacitance versusfrequency for a single conductor microstrip lines. Their 50%dispersive frequencies are higher than 5.0X 1012 Hertz (for Er =4.5).The capacitances were obtained from the empirical formulae andconformal-mapping methods of references [7, 6, 15, 11, 8]. Asshown in these figures, the capacitance varies less than 0.46%o(4.6x 10- 4), as frequency increases from 0 to 2.6X 1010 Hertz.112x10-114.8970Capacitance vs. Frequency4.8965E 4.8960L34.8955a:1z'C-.) 4.89504.89454.89400 0.5^1^1.5^2^2.5^3f - HzFig. 4.4 Capacitance of a Single Conductor Microstrip Linewith Ratio -14hL = 0.5 and Dielectric Constant Er = 4.5.In general, for the frequencies of interest for transientanalysis, the capacitance of microstrip lines can be practicallytreated as constant for VLSI studies.4.2.2 Capacitance With Frequency Dependant E(f)When the dielectric permittivity itself is a function of fre-quency, such as the E(f) associated with gallium arsenide, the 50%dispersive frequency can be located in the lower band. In thiscase, the variation of capacitance with frequency can not be ne-glected for transient analysis in microstrip lines. As frequencyincreases, the capacitance increases from the zero frequency value113to a saturated value [6, 13, 171.Furthermore, if the dielectric or substrate is anisotropic,the permittivity becomes a permittivity matrix e. Then thecapacitances of the microstrip line becomes the product of anequivalent permittivity derived from e and the capacitances withoutthe substrate, C0[3].When the capacitance is obtained with constant e, it is very easyto incorporate c(f) and E into the final solutions. Therefore, theprogram of ANS is first developed for silicon substrate microstriplines with constant E.4.3 ANS VERSUS TRADITIONAL METHOD OF SUBAREASIn the traditional method of subareas [5, 14], the externalsurface of the line is initially divided into smaller subareas.Then each subarea is subdivided into increments. Based on theseincrements, a process of optimization is further used to match theassumed boundary conditions of electric potential and to obtain thecharge densities at each end point of the subarea. The chargedistribution between end points of the subareas is assumed to belinear. Consequently, the charge accumulated on each subarea canbe evaluated and the charge on all subareas can then be integratedto obtain the total charge on each line conductor. From the totalcharge on each line, the capacitance can then be calculated. Thisprocedure is implemented as follows. Assuming an equipotentialconductor surface, the elements of the capacitance matrix {C} are114computed from the following equation with a set of initial voltagevalues:{Qi} = {CO x^ (4.3.4)where Qi is the charge on line i, Cij is the capacitance betweenline i and line J, and yi is the assumed value of the potentialon line j. Assuming a unity potential boundary condition 17j, theelement of {C} is equal to the value of the charge Q.In this procedure, the line conductors have to be excited oneat a time by the applied voltage. This process has to be repeatedN(N+1) times for N coupled lines.2In the proposed ANS technique, the conductor's surface is alsoassumed equipotential. However, the subdivision is only made onceinstead of twice. And there is no need to integrate the surfacecharge on each conductor as an intermediate step to obtain thecapacitance matrix (C). The Maxwell coefficients matrix of allsubareas, {Pij,km}, is directly evaluated from Green functions. Thenmatrix reduction techniques are applied to the subareas matrix{Pij,km}. This matrix is then reduced by a process of matrix reductionor subarea \"bundling\"[2] to {PO, the Maxwell coefficients matrixfor the full line conductors. From{vi} = {PO x {gi},^(4.3.5)the capacitance matrix WO is computed by inversion of the {TOmatrix, i.e.,fcial =^ (4.3.6)115The following sections introduce the theory of ANS. Solutionwith this technique includes subdivisions, Green functions, Maxwellcoefficients of subareas, and charge distribution assumption.4 . 4 SUBDIVISIONSSince charges tend to concentrate on the sharper ends of theconductor surfaces[16], the subdivisions should be denser at thecorners. Similarly to the subdivisions technique in LCST, thesurface of each conductor is divided into small areas by a sinusoidalfunction.The coordinates defining each subarea are determined by[14]^I X(i,j) = .Ex,(i, 1+ 1) + xv(i,l) — C osj • (xv(i,1 + 1) — xv(i,1))]^(4.4.7)Y (i, j) =^+ 1) + yv(i,l) — Cosj • (yv(1,1 + 1) — yv(i,1))]where (xvoh) points to the conductor's vertex; (i,j) means thej-th subarea of the i-th conductor; andCosj = cos ((i — 1)7)k(1)withk(1) = {nnu; 1 = 1,31 = 2,4 •(4.4.8)(4.4.9)116In the above equations, nw and nt are, respectively, the numberof subdivision on the line width and thickness. The penetrationdepth 6' can be used as a threshold in the automatic generation ofthe number of subdivisions. Other criteria could also be selected.Here, we useWidthnw = nt = 2 X^.Thickness (4.4.10)4 . 5 GREEN' S FUNCTIONSThe Green functions are defined as the electric potentialproduced by a unit charge. For the configuration of Figs.4.1and 4.2, the Green functions can be derived from the well—knownmethod of images[14]. Respectively,^G (X , y I xi , yi) = ____1 '\"--‘')^k (E2 — El471-€ z--,`^1 k=o^62 + €1( _ .1))ln(x — x') 2 + (y + y' + 2kh) 2(x — x')2 + (y — y' — 2kh)21 k+1v• ( Dk+i (€2 — El)^.47€1 kr—d=0 ‘ j^C2 + Ciln (x — x' ) 2 + (y + i + 2(k — 1)02(x — x1)2 + (y — y' — 2(k + 1)h)2andG(x,yIx' ,y1) = 147r€Sinh2[7(xl + Sin2 1r (Y+Yi ) 2h^2hIn^sinh2[7(s1 + sin2 [7(Y-Y1 )12h^ 2hwhere h is the height or thickness of the dielectric;,(4.5.11)(4.5.12)C-11€2 9 and e are, ^,, ithe permittivity of vacuum or the dielectric; (x,y) s the position117of the source charge; and (x,y) is the location of the potentialto be determined.4.6 MAXWELL COEFFICIENTS OF SUBAREASIn calculating Maxwell coefficients of subareas fpii,kml, twointegrations are involved: first the integration of the potentialcaused by the charges distributed in each subarea, and second theintegration of the potential produced by the total charge on thesurface of each conductor. The second integration is implicitlyaccomplished by the technique of subareas' bundling. The firstintegration is performed as follows.By integrating the corresponding Green functions, the potentialin position (x,y) is, with reference to Fig.4.5(b):y) = I pii(t)G(x, y/xi,i -Ft cos 0,^t sin 0)dt^(4.6.13)where Al is the value of the potential produced by the charge accu-mulated in subarea (i,/), the 1-th subarea of the i-th conductor;is the length of the subarea (i,1); pa is the assumed linearcharge density in subarea (i,/); and G is the corresponding Greenfunction. Also, in eqn.(4.6.19),Ix' = X1 + t cos 0,yi = yii + t sin 0(4.6.14)in which t is an intermediate variable.118According to the mid-value theorem of integration, eqn. (4.6.19)becomeswhereIilpii(x , y) = Ai/ • G (x , y I xi' 1, yi i) • f pii(t)dt=^Pii(xii 1,^G(x,Y 1 xi' 1,{ x1 = a + V • Ail cos 0yii =^v • Ai/ sine(4.6.15)(4.6.16)withv E [0,1]^ (4.6.17)From the assumption of linear charge density distribution^pii(t) = 1Ct C,^ ( 4.6.18)and we have v =^By putting a unit value of charge on eachsubarea, eqn. (4.6.21) givesPii(x,Y)= Ail. G(x,Y1xii^(4.6.19)On the other hand, the integral in eqn.(4.6.19) can be evaluatedby combining the trapezoidal rule with the mid-value theorem asfollows:pii(x , y) = Ai/ • pii(xi' 1, i) • f G (x , y I ,^dt1^2=^• Aii • pii(xii, yii) • (G(x , y I x ,^G(x, y I(4.6.20)where{ xi1+1 = xii + Ai/ • cos 0= Yii 4- Ail • sine (4.6.21)119with the source charge in the middle of the subareax1 = x^t' • cos 0=^t' • sin 0^ (4.6.22)and1t = v • Ai/ = 2 (4.6.23)Test cases show that the trapezoidal integration rule was better be-hayed than the mid-value theorem in the integration of eqn.(4.6.19).The Maxwell coefficients of subareas, {pw,m}, are, therefore,Pij,km = Pkm(Xij,Yij),^ (4.6.24)where (xij,yii) is the potential position of interest in subarea (i, j) .4 . 7 CHARGE DISTRIBUTIONThe line capacitance is defined as the charge on the line with aunit value of voltage applied to the line. After a matrix inversion,the Maxwell coefficients correspond to the potentials excited by aunit charge. Therefore, with ANS, the actual values of the chargedensity distribution do not need to be explicitly calculated.In fact, the charge density can be assumed to be of any shapebecause only the source position is relevant in the integrationto obtain the Maxwell coefficients of the subareas. In Fig.4.5(a), the integral corresponds to the area under the assumed curveof charge density. If the charge density is assumed to be alinear function in the subarea, the unit charge source for the120ChargeDensitymiddlevalueti (x',y')^t2(a) The values of the integral under curve 1and line 2 are equal.(Xil,Yil)(b) A subarea and its co-ordinates.Fig. 4.5 Linear Charge Density and a Schematicof Coordinates Relationship in a Subarea.Green functions will be located in the middle of the subareas. Theproposed AMS technique is, therefore, very flexible in terms of theassumed form of charge distribution. Other charge distributionscan easily be incorporated into the equations. Once the properposition of the point charge source has been determined in thesubarea, the solution procedure remains the same as for the case121of linear charge distribution.4 . 8 DISCUSSIONIn contrast with the traditional approach which uses a recursiveoptimization procedure to calculate the capacitances by firstcalculating the charge distribution and then integrating chargedensities, the proposed ANS technique computes the capacitancematrix directly by inversion of a Maxwell coefficients matrix. TheMaxwell coefficients matrix can be computed in a straightforwardmanner from Green functions. It is a shortcut to obtain the linecapacitances.In particular, the bundling procedure in AMS avoids the iterationprocedure by avoiding the direct calculations of charge accumula-tions. AMS also avoids the optimization step by directly evalu-ating the Maxwell coefficients instead of calculating the chargedensities. This strategy bypasses the time-consuming processes ofoptimization, charge integration, and iteration of the traditionalsolution approach. Considerable savings can therefore be obtainedin terms of computer memory and execution time.In addition, the Green functions used to calculate Maxwellcoefficients include the microstrip line dispersion through the\"images\" method. This is equivalent to the mapping of the effectivepermittivity and it causes AMS to be simpler in algorithm mechanismthan an equivalent method of moment[9]. Moreover, ANS also fully122accounts for the finite thickness of the line conductor, as opposedto empirical and conformal mapping methods.Finally, ANS is much simpler and easier to use than other formu-lations in the analysis of open boundary and dielectric interfaceproblems, such as the structure shown in Fig.4.1. AMS can alsobe generalized to other structures, such as the three dimensionalstructure in Fig.2.2(c), with the proper Green functions[12] andwith a convenient charge distribution assumption.4 . 9 CONDUCTANCE MATRIXHaving obtained the capacitance matrix, the conductance matrixcan be expressed asG = w tan^C (4.9.25)where^=^is the dissipation factor or loss tangent of thedielectric, with c' +jc\" = c, being the general form of the dielec-tric's permittivity. From eqn.(4.9.31), it can be seen that theconductance matrix G is strongly frequency dependent.The dissipation factor is determined for the properties of thematerials. It changes with frequency and temperature [4]. Forsilicon, tan6 is very small, about 0.001 0.004 at 106 Hertz [4]and it can be assumed constant. For other semiconductors such asgallium arsenide, it may change greatly as frequency increases[13].1234 . 10 SUMMARYIn this chapter, we have discussed the computation of microstriplines capacitance and conductance. The solution algorithm of ANSfor the capacitance proceeds as follows:O Read in the line geometry and physical data, such as the numberof line conductors, line-width, line-thickness, line-space,substrate height, and dielectric permittivity;O Form the mesh of subareas on the surface of the line conductors,i.e., calculate the coordinates enclosing each subarea;O Compute the Green functions for the Maxwell coefficients matrixof the subareas,O Bundle the subareas into equivalent line conductors, thusgiving the Maxwell coefficients matrix of the line conductors,{PO; andO Invert {TO to obtain WO.Since the explicit determination of the charge distributionis not necessary, the process of optimization and integrationfor the total charges on the line conductors is avoided. Thecalculation is performed only once instead of N(N+1) times for N2coupled lines. Furthermore, ANS can be extended to other structureswith different Green Functions and with any convenient assumptionof charge distribution.The conductance matrix is calculated from the capacitance matrixmultiplied by the product of the angular frequency and the dielectric124loss factor.125REFERENCES[1] Canadian Microeletronics Corporation. GUIDE TO THE INTEGRATEDCIRCUIT IMPLEMENTATION SERVICES OF THE CANADIAN MICROELETRONICSCORPORATION, JANUARY 1987.[2] H. W. DOMMEL. EMTP REFERENCE MANUAL. University of BritishColumbia, 1986.[3] T. C. EDWARDS. FOUNDATIONS FOR MICROSTRIP CIRCUIT DESIGN.Chichester; New York: Wiley, 1992.[4] D. G. FINK. ELECTRONICS ENGINEERS' HANDBOOK. New York: McGraw-Hill, pp6--4 to pp6--39, pp8--78, 1975.[5] D. W. KAMMLER. Calculation Of Characteristic AdmittancesAnd Coupling Coefficients For Strip Transmission Lines. IEEETransactions on Microwave Theory and Techniques, vol.MTT-16,no.11, November 1968.[6] M. KOBAYASHI. A Dispersion Formula Satisfying Recent RequirementIn Microstrip CAD. IEEE Transactions on Microwave Theory andTechniques, vol.MTT-36, no.8, AUGUST 1988.[7] M. KOBAYASHI and R. TERAKADO. Accurately Approximate FormulaOf Effective Filling Factor For MicrostripLine With IsotropicSubstrate And Its Application To The Case With AnisotropicSubstrate. IEEE Transactions on Microwave Theory and Techniques,vol.MTT-27, no.9, SEPTEMBER 1979.[8] H. KOBER. DICTIONARY OF CONFORMAL REPRESENTATIONS. New York:Dover Publications, 1957.[9] G. L. MATTHAEI, G. C. CHINN, C. H. PLOTT, and N. DAGLI. A SimpleMeans For Computation Of Interconnect Distributed CapacitancesAnd Inductances. IEEE Transactions on Computer Aided Design ofIntegrated Circuits and Systems, vol.CAD-11, no.4, APRIL 1992.[10] J. C. MAXWELL. A TREATISE ON ELECTRICITY AND MAGNETISM. NewYork: Dover Publications, 1954.126[11] H. B. PALMER. The Capacitance Of A Parallel-Plate CapacitorBy The Schwartz-Christoffel Transformation. Transactions ofthe American Institute of Electrical Engineers, vol.56, MARCH1937.[12] A. E. RUEHLI and P. A. BRENNAN. Efficient CapacitanceCalculations For Three-Dimensional Multiconductor Systems.IEEE Transactions on Microwave Theory and Techniques, vol.MTT-21, no.2, FEBRUARY 1973.[13] D. SHULMAN. Side Gating Of GaAs Semiconductors. Journal ofApplied Physics, vol.73, no.9, September, 1992.[14] W. T. WEEKS. Calculation Of Coefficients Of CapacitanceOf Multiconductor Transmission Lines In The Presence Of ADielectric Interface . IEEE Transactions on Microwave Theoryand Techniques, vol.MTT-18, no.1, JANUARY 1970.[15] H. A. WHEELER. Transmission Line Properties Of Parallel WideStrips By A Conformal Mapping Approximation. IEEE Transactionson Microwave Theory and Techniques, vol. MTT-12, no.5, MAY1964.[16] E. YAMASHITA. Variational Method For The Analysis OfMicrostrip-Like Transmission Lines. IEEE Transactions on Mi-crowave Theory and Techniques, vol.MTT-16, no.8, AUGUST 1968.[17] X. ZHANG, J. FANG, K. K. MEI, and Y. LIU. Calculations Of TheDispersive Characteristics Of Microstrips By The Time-DomainFinite Difference Method. IEEE Transactions on Microwave Theoryand Techniques, vol.MTT-36, no.2, FEBRUARY 1988.127Chapter 5Simulations and Comparisons for ANSCalculations of microstrip line capacitances and conductancesare presented in this chapter. They include the simulations withANS (advanced method of subareas) and the results from a programusing the finite element method (FEM)[3], and from A. DjordjeviC'ssoftware package (A.D.)[1].5.1 MICROSTRIP LINE CAPACITANCE WITH Er = 1In this section, the substrate permittivity is assumed to bethat of air for the microstrip line of Fig.4.1.Table 5.1 shows the line capacitances obtained with ANS, FEM1,FEM2, and A.D. The FEM cases include two approaches: FEM1 —surfacecharge method, and FEM2 — energy method. The corresponding CPUcosts are also tabulated. In addition, the savings obtained withANS, as compared with the traditional subarea method, in a 32-linetest case are presented.It can be seen in Table 5.1 that the proposed AMS method resultsin CPU savings of 76.2% and memory savings of 38.8% as compared tothe traditional subarea method. Savings of 54.2% in CPU time wereobtained when comparing AMS with the FEM techniques.128Table 5.1 Capacitance Values and CPU Time from AMS, FEMs, andA.D., [1] for the Line Configuration of Fig.4.1 (with Er = 1) .C(1,1)(Pi him)C(1,2)(Pi 1 Pm)CPU cost(ms)AMS 5.33e-4 —9.70e-5 2040FEM1 5.69e-4 —9.39e-5 4500FEM2 5.87e-4 —9.80e-5 4500A.D. 5.74e-4 —8.75e-5 —Diff. in %lofFEM16.6% 3.3% 54.2%Diff. in % of1-EM29.2% 0.9% 54.2%Diff. in % ofA.D.7.1% 9.8%Diff. in % ofConven. for32 lines_ _CPU : 76.2%Mem : 38.8%The validity of the results with the ANS technique is verifiedby comparison with the results from the other techniques. Comparedwith FEM1, the self and mutual capacitances differ by only 6.6% and3.3%. The differences are 9.2% for the selves and 0.9% for themutuals when compared with those from FEM2. When compared withthe solutions from the A.D. program, the differences are 7.1% and9.8% for the selves and for the mutuals, respectively. The overallaverage difference is 6.15%.Cl Creference129All the simulations shown above were performed on a Sun UNIXsystem, except for the simulation from A.D., which was run on a PC.5.2 MICROSTRIP LINE CAPACITANCE WITH Er = 10In this section, the substrate relative permittivity Er equals 10for the microstrip lines in Figs.4.1 and 4.2. Table 5.2 displaysthe simulation solutions for both configurations.For the microstrip lines of Fig.4.1, the differences in selfand mutual capacitances between AMS and A.D. were 11.8% and 4.3%,respectively. While for the configuration of Fig.4.2, they were9.38% and 0.43%. The overall average difference was 6.48%.5.3 COMMENTS ON THE ANS TECHNIQUEAs shown by the simulation results, ANS is a more efficient methodas compared to FEMs and the traditional subarea technique. AMSoutperforms the traditional subarea method by achieving reductionsof 76% in memory requirements and 40% in CPU time. Compared toFEM techniques, CPU time saving of 54% were obtained on the SunUNIX system. Another advantage over FEM programs in dealing withopen boundary problems is that the generation of subdivisions isautomatic.The accuracy of ANS in this case was confirmed by comparisonswith three other simulation methods: FEM1, FEM2 and A.D. An average130difference of 6.31% was obtained in the solutions.5.4 LINE CONDUCTANCE MATRIXFro a silicon dissipation factor of 0.004[2], conductance valueswere calculated and are displayed in Tables 5.3 and 5.4 for 106 Hertzand 109 Hertz, respectively. Each table contains solutions for bothconfigurations of microstrip lines.From Tables 5.3 and 5.4, it can be seen that the conductance isstrongly frequency dependent. Its values are enlarged 1000 timeswhen frequency goes from 106 to 109 Hertz, although the absolutevalues are very small.5.5 CONCLUSIONSIt can be concluded from the simulations that ANS is efficientand accurate. It results in CPU time savings from 76% to 54%, andmemory savings of 40%. Its accuracy was confirmed by comparisonswith FEM techniques and the A.D. technique. The differences werewithin 6.31% in average.In addition, with AMS subdivisions are generated automatically,which results in a user-friendly software. Moreover, its solutiontechnique can be applied to a large class of microstrip linestructures.For a constant tan& of the dielectric, the conductance is a131linear function of frequency.5.6 SUMMARYSimulations with the new technique ANS were presented in thischapter for the capacitance matrix of microstrip lines. ANS'efficiency and accuracy were verified by comparison with FEMprograms and the software package from A. Djordjevia et al.The conductance values were also calculated.^They increaselinearly as frequency increases for a constant tanbo.It is concluded that ANS is efficient and accurate.132Table 5.2 Results of AMS and of A. D. for theConfigurations of Figs. 4.1 and 4.2 (with Er = 10).C(1,1)(p.filim)C(1,2)(p.//t/m)AMS(Fig. 4.1)3.113e-3 -9.503e-4A.D.(Fig. 4.1)3.528e-3 -9.097e-4Diff.in %2 11.8% 4.3%AMS(Fig. 4.2)1.555e-2 -1.384e-7A.D.(Fig. 4.2)1.716e-2 -1.390e-7Diff.in % 9.38% 0.43%Table 5.3 Results of the Conductance Matrix with Er = 10 and atf = 106Hertz for the Configurations of Figs. 4.1 and 4.2.G(1,1) (S x 10-11/pm) G(1,2) (S x 1011/pm)Fig. 4.1 7.8238 -2.3884Fig. 4.2 39.081 -3.4784e-4Table 5.4 Results of the Conductance Matrix with Er = 10 and atf = 109Hertz for the Configurations of Figs. 4.1 and 4.2.G(1, 1) (S x 10-8/pm) G(1,2) (S x 10-8/pm)Fig. 4.1 7.8238 -2.3884Fig. 4.2 39.081 -3.4784e-42^% = ^IACI Creference133REFERENCES[1] A. R. DJORDJEVIa, R. F. HARRINGTON, T. SARKAR, and M. BA2DAR.MATRIX PARAMETERS FOR MULTICONDUCTOR TRANSMISSION LINES:Software And User's Manual. Artech House, Inc., 1989.[2] D. G. FINK. ELECTRONICS ENGINEERS' HANDBOOK. New York: McGraw-Hill, pp6--4 to pp6--39, pp8--78, 1975.[3] Y. YIN. CALCULATION OF FREQUENCY-DEPENDENT PARAMETERS OFUNDERGROUND POWER CABLES WITH FINITE ELEMENT METHOD. PhD thesis,University of British Columbia, JUNE 1990.134Chapter 6Future WorkThe numerical techniques LCST and ANS developed in this thesis aregeneral and can be extended to other transmission line structures.Some suggestions for future work are:1. LCST can be applied to other kinds of transmission lines andcable structures in power systems, and for fast networkingsimulations in telecommunications;2. ANS can be extended to calculate more complicated structuresin VLSI, such as the 3-Dimension structure in Fig.1.1(c). Itcan also be extended for microstrip lines with non-homogenoussubstrate, microstrip lines with dielectric with frequencydependent permittivity (e.g. in gallium arsenide), etc; and3. The final programs can be incorporated into the EMTP.135Chapter 7ConclusionsNew numerical techniques have been developed in this thesis forthe calculation of the series (R, L) and shunt (G, C) parametersof microstrip lines in VLSI. These techniques are the linearcurrent subconductor technique (LCST) for R and L, and the advancedmethod of subareas (ANS) for C. This work introduces for the firsttime the concept of linear interpolation for the calculation ofcurrent density distribution and charge distribution using finitesubconductor and subarea techniques. This concept results in largesavings in computer memory requirements and CPU times. The LCSTtechnique was also applied to pipe-type cables in power systems.The following conclusions can be derived from this work:0 LCST for resistance and inductance is an efficient and accuratemethod for the calculation of the resistance and inductancematrices of transmission lines. In comparison with thetraditional subconductor method and with the finite elementmethod, LCST resulted in an average difference in the valueof the parameters of 3.3% and 3%, respectively. The memorystorage requirements decreased by up to 95%99%, and the CPUtime savings were above 90%;0 LCST combines the simplicity of the traditional subconductormethod with the accuracy of the finite element method. Sincethe subdivision procedure is automatic, the procedure resultsin a much more user friendly program than FEM programs;136O LCST is general, and it can be extended to other kinds oftransmission line configurations;O ANS for capacitance is efficient and valid. It results inmemory savings of 38.8% and CPU savings of 54.2%,,,76.2% ascompared with the traditional method of subareas and with twoprograms using the finite element method. Its accuracy hasalso been validated by comparisons with three methods, FEM1,FEM2, and A.D. The resultant average difference was 6.31%;O As in the case of LCST, the subdivision procedure for ANS canbe automated and can be used for other line configurations,such as 3-D microstrip lines.^It is especially useful foropen boundary problems; andO The developed ANS program is easy to extend to accept frequencydependent dielectric constants and anisotropic dielectric per-mittivities.From the results obtained for the line parameters of microstriplines in VLSI and for pipe-type cables in power systems, it canbe concluded that:O The line resistance matrix is an increasing function of fre-quency;O The line self inductances decrease as frequency increases,while the mutual inductances increase slowly or may decrease,depending on the line structure;O In general, the line capacitances can be treated as constantversus frequency in the case of silicon substrate at the137frequencies of present interest; and0 The line conductance is strongly frequency dependent althoughits absolute value is negligible. It is a linear function offrequency with a constant dielectric loss tangent.138z1 z2 Appendix I. Current Split in Subdividedand Non-Subdivided ConductorsSuppose we have a subconductor cut into two paralleled sectors,as shown in Fig.A.1.z=r+jcolzl = rl + jcollz2 = r2 + jw12z=z1//z2Fig. IA. A subconductor in forming two systems.If a voltage V is applied to the subconductor, the current I inthe un-split subconductor isV/= r+ jw/'while in the split subconductor,^V^V-/-' =11+12 = .^^ +^. ,7'1 -Fj,uni^r2+ jwt2= (ri +r2)-Fjc4/1 + 12)V(ri +3.4.0/1)(r2-1-./0)/2)^.Since1^Z1 Z2^Z = Zli/Z2 = 1 . 1 =^1-1- -^Z1 -r Z2z1^,z2(ri + jw/i)(r2+./4.012) = r + jug,(ri + r2)-Fjo)(ii +12)then(I.2)(1.3)/=/'.^ (I.4)139The above equation means, that without mutual parameters, thewhole current in the two systems will be identical for the samevoltage source.If we now assume that there is mutual resistance and inductance,the current from the un-split subconductor system will remain as ineqn.(1), while the current in the split subconductor system will ber = 11+12^ (I.5)with[1121^{ Z21Hence,Z1 2 1 -1 [ VZ2 j^V(z1.2+^v .Z1Z2 - Z12 Z21(I . 6)(1.7)This time, the currents in the two systems are not identical.140Appendix II. Symbols And Abbreviations7i)o — One thousandth;— Area of the cross-section of segments;Across—section — Area of line conductor cross-section;A, AC — Matrix of weighting functions in frequency domain;a(t), a'(0 — Form of A, JC in the time domain;A — Angle spanned by a curved cell;ai — An intermediate variable;a,b,c,d — Current filaments;— Constants in the linear current distribu-tion function;AMS — Advanced method of subareas;B — Vector of magnetic flux density;— Element of capacitance matrix;C,C — Matrix of line capacitances;Co — Matrix of line capacitances with E = Eo;CO.A4Pr — Combination of two real number into a complex number;CPU — Central processing unit in computer;CQ — Sum representation;141C — Wave speed in air or vacuum;Cosj, cosj — Intermediate variables.D — Distance between segments;EMTP — Electromagnetic Transients Program;e — Voltage source;elon, — Voltage in subsection (k,m,n);- Vector of electric field density;FEM — Finite element method;FEM1 — Surface charge method in FEM;FEM2 — Energy method in FEM;f^Frequency;fmax — The maximum frequency under which there is only surfacecharge to exist in the conductor;frnax-model — Maximum frequency of the signals for which the trans-mission line model is valid;in — Nyquist frequency;— Linear distribution correction factors;FFT — Fast Fourier transform;G — Matrix of line conductances;Gpq - Geometric Mean Distance from element p to element q;142GMD — Geometric mean distance;G(x,y) — Green function;h — Height of dielectric;H — Henry, unit of inductance;— Vector of magnetic field density;Hz — hertz, unit of frequency;— Integer index or node number in circuits;— Linear current density in a cell in pipe-typecables;i(t) — Current function in the time domain;intel2 — Integration values of current density on two areasof a cell;— Current matrix in the frequency domain;1.I - Current values in a subsection;3 '^3— Current value in conductor;I — Current array with uniform current distribution;T — Current array with linear current distribution;3m — Imaginary part of a complex number;143J — The number of subdivisions along the width of the conductorin microstrip lines;j — Imaginary sign, j =- Vector of charge density;ikmn — Current density in subsection (k,m,n);4mn — Current density at vertices of a subsection (k,m,n);4i,ja — Current density in segment a;— Inductance correction factors;A.1 — Scaling factor for the inductance by linear current distri-bution;K — Maximum dimension of subconductors impedance matrix;- Number of subintervals from vertex 1 to vertex 1+1;K4,K2 — Integration constants;KR,K1,k1,k2,k3,k4 — Inductance correction factors shown in Tables3.1-3.3 in Chapter 3;K1,K2,k —Coefficients in functions of inductance correction factors;K,C — Constants in linear function;— Length of the line;43,L231,kmn^Inductance between segment i and segment j or between(i,j,l) and (k,m,n);144\"1kmn,kn,^— Inductance between segments (kon,n)and(lcion',70 with loopenclosed by a dummy cell;L — The number of subdivisions along the thickness of the conductorsof microstrip lines, or the number of subdivisions along the azimuthdirection in pipe-type cables;L — Inductance matrix;LCDT — Linear current distribution technique;m — Node and index number in circuits;N --- Number of conductors;▪ — Ratio of the number of subdivisions by the 6 rule to thoseby the 26 rule;ATI„ Ark, Arie ,^— Number of subdivision in the geometric dimensionsof conductor k;NR — Number of subdivision along the radius in pipe-type cables;▪ — Number of subdivision in the azimuth direction in pipe-typecables;nt — Number of subdivisions in the thickness;nw — Number of subdivisions in the width;— Maxwell coefficients between subareas (ij) and (km);Pij — Maxwell coefficients of the line conductors;145pii(x,y) — Potential value in position (x,y) with charge excitationin subarea (ij);— Co-ordinates of points in cells p and q;pr — Radius of the pipe in pipe-type cables;Q3 — Charges in conductor j;q — Surface charge;R --- Matrix of subsection resistances;^ ,r — Modified segment's resistance;t — Distance from a point a cell to a point in another;ERe — Real part of complex number;^ — Radius of a circular cell;Rkmn,k'^Segment resistance;It — Resistance matrix;T — Thickness of microstrip line conductors;TE — Transverse electromagnetic wave with E,=0;TEM — Transverse electromagnetic wave with E = Hz = 0;TM — Transverse electromagnetic wave with H., = 0;T — Period of a signal;At — Interval of subdivision on the thickness of microstrip line;146t - Time or intermediate variable;e — Intermediate variable;tan& - Loss tangent or dissipation factor of dielectric;u(t) - Step function;VLSI - Very large scale integrated circuits;V,V1 - Terminal voltage in the frequency domain;-V - Voltage array with uniform current distribution;- Voltage array with linear current distribution;AV - Voltage drop on a line segment of length Az;0,v'(0 - Terminal voltage in the time domain;W - Width of microstrip line conductors;AT/17„6.11) — Interval of subdivisions on W;,^u - Modified segment's admittance;x,y— Co-ordinates of subsections or segments in a Cartesian system;X,Y - Co-ordinates of ends of subareas;xv,yv - Positions of vertices of line conductors;X - Intermediate variable;eVe,4 — Imaginary part of inductance correction factors;II. II Y km,n,k'^Admittance of cells;147- Admittance of conductors;Y - Line admittance matrix;- Line characteristic admittance;Z - Line impedance matrix;3 - Line characteristic impedance;-Z - Impedance matrix of cells with uniform current distribution;- Impedance matrix of cells with linear current distribution;z - Position on the line;zij,Zij,Zkmn,kim'n' - Impedance of cells or segments;a - Azimuth angle;A - Signal's wave length in the air;Ap - Signal's wave length in the line;w - Angular frequency;Er -Relative permittivity or upper-limit of effective permittivity;eo - Permittivity of air or vacuum;- Permittivity of dielectric, in a general form as E ==^JE == goer ;- Real part of the complex permittivity, reflecting the displace-ment current in the dielectric;c\" - Imaginary part of the complex permittivity, reflecting theconduction current in the dielectric;148— Permittivity in the anisotropic dielectric;- Skin depth;A — Integration length;(5,3,40 - Impulse function;y — Permeability;v — A constant between 0 and 1;a - Conductivity;p - Radius of circles or arcs;Ap - Length of arc with radius p;0,AO - Angle of two edges of conductor's cross-section;0 — Magnetic flux linkages of current filaments; andT — Magnetic flux linkages of segments.SUMMARY OF REFERENCES[1] P. 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IEEE Transactions on Circuit158and Systems, vol.38, no.7, JULY 1991."@en ; edm:hasType "Thesis/Dissertation"@en ; vivo:dateIssued "1993-11"@en ; edm:isShownAt "10.14288/1.0065091"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Electrical and Computer Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Numerical methods for frequency dependent line parameters with applications to microstrip lines and pipe-type cables"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/1949"@en .