NUMERICAL METHODS FOR FREQUENCY DEPENDENT LINE PARAMETERS WITH APPLICATIONS TO MICROSTRIP LINES AND PIPE-TYPE CABLES by DAN HONG ZHOU B.Eng., The Beijing Institute of Technology, 1985 M.Eng., The Beijing Institute of Technology, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF APPLIED SCIENCE (Department of Electrical Engineering) We accept this thesis as conforming THE UNIVERSITY OF BRITISH COLUMBIA September 1993 © Dan Hong Zhou, 1993 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. (Signature) Electrical Engineering Department of ^ The University of British Columbia Vancouver, Canada Date ^ DE-6 (2/88) September 23, 1993 ABSTRACT 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 transmission 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 subconductor 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 applied 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 other line structures with open boundaries. Other potential applications are in telecommunications and computer high speed networking, as well as supercomputer packaging. ^ Table of Contents ABSTRACT ^ Table of Contents ^ iv List of Tables ^ viii List of Figures ^ Chapter 1^Introduction ^ 1 1.1^SIGNAL LINES IN VLSI ^ 1 1.2^SIGNIFICANCE OF THE THESIS WORK ^ 3 1.3^LINE MODELS ^ 1.3.1^Literature Review for Line Models ^ 1.3.2^Frequency Range of Transmission Line Model for Microstrip Lines ^ Electromagnetic Wave along Microstrip Lines ^ Transmission Line Equations For Microstrip Lines ^ 1.4^LINE CHARACTERISTIC PARAMETERS ^ 1.4.1^Literature Review for Line Parameters ^ 1.4.2^Problems with the Traditional Methods ^ Resistances and Inductances ^ Capacitances ^ 5 5 8 8 9 10 10 12 13 14 1.5^CONTRIBUTION OF THIS THESIS WORK ^ 15 1.6^ORGANIZATION OF THIS THESIS ^ 16 REFERENCES ^ Chapter 2 17 ^ Linear-Current Subconductor Technique for Matrices of Frequency Dependent Resistance and Inductance ^ 25 ^ ^2.1 GENERAL DESCRIPTION ^ 25 ^ 2.2 CONDUCTOR'S CROSS-SECTION SUBDIVISION ^28 ^ 2.2.1 Subdivision of Microstrip Lines ^ 28 ^ 2.2.2 31 Pipe-Type Cables ^ ^ 2.3 SUBCONDUCTOR RESISTANCE AND INDUCTANCE FROM TRADITIONAL FORMULAE ^ 34 ^ 2.3.1 Microstrip Lines ^ 34 Self Geometric Mean Distance (GMD) ^ 36 GMDs between Subconductors in Microstrip Lines ^ 37 iv ^ ^ 2.3.2^Pipe-Type Cables ^ Paralleled Curved Cells ^ Non-paralleled Curved Cells ^ Self Geometric Mean Distance ^ GMD for Circular Cells ^ 2.4^CURRENT DISTRIBUTION ^ 39 39 41 41 42 43 2.5^APPROXIMATION OF LINEAR CURRENT DISTRIBUTION ACROSS SUBCONDUCTORS ^ 45 2.5.1^Microstrip Lines ^ 46 2.5.2^Pipe-type Cables ^ 47 2.6^LINEAR CURRENT DISTRIBUTION CORRECTION FACTORS ^ 51 2.7^INDUCTANCE CORRECTION ^ 2.7.1^Inductance Formula • Option 1 ^ Microstrip Lines ^ Pipe-Type Cables ^ 2.7.2^Inductance Formula • Option 2 ^ 53 54 57 58 59 2.8^BUNDLING PROCEDURE ^ 2.8.1^Voltage Drops along Subconductors ^ 2.8.2^Currents in the Subconductor System ^ 2.8.3^Matrix Manipulation For Matrix Reduction . . . ^ 62 62 65 2.9^SUMMARY ^ 2.9.1^Summary of the Linear Current Subconductor Technique ^ 2.9.2^Discussion ^ 67 REFERENCES ^ 72 Chapter 3 67 67 69 ^ ^ Simulations Using LCST and Other Methods ^ 75 SOLUTIONS FOR MICROSTRIP LINES ^ 75 ^3.1.1^Current Distribution from Different Subdivisions under Uniform Distributed Current Assumption . 75 3.1.2^Inductance Correction Factors ^ 76 3.1.3^Solution and Discussion of Line Resistances and Inductances — LCST vs. Traditional Subconductor Method ^ 77 3.1 3.2^SOLUTIONS FOR PIPE-TYPE CABLES ^ 92 3.2.1^Inductance Correction Factors ^ 92 3.2.2^Solution and Discussion for the Impedance Matrix — LCST vs.FEM, and LCST vs.Traditional Subconductor Method ^ 93 ^ 3.3^CONCLUSIONS ^ 103 3.4^SUMMARY ^ 105 REFERENCES ^ 106 Chapter 4^Computation of the Capacitance and Conductance Matrices of Microstrip Lines ^ 107 ^4.1^MICROSTRIP LINE CONFIGURATIONS ^107 ^108 4.2^FREQUENCY DEPENDENCE 4.2.1^Capacitance Formed by Line Conductor and Ground Plane (With Constant 0 ^109 Solution from Maxwell equations ^109 Solution from empirical formulae ^111 4.2.2 Capacitance With Frequency Dependant E(f) . . . 113 4.3 AMS VERSUS TRADITIONAL METHOD OF SUBAREAS 114 4.4 SUBDIVISIONS ^ 116 4.5 GREEN'S FUNCTIONS ^ 117 4.6 MAXWELL COEFFICIENTS OF SUBAREAS ^ 118 4.7 CHARGE DISTRIBUTION 4.8 DISCUSSION ^ 122 4.9 CONDUCTANCE MATRIX ^ 123 4.10 SUMMARY ^ 124 ^ REFERENCES ^ 120 126 Chapter 5^Simulations and Comparisons for AMS ^ 128 5.1^MICROSTRIP LINE CAPACITANCE WITH er=1. . 128 5.2^MICROSTRIP LINE CAPACITANCE WITH sr==10 . 130 5.3^COMMENTS ON THE AMS TECHNIQUE ^ 130 5.4^LINE CONDUCTANCE MATRIX ^ 131 5.5^CONCLUSIONS ^ 131 5.6^SUMMARY ^ 132 REFERENCES ^ 134 vi Chapter 6^Future Work ^ 135 Chapter 7^Conclusions ^ 136 Appendix I Current Split in Subdivided and Non-Subdivided Conductors . .139 Appendix II Symbols And Abbreviations . ^ 141 SUMMARY OF REFERENCES ^ vi i 150 List of Tables Table 3.1 Table 3.1 Impedance Correction Factors as Functions of Frequency ^ 83 Impedance Correction Factors as Functions of Frequency ^ 84 Table 3.2 Correction Factors for Subconductors at f = 10-6 Hertz ^ 85 Table 3.2 Correction Factors for Subconductors at f= 10-6 Hertz ^ 86 Table 3.3 Correction Factors for Subconductors at f = 5 x 109 Hertz ^ 87 Correction Factors for Subconductors at f = 5 x 109 Hertz ^ 88 Table 3.3 Table 3.4 Comparison of LCST with Traditional Subconductor Method Assuming Uniform Current Distribution, for the Case of Two Conductor Microstrip Lines .89 Table 3.• Comparison of LCST with Traditional Subconductor Method, for the Case of Three Conductor Microstrip Lines ^ 90 Table 3.6 Comparison of LCST with Traditional Subconductor Method, in Memory Requirement and CPU Time Cost.91 Table 3.7 Impedance Correction Factors as Function of Frequency and Conductor Sequence ^ 96 Table 3.7 Impedance Correction Factors as Function of Frequency and Conductor Sequence ^ 97 Table 3.8 Correction Factors for Subconductors at f = 60Hz 98 Table 3.8 Correction Factors for Subconductors at f = 60Hz 99 Table 3.9^Correction Factors for Subconductors at f = 600, 000Hz ^ 100 Table 3.9^Correction Factors for Subconductors at f = 600, 000Hz ^ 101 Table 3.10^Comparisons of LCST with FEM: Resistances and Inductances at 60Hz ^ 102 viii Table 3.11 Table 5.1 ^ ^ Comparisons of LCST with FEM: Resistances and Inductances at 600,000Hz ^ 104 Capacitance Values and CPU Time from ANS, FEMs, and A.D., for the Line Configuration of Fig.4.1 (with Er =1) 129 Table 5.2^Results of ANS and of A. D. for the Configurations of Figs. 4.1 and 4.2 (with Er =10) 133 Table 5.3 Results of the Conductance Matrix with Er =10 and at f = 106Hertz for the Configurations of Figs. 4.1 and 4 2 133 Table 5.4 Results of the Conductance Matrix with cr =10 and at f = 109Hertz for the Configurations of Figs. 4.1 and 4.2 133 ix List of Figures Fig. 1.1 Typical Configurations of Microstrip Lines. ^. .^.^2 Fig. 1.2 Transmission Line Models in the EMTP ^ Fig. 1.3 Subdivision Strategy in the Subconductor and Subarea Methods. ^ 6 13 Fig. 2.1 Flow Chart for the Linear Current Subconductor Technique (LCST) ^26 Fig. 2.2 Coordinate System for Microstrip Lines ^ 30 Fig. 2.3 Pipe-Type Cable Configuration and Subdivisions. Fig. 2.4 Coordinate System for Pipe-Type Cables Fig. 2.5 Cell Index and Fields Distribution in Microstrip Lines ^ 37 Fig. 2.6 Loop Definitions Related to GMD's for Inductances 33 ^ 35 38 Fig. 2.7 Schematic of Cells in Microstrip Lines. ^. . 38 Fig. 2.8 Two Kinds of Cells and Indices of Pipe-Type Cables ^ 40 ^ 42 Fig. 2.9 Cells for GMD's in Pipe-Type Cables. Fig. 2.10 Schematic of a Cell and its Current Densities for Microstrip Lines. ^ 48 Schematic of Cells and Their Current Densities for Pipe-Type Cables. ^ 49 The Voltage Drops Related to the Subconductors. ^ 63 Current Distributions with Different Number of Subdivisions ^ 79 Current Distributions with Different Number of Subdivisions. ^ 80 Current Distributions with Different Number of Subdivisions. ^ 81 Fig. Fig. Fig. Fig. Fig. Fig. 2.11 2.12 3.1 3.1 3.1 4.1 Typical Configuration of Microstrip Lines (from Fig.1.1).^Dimensions in pm ^ 108 Fig. 4.2^Another Configuration of Microstrip Lines. Dimensions in pm ^ 108 Fig. 4.3^Capacitance of a Single Conductor Microstrip Line with Ratio^= > 0.7 and Dielectric Constant Er = 4.5. 111 Fig. 4.4^Capacitance of a Single Conductor Microstrip Line with Ratio T = 0.5 and Dielectric Constant Er = 4.5. 113 Fig. 4.5^Linear Charge Density and a Schematic of Coordinates Relationship in a Subarea ^ 121 Fig. 1.1^A subconductor in forming two systems ^ 139 xi ACKNOWLEDGMENT I wish to state my gratitude to my parents, who have inspired and supported my exploration into new worlds. I wish to express my sincere appreciation to my supervisor, Dr.Jose Marti, for his excellent guidance, kindness, assistance and patience. I wish to acknowledge the Department of Electrical Engineering at the University of British Columbia (UBC) for giving me the financial support to accomplish this venture. I would like to thank Dr.Martin Wedepohl for sharing his eigenvalue and eigenvector extracting routines, and offering help during my thesis work. I would also like to thank Dr.Yanan Yin for many inspiring discussions in the early stage of my thesis work as well as for offering his results to validate the new numerical methods in this thesis, and to Mr.David Michelson, Miss Ruth Harland, Miss Carly Wong and Miss Ellen Ho, for their helpful suggestions and English polishes regarding the writing of this thesis. Last but not least, I have also appreciated the courtesy and assistance of the staff of the Department of Electrical Engineering, UBC. Mr.Robert Ross and Mr.David Gagne have been especially kind in patiently answering my questions and facilitating my use of department computers. xi i Chapter 1 Introduction This thesis is concerned with the modelling and simulation of microstrip lines for Very Large-Scale Integrated (VLSI) circuit and computer packaging design. In this chapter, the background to the thesis work will be introduced. The significance of the work is covered along with a review of the literature in related fields, and an overview of the development of two new numerical methods. Finally, the organization of the thesis will be outlined. 1 . 1 SIGNAL LINES IN VLSI In VLSI, the signal lines can fall into two groups: interconnects of devices on a chip and interconnections between chips or components on circuit boards. The interconnects and interconnections are often implemented as microstrip lines. Three typical configurations of microstrip lines are shown in Fig.1.1. Figure 1.1(a) shows the microstrip conductors coated in a substrate or dielectric with a metal layer adhered to the other side of the substrate, which functions as a ground plane. In Fig.1.1(b), the line conductors are sandwiched between two ground planes and sit on the homogeneous dielectric. Finally, in Fig.1.1(c), two batches of paralleled lines are placed between two ground planes in 1 (c) Multilayers of microstrip lines. Fig. 1.1 Typical Configurations of Microstrip Lines. such a way that they are perpendicular to each other in homogenous dielectric. The structure displayed in Fig.1.1(a) is the most complicated because of its open boundary condition. Among these microstrip lines, the line-width, line-space and line-length of the interconnects can be as small as 0.8pm — 1.2pm on a silicon chip[9, 2 26, 61]. The dimensions of the interconnections between the chips can range from lOpm — 200pm[47]. 1.2 SIGNIFICANCE OF THE THESIS WORK Prediction of the fast signal behavior or signal transient analysis in VLSI has assumed increasing importance in recent years. Signal transients influence the lifetime of the devices, their functioning and the performance of the whole system. Undesired signal transients are mainly caused by microstrip lines. The distributed nature of microstrip lines can induce harmful transients in the other parts of the circuit, especially after power switching and circuit logic transition. In the operation of high speed electronic devices, the behavior of microstrip lines is critical for system performance. Besides introducing transients, microstrip lines are the main channels of electromagnetic interference and major sources of extra power dissipation. In VLSI operations, the pulses on a microstrip line suffer loss, distortion and dispersion[4, 15, 24, 28, 34, 35, 40, 47, 52, 53] in addition to signal delays. Circuit performance may be compromised by the decaying or coupling of signals. In high speed computers the situation is even more critical. Hence, it is essential to have microstrip lines carefully designed. Current VLSI technology reduces the size or area occupied by a device. This results both in a high circuit complexity on chips 3 and in an increase in the reacting speed of the device. There can now be up to 500,000 gates and 250 I/O pins on a single silicon chip[9, 44, 61], such as the Intel i486 chip which contains 1.2 million transistors[48]. While the ultimate clock rate of 200 400 megahertz is found in current computers[7], gate transitions or gate delays of 250ps have become common in silicon, and the values of 100ps are observed in gallium arsenide[44, 51]. This great complexity leads to an increase in the number of interconnects and interconnections while the high operational speed makes it necessary to model interconnects and interconnections as transmission lines. Microstrip line models must then be incorporated in VLSI circuit and computer packaging design procedures. From a practical point of view, electromagnetic compatibility is very important in circuit and system design. Very often a "theoretically" correct circuit does not work as expected when built, or even worse, it may exhibit intermittent and unpredictable operation, especially in VLSI[23, 44, 45, 51]. Experience has also shown that electromagnetic performance deficiencies or defects are difficult, if not impossible, to correct after fabrication. Thus, accurate modelling and simulation of microstrip lines is an extremely important aspect of the complete electromagnetic prediction of the high frequency behavior of high density chips, components, and circuit boards prior to fabrication. 4 1.3 Ion= mamas 1.3.1 Literature Review for Line Models There is a large amount of literature related to microstrip line models. 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 the electric and magnetic fields along the line, all models proposed in the literature have difficulties in their implementation into practical circuit simulation packages. They need at least to solve Poisson or Laplace equations with the complicated boundary conditions of microstrip lines. Due to the open-boundary structure of microstrip lines, these models have very large memory requirements. The transmission line approach has advantages in terms of the ease of solving only linear equations, of dealing with open-boundary problems, of the direct access it provides to circuit variables such as voltages and currents, and of the almost mature and simple numerical methods that have been developed for its description. It also requires much less computer memory. The transmission line model has been well developed in the Electromagnetic Transient Program (EMTP). For the purpose of the study of this thesis, this transmission line model has been used for microstrip lines. 5 R1 L1 V1+AV1 1/1 V2+AV2 V2 V3+ AV3 V3 C3,G3 (a) Transmission line parameters; Zeq(f) node m Vm(t) (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-varying voltage source (grounding one terminal) connected to a frequency 6 dependent impedance.^The voltage source is determined by the parameters and history of the branch current linking node k and ground. 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 these parameters, the line model can be setup for node voltages and branch currents. In comparison with the other two strategies, which focus on the electromagnetic field around the microstrip lines, there is one alleged weak point in the application of the transmission line model to microstrip lines. When the signal frequency rises to a high enough level, the transmission line theory loses its effectiveness for microstrip lines. For example, beyond a certain cutoff frequency, microstrip lines will become leaky wave guides rather than transmission lines[14, 15, 20, 64]. However, it can be proven that the transmission line model is still able to simulate the behavior of microstrip lines with dimensions in the order of microns within the frequencies of interest. It is also very likely to have a further extended frequency range as VLSI technology is upgraded. 7 1.3.2 Frequency Range of Transmission Line Model for Microstrip Lines Electromagnetic Wave along Microstrip Lines Transmission lines are defined as structures that allow guided Transverse Electromagnetic waves (TEM) to travel from one point to another[5, 14]. From this definition and from the reason that the wave on the lines is "quasi-TEM" type [15, 10, 11, 14, 20, 64], the microstrip lines are lossy (distortion) and dispersive transmission lines (signals on the lines will suffer delay, loss or distortion, dispersion, and introduce crosstalk in the circuits). They thus support a hybrid wave mode containing TEM, TE, and TM modes[6, 14, 15, 29]. The latter two modes contain the longitudinal components of electric and magnetic fields on the microstrip line and may cause the generation of eddy currents on the conductor's surface. The cutoff frequency of the lowest TE mode (TED) is given by Lutoff= 4h/Er -1' (1.3.1) while the frequency for strong coupling of TEM and TM modes[15] can be expressed as fTEIW1= C • tan E r N/2-rhVer -1 (1.3.2) Below both these frequencies, the TEM wave mode will dominate in the propagation. TT° and TM() surface wave modes only hold a very small portion of power. Then the higher order modes are neglected in the modelling and simulation of microstrip lines by transmission line model. (A schematic of the electromagnetic field in a single conductor microstrip line is shown in Fig.2.5(b) [15, 33, 54, 64].) 8 Therefore, transmission line theory will be valid for microstrip lines as long as the signal frequencies are under the thresholds indicated in eqns.(1.3.1) and (1.3.2). In eqns.(1.3.1) and (1.3.2), c is the travelling speed of the wave in vacuum or air; h is the height of the dielectric or the distance between the line conductor and the ground plane; and Er is the relative permittivity or dielectric constant, as shown in Fig.1.1(a). Transmission Line Equations For Microstrip Lines Applying transmission line theory to microstrip lines, the voltages and currents along the lines satisfy the transmission line equations (TEM mode) 1 ----0L=ziaz ^ _az = yv (1.3.3) Tz- where z is the direction of wave propagation along the line, and Z and Y are, respectively, impedance and admittance matrices formed by the line parameters. As shown in Fig.1.2(a), Z = R-FjcoL and Y G jcvC Eqn.(1.3.3) implies that on a cross-section of the line, perpendicular 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 equipotential plane. In another instance, if Across_ section <()q,, i.e., the cross- section area of the line is much less than the squared wavelength of 9 the signal on the line, the condition in eqn.(1.3.4) is satisfied, and the line is called electrically "fine"[35]. The cross-section then forms an equipotential plane and the transmission line model is valid. 1.4 LINE CHARACTERISTIC PARAMETERS To apply the transmission line model to microstrip lines, the line parameters must be calculated. In the past, RC networks have been used to approximate microstrip lines in VLSI studies, but this is not regarded as acceptable[70]. For example, among the parameters in Fig.1.2(a), L becomes very important in high frequencies. 1.4.1 Literature Review for Line Parameters Many methods have been proposed to compute the frequency dependent parameters of microstrip lines. One method is based on the assumption of zero thickness of the conductors. In this approach, C can be obtained by using conformal mapping theory and the concept of effective permittivity[2, 22, 30, 31, 32, 39, 43, 63, 62, 671. The resistance R can then be calculated by the method of Geometric Mean Distances (GMD) and Ohm's law[13]. Finally, L and G can be obtained using the relationship between the four line parameters according to electromagnetic theory[5]. This approach is simple and based on analytical formulae. The calculation for the line 10 inductance, however, is incorrect at low frequencies because the internal inductance has not been taken into account. Also, although an adjustment for the thickness can be included, according to an empirical or asymptotic formula[49, 62, 63], the actual thickness effect has not been adequately modelled for the line capacitance. This is because the surface charge concentrates on the corners of the line conductors, and thickness becomes the dominant factor for the capacitance. Besides the capacitance, the thickness assumption is also important for the resistance and inductance parameters. Taking into consideration that the current will distribute more densely towards the corners and edges of the conductors, and will change the line resistance and inductance as frequency increases, it is concluded that the assumption of zero thickness is not appropriate for modelling microstrip lines in the simulation of circuit transients. It may only be suitable to simulate situations such as the asymptotic behavior in microwave circuits at single frequencies. Another general approach to calculate transmission line parameters 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 specify for FEM techniques. In addition, there are intensive computer memory requirements for the initiation and solution process using FEM techniques. 11 Another approach to determining the line's frequency-dependent parameters is the method of subconductors for R and L, and the method of subareas or moments for C [1, 2, 8, 27, 38, 50, 58, 59]. The parameter G can be obtained from C, the signal frequency and the dielectric loss tangent. These methods share the same subdivision strategy as FEM, but avoid the main difficulties of the FEM approach. The subconductor technique accommodates the influences of dynamic current distribution on the line when a signal has a wide frequency range. As frequency changes, the current automatically redistributes itself among the subconductors. Hence, there is no need to assume zero line thickness or specific boundaries in the free space region surrounding the conductors. Similar considerations apply to the method of subareas and charge redistribution. In addition to these advantages, the subconductor and subarea methods are simple and straight forward to implement. Figures 1.3(a) and (b) illustrate the subdivision strategies adopted in the subconductor and subarea methods. 1.4.2 Problems with the Traditional Methods The traditional methods for calculating the line resistance and inductance in microstrip lines face a common problem in their applications — the large amount of memory storage required for accurate results. In addition, the traditional methods for calculating the line capacitance in microstrip lines are based on recursion and optimization procedures which are costly in CPU time. Following 12 #1 ^ #N line conductor dielectric ground plate (a)The method of subconductors; #1^#N fr3 line conductor dielectric ground 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 algorithms will be introduced. Resistances and Inductances To evaluate the frequency dependent R and L matrices in microstrip lines by the subdivision strategy, most of the methods proposed in 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 for the transient analysis, however, the uniform current distribution 13 assumption requires a very large number of subconductors to produce accurate results. The linear current subconductor technique (LCST) proposed in this work greatly reduces the number of subconductors required for accurate results. Instead of assuming a uniform current distribution across the subconductors, the proposed technique assumes a piecewise linear current distribution[69]. Typically sixteen times fewer subconductors are required with the new technique, resulting in savings of up to 98% to 99% in storage and CPU time requirements. Capacitances In most published material dealing with calculation of the C matrix (WO) of coupled microstrip lines[2, 12, 17, 20, 38, 51, 57, 58, 60, 65, 66], the matrix elements Cri.7 are calculated by repeatedly supplying a unit voltage on one line at a time until all the lines have been excited, and determining the total charge accumulated on each line. The dipole method from [12] is used to obtain the accumulated charges by putting an electric dipole on all the surfaces (conductors and dielectric). The total charge is calculated by integrating the field charge distributions induced by the voltage source or dipole. This process involves numerical integration, recursion and optimization; all are CPU time consuming. A new method for C calculation is proposed and derived from the traditional method of subareas 158]. The proposed method, which will be referred to here as the advanced method of subareas 14 (ANS), computes the Maxwell coefficients matrix and inverts it in order to obtain the C matrix. It is more efficient since it eliminates explicit calculations of charge distributions and total charge accumulations. Hence, the process of recursion and optimization involved in the conventional application of the method are avoided[68]. The method of moments used in reference[38] is equivalent to the method of subareas, but it requires the use of delta basis functions. Since the method of subareas uses Green's Functions, the underlying physical concept is simpler, and it is easier to implement as a numerical algorithm. 1.5 CONTRIBUTION OF THIS THESIS WORK The main contribution of the work in this thesis is the development of numerical methods that allow more efficient and accurate calculations of the parameters of microstrip lines. The linear current subconductor technique (LCST) has been developed and implemented to compute the frequency dependent R and L of microstrip lines. The technique has also been applied to pipe-type cables for underground power transmission. Another new algorithm, the advanced method of subareas (ANS), has been proposed to calculate microstrip line capacitances. The methods developed in this thesis are validated by comparisons with existing numerical techniques. These techniques include the 15 traditional subconductor method[59], the method of subareas[58], the finite element method[66], and the dipole method[l2] . The new methods result in savings between 38.8% and 99% in memory requirements, and between 50% and 99% in CPU time. Although the microstrip line structure in Fig.1.1(a) is used in developing the proposed numerical techniques, the techniques can be 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 THESIS This thesis is organized as follows. Chapters 2 and 3 present the theory and the proposed calculation techniques for microstrip line resistances and inductances. Chapters 4 and 5 describe the proposed technique for the computation of the capacitances. Chapter 6 makes suggestions for future research work and Chapter 7 outlines the conclusions. Finally, the appendices include division of currents in split and non-split conductors, list the symbols and abbreviations used in the thesis and provide a summarized reference list. The relevant references are included at the end of each chapter. 16 REFERENCES [1] P. ARIZON and H. W. DOMMEL. Computation Of Cable Impedance Based On Subdivision Of Conductors. IEEE Transactions on Power Delivery, vol.PWRD-2, no.1, January 1987. [2] K. G. BLACK and T. J. HIGGINS. Rigorous Determination Of The Parameters Of Microstrip Transmission Lines. IRE Transactions on Microwave Theory and Techniques, vol.MTT-3, MARCH 1955. [3] J. R. BREWS. Transmission Line Models For Lossy Waveguide Interconnections In VLSI. IEEE Trans. on ELECTRON DEVICES, vol.ED-33, no.9, SEPTEMBER 1986. [4] C. S. CHANG. Electrical Design Of Signal Lines For Multilayer Printed Circuit Boards. IBM Journal of Research and Development, vol.32, no.5, SEPTEMBER 1988. [5] R. CHATTERJEE. ELEMENTS OF MICROWAVE ENGINEERING. Chichester, West Sussex, England: Ellis Horwood; New York: Halsted Press, 1986. [6] J. CHILO, C. MONLLOR, and M. BOUTHINON. Interconnection Effects In Fasst Logic Integrated GaAs Circuits. International Journal of Electronics, vol.58, no.4, pages 671--686, April 1985. [7] R. COMERFORD. 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Improved Algorithmic Methods For The Prediction Of Wavefront Propagation Behavior In Multiconductor Transmission Lines For High Frequency Digital Signal Processors. IEEE Transactions on Computer Aided Design of 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 Properties Of Printed Circuit Boards. IBM Journal of Research and Development, vol.33, no.1, JANUARY 1989. [47] R. F. PEASE and 0. K. KWON. Physical Limits To The Useful Packaging Density Of Electronic Systems. IBM journal of Research 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 Formulae For Line Capacitance And Characteristic Impedance Of Microtrip Line. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-29, no.2, FEBRUARY 1981. [50] S. TALUKDAR R. LUCAS. 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IEEE Transactions on Microwave Theory and Techniques, vol.MTT-35, no.1, JANUARY 1987. [65] E. YAMASHITA. Variational Method For The Analysis Of Microstrip-Like Transmission Lines. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-16, no.8, AUGUST 1968. [66] Y. YIN. CALCULATION OF FREQUENCY-DEPENDENT PARAMETERS OF UNDERGROUND POWER CABLES WITH FINITE ELEMENT METHOD. PhD thesis, University of British Columbia, JUNE 1990. [67] X. ZHANG, J. FANG, K. K. MEI, and Y. LIU. Calculations Of The Dispersive Characteristics Of Microstrips By The Time-Domain Finite Difference Method. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, no.2, FEBRUARY 1988. [68] D. ZHOU and J. R. MARTI. A New Algorithm For The Capacitance Matrix Of Microstrip Lines In VLSI. BALLISTICS SIMULATION and SIMULATION WORK AND PROGRESS, Proceedings of the SCS Simulation MultiConference, New Orelean, USA, pages 60--65, APRIL 1991. [69] D. ZHOU and J. R. MARTI. A Linear Current Distribution Technique In The Computation Of Resistance And Inductance Matrices Of Microstrip Lines. MODELING AND SIMULATION, Proceedings of the 23 IASTED International Conference, Calgary, Canada, pages 47-49, JULY 1991. [70] D. ZHOU, F. P. PREPARATA, and S. M. KANG. Interconnection Delay In Very High-Speed VLSI. IEEE Transactions on Circuit and Systems, vol.38, no.7, JULY 1991. 24 Chapter 2 Linear-Current Subconductor Technique for Matrices of Frequency Dependent Resistance and Inductance In Chapter 1, we indicated that transmission line theory can be used to model microstrip lines, and that to determine the line characteristic parameters (Fig.1.2(a)) is the first step towards setting up the model. Existing techniques for microstrip line parameter calculations share similar problems of inefficiency for practical use. In this chapter, we derive and examine a new numerical method — linear current subconductor technique (LCST) — to efficiently calculate the frequency dependent R and L matrices. 2 . 1 GENERAL DESCRIPTION For signal frequencies of interest in microstrip lines, the values of resistance and inductance are strongly frequency dependent. This is due to the skin effect, which limits the wave penetration depth in the conductors along the line, and to the proximity effect, which is the current redistribution in the presence of other currents and is generated in wave propagation on coupled lines[9, 8, 12, 13, 3]. As the frequency of the signal increases, the wave penetration depth decreases and the conductor's resistance R increases. In turn, the current distributed inside the conductor changes and the inductance L varies accordingly. Traditional for25 Subdivision of line conductors + Rand L of subconductors + Current in subconductors: V =-..Z I + Modification of current with linear current distribution + Correction ofRandL + Subconductors' bundling into conductors, finalZ matrix Fig. 2.1 Flow Chart for the Linear Current Subconductor Technique (LCST). mulae for resistance and inductance of microstrip lines are based on zero frequency signals and thus of very limited value. As introduced in Chapter 1, the proposed LCST technique is used to compute the R and L parameters accurately and efficiently, with a subdivision strategy and a linear current distribution assumption. 26 A flow chart to illustrate the implementation of the LCST technique is shown in Fig.2.1. To employ LCST, firstly the microstrip line conductors, including the ground conductor, are subdivided into small segments (subconductors or cells) in their cross-sections according to a 28 rule. Here 5 represents the skin depth at a specific frequency. Secondly, the segment's resistance and inductance are calculated from traditional Ohm's law and Geometric Mean Distance (GMD). Thirdly, the current held in each cell is solved from the Telegrapher's equations defined by the initial resistances and inductances, as well as assumed initial voltages. Then new currents under assumption of linear current distribution across segment are calculated from the currents obtained in the third step. These new values of current distribution are subsequently used to modify the resistances and inductances from the second step. Meanwhile, an inductance correction factor is incorporated into the modification. Finally, a bundling procedure is applied to the complex matrix of subconductor impedances to obtain a reduced matrix with dimension /V — the number of equivalent conductors. The real part of the reduced matrix is the desired line resistance matrix, and its imaginary 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 includes subdivision, current distribution, correction factors, and bundling procedure. Microstrip lines and pipe-type cables are used as examples to illustrate the technique. 27 2 . 2 CONDUCTOR' S CROSS-SECTION SUBDIVISION The criterion for the conductor's cross-section subdivision is based on the fact that the wave propagating along the line penetrates a small depth into the conductors when the signal frequencies increase to relatively high levels. (At zero frequency, the wave penetrates the whole conductor which results in a uniform current distribution across the conductor's cross-section [5].) The skin depth is defined as _ A1^ ufa (2.2.1) where - Skin depth; y — Permeability of the conductor; a — Conductivity of the conductor; and f — Frequency of signals on the line. A sinusoidal distribution rule is used to determine the size of the subconductors, according to the line geometry and the skin depth[16]. The number of subconductors may change at different frequencies. The schematics of the subdivision for microstrip lines and pipe-type cables are shown in Fig.1.3(a) and in Fig.2.3, respectively. The formulae to determine the subdivisions are derived as follows. 2.2.1 Subdivision of Microstrip Lines In order to take into account the nature of the current distribution, the subdivisions are made denser near the conductor edges and 28 corners according to sinusoidal functions[7, 16]. The microstrip line conductors are divided according to a 28 rule. The ground plate is considered as N pseudo conductors (with N the number of line conductors in the system), as shown in Fig.2.2. An equal segments rule is used for the subdivision of the ground plate. Let J be the number of subdivisions on the line width, and L the number of subdivisions on the line thickness. The values of J and L are calculated from 7T J= 2 arcsin V Wid th (2 . 2 . 2 ) ) ( and 7r L , 25^ (2.2.3) 2 arcsin V Thickness) ( where 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 to determine the size of the subdivisions. In the line conductors, the end points (x,y) enclosing a cell are calculated from (refer to Fig.2.2) 1 x =^• [xv(i, 2) + xv(i, 1) — cos j, • (xv(i, 2) — xv(i, 1))] i= 1,...,N (2.2.4) I^1 y =^• [yv(i, 2) + yv(i,l) — cos jy • (Y.„(, 2) where 29 — yv(i, 1))yd Width Conductor MI•• thickness Yv(i,2) ^ I^I Dielectric—. ^)c,(i,1)^)s,(i,2) ••• :height Ground plate Psuedo conductor (a) Co-ordinates with even number of conductors; Conductor^ Width Y(i,2) thickness ••• Dielectric^MO)^)c,(i,2)^y,(1,1) :height Ground plate Psuedo conductor (b) Co-ordinates with odd number of conductors. Fig. 2.2 Coordinate System for Microstrip Lines cos j x = cos ((j — 1) x 7r), j = 1 . . . J,^(2.2.5) and j = 1 . . . L.^(2.2.6) cos jv = cos 30 For the ground plate, the numbers J and L are used to equally subdivide 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 area occupied 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-th conductor. 2.2.2 Pipe-Type Cables Pipe-type cables consist of different kinds of conductive materials with an arrangement as shown in Fig.2.3. Notation A, B, and C 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 is holding the cables. A similar subdivision method to that used for microstrip lines is employed to subdivide the core, sheath and pipe of pipe-type cables. To adapt to the circular geometry, arc-shaped cells or curved cells are employed[1], as shown in Fig.2.3. The outside pipe corresponds to the ground plate in microstrip lines. Following the azimuth direction, the circles are divided into equal sectors. The diameter or the annulus is sinusoidally divided along the radial direction. 31 For the core, the number of subdivisions, NRi, is Ir NRi = ^ 2 arcsin (AP11-) (2.2.8) for the sheath, Arli2, 7r NR2 = ^ 2 arcsin (V-212—) (2.2.9) and for the pipe, NR3, NR3 ^ir^ (2.2.10) 2 arcsin^pr2 -pri In 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; and pr2 — Outer radius of the pipe. Along the azimuth direction, the number of subdivisions is 27rri^ z. =^= 2 N^ 168^86 (2.2.11) with i, the sequence index of conductors, as shown in Fig.2.4. Here pri takes the place of ri in the pipe. The coordinates of the vertices of each cell are determined (refer to Fig.2.4) by I x = xo xi cos a — ysinc = Yo x sin a + cos a 32 (2.2.12) 240mm Pipe: a = 3907.7 S/mm pr =500 r1 = 24.25mm Sheath: a = 4800 S/mm r2 = 40.25mm r3 = 42.25mm r4 = 44.25mm Core : a = 34060 S/mm [is = 1 for all Fig. 2.3 Pipe-Type Cable Configuration and Subdivisions. in which coordinates (x,y) represent the position in the main coordinate system with its origin as the center of pipe circle, ,^, , , and coordinates (x ,y ) are measured with respect to three secondary coordinate systems with their origins (x00/0 at the centers of the cables. In the above equations, a is the axis' angle between the main and secondary coordinate systems. 33 ^ Additionally, in the secondary coordinate system, we have = p cos 1 y^p sin 13^ where p and 0 ( 2.2.13) are the radius and azimuth of a cell. The cell's circular size p in the core is calculated from (NRI—j r p^ri cos ^ 2)' j^ N^(2.2.14) with index j starting from the centre of the cable.^For the sheath, — 1 7r) p = r2 (r3 — r2)sin ^ = 1, . , NR^(2.2.15) NR2 2 )^j and for the pipe, 1 7r^ )^— 1^N Rq Pi = pri (pr2 — pri) sin (iNR ••• 3 2 (2.2.16) 2.3 SUBCONDUCTOR RESISTANCE AND INDUCTANCE FROM TRADITIONAL FORMULAE After the line conductors have been subdivided into small segments, cells or subconductors, the initial R and L of these subconductors can be calculated using Ohm's law and the concept of Geometric Mean Distances (GMD). 2.3.1 Microstrip Lines In Fig.2.5(a), a given segment is labeled (i,j,/), meaning row j and column 1 in the i-th conductor. (Fig.2.5(b) shows schematically 34 Fig. 2.4 Coordinate System for Pipe-Type Cables. the electromagnetic field in a microstrip line system when a static signal travels down the line[18, 15, 10, 6].) The resistance per unit length for subconductor (i,j,/), assuming uniform current distribution in its cross-section, is = 1 ai Äjil where - Cross-section area of subconductor (i,j,/); — Resistance of subconductor (i,j,/); and 35 (2.3.17) cri - Conductivity of conductor i. The inductance per unit length (see Fig.2.6) is[14, 11] (GadGbc) 42= —1n. 2r^GacGbd)^(Him) with In (Gad) = 1 ^f AaAd J.] int ditadAb (2.3.18) (2.3.19) AA b where a, b, c and d correspond to four long parallel subconductors or 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,,, is the cross-section area of cell a. Constant Ito is the permeability of air or vacuum, and pr is the relative permeability of the line conductors. Gad, for example, is the GMD between filaments a and d. As shown in Fig.2.6, loop 1 is formed by connecting the ends of 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. If loop 1 and loop 2 are connected to become one loop, 42 becomes the self inductance 41 or 122. In the microstrip line system, the line permeability is assumed to be that of the air, i.e. Ito = 4r x 10 7 (Him) and /4=1.^The - shape of the subconductors cross-section (cells) can be rectangular or square. The GMDs are calculated from the geometric positions of 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)^ 36 (2.3.20) (a) A cell or subsection denoted as (i,j,I), meaning row], column I in conductor i. (b) Schematic of magnetic and electric field in a single conductor microstrip line. Fig. 2.5 Cell Index and Fields Distribution in Microstrip Lines where Alv and At represent the cell's width and thickness, respectively[1, 14]. GMDs between Subconductors in Microstrip Lines Mutual GMDs of retangular or squared cells in microstrip lines 37 loop 1 ^ loop 2 Fig. 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 are the integral of geometric distances between two cells [17] ,s4 in (Gad) 25^(Y y ) — 4(y — yi)2(x — xi)2 + (x — x')4^, 2^, 2 in [(x — x ) + (y — y ) 1 6^ 24^ — 7--- , (x — x')(y — y) {( 3^ x—x x x ,)2 arctan Y — Y, + (y y')2 arctan ^ x — x^y — y'l 1 y2 y, X2 X2 Y1 Y1 xi xi (2 3.21) where (x,y) is located in cell a, (x',i) is in cell d and xi , x2, xi, x2 IYI,Y2,Y1,Y2 are the co-ordinates enclosing the cells. When two cells are merged, the mutual GMD between them become the self GMD. 2.3.2 Pipe-Type Cables For pipe-type cables, the cross-sections of subconductors (or elementals [14]) have different shapes from those of microstrip lines. The permeability for the core and sheath is po while that for the pipe is go x,tir = 500p0. The subconductors resistance and inductance are still determined by eqns.(2.3.17) and (2.3.18). The corresponding 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 along the annulus in the k-th conductor (core, sheath or pipe), as shown in Fig.2.8. Paralleled Curved Cells As shown in Fig.2.9(a), the cells in the conductors of the pipe-type cables are curved and close to each other. For these 39 (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 of microstrip lines, eqn.(2.3.19), except that the average arc lengths of cells, ZVI and respectively, A102, are used instead of the widths Am4 and Atv2, i.e., 4- 1 — Y‹ +1)2 In + D2)+ (.955) —2,41 ,42 /< — ><^Y< + D2 in (1q + v2)+ 2,6.1°1,42 2,41,42 2DX,^2DX , ^ arctan (N2)^ arctan AP1z-1192^D^APiAP2 in^GIN (2.3.22) 1)2^ ln ()( +D2) P1,42 \ + D2 with )(1 = ,41 — A P2 (2.3.23) Aroi AP2 (2.3.24) 2 and X,= 2 40 Non-paralleled Curved Cells AS shown in Fig.2.9(b), when two cells are located relatively far apart, each cell is subdivided into four sectors such that the sectors enclosing one straight cell edge will have one-sixth of the arc's length, while the middle sectors will have one-third of the arc length. Let A be the spanned angle of a curved cell. Then pairs of points are symmetrically placed about the edges of the sectors, at an angle distance of 0.564x 4[14]. The mutual GMD between two cells is, therefore, the geometric mean of the 36 distances between six points in one cell and six points in the other, as shown in Fig.2.9. If P is one of six points in cell p, and -0 is one of six points in cell q, the GMD between cells p and q will be Gpq^ 36 II j=II (Pi-01 i=1,6 (2.3.25) 1,6 where /5i and 01-i are coordinate variables of points P and -0, respectively. Self Geometric Mean Distance The self GMD for curved cells is given by G ^0.2315 x (Ap + At) x e1-4632(1 ur)^(2.3.26) - 41 (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 Cells When cell p is a circle, eqn.(2.3.27) becomes Gpq 6 { u, (Po 3 6 (2.3.27) — where P0 is the origin of circular cell, and Q3 ascribes the coordinates of one of the six points located as in Fig.2.9(b). 42 The self GMD of a circular cell is: (2.3.28) G PP= r x e-7"- where r is the radius of the circle. 2 . 4 CURRENT DISTRIBUTION In a linear system such as microstrip lines or pipe-type cables, when the currents are assumed to be uniformly distributed across each subconductor, their values can be obtained from solving the following linear equations:[5,^4] VI- Z11^Z12^ Z1K 172 Z21^Z22^ Z2K Zii^Zi2^ ZiK - - Ii - 12 (2.4.29) 1/‹- zic2^ or in compact notation, ZKK - VKxi = ZKxKIKx1^ (2.4.30) where Zia = i=j roo (2.4.31) with K being the dimension of subconductor's impedance matrix. For microstrip lines, K=2xNx,IxL - 1^ (2.4.32) with Ar the number of line conductors, J the number of subdivisions on the width, and L the number of subdivisions on the thickness of the line conductors and pseudo conductors. For pipe-type cables, 6 K=^Nk X Nk' Npk X N 13'^ k=1 43 3 x (Nki — 1) — 1,^(2.4.33) ^ with Ark the number of subdivisions on the radius, Aq the number of subdivisions along the azimuth for the conductors, and Arpk subdivisions on the radius and /NT the subdivisions along the azimuth in the pipe. Also, rij (i,j = 1^K)^is the subconductor's resistance from eqn.(2.3.17) and^(i,j = 1^K) the subconductor's mutual or self inductance, calculated from eqn.(2.3.18). In particular, roo =^ 1 ^ o-A000 (2.4.34) is the resistance of a reference cell which is arbitrarily chosen inside the ground plane or pipe. For a different subdivision criterion, e.g., 2(5 or S rule, there will be different current distributions. Because of the assumption of uniform current distribution across subconductors, the current densities will present "jumps" at the edges between cells. The plots in Chapter 3 show these discontinuities in the solutions of current density. To minimize the error caused by the discontinuities between sections, a relatively large number of subdivisions would be required. This, in turn, would require large amounts of computer memory and resources, especially at high frequencies. By assuming linear current distribution across each subconductor, the current density can be made continuous across line sections. This results in much smaller computer memory and resources requirements for a given accuracy of the results. 44 2.5 APPROXIMATION OF LINEAR CURRENT DISTRIBUTION ACROSS SUBCONDUCTORS Linear current densities at the cell boundary can be evaluated by averaging the values of the current densities in the cells around that boundary. Those current densities in the cells around the boundary are calculated assuming uniform current distribution. The linear current held in the cell (subconductor) will be the integral of the assumed linear current density. The procedure implemented is 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 current densities as the geometric mean of uniform current densities in the cells around the boundary obtained from Step 1; and Step 3. Integrate the linear current density in each cell to obtain a new total current value in the full conductor. Step 1 is the same for all cases. In Step 2, the values of current density on the vertices or edges of a cell are approximated by the geometric mean value of those current densities of the particular cells around the vertices or edges. They are, therefore, dependent on the line configuration. Microstrip lines and pipe-type cables are used as examples to describe in detail this procedure. 45 2.5.1 Microstrip Lines After the values of uniformly distributed current density are obtained from eqn.(2.4.31), the values of linearly distributed current density at each vertex of cell (k,m,n), mn, (2" =1,2,3,4), are estimated by the geometric mean of the uniformly distributed current 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) where ikrrin jkmn (2.5.36) 24krnn is the value of uniformly distributed current density in cell (k,m,n). The operator "±" means either "+" or "- If . Vertex index z 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 line conductors, excluding the ones containing conductor vertex, z = 1,2 or Jlzcrnn =^Jkmn X k(m±l)n,^ and z = 3,4 ^ (2.5.37) z = 1,2 or Jlzcmn =^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) 46 The new current in cell (k,m,n) is the integral of the assumed linear current density on the rectangular or squared cells: x2Y2 = lli(x, y)dx dy x2Y2 [ii (x ,^+ in (x , y)] dx dy (2.5.40) xim x.,2y2 = I f ((%' +QC' ) x ^+ 93") y (C' ^)) dy XiYi 71^ 9 72^, 73^, 9 74 = krnnl- kmn-r"^krnm. , in which (k,m,n), r. is 3 while 6 the linearly distributed current carried by cell 244mn is the area of that cell. The xi,x2,yi,y2 points are the four coordinates enclosing the cell (refer to Fig.2.10). The linear function i(x ,^= 9ix + 93y^ (2.5.41) is assumed to be the linear current density distribution across the subconductor, with JLin (z = 1,2,3,4) as the values at its four vertices. 94,93 and t can be determined by three of four vertex current densities. Then in each cell, two sets of 2t,93 and Q are obtained for the integration: { } and 2.5.2 Pipe-type Cables The same approach described above for microstrip lines is applied to approximating a linear current density in each subdivision of pipe-type cables. In these cables the subconductor's cross-section is 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. 47 J [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 number 1 to 7 as shown in Fig.2.4. Cell (k,m,n) refers to the k-th conductor, m-th layer along the radial direction and n-th angle in the azimuth direction. In Step 2 of the solution procedure, for circular cells (k,1,1), the linear current densities at the edges are estimated by 47nn = 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 cells adjacent to the circle, the linear current densities at the vertices 48 (b) A circled cell and its current densities. Fig. 2.11 Schematic of Cells and Their Current Densities for Pipe-Type Cables. are evaluated from \I T1 ,3^ dkmn =^ikll X ik2n X i k2(n±1) 1 T2,4 =^/,: ' kmn^4\1 Jkmn X ik(m+i)n X ikm(n±1) X i k(m+1)(n±1) • 49 (2.5.43) For general curved cells, those densities are calculated from 4mn = 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 a curved 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". The constants 2(1, 931, C' or Qt", 93", C" result from the following linear equations x2 Y^11[2111^[,It Y2^93 = J x3 Y3^Ci^47-nn (2 5 45 in a plane supported by cell vertices (xi,Y1), (x2,Y2), and (x3,y3), and X4 Y4 11[2t1:,1^[Jtcni X2 Y2 1^= 47nn x3 y3 nmn (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 density functions, the integration in Step 3 of the solution procedure has to be calculated on two split planes enclosed by curved cell boundaries. Then the new current value in a cell under the linear 50 current distribution assumption is 3 = Ki x intel1 + K2 X int el2 (2.5.47) where 02p2 = I (911 • p cos + 93' • p sin 0 + C') pc/0 = 00,1 1 —2- 3 3\ — Pi) [QC (sin 02 — sin 00 — '(cos 02 — COS el)] + (2.5.48) — I9?) • (0 and 02 P2 intel2 = f (21" • p cos 9 + 93" • p sin0+ C") pc/pc/0 8 P1 =^— pT)[91" (sin 02 — sin 01) — 93" (cos 92 — cos 01)] + (2.5.49) CH 2^2 2 (P2 —Pi) • ( 02 —00. In the above equations, pu p, p, 01,02 and 0 are the corresponding polar coordinates of the cell positions in the local (secondary) coordinate system, and will take the values from pl to p4 or from al to a4 as illustrated in Fig.2.11. K1 and K2 are between 0 and 1, 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 for QC, 93", and C" instead of 911, 93', and C', respectively. 2 . 6 LINEAR CuRREbrr DISTRIBUTION CORRECTION FACTORS In this section, the linear current distribution approximation is employed to modify the subconductor impedances in eqn.(2.4.33). 51 With the correction for linear current distribution, eqn. (2.4.31) is written as: V1 V2 Vi VK - Zi 1+ fl+ fl )^Z12+ (f1+ f2i z i,+(f; -14 Z1K+(.6.0 Z21+ (f;+f; )^Z22+ (f2+ /2 ) z2J+(f;+.6) Z2K+ Z1+ (f:+ f;)^;2+ (f:+ f 11; zt,+(ft+ f,)^• ) zic.,+(fK+ f.;) (f1{-1- fi) ZK2+ fk+ or (2.6.50) ZICK+ (fK+ fK)- ti'^-^Y' Kx1^KxK Kxl where 4+ and fi+ fi (2.6.51) are the linear current distribution correction factors for the impedance of cell (k,m,n), zij (i, j= (k 1) xJ x L+ - (m 1) x J n) , with - =^x f. (2.6.52) ^zii ^- and zij =—x ^„, ^ (2.6.53) fj. By rearranging eqn. (2.6.52), the linear distribution correction factors, f (i= 1,...,K), can be found by solving the equation with the supplied initial voltages, V, (i =1,...,K). That is, - V1 + CQ1 - +^112 z„ 2 V2 CQ2 z=n-1-;^a2 – CQi v, – aLr .11 I^Z22 2 laj. zjj3 •^ti +a, C QK ZKK - zKK - 12 zili_t 1, ZKK R fK Z22 - 52 ,^(2.6.54) - in which CQi =E ^(2.6.55) j=1 and (2.6.56) j=i Zzz• • The corrected value for the cell's impedance in eqn.(2.4.33) is 2/,3 =+ i;+) ( (2.6.57) . This quantity can also be written as = rii jxii (2.6.58) y 1 ^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 with the linear current distribution. 2.7 INDUCTANCE CORRECTION In the test cases using the linear distribution correction, it was found that the percentage error in the final inductance values was higher than that of the resistance. This is because the concept of GMDs is established under the assumption of an infinitely fine current loop (or uniform current distribution along the current loop), regardless of the subdivision rules. There are two options 53 to improve the correction of the inductance for linear current distribution: 1. Post-processing - Use of correction factors to compensate for the error; or 2. Pre-processing - Re-calculation of the inductance for linear current density before finding the Zjj correction factors. 2.7.1 Inductance Formula • Option 1 The parameter inductance is derived from the flux linkages of the currents. From electromagnetic theory[14], the flux linkage at a point on subconductor d due to a fine current Ja•dAa in subconductor a is, in per unit length, r2^2 ln [^+ 1) — ,\/(—) + 1 + — Di^(2.7.59) 27^ D^1 Oad = g • Ja • dAa • with D being the distance between subconductor a and the point in d, and ja the current density in a. With the assumption (>1), Oad 'Oa,/ becomes = -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 =^ Ad if A dA. 2r Ad Oadditadild •ff [111 2r) (- — 1] Ad A. 54 • (2.7.61) a • ditad4d • Integrating by parts, the linear current distribution across subconductor a will be 'F ad =^kJ. f (ja^ [ha^— Ad^Aa^ =^.^ 2ir Ad f (4, f daa — f I dj 11 fI AA [1 n^ [In ()) - 1] 4a - - a) dAd clAa • J:dAa)dAd {^f [in (i1_,) _ddAadAd A.A. Ad^Aa^ p 1 [hi^— lidAa =^ — 27r — Ad AdA„ - J: IjI [In (51,) - dAadAddAa} A. Ad An r>D p^1 •^• {1- a • 1 1 [1112i) (- - 11dAadAd - 27r Ad^ Ad A. V —Jailif[(2()-1]dAadAddAa} Aa Ad An = P 1 {J a 1 I [in (77 21) - 11dAadAd 27r Ad • Ad Aa (2.7.62) I r_() —5 — A a + In 2[1 dAadAddAa} - 2(.4,^ 2( I Aa Ad p1 27r • Ad {fa. 2() _ i]deladAd 11 [in (_ AdAa 21) - 1 + Aa + 1 -11121PAadAddAa} 2[ • Jai [ln (Aa Ad 1 (Ja^2[ • Ja')• I .1 27r • Ad • 2[) ddAadAd p 1 [in (5 J:(Aa + 1 -11120 -AaAd 27r Ad 2 [• AdA. =^ ^+ 2 jai )^ {In (^— dAad4d Ad AdA. + Li- • Aa 2[ • J: • Oa + 1- In 20. 27r From definition of GMD, 1 ln (Gad) = AaAd If Ad 1nD • actad (2.7.63) A. then, it follows that 2[ Wad = Pr • Aa • (Ja + 2L/a)• {ln (- - 1} + Li • Aa • Ja • (Aa + 1 - ln 20. (2 .7 .64) 2 Gad^27r 55 The mutual inductance between loop / and loop 2 in Fig.2.6 is, therefore, 1,^1F12^Tac — Tad — (Tbc Ibd) 12 ^1-1^ f Jad4a A. ,a Aa(J, + 2[4)^[GadGbc 1 (2.7.65) 27r^f Jadlia^Ga,Gbd..1 A. =^' 112 • In the above equation, the inductance correction factor is defined as = Aa (Ja + 2 Ucl) f JadA. (2.7.66) A„ where A.2 — <la = Ja(Aal) 21.1 Ja(Aa2) (2.7.67) is the integral of the linear current density on subconductor a, and 4 is the derivative of Ja. With our linear current distribution assumption J is constant. Because Al must be real, J represents the magnitude of the linear current density. Inductance correction factors can also be derived from circuit analysis theory. From a circuit point of view, a cell of width 26 can be split into two sectors of width 45 in parallel.^This is equivalent to splitting the current into finer current "loops". For example, a cell obtained with a 26 rule can be split into two cells that follow a 6 rule. The original cell's conductance, is two times that of the split cell's conductance. In what follows, inductance correction factors are derived for microstrip lines and pipe-type cables. 56 Microstrip Lines From circuit analysis theory, inductance correction factors can be derived from the ratio of the number of subdivisions with the 6 rule to the 26 rule as follows. With^=^jx, and y j =^=^ , after compensating with inductance correction factors, the impedances become zii = rii ^ (2.7.68) where 0 .819.(iir " [ CA4PGX (3.49(1..)'.45g:i, 443.56 [4.59.( r = 0.13 6: j X./VD x .Arc ) 1.31.(fl-4.75 [ cmpLx i=j in-24.65 CA4P L X (21.39(1)-32.75g:i, 91e Kix.Ari, x.Arz) (62.74i, 0.40( 1=27)21.1 X.Aip XArc) i j (Same Conductor) i j (Different Conductors) (2.7.69) 0.999.(i ) 14.86 te[ 91e{ C AAP X (3.49( l'1)1.459 443.56 C MP X (21.39(1)-32.75 , 0.13 X.ArD X.Arc) 5.49.(-In-35.96 1 .10.(1)[ xArvxN) i=j i j (Same Conductor) 14.06 C MP X (62.7 g:j, 0.4(0) 21.1 b:i X.ATV X.Arc) i j (Different Conductors) (2.7.70) In the above equations, CMPEX means to combine two reals into a complex number. Also, JVD is the ratio of the subdivision number with the 6 rule to that of the 26 rule on the width of the line 57 conductors, while N:c refers to this ratio on the thickness of the line conductors. Re and am mean to select real part and imaginary part of a complex number, respectively. In general, the inductance correction factors are expressed by (2.7.71) and (2.7.72) where .04. and Ax are defined as the general inductance correction factors, which are dependent on the frequency. Tables 3.1 through 3.3 in Chapter 3 show the inductance correction factors for microstrip lines. Pipe Type Cables - For pipe-type cables, the general inductance correction factors for .u z•2^z3 3• = r•• jX ^are defined by (2.7.73) r•• —^• Z3 - • m and = Ax rn where in x •^• • (2.7.74) is the conductor index as shown in Fig.2.4. Tables 3.8 through 3.10 in Chapter 3 list the corresponding correction factors for pipe-type cables. 58 2.7.2 Inductance Formula • Option 2 In Option 1, general inductance correction factors were obtained after the linear distribution correction factors. Trial and error experiments were needed for determining these factors. ^These factors are configuration dependent.^The general inductance correction factors can also be obtained through an iteration procedure using for example a Newton-Raphson algorithm. Equation (2.6.63) can be reformulated [2] as 111 ad = 27r• Ad 1• 1 I [in (T 2r) —11• AdA Ja • dAadAd „ =^1 .f [fin (_ 2C) 2r Ad j j^vs) 11 • d(41 JadAa)dAd AdA. -2-77r - • 741 I(I JadAa) • Ti (T5 2C) 11 dAd A. Ad if 1 A [In^- 11 I JadAa) • dAadAd 27r^ A4a^\ A. =^1^(Ial x2^'ail 277. Ad^2 Y 2-xY2 lqxY). 2C \/(x - x')2 (y - y')2) 11 dr dyi 1 Iff Tx2y TxY2 IqXY)^ I y'x' Y (X + y) • dxdy dx'dyi ( (2.7.75) where Ja=(X y) =9J.X+ 8y+Q.. The integral of the logarithmic function , 59 ^y in the above equation is if ^2r [ln ( s'y ,^\ = Ad • = Ad • I (x — x')2 + (y — Y' )2 [1n (2) — 1] if ln Rx — ') 2 + — — {xiy' (1n Rx Rx (y — y1)2] cividy1 x'y' [ln (20 — 1] 1 —— 2 )^11 dx1 dy' — x1) 2 + (y — y1)2] _ 4) +2 — x arctan 1 y x xi)^— arctan ( — x1) 2^— y1)2]^ y — y' )^4 Ix' — xl — (2.7.76) We then have it^ 1 ((1% 13,271H931^2 • Ad •^2 - , _ 2 xy^I lxy) • {Ad • P.(20 — 1] T^27r ad^ — {x'^ Rx — ') 2 + ( y — y1)21 _ 4) +2 — x arctan 1 " x—x\^1 y —y —— x1)2^— y1)21 arctan — y')^4 ix' — xj 2 Rx + x2 Y2 Ix 2 L 1%1 x 8 3 J xY 9 II)+Y(TY+ )1 /A.,A). (2.7.77) Here (x, Y) is located inside cell a and (x',i) belongs to cell d. Similarly, we can have Tac, Tb, and Tm• Therefore, the new inductance between loop / and loop 2 in Fig.2.6, under the linear current distribution, is Tac 112 Tad ^ = I/11 f Jadela A. 60 bd) (2.7.78) ^ For the reference cell, ^1 ^( 2 - • /000 A^ 111600 • [ in (2r) —1] — in (Goo)}. WOO =27r 4,00 (2.7.79) Because the reference cell will carry the total return current of the subconductors system, its current density is assumed to be uniform, as an exemption of linear current density distribution (This will not influence the final solutions). We can now obtain z ^3 . Substituting l into eqns . (2 .6 .52) the equations are updated. ^The linear current correction factors are re-solved from eqns. (2.6.56) . ^Hence, the subconductor impedance becomes A Z = Z' +^• 23^ z3^ LIZt.1 • (2.7.80) This procedure is based on a pre-processing strategy: inductance correction goes before linear current distribution correction. According to the results from the test cases, the subdivision pattern across line conductors plays an important role. Trial and error experiments are required for determining the proper subdivision for the final R and L values. Taking pipe-type cables as an example, it was found in test cases that increasing the number of subdivisions along the azimuth direction does not contribute much to the accuracy of the final parameters. Employing the skin depth as criterion, 18.58 to 19.58 on circles were acceptable choices. 61 After the subconductor impedance is modified, subconductor bundling is applied to eqn.(2.6.52) to reduce the rank of the system from K subconductors to N equivalent line conductors [41. 2.8 EXWMYENG PROCEDURE In the subconductors system, the reference cell (0,0,0) is chosen arbitrarily in the ground plate or pipe. This cell is assumed to carry the whole return currents in the system. In addition, the voltage drop across the cross-section of the conductors is assumed to be zero, from TEM transmission line theory. Under these conditions, the bundling procedure can be derived from circuit theory as follows[17]. 2.8.1 Voltage Drops along Subconductors The voltage drop on subconductor (k,m,n) on per unit length is (refer to Fig.2.12) eii = ekmn = 2N N N, E E E (rkmnbkiesmrrebn. + jW1 lc" mn,km'n')-1-ki m'n' • Az k=1 m' =1 n=1 (2.8.81) Z. mn,0004 00 • Az In particular, e„ = e000 = (row + icolonoo,000-1-0' 00 • Az 2N Nw Nwi^ E E E k' =1 10 00,1ern'n' m' n' • (2.8.82) Az , m'=1 n'=1 and 2N Nw Io'oo= —EEE-rienini • (2.8.83) k=1irre=17-e=1 Here, bij is the moment function, 6.^f^i=j — 'o 62 (2.8.84) The loops for the inductances, 1 Ic" mn,ki n" are enclosed by the reference cell (0, 0, 0). N I kinn v (z+Az) eo o o Fig. 2.12 The Voltage Drops Related to the Subconductors. By substituting eqn. (2.8.85) into (2.8.83) and (2.8.84), and dividing eqns. (2.8.83) and (2.8.84) by Az, eqns. (2.8.83) and (2.8.84) become ekmn AZ AZ (2.8.85) 2N^ =EEE [rknin6kommq5n7i + j411c" mn,krn're ik"mn,000)] IkiMire 7 k=1 m'=1 re=1 and = eoo^e000 Az Az 2N N,e =EEE [—r000+iw(/00,k'nen' 10" 00,000)]i ' n' (2.8.86) lc'^m1 n' =1 Let Vknin(z) be the voltage at position z of subconductor (k,m,n) to (0, 0, 0), as shown in Fig. 2 .12, then we have e000-FV/wm(z)= Vi,„,(z+Az)d- ^ (2.8.87) 63 Exchanging items on both sides of the operator "=", — Vk(z + Az) =^ — e000 = — AVi.^(2.8.88) Substituting eqn.(2.8.90) into {eqn.(2.8.87) — eqn.(2.8.88)1, we obtain 2.1V Ark, Arik, Vknin Az =E E E{Rk.n,k,fre+i4Akrrin,k}ikiirrin' k=17771=1d=1 K K (2.8.89) EE i=1 j=1 where I/ Rkmn,kim'ri I/ = 1'00 0 rkmn6kk arnmi 46nre (2.8.90) and Lkmrt,le min' = I kmn,k'm'n' ^111: mn,000^10" 00,1em'rl + 1000,000• (2.8.91) Define Zkmn,lcim'd = Rkmn,kirred^kmn,k1 mire = rii^ (2.8.92) eqn.(2.8.91) then becomes 2NN, _ v-knin E EE Az k=177i = 171=1 Zkmn,k/ rre n' m' n' (2.8.93) in which D Symbol k' is the conductor's index: k' =1,2,...,2N for microstrip lines, or k' = +1 for pipe-type cables; 64 ^Y O Index 7 17: is to indicate rows or layers: 771:=1,2, ,J - - for microstrip lines (Arki= J), and 717: =1,2, - ,Ark for pipe-type cables (Ark,^Ark); O The index of column or angle sequences is^n'= 1,2,...,L for microstrip lines (Ni, = L) , and n' -= 1,2,..., for pipe-type cables (Ni, = Ni); O When lc' = N +1 in pipe-type cables, nz' 1,2,..., Npk and 1,2,...,Npik; and O Nk,,Nk'„Npk and^refer to the number of subdivisions on corresponding dimensions of the line conductors. If the number of cells is ^=2xNxJxL for microstrip lines, the rank of impedance matrix will be K=/(' -1 with (0,0,0) located in the ground plane.^Similarly is for pipe-type cables. ^The dimension of matrix^is (I(' — 1) x (.1(' — 1) . 2.8.2 Currents in the Subconductor System From the voltages obtained in the previous section, we calculate the total current in the line conductors (what "bundling" really means). Then, the line's impedance matrix is derived. Suppose Y is the admittance matrix of the subconductors system and Z is the corresponding impedance matrix, then = Z-1.^ 65 (2.8.94) The current in subconductor (k,m,n) is 2N Nk' \ AVIe ni mn =^Ykmn,knin'^A^• LAZ k=1 ne=1 r/=1 Here, kmn,k' ^ (2.8.95) is the subconductor's admittance between (k,m,n) and (k',m',n' ). The current in conductor k is N,, N ,: m=1 n=1 (2.8.96) Nk IV,: 2N Nkl -EEEEE 11 Ykmn,k'm' n' m=1 n=1 k=1 m'zzl 7-1=1 AVierrere Az According to transmission line theory, voltage drops along the subconductors of the ground plate are assumed zero, 1p= A Vpmn = 0, N +1,N +2,...,2N or p = 2N + 1, m=1,2,•••, (2.8.97) N lc, n=1,2,•••,N;,, and the voltage drops are also zero because of equi potential planes — across conductors.^That is, f^=1,2,• • • ,Nk, = AVkmn = AVkm/ n' Therefore, ATTimin (2.8.98) only depends on the conductor index, AVkrnn = Vk fm = 1,2,•••,Nk, n= (2.8.99) Hence, the current in conductor k is expressed as IkYkk = k=1 66 AK, Az (2.8.100) where Ark NI: Ykk =EEEE m=1 n=1 n1=1 21=1 Ykmn,k min' • ^(2.8.101) Now the subconductors admittance Y becomes the line conductors' ff with elements Pk,k'^flick' • ^ (2.8.102) Inverting back the admittance matrix P., the impedance matrix Z of N conductors is obtained. For the case of pipe-type cables, the upper limit of ^2N, becomes N4-1 in the above procedure. 2.8.3 Matrix Manipulation For Matrix Reduction With the principles indicated previously, the impedance matrix in eqn.(2.6.52), ZKXK can be reduced directly through matrix manipulation. In the programs, ZicxK is directly reduced with a subroutine from reference [4]. 2.9 SUMMARY 2.9.1 Summary of the Linear Current Subconductor Technique In this chapter, the theory of the linear current subconductor technique (LCST) has been derived for the computation of the line parameters R and L (resistance and inductance). The LCST consists of six steps: 67 ^ Step /.Subdivision of the line conductors and ground conductor by the 25 rule; Step 2.Calculation of the subconductors' resistances and induc- tances with the traditional analytical formulae; Step 3.Computation of the current distribution in the subconduc- tors by solving the system of linear equations (copy from eqn.(2.4.30)) - ^VKxi = ZKxKIKx1; ^ (2.9.103) Step 4.Approximation of current's linear distribution from the results of step 3, and computation of the linear distribution correction factors; Step 5.Evaluation of the inductance correction factors and modi- fication of the equations (copy eqn.(2.6.53)) = lilfxKiKx1; ^ (2.9.104) and Step 6.Bundling of the system of subconductors into the equivalent line conductors, to form a new impedance matrix with the dimension of the number of line conductors. Step 5 and Step 4 can be exchanged if Option 2 is used to correct the inductance matrix. In Step 6, the real part of the impedance matrix is the line resistance, and its imaginary part the line admittance. 68 2.9.2 Discussion The LCST technique combines the efficiency of subconductor methods with the accuracy of finite element methods. It automatically incorporates skin effect into the line parameters [17, 19, 201. This technique can result in large savings of CPU and memory requirements. In LCST, the current does not need to be assumed uniform. As in the finite element method, the condition of current continuity on cross-sections between subconductors is naturally satisfied, while in the conventional subconductor method, the number of subdivisions must be large enough to approximate this condition. LCST can generate accurate results with at least 75% fewer number of subconductors, which is equivalent to a reduction of 16 times in the number of elements in the impedance matrix when compared to the conventional method. Moreover, due to the memory swapping mechanism in some operating systems, CPU time costs will also be reduced. As an example of the memory and time savings, for a given case of pipe-type cables at f= 600,000 Hertz (Fig.2.2), there were 618 subconductors needed with the traditional approach of assuming uniform current distribution, while only 138 subconductors were required with the LCST technique. In terms of computer storage in the SUN UNIX system, with double precision reals, there would be 618(618+1) 2 8=1.53 Mbytes necessary for matrix processing using the old method, while there were only 76 Kbytes required with LCST. This 69 corresponds to the savings of 94.99%. Also, the memory swapping in running a simulation with LCST was much less than that without LCST. The resultant time savings were estimated as at least 80% regarding CPU cost. Also, more than a 95% reduction in CPU time and in memory requirements were found for the case of microstrip lines. From these results, it can be seen that the additional manipulations required by the LCST technique are fully compensated by the reduction in memory swapping, and consequently a much faster simulation speed was obtained than with the old subconductor method. On the other hand, compared to the finite element method, the subconductor method forms "meshes" or patterns of subdivisions only in conductors, and does not require the explicit specification of magnetic boundaries in the problem solution domain. This simplifies the "mesh" generation and results in an automatic subdivision procedure. Besides savings in computer resources and convenience for users, LCST also has the following advantages: 1. At high frequencies, much more accurate results can be obtained without exceeding system memory limits; 2. It is very straightforward to implement, and does not contain any recursive procedures to cause any divergence problems; and 3. Instead of linear, the current distribution could be assumed as another convenient function, as for example an exponential function, within the same solution procedure. 70 If needed, a 36 or 46 rule could be chosen in subdividing the line conductors to obtain additional savings. In the next chapter, simulation results from LCST will be shown and the efficiency and accuracy of LCST will be demonstrated. 71 REFERENCES [1] P. ARIZON and H. W. DOMMEL. Computation Of Cable Impedance Based On Subdivision Of Conductors. IEEE Transactions on Power Delivery, vol.PWRD-2, no.1, January 1987. [2] W. H. BEYER. STANDARD MATHEMATICAL TABLES. CRC Press, Florada, c1981. [3] J. CHILO and C. MONLLOR. Magnetic Field And Current Distributions In A System Of Superconductor Microstrip Lines. IEEE Transactions on Magnetics, vol.MAG-19, no.3, MAY 1983. [4] H. W. DOMMEL. EMTP REFERENCE MANUAL. University of British Columbia, 1986. [5] L. N. DWORSKY. MODERN TRANSMISSION LINE THEORY AND APPLICATIONS. New York: Wiley, 1979. [6] T. C. EDWARDS. FOUNDATIONS FOR MICROSTRIP CIRCUIT DESIGN. Chichester; New York: Wiley, 1992. [7] D. W. KAMMLER. Calculation Of Characteristic Admittances And Coupling Coefficients For Strip Transmission Lines. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-16, no.11, November 1968. [8] M. KOBAYASHI. Longitudinal And Transverse Current Distributioin On Microstriplines And Their Closed-Form Expression. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-33, no.9, SEPTEMBER 1985. [9] M. KOBAYASHI and H. MOMOI. Longitudinal And Transverse Current Distribution On Coupled Microstrip Lines. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, no.3, MARCH 1988. [10] S. KOIKE, N. YOUSHIDA, and I. FUKAI. Transient Analysis Of Microstrip Line On Anisotropic Substrate In Three-Dimensional Space. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, no.1, JANUARY 1988. 72 [11] V. D. OEDING and K. FESER. Mittlere Geometrische Abstande Von Rechteckigen Leitern (in German). DK 621. 316. 35 : 538. 311, ETZ-A, Bd. 86, H. 16, pages 525--533, 1965. [12] B. D. POPOVIa and D. N. FILIPOVIL Theory Of Power-Frequncy Proximity Effect For Strip Conductors. Proceedings of IEE, vol.122, no.8, AUGUST 1975. [13] R. A. PUCEL, D. J. MASSE, and C. P. HARTWIG. Losses In Microstrip. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-16, no.6, JUNE 1968. [14] S. TALUKDAR R. LUCAS. Advances In Finite Element Techniques For Calculation Cable Resistance And Inductance. IEEE Transactions on Power Apparatus and Systems, vol. PAS-97, no.3, MAY/JUNE 1978. [15] T. SHIBATA, T. HAYASHI, and T. KIMURA. Analysis Of Microstrip Circuits Using Three-Dimensional Full-Wave Electromagnetic Field Analysis In The Time Domain. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, no.6, JUNE 1988. [16] W. T. WEEKS. Calculation Of Coefficients Of Capacitance Of Multiconductor Transmission Lines In The Presence Of A Dielectric Interface . IEEE Transactions on Microwave Theory and Techniques, vol.MTT-18, no.1, JANUARY 1970. [17] W. T. WEEKS, L. L. WU, M. F. MCALLISTER, and A. SINGH. Resistance And Inductance Skin Effect In Rectangular Conductors. IBM Journal of Research and Development, vol.23, no.6, NOVEMBER 1979. [18] J. F. WHITTAKER, T. B. NORRIS, G. MOUROU, and T. Y. HSIANG. Pulse Dispersion And Shaping In Microstrip Lines. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-35, no.1, JANUARY 1987. [19] D. ZHOU and J. R. MARTI. A New Algorithm For The Capacitance Matrix Of Microstrip Lines In VLSI. BALLISTICS SIMULATION and SIMULATION WORK AND PROGRESS, Proceedings of the SCS Simulation MultiConference, New Orelean, USA, pages 60--65, APRIL 1991. [20] D. ZHOU and J. R. MARTI. The Application Of Linear Current Distribution Technique In Pipe-Type Cables. Power Systems 73 And Engineering, Proceedings of the _TASTED International Conference, Vancouver, Canada, pages 159--163, AUGUST 1992. 74 Chapter 3 Simulations Using LCST and Other Methods With the algorithm developed in the previous chapter, the frequency dependent resistance and inductance matrices are obtained for simulations of microstrip lines and pipe-type cables. Section 3.1 of this chapter will show results and comparisons for the case of microstrip lines. Section 3.2 will present the solution results for the case of pipe-type cables. Conclusions will be made in section 3.3. Finally, a summary will be outlined in Section 3.4. 3.1 SOLUTIONS FOR MICROSTRIP LINES The multi-conductor microstrip lines in the test cases have conductor dimensions of 30pm width, 2/fm thickness, and 2pm dielectric height with 20/nn linespace. The conductors location is shown in Fig.2.2. 3.1.1 Current Distribution from Different Subdivisions under Uniform Distributed Current Assumption For the same applied voltages, different current distributions in the subconductors are obtained when using different number of subdivisions. As shown in Fig.3.1(a) and (b), the distribution of the magnitude of the current density using 25x 3 (25 on width, 3 on thickness) subdivisions is quite different from that with 5x 3 75 subdivisions. The phase angles are also quite different. However, the ratio of maximum difference to maximum value in densities is approximately the same, 83% to 85%. The main effect of a larger number of subdivisions is to improve the current continuity at the edges of the cells. The larger density values with larger number of subdivisions are due to smaller cross-section areas. On the other hand, the total current carried in a line conductor will not be the same with a different number of subdivisions, because of the mutual inductances and resistances. Appendix I demonstrates this point. From the line parameter point of view, it is obvious that the values of current obtained with a larger number of subdivisions are more significant with a smaller number of subdivisions. The calculated results of current densities are consistent with those from references [1, 2, 3]. 3.1.2 Inductance Correction Factors Inductance correction factors for the two-conductor lines in Fig.1.1(a) are formulated as exponential or linear functions of frequency in Table 3.1. Tables 3.2 and 3.3 showe the values of the inductance correction factors at different frequencies. At 10-6, the correction factors are ones and zeros. The factors are calculated from eqns.(2.7.71)-(2.7.72) in Chapter 2. 76 The headings in Tables 3 .1 to 3 .3 are shown as KR, K1, k1 k , , k2, k, k3,k3' ,k4 and CI, according to the following relationships 1C7e = 1)14 ^ (k1 • gii^Agii) j(k2 • bii^k2' • Abij)} (3.1.1) = jrn{ ^ (3.1.2) and u (k3 • gii 14• Agii) j(k4 • bii^Abii)}. It can be seen from the tables that the inductance correction factors are complex numbers, and that they are dependent on both frequency and geometric configuration. 3.1.3 Solution and Discussion of Line Resistances and Inductances — LCST vs. Traditional Subconductor Method Tables 3.4 and 3.5 display the resistances and inductances obtained from LCST and from the traditional subconductor method. These solutions were found for the microstrip lines in Fig.1.1(a) at both 10-6 and 5x109 Hertz. Table 3.4 shows the results of the two-conductor case, while Table 3.5 displays the results of the three-conductor case. The results in these tables show that the self and mutual resistances increase as frequency increases, and the self and mutual inductances decrease with frequency. LCST makes a great difference from the traditional subconductor method in the required number of subdivisions. For the two conductor lines, the number of subdivisions from LCST was 60 at both 10-6 Hertz and 5x109 Hertz. Yet with the traditional subconductor 77 method for a similar accuracy, 540 subconductors were needed at 5x10 9 Hertz. The resulting difference in resistances and inductances is less than 3.72% in average. For the three conductor lines, 90 subconductors were used by LCST, compared to 1620 subconductors required by the traditional method. The average difference in the results was about 3.58%. The overall differences, including both the two-conductor case and the three-conductor case, averages 3.65%. In terms of CPU time and memory storage, the new LCST technique also results in large gains. Table 3.6 lists the CPU time costs and memory storage requirements for the calculation of the microstrip line impedances. Taking the two-conductor line as an example, the memory requirement by LCST was 1.83 Kbytes with double precisions reals, compared to 6595 Kbytes needed by the traditional subconductor method. The resultant savings are 97% to 98%. 78 AMPLITUDE in LINES from 0.6322 to 3.7326 (1.e4) ANGLE in LINES from -1.4212 to -1.5855 (rad) (a) Current distribution with 5x3 subdivisions; Fig. 3.1 Current Distributions with Different Number of Subdivisions. 79 AMPLITUDE in LINES from 0.3076 to 2.0625 (xl.e5) ANGLE in LINES from -0.4281 to -1.4331 (rad) (b) Current distribution with 25x3 subdivisions; (Cont.) Fig. 3.1 Current Distributions with Different Number of Subdivisions. 80 (c) Contours of current distribution with 25x3 subdivisions; (Cont.) Fig. 3.1 Current Distributions with Different Number of Subdivisions. 81 For the three-conductor microstrip line, at 5x109 hertz, ,4.10 Kbytes were used by LCST, while up to ,,146 Kbytes were required by the traditional method. Savings of 97.2% were obtained. At 10-6 hertz, the savings were about 96%. As frequency increases, the number of subconductors has to be increased to account for the finer changes in current distribution. However, the increase needed with LCST is much less than with the traditional method. Zero increase is found for our cases as frequency goes from 10-6 Hertz to 5 gigahertz. While using the traditional method, rs,89%-95% extra subconductors had to be supplied for accurate solutions at the higher frequencies. As to the CPU time costs (on a SUN UNIX system, OS Sun4), the savings of LCST were also quite significant. At 5 gigahertz using the traditional subconductor method, minutes (268,960ms) of CPU time were needed for the two-conductor case. For the threeconductor case, CPU time was about r,,14 minutes (824,610ms). With LCST, the CPU time costs were only from rs,3 seconds (3190ms) to ,•,10 seconds (9,709ms ) in those instances. This corresponds to savings of 97%-99%. Therefore, the CPU time used for the extra manipulations in LCST was fully compensated by the reduction in the number of operation and of memory requirements. To summarize, LCST is efficient, accurate and practical for frequency dependent line parameters of microstrip lines. 82 ^ Table 3.1 Impedance Correction Factors as Functions of Frequency (a) For the real part of the impedance, Index IC22 KT ki k k2 k; Same Conductor 7.172800 • 10' • f + 1.0 8.98900 • 10" • f + 1.0 4.077012. 10 ' f + 1.0 4.277012. 10 ' • f^ — 4.277012 10—" 1.906484 - 10' • f + 1.0 3.906484 • 10—" f — 3.906484. 10—" Different Conductors 6.20800 - 10" • 1.97800 • 10' • 1.234034 • 10' • f + 1.0 f + 1.0 f + 1.0 0.1810002. 0.9001009 • 3.523076 - 10" • i.i i=j e —f + 0.81900 e - 1 + 0.09990 f + 1.0 1.254034. 10' • f — 1.254034. 10' 5.523076 - 10-1° • f — 5.523076 10" 1.000035 • 10' - f + 1.0 1.340832^10" • • f + 1.0 1.200035 • 10' • f — 1.200035. 10" 1.341032^10-6 • • f — 1.341032 10-12 (Cont.) Table 3.1 Impedance Correction Factors as Functions of Frequency (b) For the imaginary part of the impedance, Index KR /C2- k3 k k4 Same Conductor 7.172800 • 10-1° • f+1.0 8.98900 - 10-1° • f+ 1.0 4.077012 • 10' • f+ 1.0 4.277012^10 • ' • f - 4.277012. 10-15 1.906484 - 10-1° • 1.0 Different Conductors 6.20800 • 10-11 f + 1.0 1.97800 • 10^1 f+ 1.0 1.234034 • 10" f + 1.0 1.254034^10' • • f^• - 1.254034 10-14 1.000035 • 10' • f+ 1.0 1.200035 • 10' • - 1.200035 f^ 10-15 0.1810002 • e-f + 0.81900 0.9001009 • e-f + 0.09990 3.523076 - 10-1° • f+ 1.0 5.523076 - 10 -1° • f- 5.523076 • 10' 1.340832 • 10' • f+ 1.0 1.341032 - 10 ' f- 1.341032 • 10-12 3.906484 • f - 3.906484. 10-16 i.i • =j Table 3.2 Correction Factors for Subconductors at f = 10-6 Hertz (a) For the real part of the impedance, i=j Location , , JCR ki- k1 , k1 k2 k2 KR, ki ki k1 k2 k2 Same Conductor 1 1 1 0 1 0 1 1 1 0 1 0 Different Conductors 1 1 1 0 1 0 - - - - - - (Cont.) Table 3.2 Correction Factors for Subconductors at f = 10-6 Hertz (b) For the imaginary part of the impedance, i=j i0i Location , KR, Ki- k3 k3 k4 k4 K7 Ky k3 k3 k4 k4 Same Conductor 1 1 1 0 1 0 1 1 1 0 1 0 Different Conductors 1 1 1 0 1 0 - - - - - - Table 3.3 Correction Factors for Subconductors at f = 5 x 109 Hertz (a) For the real part of the impedance, i0j Location /CR, ki ki = ki. Same Conductor 4.586400 5.494500 21.38506 Different Conductors 1.310400 1.098900 62.70169 i=j k2 = k; 1Ciz 1C1 1.953242 .819 .0999 6.000175 - - k1 = ki. k2 = k2 2.761538 6705.159 - - (Cont.) Table 3.3 Correction Factors for Subconductors at f = 5 x 109 Hertz (b) For the imaginary part of the impedance, /CR, 1C/ k3 = k3 Same 4.586400 5.494500 21.38506 Conductor Different Conductors t=j ij Location 1.310400 1.098900 62.70169 k3 = k3f k4 = k4' KR, 1.953242 .819 .0999 6.000175 - - k4 = 14 2.761538 6705.159 - - Table 3.4 Comparison of LCST with Traditional Subconductor Method Assuming Uniform Current Distribution, for the Case of Two Conductor Microstrip Lines Frequency in (Hertz) 10-6 5 x 109 Diff in %I. Linear Current Distribution (LCST) Uniform Current Distribution* R11 R12 L11 L12 R11 R12 L11 L12 4.607870 0.124537 7.129619 3.046808 4.607870 0.124537 7.129619 3.046808 18.075086 2.532505 R11 : 0.17% * Unites: R in C2/cm, L in nHkm %(R) %(L) 0.821682 0.0699571 18.036287 2.4921622 1.024858 0.0730512 R12 : 1.52% L11 : 11.0% L12 : 2.16% Table 3.5 Comparison of LCST with Traditional Subconductor Method, for the Case of Three Conductor Microstrip Lines Frequency (Hertz) Uniform Current Distribu- tion t Linear Current Distribution (LCST) R11 R12 R13 L11 L12 L13 10-6 4.602360 0.119027 0.119027 7.354618 3.261350 1.357657 5 x 109 12.578624 0.641673 0.113803 1.490643 0.0444916 0.00858456 10-6 4.602360 0.119027 0.119027 7.354618 3.261350 1.357657 5 x 109 12.902777 0.51273834 0.12479961 1.560757 0.0463622 0.00857430 R11 : 1.27% R12 LH : 2.3% L12 : 11.17% R13 2.06% L13 : 4.61% Diff. in (0 Units: R in C2/cm, L in nH/cm § %(R) = E^AL,, E ^_ %(L)— E E^, : : 0.06% Table 3.6 Comparison of LCST with Traditional Subconductor Method, in Memory Requirement and CPU Time Cost Frequency (Hertz) Lines I Savings 2 cond. , lines 5 x 10' 3 cond. lines Savings LCST Memory (Kbytes) CPU (ms) Traditional Method Memory (Kbytes) CPU (ms) 3190 1.83 268,960 93 9389 4.10 824,610 146 CPU: 98.81% -, 98.86%^I Mem: 97.19% r,-, 98.03% 3.2 SOLUTIONS FOR PIPE-TYPE CABLES LCST was also used to compute the R and L parameters of more complicated structures such as pipe-type cables. Exact analytical formulae are not available for these arrangements and approximated techniques such as the finite element method of [4] have to be used for the calculations. Pipe-type cables are formed by seven conductors. ^They contain heterogeneous structures - core, sheath and pipe, which are built with materials with different conductivity and permeability. (Physical arrangement is shown in Fig.2.3). 3.2.1 Inductance Correction Factors The inductance correction factors for pipe-type cables, Asm Arm and in eqns.(2.7.75) and (2.7.76), are shown in Table 3.8, expressed as linear functions and exponential functions of frequency. Their values at 60 and 600,000 Hertz are separately listed in Tables 3.9 and 3.10. They were calculated according to the ratios of subdivision numbers mentioned in Chapter 2, and determined by trial experiments. The coefficients of the linear interpolating functions in Table 3.8(a), KI,K2 , are determined from the equations f Kifl + K2 = Arm(f1) K1f2 + K2 = Ar.(h) (3.2.3) The coefficients of the exponential functions in Table 3.8(b), 92 K1,K2,k , are solved from Kie—kfl -I- K2 = Kie—icf2 + K2 = + K2 = 0 As.(f1) ^ Ax.(h) (3.2.4) where f is frequency and A is the inductance correction factor. As shown in Tables 3.8 through 3.10, the inductance correction factors are complex numbers to account for the influence of current distribution on the line parameters, and are dependent upon both geometric configuration and signal frequency. 3.2.2 Solution and Discussion for the Impedance Matrix — LCST vs.FEM, and LCST vs.Traditional Subconductor Method The R and L matrices at 60 Hertz and 600,000 Hertz are shown in Tables 3.10 and 3.11, respectively. In these tables, LCST corresponds to the results with the new algorithm, while FEM corresponds to those obtained with a finite element method. The results of the finite element method have been verified in another thesis work[4]. The solutions of [4] are expressed in terms of the final line impedance matrix, partitioned by conductor groups, as^indicated in Fig.2.3, A zi 1 Z12 Z13 Z14 Z15 Z16 Z21 Z22 Z23 Z24 Z15 Z16 ZAA^ZAB^ZAC Z31 Z32 Z33 Z34 Z35 Z36 ^ZBA^ZBB^ ZBC [Z]NN = Z43 Z44 Z45 Z46 Z41 Z42 ZCA^ZCB^ZCC ^1 Z51 Z52 Z53 Z54 Z55 Z56 Z61 Z62 Z63 Z64 Z65 Z66 • (3.2.5) 93 The conductors matrix [Zb\TN is symmetric with -AB = ZBC and Zcc = ZAA • The difference in the solutions with LCST and those with FEM averages less than 3%. This verifies the consistency of the two approaches. In the case of pipe-type cables, LCST also results in large savings in memory resources, as opposed to the traditional subconductor method. With LCST, the number of subdivisions was 114 (52.44 Kbytes) for the 60 Hertz parameters, and was 138 (,76.728 Kbytes) for the 600,000 Hertz parameters. Employing the traditional subconductor method, the number of subdivisions to obtain the accurate results was estimated to be 522 (1.1 Mbytes) at 60 Hertz and 618 (1.53 Mbytes) at 600,000 Hertz. The memory required by LCST was only 4.8% to 5.01% of the memory required by the traditional subconductor method. Also, as frequency increases from 60 Hertz to 600 KHz, the number of subconductors by LCST increased only 24 (2.4 Kbytes) while it had to be raised to an estimated 96 (37.248 Kbytes) for the traditional subconductor method. The CPU time consumed by the traditional method could not be exactly reported for pipe-type cable simulations, since the program using the traditional method was not fully implemented. Its cost was estimated by analogy between pipe-type cables and microstrip lines, through the relationship of CPU time cost to correspondent memory storage requirement. The CPU time cost with LCST was only 30,-,94 seconds (29,980,-,93,970 ms), compared to 1031 minutes that were required by the traditional 94 subconductor method. Similar conclusions were observed as for the microstrip line. LCST showed its efficiency and accuracy as well as its ease of use for the case of pipe-type cables. 95 Table 3:7 Impedance Correction Factors as Function of Frequency and Conductor Sequence (a) For the real part of the impedance Conductor Sequence Index (C.S.I.) 1 1 f 2.782 • 10-5 +.1285^(i 0 j) f • 2.254 • 10-4 +.1167^(i = j) 2 • 2 f• 3 1.036 - 10-5 +.0672 f • 1.296 • 10-5 +.1294^(i A j) 1 • 1.694 • 10-5 +.1292^(i = j) f• 1.298 • 10-6 +.0994 f• 8.563 • 10-6 +.0672 1• 1.848 • 10-5 +0666 f- 1• f• 1.848 • 10-5 +.0679 f• 1.298 • 10-5 +.0994 f• 1.036 . 10-5 +.0670 f • 2.904 • 10-6 +.1300^(i 0 j) f • 3.858 • 10-5 +.1279^(i = j) 1• 6.761 • 10-6 +.0673 f- 4.882 . 10-6 +.0520 f • 1.848 • 10-5 +0679 f• 1.298 • 10-5 +0994 f• 8.563 • 10-6 +.0672 1• f• 1M36 . 10-5 +.0670 1 . 5.750 . 10-6 +0519 f• 9.910 .10_6 +0782 f• 3 C 4 5 4 5 2434 - 10-6 +.0375 1.298 • 10-5 +.0994 f• 6.761 • 10" +.0673 f• 9.910 • 10-6 +.0782 4.882 • 10-6 +0520 f• 6.761 • 10-5 +.0673 f• 5.750 • 10-6 +.0519 f• 2.149 • 10-5 +.1289 1.036 .945 +.0672 f• 1.298 • 10" +0994 f• 8.563 • 10-6 +0672 f• 9.910 • 10-6 +.0782 f• 4.882 • 10-6 +.0520 f• 2.149 • 10-5 +.1289 f• 1.036 • 10-5 +.0672 f• 9.910 . 10-6 +.0782 f• 2.149 • 10" +.1289 f • 1.296 • 10-5 +.1294^(i 0 j) f. 1.848 • 10-5 f • 1.694 • 10+.0666 +.1292^(i = j) f • 2.782 • 10-5 +.1285^(i 0 j) f • 2.254. 10-5 +.1167^(i = j) 1.298 • 10-5 +.0994 f• 1.036 . 10-5 +.0670 f • 6.761 .10-6 +.0673 f• 4.882 • 10-6 +0520 f• 1.848 . 10-5 +.0679 9.910 • 10-6 +0782 f• 2A34 . 10-6 +.0375 f• 9.910 .10_6 +0782 f• 7 f. 6 7 6 1. 1.296 • 10-5 +.1294^(i. 0 j) f. 1.694 - 10-5 +.1292^(i=.0 f• 2A34 • 10-6 +0375 f • 2.219 - 10' +0945^(i 0 j) 1 - 1.932 • 10-5 +.1291^(i = j) (Cont) Table 3:7 Impedance Correction Factors as Function of Frequency and Conductor Sequence (b) For the imaginary part of the impedance Conductor Sequence Index (C.S.I.) 1 1 -9,7341+9.7404^(1^j) -56.4139- e- MI05 +56.4155^(1 = j) 2 -14.0848 - e- 60400 +14.0851 C 2 -3.9500- e- r5165 +3.9511 .6744 • e - .1-0715-60 :1_44.6770^(i $ if) -138.9083- e^600000 +138.8975^(1 = j) 3 4 -3.4615 • e- .0400 +3.4636 5 6 7 -1.2351- e- 60400 +1.2364 -34615. e- 601000 +3.4636 -1.9098 • e000000 +1.9111 - 3.7576 e 60400 +3.7591 -14.08516- e- Mint, -0.4507- e- 600000 +14.08524 +0.4518 -1.9098 . e-1,1400 +1.9111 -1.8599 e 60400 +1.8608 - 8.1237 e 00400 +8.1260 -3.4615- e- 60400 +3.4636 - 1.2351 • e miss6 +1.2364 - 3.7576. e 601000 +3.7591 14 08516^- sajou -^'^ -e +14.08524 - 0.4507. e-r0405U +0.4518 - 8.1237 e +8.1260 -3.9497. e bud000 +3.9510 - 3.7576- e 00(1.000 +3.7591 3 -3.4615 - e- 60400 +3.4636 S i -3.9500 • e- so& ° +3.9511 349811^raicw .^+34.9848^(i^j) -59.6654- e am-, +59.6667^(1 = j) -3.9497 . e- 601000 +3.9510 4 -1.9098 - e- 60400 +1.9111 -0.4507 - e-601000 +0.4518 -14.0851 • e^60400 +14.0852 -+4.4667747 600009 4c ^ +4.6770(..7,2 -138.9083 - 0-60000 +138.8975^(ti = j) 5 -3.4615 • e- rr04056 +3.4636 -1.2351- e- 601000 +1.2364 I -3.4615- e- 60000o +3.4636 f -3.9500- e- 600000 +3.9511 9 7341 e- a-LI00o o --1-9. 7404-^(1^j) -56.4139+56.4155^(1 = j) 6 -3.9516- e- 60400 +3.9527 -1.8599 - e- 601000 +1.8608 -1.9098- e- 60400 +1.9111 -0.4507. e- 60400 +0.4518 - e- 60400 +14.0852 -140851 -4.6744- e zz457■71 +4.6770 (i 9 138.9083 • e soocioc +138.8975 (1 j) - 8.1237 • e- TUZI6 +8.1260 7 -3.7576- e- areass +3.7591 e-0.05887-^60400 +0.05966 -3.7576- e-601000 +3.7591 -0.05887. e^600000 +0.05966 -37576- e-601000 +3.7591 0.05887 - e-r405 +0.05966 - 6.4729- e 60400 +6.4744 (i j) - 2.8171- e 60400 +2.8188 (i = Table 3.8 Correction Factors for Subconductors at f == 60Hz (a) For the real part of the impedance Conductor Index 1 2 3 4 5 6 7 1 .13021 .06778 .10013 .06767 .10013 .06767 .07868 C 2 .06901 .13021 .06767 .05229 .06767 .05229 .13021 3 .10013 .06767 .13021 .06778 .10013 .06767 .07868 4 .06767 .05229 .06901 .13021 .06767 .05229 .13021 5 .10013 .06767 .10013 .06767 .13021 .06778 .07868 6 .06767 .05229 .06767 .05229 .06901 .13021 .13021 7 .07868 .03763 .07868 .03763 .07868 .03763 .13021^i .09583,0i 0 a u c t o r I n d e X (Cont.) 'rade 3.8 Correction Factors for Subconductors at f == 60Hz (b) For the imaginary part of the impedance Conductor Index 1 2 3 4 5 6 7 1 .00726 .00152 .00244 .00149 .00244 .00149 .00192 C 2 0 .00173 .00310 .00149 .00112 .00149 .00112 .00317 3 c t o 4 r .00244 .00149 .00726 .00173 .00244 .00149 .00192 .00149 .00112 .00152 .00310 .00149 .00112 .00317 .00244 .00149 .00244 .00149 .00726 .00173 .00192 .00149 .00112 .00149 .00112 .00152 .00310 .00317 .00192 .00079 .00192 .00079 .00192 .00079 .00192i=1 .00221,56 3u I n 5 d e X 6 7 * Dataxfrequency are the true values. Table 3.9 Correction Factors for Subconductors at f == 600,000I/z (a) For the real part of the impedance Conductor Index 2 3 4 5 6 7 16.821,03 135.38, 6.2855 7.888 5.2047 7.888 4.1240 6.0240 11.1558 7.9031,03 10.293,-.1 11.1558 2.9812 4.1240 3.5021 13.021 7.888 6.2855 1.8727,03 23.275,=3 6.2855 7.888 5.2047 6.0240 4 4.1240 2.9812 11.1558 7.9031,01 10.2933 11.1558 2.9812 13.021 5 7.888 5.2047 7.888 6.2855 16.821iO3 135.38,=3 6.2855 6.0240 6 6.2828 3.5021 4.1240 2.9812 11.1558 7.9031i0I 10.293i=3 13.021 7 6.0240 1.4976 6.0240 1.4976 6.0240 1.4976 13.410iO3 11.719i=3 1 1 C 0 n 2 d u c t 0r 3 I 3e x (Cont.) Table 3.9 Correction Factors for Subconductors at f == 600,000Hz (b) For the imaginary part of the impedance t Conductor Index 2 3 4 5 6 7 6.1594103 35.6621 2.4980 2.1902 .78208 2.1902 1.2085 2.3768 8.9036 2.957420, 87.796,=3 8.9036 .28597 1.2085 2.1766 5.1375 2.1902 2.4980 22.116.03 37.717,=3 2.4980 2.1902 .78208 2.3768 1 1 C n° 2 d U C t 3 o r I n 4 d e x 5 1.2085 .28597 8.9036 2.9574202 87.796i=3 8.9036 .28597 5.1375 2.1902 .78208 2.1902 2.4980 6.1594,0, 35.662,=, 2.4980 2.3768 6 2.4990 2.1766 1.2085 .28597 8.9036 2.9574,0i 87.796,=, 5.1375 7 2.3768 .0380 2.3768 .0380 2.3768 .0380 4.0932i01 1.7824i=, 1- Data x frequency x 10-9 are the true values. Table 3.10 Comparisons of LCST with FEM: Resistances and Inductances at 60Hz L(mHlkm) R(C2 I km) R11 R12^R21 R22 L11 L12 L21 L22 [Z AA] LCST FEM .2844 .2856 .2599 .2642 .2599 .2642 .6691 .6663 1.0239 1.0498 .8946 .9019 .8946 .9019 .8809 .9002 [ZBB] LCST FEM .2845 .2875 .2599 .2661 .2599 .2661 .6692 .6681 1.0246 1.0428 .8955 .8948 .8955 .8948 .8798 .8932 [Z AB] LCST FEM .2543 .2513 .2518 .2513 .2518 .2513 .2519 .2513 .7194 .7003 .7251 .7003 .7131 .7003 .7046 .7003 [Z Ad LCST FEM .2543 .2446 .2511 .2446 .2511 .2446 .2519 .2446 .7189 .6115 .6965 .6115 .6985 .6115 .7045 .6115 0.79 0.76 0.76 0.32 2.48 1.86 1.71 Difference in %* ' and % in L = %inR= ^ 14; 2.07 3 . 3 CONCLUSIONS A new technique, the linear current subconductor technique (LCST), has been developed to model the skin effect and proximity effect in microstrip lines (case A) and was also adapted to compute the impedance matrix of pipe-type cables (case B). Extensive simulations were run on the SUN UNIX system (OS Sun4). The simulation solutions were compared with results from FEM and the traditional subconductor method. The results were within 3.65% and 3.0% of those from the traditional method for cases A and B, respectively. The computer resources requirements for LCST are considerably smaller than those required by the traditional subconductor method. For the frequency range considered, LCST resulted in savings of more than 95% in computer memory storage. The CPU time cost was also reduced drastically, with savings of 95% as well. As a result of the reduction in computer resources, LCST allows accurate calculations at much higher frequencies than those that are possible with the conventional approach. Moreover, as opposed to the FEM technique, the process of subdivisions in simulations is automatic, without user intervention. It can be concluded that LCST is an efficient and accurate method for the computation of frequency dependent parameters of general transmission lines. 103 Table 3.11 Comparisons of LCST with PEM: Resistances and Inductances at 600,000Hz L(rnH I km) R(-21km) RH R12 R21 R22 L11 L12 L21 L22 [Z AA] LC ST FEM 75.50 73.04 68.16 68.55 68.16 68.55 69.28 68.55 .1751 .1766 .0735 .0741 .0735 .0741 .0750 .0741 [Z BB] LCST FEM 74.23 64.42 68.16 59.93 68.16 59.93 68.72 59.94 .1700 .1682 .0735 .0657 .0735 .0657 .0755 .0657 [Z AB] LC ST FEM 33.27 35.39 36.07 35.39 36.11 35.39 35.44 35.39 .0206 .0208 .0204 .0208 .0204 .0208 .0210 .0208 [Z Ad LCST FEM 20.48 21.59 21.86 21.59 21.86 21.59 21.37 21.59 .0097 .0094 .0097 .0094 .0097 .0094 .0098 .0094 3.90 2.52 2.52 2.57 0.51 2.62 2.62 3.22 Difference in %t ' and % in L = ^ % in R =Td-IARiji LRi; 3 . 4 SUMMARY In this chapter, simulations and comparisons for the LCST technique were presented. The test cases included two-conductor and three-conductor microstrip lines, as well as a seven-conductor pipe-type cable. The test cases validated the LCST technique. 105 REFERENCES [1] M. KOBAYASHI. Longitudinal And Transverse Current Distributioin On Microstriplines And Their Closed-Form Expression. IEEE Transactions on Microwave Theory and Techniques, vo1.MTT-33, no.9, SEPTEMBER 1985. [2] M. KOBAYASHI and H. MOMOI. Longitudinal And Transverse Current Distribution On Coupled Microstrip Lines. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, no.3, MARCH 1988. [3] M. KOBAYASHI and H. MOMOI. Normalized Transverse Current Distributions Of Microstrip Lines On Anisotropic Substrates. IEEE Transactions on Microwave Theory and Techniques, vol.MTT36, no.10, OCTOBER 1988. [4] Y. YIN. CALCULATION OF FREQUENCY-DEPENDENT PARAMETERS OF UNDERGROUND POWER CABLES WITH FINITE ELEMENT METHOD. PhD thesis, University of British Columbia, JUNE 1990. 106 Chapter 4 Computation of the Capacitance and Conductance Matrices of Microstrip Lines The calculation of the series parameters of microstrip lines, resistance (R) and inductance (L) was presented in the previous chapters. In this chapter, the shunt parameters, capacitance (C) and conductance (G), will be calculated. Anew algorithm— advanced method of subareas (ANS) is developed for this purpose. The ANS technique is derived from the method of subareas[14]. The following sections will first discuss the configuration of microstrip lines and the frequency dependence of their capacitances. Then, the ANS technique will be presented. Finally, the calculation of the conductance is discussed, followed by a discussion and a summary. Simulation results are presented in the next chapter. 4 . 1 MICROSTRIP LINE CONFIGURATIONS The diagrams in Figs.4.1 and 4.2 (re-drawn from Figs.1.1 (a) and (b)) show two structures that are commonly used in microstrip lines. In Fig.4.1, the line conductors are mounted above a ground plane sandwiched by a slice of dielectric, while in Fig.4.2 they are sandwiched between two ground planes in a homogeneous dielectric. The ground plane in these figures is assumed infinite in comparison with the surface area of the line conductors. The dimensions shown in the diagrams are in pm. 107 Fig. 4.1 Typical Configuration of Micro strip Lines (from Fig.1.1). Dimensions in ,um. Fig. 4.2 Another Configuration of Microstrip Lines. Dimensions in ,um. 4 . 2 FREQUENCY DEPENDENCE Several questions are of concern regarding the frequency dependance of capacitance of microstrip lines. 1. Are the charge distribution and the capacitance frequency dependent for a constant dielectric permittivity c, as in the case of silicon? 2. Is the charge limited to surface charge only in the microstrip line system in the frequency range of interest for transient analysis? 108 3. What will happen if the dielectric permittivity were frequency dependent such as in gallium arsenide? The three questions above cover two different aspects. The first two questions deal with the electrostatic mechanism in conductors, while question 3, particularly, is related to the physics of semiinsulating dielectrics. 4.2.1 Capacitance Formed by Line Conductor and Ground Plane (With Constant E) In this section, a general solution for the frequency dependence of a conductor's capacitance will be derived first. Then a special property of the microstrip line structure (Eeff) will be discussed. Solution from Maxwell equations From electrostatics, if we answer "Yes" to question 2, the reply to question 1 will be "No". That means, if there is only surface charge on a conductor, the capacitance is independent of frequency for a constant dielectric permittivity. This is because, from electrostatic theory, capacitance is defined as the ability to accumulate charges [10]. If a conductor is an equi-potential object or there is only surface charge on a conductor, the capacitance will be constant. Line capacitance actually reflects the electrostatic property of transmission lines. In a transmission line system, there is only surface charge associated with the conductors. The proof is as follows. 109 From Maxwell's equations, V 1^ a^ •e=— = —V •j = 1 dq €^a^a dt — — (4.2.1) and V x V x H = — jaw)B (4.2.2) describe the electromagnetic relationships inside a conductor. Here c is the dielectric permittivity, a is the conductivity of the metal, q is the charge density in the conductor, w is the angular frequency, E is the vector electric field density, 1-1 is the vector magnetic field density, B is the vector magnetic flux density, J is the vector current density, and finally, t refers to time. The solution of eqn.(4.2.1) is q= (4.2.3) where qo is the initial value of the charge density inside a conductor. According to electrostatic theory, a free charge cannot exist inside a conductor [10]. From eqn.(4.2.2), one arrives at the conclusion that only surface charge will exist in the microstrip line conductors as long as the transmission line model is valid. For example, by taking the values of a and c as those of aluminium (3.717x 107(S/m) ©20°C[4] ) and silicon (Er= 10 from a range of 8-16[4]) respectively, the limit frequency of the model is much lower than the signal frequency beyond which the charges induced by signals will not only present on the conductor surface. 110 ^ Therefore, as long as the transmission line model is valid, the charge densities and the capacitance of microstrip lines will not be affected by magnetic fields. They will only be influenced by the geometry of the conductors and the physical properties of the dielectric. The capacitance can then be computed from the surface charge accumulation on the conductors. Solution from empirical formulae x10-11 9.618 ^ Capacitance vs. Frequency 0.5 ^ 1.5^2^2.5^3 f - Hz^ x1010 Fig. 4.3 Capacitance of a Single Conductor Microstrip Line with Ratio > 0.7 and Dielectric Constant Er = 4.5. -T = 3 Since microstrip line conductors contain a very small surface area compared to the ground plane, the dielectric constant for the 111 capacitance will not always be the exact physical value Er. An effective permittivity, eeff, can be defined for the structure [6, 7]. As frequency increases, seff to cr. The frequency dependence of goes from an initial value up Eeff is one of the reasons for wave dispersion on microstrip lines and causes the capacitance to change as frequency changes. However, within the frequency range of transient analysis and the range of validity of the transmission line model, the variation of Eqf is very small for silicon. For example, with 1.2pm CMOS technology in VLSI, the width to height ratio of microstrip lines W 1.2m is generally about -= 3.5417g = 3388 [1]. Its 50% dispersive h frequency is then around 5.0x 1012 Hertz the empirical formulae with Eeff (Er =4.5), calculated from and the assumption of an infinite thin line conductor[6, 7]. Beyond the frequency of 5.0x 1012 Hertz, the capacitance will begin to eventually saturate to the asymptotic solution calculated from the dielectric constant E. Below this frequency, 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 versus frequency 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 and conformal-mapping methods of references [7, 6, 15, 11, 8]. As shown 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. 112 4.8970 x10-11 Capacitance vs. Frequency 4.8965 E 4.8960 L34.8955 a:1z' C-.) 4.8950 4.8945 4.8940 0 0.5 ^^ ^ ^ ^ 1 1.5 2 2.5 3 f - Hz Fig. 4.4 Capacitance of a Single Conductor Microstrip Line with Ratio -14hL = 0.5 and Dielectric Constant Er = 4.5. In general, for the frequencies of interest for transient analysis, the capacitance of microstrip lines can be practically treated as constant for VLSI studies. 4.2.2 Capacitance With Frequency Dependant E(f) When the dielectric permittivity itself is a function of frequency, such as the E(f) associated with gallium arsenide, the 50% dispersive frequency can be located in the lower band. In this case, the variation of capacitance with frequency can not be neglected for transient analysis in microstrip lines. As frequency increases, the capacitance increases from the zero frequency value 113 to a saturated value [6, 13, 171. Furthermore, if the dielectric or substrate is anisotropic, the permittivity becomes a permittivity matrix e. Then the capacitances of the microstrip line becomes the product of an equivalent permittivity derived from e and the capacitances without the substrate, C0[3]. When the capacitance is obtained with constant e, it is very easy to incorporate c(f) and E into the final solutions. Therefore, the program of ANS is first developed for silicon substrate microstrip lines with constant E. 4.3 ANS VERSUS TRADITIONAL METHOD OF SUBAREAS In the traditional method of subareas [5, 14], the external surface of the line is initially divided into smaller subareas. Then each subarea is subdivided into increments. Based on these increments, a process of optimization is further used to match the assumed boundary conditions of electric potential and to obtain the charge densities at each end point of the subarea. The charge distribution between end points of the subareas is assumed to be linear. Consequently, the charge accumulated on each subarea can be evaluated and the charge on all subareas can then be integrated to obtain the total charge on each line conductor. From the total charge on each line, the capacitance can then be calculated. This procedure is implemented as follows. Assuming an equipotential conductor surface, the elements of the capacitance matrix {C} are 114 computed from the following equation with a set of initial voltage values: {Qi} = {CO x where Qi is the charge on line i, ^ Cij (4.3.4) is the capacitance between line i and line J, and yi is the assumed value of the potential on line j. Assuming a unity potential boundary condition 17j, the element of {C} is equal to the value of the charge Q. In this procedure, the line conductors have to be excited one at a time by the applied voltage. This process has to be repeated N(N+1) 2 times for N coupled lines. In the proposed ANS technique, the conductor's surface is also assumed equipotential. However, the subdivision is only made once instead of twice. And there is no need to integrate the surface charge on each conductor as an intermediate step to obtain the capacitance matrix (C). The Maxwell coefficients matrix of all subareas, {Pij,km}, is directly evaluated from Green functions. Then matrix reduction techniques are applied to the subareas matrix {Pij,km}. This matrix is then reduced by a process of matrix reduction or subarea "bundling"[2] to {PO, the Maxwell coefficients matrix for the full line conductors. From {vi} = {PO x {gi},^(4.3.5) the capacitance matrix WO is computed by inversion of the {TO matrix, i.e., fcial = ^ 115 (4.3.6) The following sections introduce the theory of ANS. Solution with this technique includes subdivisions, Green functions, Maxwell coefficients of subareas, and charge distribution assumption. 4 . 4 SUBDIVISIONS Since charges tend to concentrate on the sharper ends of the conductor surfaces[16], the subdivisions should be denser at the corners. Similarly to the subdivisions technique in LCST, the surface of each conductor is divided into small areas by a sinusoidal function. The coordinates defining each subarea are determined by[14] ^I X(i,j) = E x,(i, 1+ 1) + xv(i,l) — C osj • (xv(i,1 + 1) — xv(i,1))]^ . Y (i, j) =^+ 1) + yv(i,l) — Cosj • (yv(1,1 + 1) — yv(i,1))] (4.4.7) where (xvoh) points to the conductor's vertex; (i,j) means the j-th subarea of the i-th conductor; and Cosj = cos ((i — 1)7) k(1) (4.4.8) with k(1) = {nnu; 116 1 = 1,3 1 = 2,4 • (4.4.9) ^ In the above equations, nw and nt are, respectively, the number of subdivision on the line width and thickness. The penetration depth 6' can be used as a threshold in the automatic generation of the number of subdivisions. Other criteria could also be selected. Here, we use Width nw = nt = 2 X^. Thickness (4.4.10) 4 . 5 GREEN' S FUNCTIONS The Green functions are defined as the electric potential produced by a unit charge. For the configuration of Figs.4.1 and 4.2, the Green functions can be derived from the well—known method of images[14]. Respectively, ^G (X ( _ .1) k (E2 — El ')^ 1 '"--‘ , y I xi , yi) = ____ 471-€ z--,` ) 1 k=o^62 + €1 ln (x — x') 2 + (y + y' + 2kh) 2 (x — x')2 + (y — y' — 2kh)2 k+1 1 v• ( Dk+i (€2 — El)^. 47€1 ln =0 ‘ j^ kr—d (4.5.11) C2 + Ci (x — x' ) 2 + (y + i + 2(k — 1)02 (x — x1)2 + (y — y' — 2(k + 1)h)2 , and G(x,yIx' ,y1) = 1 47r€ + Sin2 1r (Y+Yi ) 2h^2h In ^ Sinh2[7(xl 2h^ sinh2[7(s1 + sin2 [7(Y-Y1 2h (4.5.12) )1 where h is the height or thickness of the dielectric; C-11€2 9 and e are , ^,, i the permittivity of vacuum or the dielectric; (x,y) s the position 117 of the source charge; and (x,y) is the location of the potential to be determined. 4.6 MAXWELL COEFFICIENTS OF SUBAREAS In calculating Maxwell coefficients of subareas fpii,kml, two integrations are involved: first the integration of the potential caused by the charges distributed in each subarea, and second the integration of the potential produced by the total charge on the surface of each conductor. The second integration is implicitly accomplished by the technique of subareas' bundling. The first integration is performed as follows. By integrating the corresponding Green functions, the potential in 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 accumulated in subarea (i,/), the 1-th subarea of the i-th conductor; is the length of the subarea (i,1); pa is the assumed linear charge density in subarea (i,/); and G is the corresponding Green function. Also, in eqn.(4.6.19), Ix' = X1 + t cos 0 yi = yii + t sin 0 in which t is an intermediate variable. 118 , (4.6.14) According to the mid-value theorem of integration, eqn. (4.6.19) becomes Iil pii(x , y) = Ai/ • G (x , y I xi' 1, yiii) • f pii(t)dt (4.6.15) =^Pii(xii 1,^G(x,Y 1 xi' 1, where { x1 = a + V • Ail cos 0 yii =^v • Ai/ sine (4.6.16) with v 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 each subarea, eqn. (4.6.21) gives ^ Pii(x,Y)= Ail. G(x,Y1xii (4.6.19) On the other hand, the integral in eqn.(4.6.19) can be evaluated by combining the trapezoidal rule with the mid-value theorem as follows: pii(x , y) = Ai/ • pii(xi' 1, i) • f G (x , y I ,^dt 1^2 =^• Aii • pii(xii, yii) • (G(x , y I x ,^G(x, y I where (4.6.20) { xi1+1 = xii + Ai/ • cos 0 = Yii 4- Ail • sine 119 (4.6.21) with the source charge in the middle of the subarea x1 = x^t' • cos 0 ^ =^t' • sin 0 (4.6.22) and t = v • Ai/ = 1 2 (4.6.23) Test cases show that the trapezoidal integration rule was better behayed 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 DISTRIBUTION The line capacitance is defined as the charge on the line with a unit value of voltage applied to the line. After a matrix inversion, the Maxwell coefficients correspond to the potentials excited by a unit charge. Therefore, with ANS, the actual values of the charge density distribution do not need to be explicitly calculated. In fact, the charge density can be assumed to be of any shape because only the source position is relevant in the integration to obtain the Maxwell coefficients of the subareas. In Fig.4.5 (a), the integral corresponds to the area under the assumed curve of charge density. If the charge density is assumed to be a linear function in the subarea, the unit charge source for the 120 Charge Density middle value ti (x',y')^t2 (a) The values of the integral under curve 1 and line 2 are equal. (Xil,Yil) (b) A subarea and its co-ordinates. Fig. 4.5 Linear Charge Density and a Schematic of Coordinates Relationship in a Subarea. Green functions will be located in the middle of the subareas. The proposed AMS technique is, therefore, very flexible in terms of the assumed form of charge distribution. Other charge distributions can easily be incorporated into the equations. Once the proper position of the point charge source has been determined in the subarea, the solution procedure remains the same as for the case 121 of linear charge distribution. 4.8 DISCUSSION In contrast with the traditional approach which uses a recursive optimization procedure to calculate the capacitances by first calculating the charge distribution and then integrating charge densities, the proposed ANS technique computes the capacitance matrix directly by inversion of a Maxwell coefficients matrix. The Maxwell coefficients matrix can be computed in a straightforward manner from Green functions. It is a shortcut to obtain the line capacitances. In particular, the bundling procedure in AMS avoids the iteration procedure by avoiding the direct calculations of charge accumulations. AMS also avoids the optimization step by directly evaluating the Maxwell coefficients instead of calculating the charge densities. This strategy bypasses the time-consuming processes of optimization, charge integration, and iteration of the traditional solution approach. Considerable savings can therefore be obtained in terms of computer memory and execution time. In addition, the Green functions used to calculate Maxwell coefficients include the microstrip line dispersion through the "images" method. This is equivalent to the mapping of the effective permittivity and it causes AMS to be simpler in algorithm mechanism than an equivalent method of moment[9]. Moreover, ANS also fully 122 accounts for the finite thickness of the line conductor, as opposed to empirical and conformal mapping methods. Finally, ANS is much simpler and easier to use than other formulations in the analysis of open boundary and dielectric interface problems, such as the structure shown in Fig.4.1. AMS can also be generalized to other structures, such as the three dimensional structure in Fig.2.2(c), with the proper Green functions[12] and with a convenient charge distribution assumption. 4.9 CONDUCTANCE MATRIX Having obtained the capacitance matrix, the conductance matrix can be expressed as G = w tan^C^ (4.9.25) where^=^is the dissipation factor or loss tangent of the dielectric, with c' +jc" = c, being the general form of the dielectric's permittivity. From eqn.(4.9.31), it can be seen that the conductance matrix G is strongly frequency dependent. The dissipation factor is determined for the properties of the materials. It changes with frequency and temperature [4]. For silicon, tan6 is very small, about 0.001 0.004 at 106 Hertz [4] and it can be assumed constant. For other semiconductors such as gallium arsenide, it may change greatly as frequency increases[13]. 123 4 . 10 SUMMARY In this chapter, we have discussed the computation of microstrip lines capacitance and conductance. The solution algorithm of ANS for the capacitance proceeds as follows: O Read in the line geometry and physical data, such as the number of 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 matrix of the subareas, O Bundle the subareas into equivalent line conductors, thus giving the Maxwell coefficients matrix of the line conductors, {PO; O Invert and {TO to obtain WO. Since the explicit determination of the charge distribution is not necessary, the process of optimization and integration for the total charges on the line conductors is avoided. The calculation is performed only once instead of N(N+1) 2 times for N coupled lines. Furthermore, ANS can be extended to other structures with different Green Functions and with any convenient assumption of charge distribution. The conductance matrix is calculated from the capacitance matrix multiplied by the product of the angular frequency and the dielectric 124 loss factor. 125 REFERENCES [1] Canadian Microeletronics Corporation. GUIDE TO THE INTEGRATED CIRCUIT IMPLEMENTATION SERVICES OF THE CANADIAN MICROELETRONICS CORPORATION, JANUARY 1987. [2] H. W. DOMMEL. EMTP REFERENCE MANUAL. University of British Columbia, 1986. [3] T. C. EDWARDS. FOUNDATIONS FOR MICROSTRIP CIRCUIT DESIGN. Chichester; New York: Wiley, 1992. [4] D. G. FINK. ELECTRONICS ENGINEERS' HANDBOOK. New York: McGrawHill, pp6--4 to pp6--39, pp8--78, 1975. [5] D. W. KAMMLER. Calculation Of Characteristic Admittances And Coupling Coefficients For Strip Transmission Lines. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-16, no.11, November 1968. [6] M. KOBAYASHI. A Dispersion Formula Satisfying Recent Requirement In Microstrip CAD. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, no.8, AUGUST 1988. [7] M. KOBAYASHI and R. TERAKADO. Accurately Approximate Formula Of Effective Filling Factor For MicrostripLine With Isotropic Substrate And Its Application To The Case With Anisotropic Substrate. 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 Simple Means For Computation Of Interconnect Distributed Capacitances And Inductances. IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, vol.CAD-11, no.4, APRIL 1992. [10] J. C. MAXWELL. A TREATISE ON ELECTRICITY AND MAGNETISM. New York: Dover Publications, 1954. 126 [11] H. B. PALMER. The Capacitance Of A Parallel-Plate Capacitor By The Schwartz-Christoffel Transformation. Transactions of the American Institute of Electrical Engineers, vol.56, MARCH 1937. [12] A. E. RUEHLI and P. A. BRENNAN. Efficient Capacitance Calculations For Three-Dimensional Multiconductor Systems. IEEE Transactions on Microwave Theory and Techniques, vol.MTT21, no.2, FEBRUARY 1973. [13] D. SHULMAN. Side Gating Of GaAs Semiconductors. Journal of Applied Physics, vol.73, no.9, September, 1992. [14] W. T. WEEKS. Calculation Of Coefficients Of Capacitance Of Multiconductor Transmission Lines In The Presence Of A Dielectric Interface . IEEE Transactions on Microwave Theory and Techniques, vol.MTT-18, no.1, JANUARY 1970. [15] H. A. WHEELER. Transmission Line Properties Of Parallel Wide Strips By A Conformal Mapping Approximation. IEEE Transactions on Microwave Theory and Techniques, vol. MTT-12, no.5, MAY 1964. [16] E. YAMASHITA. Variational Method For The Analysis Of Microstrip-Like Transmission Lines. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-16, no.8, AUGUST 1968. [17] X. ZHANG, J. FANG, K. K. MEI, and Y. LIU. Calculations Of The Dispersive Characteristics Of Microstrips By The Time-Domain Finite Difference Method. IEEE Transactions on Microwave Theory and Techniques, vol.MTT-36, no.2, FEBRUARY 1988. 127 Chapter 5 Simulations and Comparisons for ANS Calculations of microstrip line capacitances and conductances are presented in this chapter. They include the simulations with ANS (advanced method of subareas) and the results from a program using the finite element method (FEM)[3], and from A. DjordjeviC's software package (A.D.)[1]. 5.1 MICROSTRIP LINE CAPACITANCE WITH Er =1 In this section, the substrate permittivity is assumed to be that 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 —surface charge method, and FEM2 — energy method. The corresponding CPU costs are also tabulated. In addition, the savings obtained with ANS, as compared with the traditional subarea method, in a 32-line test case are presented. It can be seen in Table 5.1 that the proposed AMS method results in CPU savings of 76.2% and memory savings of 38.8% as compared to the traditional subarea method. Savings of 54.2% in CPU time were obtained when comparing AMS with the FEM techniques. 128 Table 5.1 Capacitance Values and CPU Time from AMS, FEMs, and A.D., [1] for the Line Configuration of Fig.4.1 (with Er = 1) . C(1,1) C(1,2) (Pi him) (Pi 1 Pm) AMS 5.33e-4 —9.70e-5 2040 FEM1 5.69e-4 —9.39e-5 4500 FEM2 5.87e-4 —9.80e-5 4500 A.D. 5.74e-4 —8.75e-5 — Diff. in %lof FEM1 Diff. in % of 1-EM2 6.6% 3.3% 54.2% 9.2% 0.9% 54.2% 7.1% 9.8% _ _ Diff. in % of A.D. Diff. in % of Conven. for 32 lines CPU cost (ms) CPU : 76.2% Mem : 38.8% The validity of the results with the ANS technique is verified by comparison with the results from the other techniques. Compared with FEM1, the self and mutual capacitances differ by only 6.6% and 3.3%. The differences are 9.2% for the selves and 0.9% for the mutuals when compared with those from FEM2. When compared with the solutions from the A.D. program, the differences are 7.1% and 9.8% for the selves and for the mutuals, respectively. The overall average difference is 6.15%. Cl Creference 129 All the simulations shown above were performed on a Sun UNIX system, except for the simulation from A.D., which was run on a PC. 5.2 MICROSTRIP LINE CAPACITANCE WITH Er = 10 In this section, the substrate relative permittivity Er equals 10 for the microstrip lines in Figs.4.1 and 4.2. Table 5.2 displays the simulation solutions for both configurations. For the microstrip lines of Fig.4.1, the differences in self and mutual capacitances between AMS and A.D. were 11.8% and 4.3%, respectively. While for the configuration of Fig.4.2, they were 9.38% and 0.43%. The overall average difference was 6.48%. 5.3 COMMENTS ON THE ANS TECHNIQUE As shown by the simulation results, ANS is a more efficient method as compared to FEMs and the traditional subarea technique. AMS outperforms the traditional subarea method by achieving reductions of 76% in memory requirements and 40% in CPU time. Compared to FEM techniques, CPU time saving of 54% were obtained on the Sun UNIX system. Another advantage over FEM programs in dealing with open boundary problems is that the generation of subdivisions is automatic. The accuracy of ANS in this case was confirmed by comparisons with three other simulation methods: FEM1, FEM2 and A.D. An average 130 difference of 6.31% was obtained in the solutions. 5.4 LINE CONDUCTANCE MATRIX Fro a silicon dissipation factor of 0.004[2], conductance values were calculated and are displayed in Tables 5.3 and 5.4 for 106 Hertz and 109 Hertz, respectively. Each table contains solutions for both configurations of microstrip lines. From Tables 5.3 and 5.4, it can be seen that the conductance is strongly frequency dependent. Its values are enlarged 1000 times when frequency goes from 106 to 109 Hertz, although the absolute values are very small. 5.5 CONCLUSIONS It can be concluded from the simulations that ANS is efficient and accurate. It results in CPU time savings from 76% to 54%, and memory savings of 40%. Its accuracy was confirmed by comparisons with FEM techniques and the A.D. technique. The differences were within 6.31% in average. In addition, with AMS subdivisions are generated automatically, which results in a user-friendly software. Moreover, its solution technique can be applied to a large class of microstrip line structures. For a constant tan& of the dielectric, the conductance is a 131 linear function of frequency. 5.6 SUMMARY Simulations with the new technique ANS were presented in this chapter for the capacitance matrix of microstrip lines. ANS' efficiency and accuracy were verified by comparison with FEM programs and the software package from A. Djordjevia et al. The conductance values were also calculated. ^They increase linearly as frequency increases for a constant tanbo. It is concluded that ANS is efficient and accurate. 132 Table 5.2 Results of AMS and of A. D. for the Configurations of Figs. 4.1 and 4.2 (with Er = 10). C(1,2) (p.//t/m) C(1,1) (p.filim) AMS (Fig. 4.1) 3.113e-3 -9.503e-4 A.D. (Fig. 4.1) 3.528e-3 -9.097e-4 Diff.in %2 11.8% 4.3% AMS (Fig. 4.2) A.D. (Fig. 4.2) 1.555e-2 -1.384e-7 1.716e-2 -1.390e-7 9.38% 0.43% Diff.in % Table 5.3 Results of the Conductance Matrix with Er = 10 and at f = 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 -2.3884 Fig. 4.2 39.081 -3.4784e-4 Table 5.4 Results of the Conductance Matrix with Er = 10 and at f = 109Hertz for the Configurations of Figs. 4.1 and 4.2. G(1, 1) (S x 10-8/pm) 2^ G(1,2) (S x 10-8/pm) Fig. 4.1 7.8238 -2.3884 Fig. 4.2 39.081 -3.4784e-4 % = ^IACI Creference 133 REFERENCES [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: McGrawHill, pp6--4 to pp6--39, pp8--78, 1975. [3] Y. YIN. CALCULATION OF FREQUENCY-DEPENDENT PARAMETERS OF UNDERGROUND POWER CABLES WITH FINITE ELEMENT METHOD. PhD thesis, University of British Columbia, JUNE 1990. 134 Chapter 6 Future Work The numerical techniques LCST and ANS developed in this thesis are general 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 and cable structures in power systems, and for fast networking simulations in telecommunications; 2. ANS can be extended to calculate more complicated structures in VLSI, such as the 3-Dimension structure in Fig.1.1(c). It can also be extended for microstrip lines with non-homogenous substrate, microstrip lines with dielectric with frequency dependent permittivity (e.g. in gallium arsenide), etc; and 3. The final programs can be incorporated into the EMTP. 135 Chapter 7 Conclusions New numerical techniques have been developed in this thesis for the calculation of the series (R, L) and shunt (G, C) parameters of microstrip lines in VLSI. These techniques are the linear current subconductor technique (LCST) for R and L, and the advanced method of subareas (ANS) for C. This work introduces for the first time the concept of linear interpolation for the calculation of current density distribution and charge distribution using finite subconductor and subarea techniques. This concept results in large savings in computer memory requirements and CPU times. The LCST technique 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 accurate method for the calculation of the resistance and inductance matrices of transmission lines. In comparison with the traditional subconductor method and with the finite element method, LCST resulted in an average difference in the value of the parameters of 3.3% and 3%, respectively. The memory storage requirements decreased by up to 95%99%, and the CPU time savings were above 90%; 0 LCST combines the simplicity of the traditional subconductor method with the accuracy of the finite element method. Since the subdivision procedure is automatic, the procedure results in a much more user friendly program than FEM programs; 136 O LCST is general, and it can be extended to other kinds of transmission line configurations; O ANS for capacitance is efficient and valid. It results in memory savings of 38.8% and CPU savings of 54.2%,,,76.2% as compared with the traditional method of subareas and with two programs using the finite element method. Its accuracy has also 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 can be automated and can be used for other line configurations, such as 3-D microstrip lines. ^It is especially useful for open boundary problems; and O The developed ANS program is easy to extend to accept frequency dependent dielectric constants and anisotropic dielectric permittivities. From the results obtained for the line parameters of microstrip lines in VLSI and for pipe-type cables in power systems, it can be concluded that: O The line resistance matrix is an increasing function of frequency; 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 constant versus frequency in the case of silicon substrate at the 137 frequencies of present interest; and 0 The line conductance is strongly frequency dependent although its absolute value is negligible. It is a linear function of frequency with a constant dielectric loss tangent. 138 ^ Appendix I. Current Split in Subdivided and Non-Subdivided Conductors Suppose we have a subconductor cut into two paralleled sectors, as shown in Fig.A.1. zl = rl + jcoll z2 = r2 + jw12 z=r+jcol z1 z2 z=z1//z2 Fig. IA. A subconductor in forming two systems. If a voltage V is applied to the subconductor, the current I in the un-split subconductor is /= V r+ jw/' while in the split subconductor, V^V -/-' =11+12 = ^ .^ +^. , 7'1 -Fj, uni^r2+ jwt2 = Since ^Z (ri +r2)-Fjc4/1 + 12)V (I.2) (ri +3.4.0/1)(r2-1-./0)/2) ^. 1^Z1 Z2 = Zli/Z2 = 1-1-. 1-^Z1 =^-r 1 Z2 z1^,z2 (ri + jw/i)(r2+./4.012) (ri + r2)-Fjo)(ii +12) (1.3) = r + jug, then /=/'.^ 139 (I.4) The above equation means, that without mutual parameters, the whole current in the two systems will be identical for the same voltage source. If we now assume that there is mutual resistance and inductance, the current from the un-split subconductor system will remain as in eqn.(1), while the current in the split subconductor system will be r = 11+12^ (I.5) with Z1 2 [1121^{ Z21 1 -1 [ V Z2 j^V (I . 6) Hence, (z1.2+ ^v Z1Z2 - Z12 Z21 . (1.7) This time, the currents in the two systems are not identical. 140 Appendix II. Symbols And Abbreviations 7i)o — One thousandth; — Area of the cross-section of segments; Across—se ction — A, AC a(t), — Area of line conductor cross-section; Matrix of weighting functions in frequency domain; 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; 141 C — 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; — The maximum frequency under which there is only surface fmax charge 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; 142 GMD — 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-type cables; i(t) — Current function in the time domain; intel2 — Integration values of current density on two areas of a cell; — Current matrix in the frequency domain; 1.I 3 '^3 - Current values in a subsection; — 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; 143 J The number of subdivisions along the width of the conductor — in 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 Tables 3.1-3.3 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 " ,^— Inductance between segments (kon,n)and(lcion',70 with loop 1 kmn,kn enclosed by a dummy cell; L — The number of subdivisions along the thickness of the conductors of microstrip lines, or the number of subdivisions along the azimuth direction 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 those by the 26 rule; ATI„ Ark, Arie ,^— Number of subdivision in the geometric dimensions of conductor k; NR — Number of subdivision along the radius in pipe-type cables; ▪ — Number of subdivision in the azimuth direction in pipe-type cables; 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; 145 pii(x,y) — Potential value in position (x,y) with charge excitation in 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; 146 t - 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 displacement current in the dielectric; c" - Imaginary part of the complex permittivity, reflecting the conduction 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; and T — Magnetic flux linkages of segments. SUMMARY OF REFERENCES [1] P. ARIZON and H. W. DOMMEL. Computation Of Cable Impedance Based On Subdivision Of Conductors. IEEE Transactions on Power Delivery, vol.PWRD-2, no.1, January 1987. [2] W. H. BEYER . STANDARD MATHEMATICAL TABLES. CRC Press, Florada, c1981. [3] K. G. BLACK and T. J. HIGGINS. Rigorous Determination Of The Parameters Of Microstrip Transmission Lines. IRE Transactions on Microwave Theory and Techniques, vol.MTT-3, MARCH 1955. [4] J. R. BREWS. Transmission Line Models For Lossy Waveguide Interconnections In VLSI. IEEE Trans. on ELECTRON DEVICES, vol.ED-33, no.9, SEPTEMBER 1986. [5] C. S. CHANG. Electrical Design Of Signal Lines For Multilayer Printed Circuit Boards. IBM Journal of Research and Development, vol.32, no.5, SEPTEMBER 1988. [6] F. Y. CHANG. Transient Analysis Of Lossy Transmission Lines With Arbitrary Initial Potential And Current Distributions. IEEE Transactions on Circuits and Systems — I: Fundamental Theory and Applications, vol.39, no.3, March 1992. 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Numerical methods for frequency dependent line parameters with applications to microstrip lines and pipe-type… Zhou, Dan H. 1993
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Title | Numerical methods for frequency dependent line parameters with applications to microstrip lines and pipe-type cables |
Creator |
Zhou, Dan H. |
Date Issued | 1993 |
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. |
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Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2008-09-15 |
Provider | Vancouver : University of British Columbia Library |
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. |
DOI | 10.14288/1.0065091 |
URI | http://hdl.handle.net/2429/1949 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1993-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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