@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix dc: . @prefix skos: . vivo:departmentOrSchool "Applied Science, Faculty of"@en, "Electrical and Computer Engineering, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:degreeCampus "UBCV"@en ; dcterms:creator "Kashani, Mehrdad M."@en ; dcterms:issued "2008-10-10T17:39:16Z"@en, "1993"@en ; vivo:relatedDegree "Master of Applied Science - MASc"@en ; ns0:degreeGrantor "University of British Columbia"@en ; dcterms:description """Accurate determination of frequency dependent resistance of stranded conductors in transmission lines has not been thoroughly investigated for a wide frequency range of DCto 10 MHz. The main objective of this thesis project has been to write a computer program that calculates the resistance and inductance of stranded conductors in the indicated frequency range. The fundamental idea in the implemented technique is to subdivide the cross section of each strand with circular and straight line segments forming circular and elemental shape subconductors (or elements). Then, assuming uniform current density within each area, resistance, self inductance, and mutual inductances for all subconductors are calculated. The resistances and inductances are placed into a complex impedance matrix which is then reduced through mathematical manipulations to obtain a single complex number that represents the equivalent resistance and inductance of the whole conductor. The advantage of this method is that it automatically considers the skin effect and proximity effect of all of the subconductors in the conductor. Elemental shape subdivisions are more efficient than circular or rectangular shape subdivisions. In addition, the positioning of the elements within each strand is optimized for accuracy and CPU time. The electromagnetic transients program EMTP requires the values of the line parameters in the frequency range of DC to 10 MHz, and it uses the "TUBE" approximation to calculate them. As shown in this thesis, the TUBE approximation does not provide accurate results for resistance of stranded conductors above 5 kHz. The proposed program needs an IBM compatible (80386 or above) personal computer and runs with minimum user intervention. The results obtained in this research with the Subdivision method suggest that the log(R) versus log (frequency) graph can be approximated by a straight line in its high frequency range. Slopes of graphs for conductors with different number of strands have been calculated, and an exponential formula for resistance calculations above 5 kHz is devel-oped. The resistance value at 5 kHz is found by the TUBE formula, and then using the appropriate slope, the resistance of the stranded conductor is estimated at higher frequencies."""@en ; edm:aggregatedCHO "https://circle.library.ubc.ca/rest/handle/2429/2548?expand=metadata"@en ; dcterms:extent "3926236 bytes"@en ; dc:format "application/pdf"@en ; skos:note "Frequency Dependent Impedance of StrandedConductors Using the Subdivision MethodbyMehrdad Manouchehri KashaniB.A.Sc.(E.E.) The University of British Columbia, Canada, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF ELECTRICAL ENGINEERlNGWe accept this thesis as conforming to the required standardTHE UNIVERSITY OF BRITISH COLUMBIAFebruary 1993© Mehrdad Manouchehri Kashani, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of 64C 7/7c.\\- eThe University of British ColumbiaVancouver, CanadaDate Fes 2_‘, 0,3DE-6 (2/88)AbstractAccurate determination of frequency dependent resistance of stranded conductors intransmission lines has not been thoroughly investigated for a wide frequency range of DCto 10 MHz. The main objective of this thesis project has been to write a computer pro-gram that calculates the resistance and inductance of stranded conductors in the indicatedfrequency range. The fundamental idea in the implemented technique is to subdivide thecross section of each strand with circular and straight line segments forming circular andelemental shape subconductors (or elements). Then, assuming uniform current densitywithin each area, resistance, self inductance, and mutual inductances for all subconductorsare calculated. The resistances and inductances are placed into a complex impedancematrix which is then reduced through mathematical manipulations to obtain a single com-plex number that represents the equivalent resistance and inductance of the whole con-ductor. The advantage of this method is that it automatically considers the skin effect andproximity effect of all of the subconductors in the conductor.Elemental shape subdivisions are more efficient than circular or rectangular shapesubdivisions. In addition, the positioning of the elements within each strand is optimizedfor accuracy and CPU time.The electromagnetic transients program EMTP requires the values of the line parame-ters in the frequency range of DC to 10 MHz, and it uses the \"TUBE\" approximation tocalculate them. As shown in this thesis, the TUBE approximation does not provide accu-rate results for resistance of stranded conductors above 5 kHz.The proposed program needs an IBM compatible (80386 or above) personal computerand runs with minimum user intervention.The results obtained in this research with the Subdivision method suggest that thelog(R) versus log(frequency) graph can be approximated by a straight line in its high fre-• •11quency range. Slopes of graphs for conductors with different number of strands have beencalculated, and an exponential formula for resistance calculations above 5 kHz is devel-oped. The resistance value at 5 kHz is found by the TUBE formula, and then using theappropriate slope, the resistance of the stranded conductor is estimated at higher frequen-cies.Table of ContentsAbstract^Table of Contents^ ivList of Tables viiList of Figures^ viiiDefinitions, Abbreviations, Symbols, and Units^Acknowledgment^ xiiiDedication xiv1. Introduction and Literature Search^ 11.1. Scope of the Thesis 51.2. Historical Background^ 92. Theoretical Background 112.1. Resistance Calculations^ 112.1.1. Spiraling 112.1.2. Temperature^ 122.1.3. Frequency 132.1.4. Current Magnitude^ 132.2. Inductance Calculations 142.2.1. Internal Inductance of a Solid Conductor^ 152.2.2. External Inductance^ 162.2.3. Total Inductance (or Self Inductance)^ 172.2.4. Mutual Inductance^ 182.2.5. Internal Inductance of the Entire Conductor^ 193 Program Design^ 203.1. Finding Equal Distance and Equal Angle Strand Pairs^ 223.2. Finding Symmetric Strand Pairs^ 23iv3.3. Combining Equal-Distance-Angle and Symmetric Strand Pairs^ 233.4. Equal Current Criteria for Location of Circular Subdivisions^ 243.5. GMD and GMR Calculations^ 273.5.1. Positioning Points within Elementals^ 273.5.2. GMD Calculations for Elementals Not Within the SameStrand^ 293.5.3. GMD Calculations for Elementals Within the Same Strand ^ 293.5.4. GMR Calculations^ 323.6. Using Symmetry in Finding R and L 323.7. Row-Column Subtraction and Reduction Procedures^ 334. Improvements in the Basic Program Design^ 344.1. Automatic Choice of RPoints and THETPoints^ 344.2. Automatic Choice of M and N^ 355. Results^ 396. Exponential Formula for the Resistance Based on the Subdivision Results^ 487. Conclusions^ 52References 54Appendices^ 57A. Package of Programs to Implement the Subconductors Technique for StrandedConductors^ 57A.1. Finding Strand Pairs with Equal Distances and Angles (37DIST.EXE) ^ 57A.1.1. 37DIST Sample Output^ 58A.2. 37XY Output^ 59A.3. Finding Symmetric Strand Pairs (37PAIR.EXE)^ 59A.3.1. 37PAIR Sample Output^ 60A.4. Combining Equal-Distance-Angle Pairs with Symmetric Pairs(37DSPR.EXE)^ 61VA.4.1. 37DSPR Sample Output^ 61A.5. Before Constructing the Huge Z Matrix (37HZR13.EXE)^ 62B. Technical Details of 37HZR13^ 63B.1. Locate_Points Function 63B.2. Find _ One_ GMD Function^ 63B.3. Find GMR Function 64_B.4. Find SelfGMD_Ls_And_Resistances Function^ 64B.5. The GMD and SelfUMD Matrices^ 64B.5.1. How the GM]) Matrix is Constructed^ 65B.5.2. How the SelfGMD Matrix is Constructed 66B.6. Constructing the Huge Z Matrix^ 66B.7. Find_R_L_For_HUGEZ Function 67B.8. Row-Column Subtraction and Symmetry^ 68B.9. Reducing the Huge Z Matrix and Saving the Results^ 68B.9.1. 37HZR13 Sample Output^ 69C. Analytic GM]) Formulas^ 72viList of Tables1. Characteristics of Copper Conductors, Hard Drawn, 97.3% Conductivity^ 32. Characteristics of Aluminum Cable, Steel Reinforced^ 43. Stranding Factors^ 124. Percent Conductivity, Resistivity, and Temperature Constant of ConductorMaterials^ 125. THETPoints for M =4 at Different Frequencies^ 346. RPoints for N = 2 and 3 at Different Frequencies 357. The N Variable for M = 4, 10, and 14^ 368. Case A with 4% Tolerance^ 379. Case B with 10% Tolerance 3710. Case C with 13% Tolerance^ 3811. Case D with 21% Tolerance 3812. High-Frequency Asymptotic Slopes for Different Numbers of Strands^ 5014. Contents of the GMD Matrix for M N = 4_2^ 6515. Lower Triangular SelfGMD Matrix Saved as a One Dimensional Matrix^ 6616. Sample Text Output of 37hzr13.exe^ 69viiList of Figures1. Four Basic Transmission Line Parameters^ 12. Cardinal 54/7 ACSR^ 23. Concentric Stranding 24. Circular and Elemental Shape Elements^ 55. Two Examples of Subdivided Strands 56. Equivalent Circuit of the Subdivided Conductor with Identical Return Path^ 67. Positioning R and L in the Impedance Matrix^ 78. Calculation of the Internal Inductance of a Solid Conductor^ 159. Calculation of the External Inductance of a Solid Conductor 1610. Calculation of Mutual Inductance^ 1811. A 54/7 ACSR transmission line with M=1, N=1^ 2112. A 30/7 ACSR transmission line with M=4, N=2 2213. Total Current as the Volume Under the Current Density Curve^ 2514. M_N_R_O = 4_2_1_5 for Each Strand^ 2715. M_N_R_O = 4_2_1_5 with Equal Angular Distance between THETPoints^ 2816. Position of RPoints within Elementals^ 2817. Two Elements with THETPoints =4 and RPoints = 1 for GMD Calculations^ 2918. Calculations of GMD for Elements within One Strand for M N = 8 4^ 30_^_19. Calculating the GMR of an Elemental^ 3220. Options A to D in the Automatic Mode of 37hzr13.exe for the Resistance^ 3921. Options A to D in the Automatic Mode of 37hzr13.exe for the Inductance^ 4022. Mtline and Subdivisions Method (Case A, 4% Tolerance) Results for theResistance^ 4223. Mtline and Subdivisions Method (Case A, 4% Tolerance) Results for theInductance^ 42viii24. Inductance by Subdivision-Case-A, TUBE, and 7-TUBE for a 7-StrandConductor^ 4325. Inductance by Subdivision-Case-A, TUBE, and 37-TUBE for a 37-StrandConductor^ 4326. Percent Difference between Subdivision and Galloway for the Resistance^ 4427. Percent Difference between Subdivision and Galloway for the Inductance^ 4528. Percent Difference between Subdivision and TUBE for the Resistance^ 4529. Percent Difference between Subdivision and TUBE for the Inductance^ 4630. Percent Difference between Subdivision and 37-TUBE for the Resistance^ 4631. Percent Difference between Subdivision and 37-TUBE for the Inductance^ 4732. Percent Difference between Subdivision and TUBE results of 1, 7, 19, and 37-Strand Conductors for the Resistance^ 4833. Normalized 1, 7, 19, and 37-Strand Subdivision Results for the Resistance^ 4934. Subdivision and Exponential Formula Results of a 19-Strand Conductor for theResistance^ 5135. Percentage Difference between the Subdivision and Exponential FormulaResults of 1, 7, 19, and 37-Strand Conductors for the Resistance^ 5136. Lower Triangular HugeZ Matrix^ 6637. Inside a Rectangular and a Triangular Block of HugeZ^ 6738. Geometric Mean Distance (GMD) Analytic Formula between Two Rectangles ^ 7239. Geometric Mean Distance Analytic Formula between Two Elementals withinthe Same Strand.^ 73ixDefinitions, Abbreviations, Symbols, and UnitsSymbol for \"less than or equal to.\"a^Attenuation Constant (Np/m).A^Area (m2).AAAC All Aluminum Alloy Conductor.AAC All Aluminum Conductor.ACAR Aluminum Conductor Alloy Reinforced.ACSR Aluminum Conductor Steel Reinforced (see Figure 2 on page 2).Phase Constant (rad/m).• Magnetic Flux Density (Wb/m2 or T for Tesla• Capacitance (F).cmil circular mil = cross sectional area of a wire of 0.001 inch diameter (n/4 squareCPU Central Processing Unit of a computer.8^Depth of Penetration (see Skin Depth) (m).ds^Differential Area (m2 ).Depth of Penetration (see Skin Depth).Pennitivity = E0 e, (F/m).E Electric Field Intensity (V/m).Element^A hypothetical conductor within an actual conductor (the same assubconductors, filaments).Elemental^Non-circular shapes obtained by using circles and straight lines to subdividea circle (see Figure 4 on page 5).emf (or electromotive force) Rate of change of flux linkages with respect to time(Wb.turn/sec).(1)^Magnetic Flux through a surface (Wb).Xf^Frequency (Hz).Filament^A hypothetical conductor within an actual conductor (the same assubconductors, elements).Propagation Constant = a + j fl (complex).Conductance (Mho, Simens (S), or one over ohm (1/S/)).GMD Geometric Mean Distance = the geometric mean of all distances between all pointswithin two areas (m).GMR Geometric Mean Radius or geometric mean distance of an area with itself = thegeometric mean of all distances between all points within one area (m).Magnetic Field Intensity (A/m).Current (A).Current Density (A/m 2 ).Number of Magnetic Flux Linkages or Rings in a circuit = \"summation or integralof all the elements of flux multiplied by the fraction of the total current linked byeach flux element DI\" (VVb turn).Length (m).L Inductance (H for Henrys).11^Permeability = ,u0 Pr (H/m)./40^Permeability of Free Space = 47E10-7 (H/m).Pr^Relative Permeability (unitless).M^Number of straight line divisions within each strand.N Number of circular divisions within each strand.• Power (W).Percent Difference between a and b: 100(difference)/(average) = 200(a-b)/(a+b).Proximity Effect^\"Tendency of the current to concentrate in the regions of theconductors where the magnetic fields that produce the currents arestrongest 121.\"xiResistivity (a.m) reciprocal of conductivity a.Resistance (a).RPoints^Points within each elemental subconductor in the radial direction.Magnetic Flux through a surface (Wb).Conductivity (S/m) reciprocal of resistivity p.Skin Depth (or Depth of Penetration) Distance from the surface towards the centre of aconductor where the current density decreases to 36.8% of its value at thesurface of the conductor (m).Skin Effect Tendency of the current to crowd towards the conductor surface asfrequency increases.Stranding Factor^A constant that the final resistance value is multiplied by to accountfor increase in the length of the conductor due to stranding (seeTable 3, page 12).Subconductor A hypothetical conductor within an actual conductor (the same aselements, filaments).Subdivision The method of dividing an area into smaller areas.THETPoints Points within each elemental subconductor in the angular direction.V^Potential (V).o.)^Radian Frequency = 27cf (rad/s).Impedance = R + jcoL (a).xiiGod is the Light of the heavens and theearth. The likeness of His Light is a Nichewherein is a Lamp: The Lamp in a glass: theglass as it were a glittering star: kindled bya blessed Tree, an Olive that is neither of theEast nor of the West, whose oil well-nighwhich shine even if no fire touched it: Lightupon Light! God guides to His Light whomHe will (Quran).AcknowledgmentThis work could not be completed without the moral and financial support of my lov-ing parents to whom I owe all my life.My appreciation is extended to my supervisor Dr. J.R. Marti who kindly contributedhis time and efforts in enhancing the scientific value of this thesis.I would like to express my gratitude to my undergraduate and graduate professors:M.P. Beddoes, D.S. Camporese, M.S. Davies, H.W. Dommel, R.W. Donaldson, W.G.Dunford, A. Ivanov, E.V. Jull, S. Kallel, M.M.Z. Kharadly, C.A. Laszlo, P.D. Lawrence,C.S. Leung, C.C.H. Ma, J.R. Marti, A.D. Moore, D.L. Pulfrey, A.C. Soudack, R.K.Ward, L.M. Wedepohl, and L. Young.Thanks are due to my friend and my brother in law Kamran, my fiancé, Minoo, and mysister, Mahtab who provided kind support during my entire graduate studies.I appreciate the efforts of my dear friend Dr. Morteza Ghomshei who provided usefulcomments during the drafting of the manuscript.Finally, I would like to thank Dr. H.W. Dommel and Dr. G.E. Howard for their carefulexamination of this thesis.DedicationThis work is dedicated to all the people who work towards the education and aware-ness of others.xiv1. Introduction and Literature SearchThe transmission line parameters per unit length are the series resistance R, series in-ductance L, shunt capacitance C, and shunt conductance G, as shown in Figure 1.z + A zFigure 1. Four Basic Transmission Line Parameters.The series resistance accounts for the ohmic (RI2) losses, and the series impedance,including resistance and inductive reactance, gives rise to series voltage drops along theline. Transient analysis in general and fault and surge propagation studies in particular, inaddition to design studies of stranded transmission line conductors, all require accurateimpedance calculations.The goal of this thesis project is to write a computer program that calculates the seriesresistance and inductance of stranded conductors (with equal strand diameters), such asACSR, AAC, AAAC, and ACAR at any given frequency.Because Aluminum has lower cost and lighter weight than copper, it has replaced cop-per as the most common conductor metal for overhead transmission lines. One of themost common conductor types is ACSR which consists of layers of aluminum strandssurrounding a central core of steel strands (see Figure 2). Each layer is spiraled in op-posite direction to its overlaying layer to hold the strands together.124 30 36 42Numberof winsoft LowFigure 2. Cardinal 54/7 ACSR [3].Stranded conductors are easier to manufacture since larger conductor sizes can beobtained by simply adding successive layers of strands. They are also easier to handle andmore flexible than solid conductors, especially for the storage of long lengths of cable onreels. The use of steel strands gives ACSR conductors a high strength to weight ratio [3].Different types of ACSR conductors for transmission lines are shown in Tables 1 and 2.The most common form of strand construction for high voltage cables is the concetricallystranded conductor shown in Figure 3.Figure 3. Concentric Stranding [8].20-3CDPr20CD•0or*,0CD0\"8\"Ca.ta)1/40c.)0<.Sea 0_Num ofConductorChoral.D. or A uirooGaornourcbOwnf„ Roustancro1011rm oat Conductor, ow 1.4•1o)_...Inductive Nactancaa;CroculoA W GetbreofIndrodualShawlsOwnowl&milkingSatortothWAGAIMoundspotCwtontConondCapacw.AWLWSat 60H.29C (77.5) 60•C (um(OhMS SW ConductorN. 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Scope of the ThesisThe main objective of this thesis is to write a computer program on an IBM compat-ible personal computer that calculates the resistance and inductance of stranded conduc-tors in transmission lines. For the calculations, the cross section of each strand is dividedinto circular and elemental shape subconductors or elements (see Figure 4).Cross Sectionof One StrandCircular^ElementalElement^ElementFigure 4. Circular and Elemental Shape Elements.Figure 5 shows two examples of subdivided strands.Figure 5. Two Examples of Subdivided Strands.Figure 6 shows how each element is considered as an independent conductor. Eachelement has its own resistance R = 1/(a A), where a = conductivity (S/m) and A = area of2hthe subconductor (m2), and self inductance Lself =^ln(^) , with ,u =^,10 = Per-27t GMR5meability, h = average height of the transmission line, and GMR = Geometric Mean Ra-dius of the elemental (GMR and GMD are defined later in the thesis).Each element has also a mutual inductance to other elements in the same strand and in, 2hall other strands utual = p k^ where GMD = Geometric Mean Distance be-2 r GMDtween two subconductors.Figure 6. Equivalent Circuit of the Subdivided Conductor with Identical Return Path.Considering the ground as a perfect conductor, the return path is an identical conductorwith depth equal to the height of the conductor. And assuming the height of the conduc-tor is much larger than the conductor diameter, there will be no coupling between the go-ing and the returning paths. The loop equations for the path of the first current andthrough the identical return path in Figure 6 is6becomes2V1 = 2 (R1 + j^+ j 2 co 42/2 + j 2 co 43/3 +... +j 2 co LininDividing both sides of the above equation by 2,= (RI + jo L11)11 + j co 42/2 + j 4313 +... +j coFrom which,=^+ j LH , Z12 =^Li2 , 113 = 0.) L13^Z1n = CO Lin .These values constitute the first row of the impedance matrix. By similar relations for the/2 to In current paths, impedance values for other rows are derived. These impedancevalues are placed into a huge complex impedance matrix [Z] (Figure 7) to form [V] = [Z][I].Figure 7. Positioning R and L in the Impedance Matrix.Then the first row of [Z] is subtracted from all other rows so that the equation7The voltage vector has now one nonzero element only. The first column of [Z'] is thensubtracted from all other columns.^This subtraction, in effect, replaces /1 by= /, + L, + I, +... +/.. and the above equation becomesCurrent /„ is now exchanged with the nth element (nth zero) of vector [V], with the resultCurrent / is now exchanged with the (n-1)th element of vector [V], and this processcontinues until the following result is obtained 141:The first row of [Z„e„, ] times the [I] vector is in effect element (1,1) of [Z,,e,„ ] times It„,which is equal to V.Impedance element (1,1) of [Z„ew ] is exactly the complex number Z that is wanted.Zi R11 ± /0/11 where R11 is the resistance per metre of the conductor, and L11 is the in-ductance per metre.8During the above procedure, the [Z] matrix is changed in each step. In the computerprogram, those changes are applied to the stored [Z] matrix step by step, and there is noneed to carry along the [V] and [I] vectors. Detailed explanations can be found in Ap-pendix B.9. on page 68.1.2. Historical BackgroundThe \"Line Constants\" support routine of the electromagnetic transients programEMTP [51 calculates the series resistance and internal inductance of overhead transmissionlines. According to Marti [6], these line parameters have to be calculated typically fromDC to 10 MIL for frequency-dependent line models.Lewis and Tuttle 171 use the same TUBE approximation as in the Line Constantsroutine to simulate the skin effect in stranded conductors. In this approximation, thealuminum or copper strands are replaced by a solid conductor, and in the case of ACSRconductors, the inner steel strands are ignored. To calculate the impedance of a solid an-nular conductor as a function of frequency, analytical skin effect formulas are used.According to Graneau [8], because no closed-form analytical solutions are availablefor impedance calculations of stranded conductors considering the skin effect, cablemanufacturers have to rely on measurements of sample conductors . Some measurementsare reported in Barrett, Nigol, Fehervari, and Findlay [9] and Morgan and Findlay[10] for stranded conductors at power frequencies, but no data on wide frequency rangemeasurements is available.Approximate formulas are presented by Malkov and Pavlov [11] for the impedanceof multiwire conductors at frequencies from 10 MHz to 104 MHz, but this is beyond therange of interest for power system studies.9Galloway, Shorrocks, and Wedepohl [12] have presented an asymptotic formula forstranded conductors, but no verification of this formula for an extended frequency rangehas been reported. In this thesis, it is shown that the Galloway's formula does not give ac-curate results for an extended frequency range, and the TUBE formula is satisfactory onlyup to 5 kHz.The concept of conductor subdivisions adopted in this work was earlier proposed byComellini, Invernizzi, and Manzoni [13] who obtained relatively good results usingcircular subconductors. The technique was also used by Weeks, Wu, McAllister, andSingh [14] for rectangular conductors with good agreement between calculated andmeasured values over a wide frequency range from 5 kHz to 500 kHz (with a maximumerror of 21% for the resistance at 500 kHz). Lucas and Talukdar [15] showed that theuse of elemental-shaped subconductors (Fig. 4 on page 5) was much more efficient thanusing simple circular filaments (for frequencies up to 100 Hz, 36 elementals give the sameaccuracy as 450 circular filaments and for frequencies up to 10 kHz, 36 elemental s givethe same accuracy as 1007 circular filaments). Arizon and Dommel [16] used circular,rectangular, and elemental shape subconductors. They also concluded that for a givenaccuracy, the elemental shape subdivision method was more efficient in terms of computerCPU time and memory requirements than the other two subdivision methods. Thiselemental shape is the one used in the program presented in this work. Wu, Kuo, andChang [17] used circular and rectangular subdivision for impedance computations ofthree dimensional multiconductor interconnection structures. Their numerical results arein good agreement with the measurement data and the available results in the literature.102. Theoretical BackgroundThe following assumptions are made in all calculations:1. The current flows longitudinally in all elements.2. The current density is constant within each element.3. The conductivity of each element is constant and in the case of ACSR conductors, thesteel strands may have a different conductivity from the aluminum strands.4. All strands have equal diameters.5. All strands have equal relative permeabilities.2.1. Resistance CalculationsThe formula for dc resistance of a solid conductor with circular cross section andlength t metres is :Rd,' p t /A, where p = resistivity (D m), t = length (m), A = cross sectional area (m2),and a = 1/p = conductivity (S/m).Rdc I t p/A (n/m) = dc resistance per metre of the conductor.Conductor resistance depends on four factors: 1. Spiraling, 2. Temperature, 3. Fre-quency (skin effect), and 4. Current Magnitude (for magnetic conductors).2.1.1. SpiralingBecause of the spiraling of the strands, the current flows through a helical path in eachstrand. This makes the strands longer and slightly increases their dc resistance (because oftough oxide coating, there is no interwire conduction). In normal resistance calculations,the helical path is ignored, and the conductor's longitudinal length and normal cross sec-tional areas of the wires are used. The final result is then multiplied by a stranding factor11that depends on the size of the conductor 181 (Table 3). In the computer program, the fi-nal resistance value is multiplied by the appropriate stranding factor. However, for com-parison purposes, in none of the graphs of the subdivision method shown in the resultssection, stranding factors are used.Table 3. Stranding Factors.No. of Wiresin Conductor7 19 37 61 91 127 169Stranding 1.145 1.054 1.028 1.017 1.011 1.008 1.006Factor2.1.2. TemperatureThe resistivity varies with temperature as1),,= pii( + T ) where PT2 and pTi are resistivities at temperatures 12 and 1'1 in °C, re-spectively, and T is a temperature constant that depends on the conductivity of the mate-rial, as listed in Table 4.Table 4. Percent Conductivity, Resistivity, and Temperature Constant of ConductorMaterials [31.Material PercentConductivityResistivity^at20 °C in C2 m x1 0-8Resistivity^at20^°C^in^nmil/ftT TemperatureConstant °CAnnealed^Cop-per100% 1.72 10.37 234.5Hard-drawnCopper97.3% 1.77 10.66 241.5Hard-DrawnAluminum61% 2.83 17.00 228.112Brass 20-27% 6.4-8.4 38-51 480Bronze 9-13% 13-18 78-108 1980Iron 17.2% 10 60 180Silver 108% 1.59 9.6 243Sodium 40% 4.3 26 207Steel 2-14% 12-88 72-530 180-980In the Subdivision program, the user has to enter the correct conductivity of aluminumand steel at the desired temperature.2.1.3. FrequencyThe AC resistance (or \"effective resistance\") of a conductor can be defined asP,Ra c = .oss ( f2), where Ploss is the conductor's real power loss in watts and I is the rms(root mean square) of the current. As frequency increases, because of the skin effect, theconductor loss increases and by the above equation, the ac resistance increases (A mathe-matical proof of this phenomena is given in Graneau [8] book). So, one expects the re-sistance of the entire stranded conductor to increase as frequency increases, and this iswhat is observed from the obtained results.2.1.4. Current MagnitudeFor magnetic conductors such as steel conductors used for shield wires, resistance de-pends on the magnitude of the current. The internal flux linkages (and therefore the lossesin the iron) depend on the current magnitude. For ACSR conductors, the steel core has arelatively low conductivity compared to aluminum strands (steel conductivity = 1.0e613(S/m), aluminum conductivity = 34.7e6 (S/m)) and therefore, the effect of the currentmagnitude on the resistance of the steel strands is small.In the computer program, the steel conductivity is used to calculate the resistance ofsteel strands and the effect of the current magnitude is not considered. In Table 2 (onpage 4) resistance values at two current levels are shown for frequencies below 60 Hz.2.2. Inductance CalculationsInduced voltage (or electromotive force) emf in a circuit is equal to the rate of changeof flux linkages with respect to time, i.e., e = emf = dA / dtwhere e = induced voltage(volts), and 2 = flux linkages (Wb.turn).The induced voltage e is also proportional to the rate of change of current with respectto time and the constant of proportionality is the inductance, i.e.,emf oc di / dt^e = L di / dt;emf = dX / dt & emf = L di / dt^dX / dt = L di / dt^L = dX / di.Inductance is the rate of change of flux linkages with respect to current. For constantpermeability p, L = X / i is constant.To find the inductance of a magnetic circuit with constant permeability, one has tofind:1. Magnetic field intensity H (A/m) from Ampere's law: -1-1 • di 7-- I enclosed •2. Magnetic flux density B = j,t H = pop, H.3. Flux linkages 2 (Wb.turn) using f ri•eg = y„, (through surface S).4. Inductance using L = X /14To derive the formula used in the inductance calculations in the computer program,the internal, external, and the total (or self) inductance of a solid cylindrical conductorhave to be calculated.2.2.1. Internal Inductance of a Solid ConductorFigure 8 shows a one metre section of a solid cylindrical conductor with radius r car-rying current I. For simplicity, the following assumptions are made:(1) the conductor is sufficiently long that the end effects are neglected,(2) the conductor is non magnetic (p = pc, = 4 7r x 10-7 Him), and(3) the conductor has a uniform current density (skin effect is neglected).Figure 8. Calculation of the Internal Inductance of a Solid Conductor.The magnetic field intensity at a distance x from the centre of the cylindrical conduc-tor can be calculated from Ampere's law as FI • di = i enclosed •The enclosed current is equal to the total current times the ratio between the area ofcircle c and the total area of the conductor. Therefore,2^( X ) 2 /7C X /2enclosed = 2^ = fcH tangdl = 2n x^= r^=^7C r 2n x^27c r Bx = 404r1 x ; kir m= f sB • ds .15The differential flux through ds = dyt = Bxlcbc =^ ld:c27rr2Since only a fraction of I is linked by the flux through ds, the differential flux linkage (12is ratio of areas times the differential flux through ds. That is,^7C X 2 \\^X2 11 X/^mie lilt^X3 niv7C rdA-^2 114W = r 2 27c r 2^— —27c^r4^T pit x3^pit2.„,t ===,^27r r^87rx=0,ut^LintLint _ 2 int I^8rp PoProxit) =^=^ (H /m) •87r^87r2.2.2. External InductanceThe external inductance of a solid conductor of radius r up to a point P at a distance Dfrom the centre of the conductor can be calculated as (Figure 9):Figure 9. Calculation of the External Inductance of a Solid Conductor..1-lx 2g x = I; x = I^ ; Bx = pHx = 111 ;• 're^enclosed ; 27r x 2 ir xfsB • ds= Wth Bedx = dyi16Here, because path c encloses the entire conductor, the differential flux linkage (d2) andthe differential flux through ds (th,v) are equal. Then,x=D^x=D B r= dig = Hedx = p dx ext f^f i,t66c^p dx =27r x^ 27-C Xx=r^x=rtill fx=D 1^,D=^ax =^ in (—),27c x=r x^27r^rand= 2\"'ext = _!_ln (D) = 1-t011r(externao t^e^27c^r^27c^r2.2.3. Total Inductance (or Self Inductance)Combining the internal and external inductances,POI-tr(int) 87c°--111-kr(ext)^in (—D) .Lex,27cThe total inductance of a solid circular conductor is given by, ,D,]=L +L^4° r(mt)^r(ext) mI\" total =L self =^–Int^ext 27r^4^)D^ I-to 4,-(w)^(e^) +In41-1r0.1)1-to^r(t)^rr(int) ln (^)=27c^41-tr(ext)^r j^27t4r(ext)^ I-to 4r(ext)^2hin (^) =^ in (^) .27r (-A,0.0)^27c^GMRre-1Since ,u,(ext) = pr(nt) = 1.0, then GMR = r e 4 .LintFor an elemental shape filament, the GMR is calculated by positioning points withinthe elemental and finding the Geometric Mean Distance between these points. This is ex-plained in section 3.5.4.17inperfect roundq.l out^q.S i outFigure 10. Calculation of Mutual Inductance.2.2.4. Mutual InductanceFigure 10 shows two independent loops with their return paths in the ground at adepth equal to the height of the conductor.Applying Ampere's law:1•et =i,„closed •27-cxH - I • H -^ • B - pH •2 C X^B• ds -^- dy = xdx = f p H xdx PIf dx ;^=^dyi ;- thru271- Xxr fp, if?t, if = 27c xx=d4tL D^in (—d) = flux linkages in loop iq, due to current in loop27CLIf^I j ln (—D) = — ln (^) (1 1 I m)t^/ 27c^d 27c^GMDwhere D= distance of conductor j to the return path of conductor i, and d= GMD be-tween conductors i and j.18Similarly, \" =^in (—D) since the distance between conductor j and the return27t^dpath of i is the same as the distance between conductor i and the return path of j (Figure10, page 18).The above formula is used in the computer program to calculate the mutual inductancebetween stranded conductors. Stevenson [18] considers the mutual inductance caused byboth the going and the returning paths, and his formula is twice this formula.2.2.5. Internal Inductance of the Entire ConductorThe inductance value provided by the program represents the total inductance of thestranded conductor. To obtain the internal inductance only, the external inductance of theentire conductor has to be subtracted from the final result. In section 2.2.2. the externalinductance of a solid conductor to a point at distance D was derived asLex,^1101-1,(ext)^(D)27cThe external inductance of the entire conductor is_ 0-1,,(ex^2ht)^(^)Lext 2TC^rtotalwhere h is the height of the conductor, and r1011 is the external radius of the strandedconductor. In the program, the external inductance is subtracted from the final value ofinductance and the result is the internal inductance of the entire conductor. However, thegraphs shown in this thesis comparing the inductance obtained with the subdivisionmethod with the results from TUBE and Galloway's formulas from the EMTP Line Con-stants program are in terms of the total inductance for a certain height.193. Program DesignThe computer program is written in the C++ language. The program considers thecase of stranded conductors in transmission lines. Figure 11 (on page 21) shows a 54/7Canary ACSR transmission line conductor with no subdivisions within each strand. Fig-ure 12 (on page 22) shows a 30/7 Eagle ACSR transmission line with 4 straight line divi-sions (M=4) and 2 circular divisions (N=2).The Zortech C++ compiler for personal computers [19] was used to compile the vari-ous C++ programs. All the *.CPP files were compiled with the X memory model. Thismodel needs an 80386 or an 80486 PC to work. The advantage of the X memory modelis that it can use up to 16 MBytes of extended memory if the program needs to. For somevery large cases (such as the M_N = 49 case), the program needs up to 13.4 MBytes ofmemory.The conductor strands (Figure 11) are numbered starting from zero in a counter-clockwise direction, and the subconductors (elements) within each strand are numberedthe same way. To refer to a specific subconductor, the notation \"element numberstrand number\" is used. For example, element number 2 of strand 12 in Figure 12 is re-ferred to as 2, 12 and element 4 of strand 36 as 4, 36.Advantage is taken of the symmetry of the strands in two ways:1. By finding strand pairs of equal distance and angle.2. By finding strand pairs that are symmetric with respect to x and y axis, and with respectto origin.20Figure 11. A 54/7 ACSR transmission line with M=1, N=1.Taking advantage of this symmetry, it results that instead of finding 666 GMD's be-tween all strand pairs in a 30/7 ACSR, only 161 GMD's are calculated, and all the othersare equal to one of these 161 values.21Figure 12. A 30/7 ACSR transmission line with M=4, N=2.3.1. Finding Equal Distance and Equal Angle Strand PairsLooking at Figure 11 on page 21, one observes that strands 1 to 6 are at radius 2r,and that their angle increases from zero to 57c/3 (radians) in 7c/3 increments. Strands 7 to18 are at radius 4r, and their angle increases from zero to 117c/6 in 7c/6 increments.Strands 19 to 36 are at radius 6r, and their angle increases from zero to 177c/9 in 7c/9 in-crements. Strands 37 to 60 are at radius 8r, and their angle increases from zero to 237c/12in 7012 increments. These strand positions are passed on to the program.22Appendix A.1. on page 57 describes the 37DIST.EXE program to exploit this sym-metry. Appendix A.1.1. on page 58 shows a sample screen output from the program.Another program, 37xy.exe, stores the strand numbers for the x, y, and origin sym-metric pairs in a binary file to be used for subsequent processing. Appendix A.2. on page59 has the screen output of this file.3.2. Finding Symmetric Strand PairsA program was written to take one pair (out of 666) at a time, and find its x, y, andorigin symmetric pairs. It creates some groups and puts the symmetric strand names inthose groups. When finding symmetric pairs, if a pair is already found in one of the previ-ous groups, that pair is ignored, and the next pair is considered. If a pair has not alreadybeen found and if it is a new pair, it is placed in a new group, or in a group with x, y, ororigin symmetry. The advantage of this program is that considering any of the 666 strandpairs in a 37-strand conductor, that pair itself, or its x, or y, or origin symmetric pair willbe the first pair of one of the above groups. These first pairs are placed in an array andsaved in a binary file on disk to be used by the 37DSPR.EXE program.Appendix A.3. on page 59 includes a detailed explanation of this program, and Ap-pendix A.3.1. on page 60 shows a sample screen output from the program.3.3. Combining Equal -Distance -Angle and Symmetric Strand PairsThe program 37DSPR.EXE takes the output from 37DIST.EXE program (the file37DIST.DAT that has all the strand pairs with equal distant and angle with each other)and the output from program 37PAIR.EXE (the file 37PAIR.DAT with all of the firststrand pairs of groups with x, y, or origin symmetry) and combines the two. For eachgroup in 37DIST.DAT, the program creates a new group, if necessary, by checking if the23pair in 37DIST.DAT is in 37PAIR.DAT or not. If it is, it puts that strand pair in a newgroup, say group el; if it is not, it ignores that pair and goes on to the next pair in37DIST.DAT. If none of the strand pairs in 37DIST.DAT is in 37PMR.DAT, the pro-gram does not do anything.The output from 37DSPR.EXE (the binary file 37DSPR.DAT) contains groups ofstrand pairs such that the pairs in each group have equal distance and angle with eachother, and because they are derived from the 37PA1R.DAT file, all other strand pairs arex, y, or origin symmetry of these pairs.The final program, 37HZR13.EXE needs to find the Geometric Mean Distance(GMD) between the first pairs of the 37DSPR.DAT groups. Due to the indicated sym-metry pre-processing, for a 37-strand conductor, 37HZR13.EXE program has to calculateonly 161 GMD's instead of 666 total values..Appendix A.4. on page 61 includes a detailed explanation of the 37DSPR.EXE pro-gram, and Appendix A.4.1. on page 61 shows a sample screen output from this program.Appendix A.5. on page 62 briefly explains what the 37HZR13.EXE program doesbefore constructing the HUGEZ matrix. Appendix B. on page 63 and its subsectionshave detailed explanations on the different functions of the 37HZR13.EXE program.3.4. Equal Current Criteria for Location of Circular SubdivisionsIt is straight forward for the 37HZR13.EXE program to divide the cross section ofeach strand by even number of straight line divisions (M): it positions each straight line at360/M degrees. But for circular divisions, there is no straight forward method.The criteria adopted to position N circles within each strand is that of equal current.First, the total current passing through the conductor is calculated, and then, the N circles24currentdensity2rare positioned such that the amount of current going trough the central circle and througheach annular section is equal to (1/N) of the total current 124The electric field intensity in a conductor is E= Eoe-a z (w '13 z)a. where a is the at-tenuation constant, 13 is the phase factor, and z is the direction of propagation. In a goodconductor (i.e., a >> co 6), a =13 = IhrAtcy = 1/5 1211, where a is the conductivity of theconductor, co = 27cf, E is the permitivity, p the permeability, and 5 is the skin depth (ordepth of penetration) equal to 1 -z^z-z— j (. t - -)^-.--In terms of the depth of penetration 5, E = E0 e 8 e^8 a, and 1E1= Eoe 6 . The-z^-zcorresponding current density is given by J = a E ; 1J1= a Eoe 8 = Joe 8 . The totalcurrent that passes through the cross section of each strand is equal to the volume createdby rotating the above current density vector 1J1 around the strand's radius axis z = r (seeFigure 13).From Figure 13 and using the method of cylindrical shells [22] to find the volume,=r^ z=rvolume = 1(2 7C radius)(height)(thickness) = lz_027c(r - z)(1J1)(dz) = to 27c(r - z)J oe 8 dz-r= total current = 2 it Jo 8 ( r + 5 e 5 -6).PIspace^conductorFigure 13. Total Current as the Volume Under the Current Density Curve.25The amount (1/n) of the total volume is confined in the region from z = 0 to z = zl (zlis between zero and r), and the radius of the circle through which (total current)/n is pass-ing will be r — zl. Therefore,-z=(1/n) total current = fzzi 27( r — z) Joe dz = 2n Jo 5 r +( zl +5 —r) e 8 —5 .z=oLetting (1/n)th of the total current be equal to the above result by combining the above-r^ -zltwo equations, one obtains: (n-1) (r — 8) — 8 e 8 + n(z1 + 8 — r) e 8 = 0.Because zl is in polynomial form as well as exponential, iteration methods have to be usedto solve for it. Using Newton's iteration formula [22], z 1 is found to be:-r( r —8)( n— 1)-8 e 5 +plzl. = zloid—2101dwhere el = e 8 , and pl = n(zlold + 6 — Wel).With three to four iterations, identical answers for z1 up to seven decimal points areobtained.After finding zl, the radius of the circle inside each strand is strand_radius — zl. Theconstant n is replaced byN/2^N/3 N/(N-4) N/(N-3) N/(N-2) NI(N-1)to find zl first. The r — z I value will then be equal toR[N-2] R[N-3] R[N-4] ^R[3]^R[2]^R[1]^R[0]respectively. For N=4, there are a total of four circles including the body of the strand.The radius of the innermost circle is R[0], and the radius of the outermost circle is R[N-1]= strand_radius.This method of positioning the circles automatically makes the distance between twoadjacent circles smaller as one moves towards the surface of the strand.n( el) — Pi5263.5. GMD and GMR CalculationsTo calculate the GMD between different elements, points are positioned within thoseelements and the geometric mean distances between those points are calculated.Figure 14 shows the cross section of one strand with M =4 (number of straight line divi-sions within each strand), N = 2 (number of circular divisions within each strand),RPoints = 1(number of points in radial direction within each elemental), and THET-Points = 4(number of points in angular direction within each elemental).Figure 14. M_N_R_O = 4_2_1_5 for Each Strand.Total number of elements within each strand is called strnd_els and is equal toM(N-1)+1. There are a total of (strnd_ els) 2 number of GMD's between any two strands.3.5.1. Positioning Points within ElementalsWithin elementals, THETPoints are positioned such that the angular distances betweenall adjacent points are equal (Figure 15).278°M_N_R_o=4_2_1_5360/M=90°9010Figure 15. M_N_R_O = 4_2_1_5 with Equal Angular Distance between THETPoints.Figure 16 shows that RPoints are positioned in each elemental such that the distancebetween any two adjacent points is the length of the elemental / RPoints. And the distancebetween the points adjacent to the edges of the elemental is the length of the elemen-tal / (2 * RPoints).13/6I3/3bi3RPoints = 3Figure 16. Position of RPoints within Elementals.Appendix B.1. on page 63 describes the function that positions points withinelements.283.5.2. GMD Calculations for Elementals Not Within the Same StrandFigure 17 shows two elements with THETPoints = 4 and RPoints = 1 for GMD cal-culations.In Appendix C on page 72, the analytic formula for GMD calculations for elementswithin the same strand is derived. Because of the complexity of the analytical formula, anumerical approach is used for positioning points within each elemental.Figure 17. Two Elements with THETPoints =4 and RPoints = 1 for GMD Calculations.In Figure 17, 16 distances are calculated and multiplied together, and the 16th root ofthe product is taken. The resultant is the GMD between the two elementals. AppendixB.2. on page 63 describes the ffinction that calculates the GMD.3.5.3. GMD Calculations for Elementals Within the Same StrandFigure 18 shows a strand with M N = 8_4. For such cases, the functionFind SelMMDs _ Ls _ And _Resistances goes through different loops, as described nextwith reference to Figure 18.29Figure 18. Calculations of GMD for Elements within One Strand for M_N = 8_4.The function Find_SelfUMDs_Ls_And_Resistances goes through one loop to assignthe GMD between 0,a-1,a to0,a-2,a^0,a-3,a^ 0,a-8,aand to assign the GMD between 0,a-9,a to0,a-10,a^0,a-11,a^ 0,a-16,aand to assign the GMD between 0,a-17,a to0,a-18,a^0,a-19,a^ 0,a-24,aIt then goes through another loop to assign the GMD between 1,a-2,a to2,a-3,a^3,a-4,a^ 7,a-8,a^8,a-1,aand to assign the GMD between 1,a-3,a to2,a-4,a^3,a-5,a^4,a-6,a^ 8,a-2,aand to assign the GMD between 1,a-4,a to2,a-5,a^3,a-6,a^4,a-7,a^ 8,a-3,aand to assign the GMD between 1,a-5,a to302,a-6,a^3,a-7,a^4,a-8,a.Within the same loop, the program repeats the above procedure for layer two andlayer three elements.If there is more than one layer (i.e. N> 2), the program goes through another loop todetermine the GMD between elements of layer p and layer p+1. For the example of Fig-ure 18, the program assigns the GMD between 1,a-9,a to2,a-10,a^3,a-11,a^4,a-12,a^ 8,a-16,aand assigns the GMD between 1,a-10,a to2,a-11,a^3,a-12,a^4,a-13,a1,a-16,a^2,a-9,a^3,a-10,aand assigns the GMD between 1,a-11,a to2,a-12,a^3,a-13,a^4,a-14,a1,a-15,a^2,a-16,a^3,a-9,aand assigns the GMD between 1,a-12,a to2,a-13,a^3,a-14,a^4,a-15,a1,a-14,a^2,a-15,a^3,a-16,aand assigns the GMD between 1,a-13,a to8,a-9,a8,a-15,a8,a-10,a8,a-14,a8,a-11,a8,a-13,a2,a-14,a 3,a-15,a 4,a-16,a 8,a-12,aAt this point, the GMD values between elements of layer one and layer two are found.The same procedures is repeated to find the GMD's for elements of layer one and layerthree and to find the GMD's of the elements of layer two and layer three.313.5.4. GMR CalculationsThe geometric Mean Radius (GMR) of an area is defined as the geometric mean dis-tance of all points within that area with respect to each other [23]. Figure 19 shows eightout of 992 distances between points within one elemental.Figure 19. Calculating the GMR of an Elemental.For P total number of points within one elemental, there will be P(P-1) number ofdistances to be calculated. Therefore, in Figure 19, there will be 32 x 31 = 992 distances.For N number of circles within each strand, the program calculates N-1 GMR's only.Appendix B.3. on page 64 describes the function that calculates the GMR.3.6. Using Symmetry in Finding R and LWhen the program constructs the impedance matrix (HUGEZ), it needs to calculatethe mutual inductance between all of the elements and place these values in the off-diago-nal locations of the matrix, and resistance and self inductance values to place in diagonallocations of HUGEZ. Appendices B.4. to B.7. on pages 64 to 67 describe functions inthe program that find the GMD of the elements within a strand, find inductances and resis-tances, and construct the GMD, SelfGMD, and HUGEZ matrices.32The resistance and self inductance values need to be calculated only for the elementsof a single strand. These values will be repeated in the other diagonal positions ofHUGEZ.Mutual inductances have to be calculated only between the elements of the strand pairslisted in the output of the 37DSPR.EXE.In addition, there is another symmetry between the elements of the strand pairs usedby the program. The rule is to find the x-symmetry of an element every time the programis finding the x-symmetry of a strand, and similarly for y-symmetry and origin symmetry.For example, looking at Figure 12 on page 22 (and remembering that the notation a,b-c,dis used to refer to the GMD between element a of strand b with element c of strand d), theGMD for 2,5-3,8 is the same as the GMD for (x-symmetry) 3,3-2,18, and as the GMD for(y-symmetry) 4,2-1,14, and as the GMD for (x-symmetry again) 1,6-4,12.3.7. Row -Column Subtraction and Reduction ProceduresIn section 1.1. on page 5, it was explained how the row-column subtraction affectsthe impedance matrix and how the reduction process is carried on. Appendices B.8. andB.9. on page 68 describe in more detail the functions that perform these tasks.334. Improvements in the Basic Program Design4.1. Automatic Choice of RPoints and THETPointsFor a 7-strand all aluminum conductor, many runs have been performed with constantM, N, and RPoints. Only the THETPoints variable (or 0 the number of points in the angu-lar direction within each elemental) is changed. For each constant M (2, 4, 6, ..., 18, 20),a series of runs at frequencies 0.01, 0.05, 0.1, 0.5, 1.0, 5, 10, 50, 60, 100, 500, 1k, 5k,10k, 50k, 100k, 500k, 1M, 5M, and 10MHz have been performed for different 0 values.All these runs were done with N = 2 and RPoints = 1. At each frequency, the 9 variablewas increased starting from one. To find the best 0 at a certain frequency, runs with con-secutive 0 values were compared with each other until the difference was less than 2%.That run is the one with the best 0. Then, all runs with all different 0 values were com-pared with the best run, and if the percent difference between a run and the best run wasless than 2%, that 0 value is good for that frequency.For example, M_N_R_O = 4_2_1_1 is compared with M_N_R_O = 4_2_1_2 for all ofthe above 20 frequencies. Then 4_2_1_2 is compared with 4_2_1_3, and 4_2_1_3 with4_2_1_4, etc. It turns out that the difference in the resistance between 4 _ _2 1 _ 10 and4 2 1 11 is less than 2% at all frequencies. Therefore, all cases are compared with_ _ _4 _ 2 _ 1 _11 to find 0 for different frequencies. The final result for the M = 4 case is shownin Table 5.Table 5. THETPoints for M = 4 at Different Freciuencies.choose 0 1 2 3 7 8for Freq Hz <=500 <=5k <=100k <=500k <=10M34Only when M changes, the criteria for choosing 0 (or THETPoints) also changes, andthis is because when M increases, the number of divisions in the angular direction also in-creases; whereas, a change in N does not affect the criteria for choosing B.For all other values of M, the same procedure explained above was performedThe above rules for choosing 0 automatically were implemented into the program, anda similar method was used to make the choice of the R (RPoints) variable automatic. Thecriteria for choosing R changes with a change in the N variable.For example, the case M_N_R = 4_2_1 was compared with the case M_N_R =4_2_2, and 4 _ 2 _ 2 with 4 _ 2 _3. Three percent was the maximum tolerance allowed forresistance values. Because the difference between 4_2_2 and 4_2_3 was less than 3%, allcases of M N = 4_2 were compared with 4_2_3. After going through the same procedurefor M N = 4_3 and M N = 4_4, the following conclusions were reached for the choice ofthe RPoints (R) variable:Table 6. RPoints for N = 2 and 3 at Different Frequencies.For N =2choose R 1 2for Freq (Hz) <= 100k <= 10MFor N = 3choose R 1 2for Freq (Hz) <= 1M <= 10MFor all larger N values, RPoints is chosen as one.4.2. Automatic Choice of M and NA program (for a 7-strand all aluminum conductor) was written that chooses theRPoints and THETPoints variables automatically based on the criteria explained in section4.1. This program requires the user to input both the M and N variables. The program35keeps the M variable constant, and it changes the N variable for all of the 20 different fre-quencies. Allowing a maximum 4% error for the resistance, a set of appropriate N valueswere derived for different frequencies for each M equal to an even number between 2 and20. Only three results are shown here, for M = 4, 10, and 14 in, Table 7.Table 7. The N Variable for M =4, 10, and 14.When M =4for Freq (Hz) <=500 <=10k <=50k <=10Mchoose N 1 3 4 5When M = 10for Freq (Hz) <=500 <=10k <=1M <=10Mchoose N 1 3 5 7When M = 14for Freq (Hz) <=500 <=10k <=10Mchoose N 1 3 4A new program was then written for a 7-strand all aluminum conductor that uses theabove rule to choose the variable N based on M. This program only requires the user toinput the M variable.The percent difference between adjacent results for resistances were obtained. Theconclusion was that M = 16 has less than 4% difference with M = 18 at all frequencies.Therefore, M = 18 was taken as the best result and all other cases of M were comparedwith M = 18 case. It was then determined what values of M should be chosen for thevarious frequency ranges.A program was then written that asks the user if he or she would like to enter M andN manually or automatically. In the manual mode, the user enters M and N. In the auto-36matic mode, the M and N variables are chosen automatically by the program, after the userenters the tolerance as shown in Tables 8 to 11.Table 8. Case A with 4% Tolerance:M_N Freq (Hz) Elements 7-strandBytes7-strandhr:min:sec37-strndBytes37-strandhr:min:sec1_1 <=500 1 872 00:00:01 19.3k 00:00:032 3 <=1k 5 12.5k 00:00:02 317k 00:01:1643 <=10k 9 38.5k 00:00:04 1.01M 00:05:326_5 <=100k 25 288k 00:00:36 7.69M 01:27:0014 4 <=500k 43 844k 00:02:46 22.7M Impossible16_4 <=10M 49 1.1M 00:04:00 29.5M ImpossibleNote that the sign \"<=\" means less than or equal to, \"Elements\" means total number ofelements within one strand, \"Bytes\" stands for minimum required extended memory, and\"hrmin: sec\" stands for run time in hours:minutes:seconds on a 80486-33MHz IBM com-patible computer. Because up to a maximum of 16 MBytes of extended memory can beused by the program, when the required memory is more than 16 MBytes, it is\"impossible\" to run that case.Table 9. Case B with 10% Tolerance:M N Freq (Hz) Elements 7-strandBytes7-strandhrmin:sec37-strndBytes37-strandhr:min: sec1^1 <=1k 1 872 00:00:01 19.3k 00:00:034 3 <=10k 9 39k 00:00:03 1.01M 00:05:314_4 <=50k 13 79k 00:00:08 2.09M 00:14:334 5 <=100k 17 134k 00:00:16 3.56M 00:30:0510 5 <=1M 41 768k 00:02:34 20.7M Impossible14 4 <=10M 43 844k 00:03:09 22.7M Impossible37Table 10. Case C with 13% Tolerance.M_N Freq (Hz) Elements 7-strandBytes7-strandhr:min:sec37-strndBytes37-strandhr:min:sec1^1 <=1k 1 872 00:00:01 19.3k 00:00:032 3 <=5k 5 12.5k 00:00:02 317k •^00:01:164 3 <=10k 9 39k 00:00:04 1.01M 00:05:314 4 <=50k 13 79k 00:00:08 2.09M 00:14:334 5 <=100k 17 134k 00:00:16 3.56M 00:30:056 6 <=500k 31 441k 00:01:54 11.9M 02:53:228 6 <=1M 41 768k 00:03:03 20.7M Impossible14 4 <=10M 43 844k 00:03:09 22.7M ImpossibleTable 11. Case D with 21% Tolerance:M_N Freq (Hz) Elements 7-strandBytes7-strandhr:min:sec37-strndBytes37-strandhr:min:sec1^1 <=1k 1 872 00:00:01 19.3k 00:00:032 3 <=10k 5 12.5k 00:00:02 317k 00:01:234 4 <=50k 13 79k 00:00:08 2.09M 00:14:334 5 <=100k 17 134k 00:00:16 3.56M 00:30:056 6 <=10M 31 441k 00:03:06 11.9M 03:14:51The programs for 1, 7, and 19-strand conductors (lhzr13.exe, 7hzr13 exe, and19hzr13.exe) give all of the 4 options to the user in the automatic mode for all frequen-cies. On the other hand, the program for a 37-strand conductor (37hzr13.exe) gives 4 op-tions to the user only for frequencies less than or equal to 100 kHz. For frequencies largerthan 100 kHz the program runs with M_N = 6_6 with no options for the user..381^111111^1^1^111111^1^1^1^11111^1^1^1^11111^1^1^1 11111^1^1^111111^1^1^111111^f^1^1^11111 ITT15. ResultsThe results of runs with the 37hzr13.exe program in the automatic mode for all fre-quencies are presented. Options A to D of the automatic mode are used. Figures 20 and21 show the results of runs with the program 37hzr13.exe for a 37 all aluminum strandedconductor for the resistance and the inductance, respectively. The strand's parameters are:strand radius = 1.72974 (mm), aluminum conductivity = 3.4662e7 (S/m), aluminumrelative permeability = 1.0, and height of the transmission line = 20 (m).3 7 —Strand Conductor (Subdivision)Resistance A—D_R10000.011E-02 1E+021E+01^1E+03Frequency (Hz)1E+04 1E+061E+05^1E+071E+001E-01—IN— A(4% tolrnc)^B(10%) C(1 3% )^D^ D(2 1 %)Figure 20. Options A to D in the Automatic Mode of 37hzr13.exe for the Resistance.Note that the Subdivision program outputs the internal inductance of the strandedconductor. This is the final result from the Subconductors matrix reduction minus the ex-ternal inductance (Lex, = —In ( 2h ), where r10(1 is the external radius of the conductor).2 7T 'totalIn the graphs presented; however, the inductance is the total inductance of the conductor39with an ideal ground return. This has been done to compare directly with the output fromthe Line Constants routine of the EMTP (Mtline version).3 7 —Strand Conductor (Subdivision)Inductance A—D_L1.675E-031.670E-03E 1.665E-031.660E-03z; 1.655E-03_2 1.650E-03-g 1.645E-031.640E-031.635E-031 E-0 2^1E+00^1E+02^1E+04^1E+061E-01^1E+01^1E+03^1E+05^1E+07Frequency (Hz)—0— A(4% tolrnc)^B(10%)^C(13%)^0^ D(21%)Figure 21. Options A to D in the Automatic Mode of 37hzr13.exe for the Inductance.Three different runs were done with the Mtline program which is part of the Microtranpackage for personal computers (UBC-EMTP version). One run was performed usingGalloway's formula (el) for the 37 all aluminum conductor described above, one using the\"TUBE\" formula for the entire 37-strand conductor (e6), and one run using the TUBEformula for each of the 37 strands in the conductor. Figures 22 and 23 compare the re-sults of case A (4% tolerance) in the automatic mode of the Subdivision method withthose of Mtline for the resistance and the inductance respectively.1/111111^111111 II^11111111^II111111^II111 111^1111^1^1 1111114037—Sfrnd 3a owcy, LB, 37 TZ,& Subdivision for Resistance el e6 el 4A_Rel (Galloway) --4— e6(TUBE)^x^ el 4(37TUBE) x^ A(Subdiv)Figure 22. Mtline and Subdivisions (Case A, 4% Tolerance) Results for the Resistance.4137—Strnd Galloway, TUBE, 37 TUBE,& Subdivision for Inductance el e6 el 4A_L1.40E-021.20E-02_Y 1.00E-028.00E-03a)(g 6.00E-034.00E-03-o— 2.00E-030.00E+00 1^1^1 11111^1^1 1 1111^I^1 1 11 111^1^1 1 1 1111^1^1^1 1 1111^1^1^1 1 1111^1^1 IIIII^1^I I 1 1111^I^1^I1 MI1E-02 1E+00 1E+02 1E+04 1E+061E-01^1E+01^1E+03^1E+05^1E+07Frequency (Hz)0^ el (Galloway) --1-- e6(TUBE)^›(^ el 4(37TUBE)^A(Subdiv)Figure 23. Mtline and Subdivisions Method (Case A, 4% Tolerance) Results for theInductance.Figure 23 shows that Galloway's formula (el) does not give correct answers for theinductance for frequencies below 500 Hz. Therefore, in Figures 24 and 25, Galloway'sresults are deleted, and only the Subdivision, TUBE and 7-TUBE (and 37-TUBE) casesfor inductance of a 7-strand (and 37-strand) conductors are shown.In Figures 21 and 25, one observes that the inductance curve does not decreasemonotonically. This is due to the fact that the number of subdivisions required for fre-quencies above 100 kHz should be greater than M_N = 66 and this was not possible be-cause the program can handle a maximum of 16 MBytes. Figure 24 shows that for the 7-strand case, where there is enough memory to run the appropriate case, there is a con-tinuous decrease in the value of the inductance.42Figure 24. Inductance by Subdivision-Case-A, TUBE, and 7-TUBE for a 7-StrandConductor.37 —Strand Subdiv, TUBE, 37 TUBEInciiirtnnrp ApRpl zLI7—Strand Subdivision, TUBE, 7 TUBEIncitictnnr.P Ap6e1 4 IFigure 25. Inductance by Subdivision-Case-A, TUBE, and 37-TUBE for a 37-StrandConductor.43Figures 26 and 27 show the percent difference (difference divided by the averagetimes 100) between the Subdivision results Case A, and Galloway's results from Mtline forthe resistance and inductance values respectively. Figures 28 and 29 show the percentdifference between the Subdivision results and the TUBE's results, and Figures 30 and 31show the percent difference between the Subdivision and the 37-TUBE case of the Mtline.37—Strand^Diff. Subdiv. 8c Galloway7. Difference in Resistance PAel _RFigure 26. Percent Difference between Subdivision and Galloway for the Resistance.4437 —Strand^Diff. Subdiv. & Galloway% Difference in Inductance PAe 1 _LFigure 27. Percent Difference between Subdivision and Galloway for the Inductance.37 —Strand % Dff. Subdiv. & TUBE7 mcfCirCirlf^Ci^PC1C;Ch-lrirsC3 P Ank Pr requency 1-1z)Figure 28. Percent Difference between Subdivision and TUBE for the Resistance.4537—Strand 70 Diff. Subdiv. & TUBE7a Difference in Inductance PAe6_LFigure 29. Percent Difference between Subdivision and TUBE for the Inductance.37—Strand 70 Diff. Subdiv. & 37 TUBE70 Difference in Resistance PAe1 4_Rr requency riz)Figure 30. Percent Difference between Subdivision and 37-TUBE for the Resistance.4637 —Strand % Diff. Subdiv. & 37 TUBE% Difference in Inductance PAel 4_LFigure 31. Percent Difference between Subdivision and 37-TUBE for the Inductance.47a)50a)0–501E-02I^1^1^11111^1^I^1 1 1 111^I^1^1^11111^1^I^1^I 1111^I^1^1 1 1111^1^1^1^11111^I^1^1^1 11111 E + 0 21 E + 0 3Frequency (Hz)1E+00 1E+061 E + 0 41E+011 E-0 1 1E+05 1 [+07200a)ITT—x— 1–Strand^-'- ^x^ 19–Strand 0^ 37–Strand6. Exponential Formula for the Resistance Based on the SubdivisionResultsFigure 32 shows that the Subdivision and TUBE results for the resistance are veryclose to each other for frequencies up to 5 kHz.Diff. 1, 7, 1 9, & 37—StrandSubdiv. & TUBE for Resistance PAalle6_RFigure 32. Percent Difference between Subdivision and TUBE results of 1, 7, 19, and 37-Strand Conductors for the Resistance.A best-fit straight-line (in a least mean squared sense 124]) can be found that passesthrough the 5 kHz point in the log(R) versus log(freq) plot of the Subdivision results inFigure 22 on page 41. One could then use the TUBE formula to calculate the value ofthe resistance at 5 kHz and use the above calculated slope to estimate the resistance athigher frequencies.To obtain the log-log slope, the Subdivision results for 1, 7, 19, and 37-strand conduc-tors are normalized by dividing the final resistance values by the stranding factor and481 1111111^1^1^1 1 1 1 11^1^1^1 1^111^1^1^1 1 1 111^1^1^1 1 1 111^I^1^1^11multiplying the result by the number of strands in the conductor. The stranding factor(Table 3, page 12) depends on the total number of strands in the conductor. In the plotspresented in this thesis, the values of the resistance obtained by the Subdivision methodare divided by the stranding factor, since this factor is not included in the TUBE and otherapproximations.Figure 33 shows the normalized 1, 7, 19, and 37 -strand results for the resistance bythe Subdivision method.1, 7, 19, 37 Strand abdiV.Normalized for Resistance nAall_R1E+041E+03co 1E+02co-06 1E+011E+001E-021E+051E+00 1E+02 1E+041E-01^1E+01^1E+03^1E+05Frequency (Hz)1E+061E+07—IN— 1 strand 7 strand^A<^ 19 strand ^ 37 strandFigure 33. Normalized 1, 7, 19, and 37-Strand Subdivision Results for the Resistance.Slopes of the four straight lines of Figure 33 are shown in Table 12.49Table 12. High-Frequency Asymptotic Slopes for Different Numbers of StrandsNumber of Strands^Slope1 0.4187 0.74819 1.0937 0.922The slope of the 37-strand conductor does not follow the increasing pattern in theslope as the previous cases do. This is probably due to the fact that there is a computermemory limitation of 16 MBytes for a 37-strand conductor, as explained in the previoussections.Figure 33 shows that the 7-strand resistance result has a slope 0.33 more than the 1-strand case, and the 19-strand case has a slope 0.34 more than the 7-strand slope. How-ever, because of the memory limitation in the 37-strand case above 100 kHz, one cannotconclude that the true slope for the 37-strand case would also be about 0.33 more than theslope of the 19-strand case.With the resistance at 5 kHz, calculated with TUBE, the resistance at higher frequen-cies can then be approximated by this exponential formula:R= k f s^ (1)where f is the frequency in Hz, S is the slope in Table 12, and k is given byk= R(at 5kHz by TUBE) 5000sFigure 34 shows the resistance obtained by the exponential formula for the 19-strandconductor and the Subdivision results. The starting point of the exponential formula is theresistance obtained by the TUBE method at 5 kHz.Figure 35 shows the percentage difference between the Subdivision and the Exponen-tial formula results for 1, 7, 19, and 37-strand conductors.501E+0611E+04100001000100101E+05 1E+07a)o• 20'(7)a)^10CY• 0—10a)1 9 —Strand Subdiv. 8c ExponentialFor Resistance Al 9 x_R1E+05^1 E±07Frequency (Hz)Subdiv.^Exponen.Figure 34. Subdivision and Exponential Formula Results of a 19-Strand Conductor for theResistance.7.Diff. 1^7, 1 9, 8c 37—Strand Subdiv. 8cExponential for Resistance PAallx_RFrequency (Hz)0^ 1—Strand^7—Strand ^ 19—Strand —>c— 37—StrandFigure 35. Percentage Difference between the Subdivision and Exponential FormulaResults of 1, 7, 19, and 37-Strand Conductors for the Resistance.517. ConclusionsFor the resistance, Galloway's formula has a maximum difference of 27% with respectto the Subdivision results for frequencies in the range [100Hz, 10kHz]. TUBE's formulagives satisfactory results in the range [DC, 5kHz] up to a maximum of 9% difference. The37-TUBE formula gives good results in the frequency range of [DC, lkHz] up to a maxi-mum 2% difference with the Subdivision results. Outside these frequency ranges, none ofthe methods (except the subdivision method) is appropriate for resistance calculations, asshown in Table 13.11d-lz 5kHz 10kHz 100kHz 1MHzGalloway -28% -24% 8% 102% 140%TUBE -10% 1% 35% 122% 154%37-TUBE 1% 43% 84% 150% 171%Table 13. Galloway, TUBE, and 37-TUBE Difference with the Subdivision Results for theResistance.For inductance, Galloway's formula has a maximum difference of 4% with respect tothe Subdivision results for frequencies in the range [100Hz, 10MHz]. TUBE's formulagives satisfactory results at all frequencies with a maximum difference of 1.3%. The 37-TUBE formula also gives good results at all frequencies up to a maximum of 0.42% dif-ference with the Subdivision results. Therefore, using one of these methods for induc-tance calculations alone would be more efficient in terms of CPU time than the subdivisionmethod.An exponential formula is also presented that satisfactorily approximates the resistanceof stranded conductors at frequencies from 5 kHz to 10 MHz. This formula is developedfor 1, 7, and 19-strand conductors of equal strand diameters. The maximum error of theformula at all frequencies in the above range was 26.5%. For frequencies less than orequal to 5 kHz, the TUBE formula provides accurate answers.52To use the exponential formula, the TUBE formula is used to calculate the resistanceof a 1, 7, or 19-strand conductor at 5 kHz. Formula (1) on page 50 with the slopes indi-cated in Table 12 on page 50 can then be used to calculate the resistance beyond thisfrequency point.53References[1] L. F. Woodruff Principles of Electric Power Transmission, Second Edition, JohnWiley and Sons, New York, p. 4, December 1962.[2] Lawrence N. Dworsky, Modern Transmission Line Theory and Applications, JohnWiley and Sons, New York, p. 97, 1979.[3] J. Duncan Glover and Mulukutla Sarma, Power System Analysis and Design WithPersonal Computer Applications, PWS-KENT Publishing Company, Boston,1989.[4] Dr. H. W. Dommel, personal communications, Department of Electrical Engineer-ing, The University of British Columbia, 2356 Main Mall, Vancouver, B.C., V6T1Z4, Canada.[5] H. W. Dommel, EMTP Theory Book, Second Edition, MicroTran Power SystemAnalysis Corporation, 4689 West 12th Ave., Vancouver, B.C., V6R 2R7 Canada,May 1992.[6] J. R. Marti, \"Accurate Modeling of Frequency-Dependent Transmission Lines inElectromagnetic Transient Simulations,\" IEEE Transactions on Power Apparatusand Systems, vol. PAS 101, no. 1, pp. 147-157, January 1982.[7] W. A. Lewis and P. D. Tuttle, \"The Resistance and Reactance of Aluminum Con-ductors, Steel Reinforced,\" AIEE (American Institute of Electrical Engineering)Transactions, vol. 77, PAS, pp. 1189-1215, 1958.[8] P. Graneau, Underground Power Transmission, New York: John Wiley & Sons,pp. 97-107, 1979.[9] J. S. Barrett, 0. Nigol, C. J. Fehervari, and R. D. Findlay, \"A New Model of ACResistance in ACSR Conductors,\" IEEE Trans. on Power Systems, vol. PWRD-1,no. 2, pp. 198-208, April 1986.[10] V. T. Morgan and R. D. Findlay, \"The Effect of Frequency on the Resistance andInternal Inductance of Bare ACSR Conductors,\" IEEE Trans. on Power Delivery,vol. 6, no. 3, pp. 1319-1326, July 1991.[11] B. V. Mal'kov, and A. A. Pavlov, \"Calculating the High-Frequency Impedance ofMultiwire Conductors,\" Soviet Electrical Engineering, Elektrotekhnika, vol. 60,no. 1, pp. 74-78, 1989.54[12] R. H. Galloway, W. B. Shorrocks, and L. M. Wedepohl, \"Calculation of ElectricalParameters for Short and Long Polyphase Transmission Lines,\" Proc. IEE, vol.111, no. 12, pp. 2051-2059, December 1964.[13] Enrico Comellini, Angelo Invernizzi, and Giancarlo Manzoni, \"A Computer Pro-gram for Determining Electrical Resistance and Reactance of any TransmissionLine,\" IEEE Trans. on Power Apparatus and Systems, vol. PAS-92, pp. 308-314,Jan./Feb. 1973.[14] W. T. Weeks, L. L. Wu, M. F. McAllister, and A. Singh, \"Resistive and InductiveSkin Effect in Rectangular Conductors,\" IBM Journal of Research and Devel-opment, vol. 23, no. 6, pp. 652-660, November 1979.[15] R. Lucas and S. Talukdar, \"Advances in Finite Element Techniques for CalculatingCable Resistances and Inductances,\" IEEE Trans. on Power Apparatus and Sys-tems, vol. PAS-97, no. 3, pp. 875-833, May/June 1978.[16] Paloma de Arizon and Hermann W. Dommel, \"Computation of Cable ImpedancesBased on Subdivision of Conductors,\" IEEE Trans. on Power Delivery, vol.PVVRD-2, no. 1, pp. 21-27, January 1987.[17] Ruey-Beei Wu, Chien-Nan Kuo, and Kwei K. Chang, \"Inductance and ResistanceComputations for Three Dimensional Multiconductor Interconnection Structures,\"IEEE Trans. on Microwave Theory and Techniques, vol. 40, no. 2, pp. 263-271,February 1992.[18] William D. Stevenson, Jr., Elements of Power System Analysis, Fourth Edition,McGraw—Hill Book Company, pp. 43-56, 1982.[19] Zortech C++ Compiler V3.0, Symantec Corporation, 10201 Tone Ave., Cuper-tino, CA, 95014, USA, 1991.[20] Dr. J.R. Marti, personal communications, Department of Electrical Engineering,The University of British Columbia, 2356 Main Mall, Vancouver, B.C., V6T 1Z4,Canada.[21] William H. Hayt, Engineering Electromagnetics, McGraw-Hill Inc., pp. 398-401,1981.[22] Robert A. Adams, Single-Variable Calculus, Revised Edition, Addison-WesleyPublishers Ltd., 1986.[23] Frederick W. Grover, Inductance Calculations Working Formulas and Tables,Second Printing, D. Van Nostrand Company, Inc., New York, pp. 17-25, May1947.55[24] Gilbert Strang, Linear Algebra And Its Applications, Third Edition, HarcourtBrace Jovanovich Publishers, pp. 153-165, 1988.56AppendicesA. Package of Programs to Implement the Subconductors Technique forStranded Conductors For the case of a 37-strand conductor, five *.exe were written. The programs are run in the follow-ing order:1. The 37DIST.EXE program finds all strand pairs that have the same distance and angle from eachother and stores the final results in an array of structures on disk as the 37dist.dat binary file.2. The 37XY.EXE program stores the symmetric strand pairs with respect to the x and y axes on diskin a binary file called 37xy.dat. The text output of this program is included in Appendix A.2., page 59.3. The 37PAIR.EXE program uses the 37xy.dat and finds all strand pairs with x or y or x—y (origin)symmetry and stores the final results as an array of strings on disk as the 37pair.dat binary file.4. The 37DSPR.EXE program uses the 37dist.dat and 37pair.dat and combines these two files in sucha way that only 161 instead of 666 GMD's have to be calculated. It stores the final result as an array ofstructures on disk in the binary file 37dspr.dat.5. The 37HZR13.EXE program uses the 37dspr.dat and 37xy.dat and creates a huge [Z] matrix. Itthen reduces this matrix to calculate the final R and L values of the conductor and stores the results in thetext file 37hzr13.out with all the important information relating to that specific run.A.1. Finding Strand Pairs with Equal Distances and Angles (37DIST.EXE)The 37DIST.EXE program is compiled for a 37-strand conductor that looks the same as in Figure 11(on page 21) but without the fourth layer of strands.In this program, an array of structures is defined such that each of the 37 elements of the array (A)contains a double precision number as the strand's magnitude and a double precession number as thestrand's angle. The magnitude (in terms of the radius) and the angle (in radians) of each strand is as-signed to each element of this array. For example, A[14].mag = 4, A[14].ang = 77r/6; and A[351. mag = 6,A[35] .ang = 167c/9.An array of structures B is defined with 666 elements (total number of possible pairs in a 37-strandconductor = 36 + 35 + 34 + ...+2 + 1 = 666). Each element of this array contains and integer as the firststrand number, an integer as the second strand number, a double (precision number) as the distance be-tween the first and the second strands, and a double precision number as the angle between the twostrands. In a loop, all 666 pairs of distances and angles are calculated and assigned to each element ofarray B. Angles smaller than 0 0001 radians (0.0057 degrees) are considered as zero, and all angles arein the [0, it) region. For example, the strand pair 5-31 and the pair 2-22 both have distance 4r and angle7t/3 (strand pairs are refereed to as \"smaller strand number — larger strand number\").An array of structures C is defined with an unknown number of elements (maximum 666) such thateach element has a double precision number as the distance and a double precision number as the anglebetween a pair of strands, an integer for the total number of pairs with equal distance and angle, and astring array that keeps the names of the pairs with equal distance and angles. The program goes througha loop starting at strand pair a—b = 0-1 then 0-2, 0-3, 0-4, ..., 0-36; 1-2, 1-3, ..., 1-36; 2-3. ; 35—36. In each iteration, the program checks if the string a—b is already stored in the string array of C. Forexample, for a—b = 0-1, there is no data in the string array of C[0], so 0-1 is put as the first string ofC[0]. Then the program compares the distance and angle of 0-1 with all other strand pairs, if it findsother pairs with equal distance and angle, it then puts them into C[0].equal_dist_array[i].57A new group is defined for each strand pair that has a different distance and angle from the previousgroups. For instance, group 1 contains the following strand pairs : 0-1, 0-4, 1-7, 2-3, 4-13, 5-6, 7-19,13-28 that have a distance 2r and an angle of 0.0 radians from each other (see Figure 11 on page 21).Some groups have one pair, such as group 269 with distance = 12r, angle = 80 degrees, and pair = 23-32.And some groups have eight pairs, such as group 2 with distance 2r, angle 60 degrees, and pairs: 0-2, 0—5, 1-6, 2-9, 3-4, 5-15, 9-22, 15-31 (see Appendix A.1.1. on page 58).After running the 37DIST.EXE program, the user realizes that there are a total of 288 groups, so thatthe size of the array of structures C (that was unknown at the beginning) should be chosen as 288.The contents of the structure array C are stored on the hard disk as the binary file \"37dist.dat.\" Thisfile will be used later by the program 37DSPR.EXE.A.1.1. 37DIST Sample OutputThe contents of the structure array C are:Group: 1 dist:^2.00r^angle:^0.00 deg, pairs:^0-1^0-4^1-7^2-3^4-135-6 7-19 13-28 total pairs = 8Group: 2 dist:^2.00r^angle:^60.00 deg,^pairs:^0-2^0-5^1-6^2-93-4^5-15 9-22 15-31 total pairs = 16Group: 3 dist:^2.00r^angle:^120.00 deg,^pairs:^0-3^0-6^1-2^3-11^4-56-17 11-25 17-34 total pairs = 24Group: 4 dist:^4.00r^angle:^0.00 deg, pairs:^0-7^0-13^1-4^1-19^4-28^9-11 15—17 total pairs = 31Group: 5 dist:^4.00r^angle:^30.00 deg, pairs:^0-8^0-14 10-12 16-18 total pairs = 35Group: 6 dist:^4.00r^angle:^60.00 deg,^pairs:^0-9^0-15^2-5^2-22^5-317-17 11-13 total pairs = 42Group: 7 dist:^4.00r^angle:^90.00 deg, pairs:^0-10^0-16^8-18 12-14 total pairs = 46Group: 8 dist:^4.00r^angle:^120.00 deg,^pairs:^0-11^0-17^3-6^3-25^6-347-9 13-15 total pairs = 53...Group: 46 dist:^4.18r^angle:^150.57 deg,^pairs:^1-36^4-27 total pairs = 184Group: 47 dist:^2.48r^angle:^6.21 deg, pairs:^2-8^5-14 total pairs = 186Group: 48 dist:^2.48r^angle:^113.79 deg,^pairs:^2-10^5-16 total pairs = 188Group: 49 dist:^4.47r^angle:^176.57 deg,^pairs:^2-12^5-18 total pairs = 190Group: 50 dist:^5.29r^angle:^19.11 deg, pairs:^2-13^3-28 5-7^6-19 11—22 17-31 total pairs = 196Group: 267^dist:^4.10r^angle:^10.00 deg, pairs:^23-25^32-34 total pairs = 629Group: 268 dist:^7.71r^angle:^30.00 deg, pairs:^23-27^32-36 total pairs = 631Group: 269^dist:^12.00r^angle:^80.00 deg,^pairs: 23-32^total pairs = 632Group: 270 dist:^11.82r^angle:^90.00 deg, pairs:^23-33^24-32 total pairs = 634Group: 271^dist:^11.28r^angle:^100.00 deg,^pairs:^23-34^25-32 total pairs = 636Group: 286^dist:^12.00r^angle:^140.00 deg,^pairs:^26-35^total pairs = 663Group: 287^dist:^11.82r^angle:^150.00 deg,^pairs:^26-36^27-35 total pairs = 665Group: 288^dist:^12.00r^angle:^160.00 deg,^pairs:^27-36 total pairs = 666Good Bye 37dist!58A.2. 37XY OutputHere is part of the screen output of 37XY.EXE (see Figure 11 on page 21).This is 37XY program that stores the strand numbers symmetric with respect to x and y axes on harddisk and shows them on the screen.Strand :0 1 2 3 4 5 6 7 8 9 10X-symmetric : 0 1 6 5 4 3 2 7 18 17 16Y-symmetric : 0 4 3 2 1 6 5 13 12 11 10Strand: 11 12 13 14 15 16 17 18 19 20X-symmetric : 15 14 13 12 11 10 9 8 19 36Y-symmetric : 9 8 7 18 17 16 15 14 28 27Strand: 21 22 23 24 25 26 27 28 29 30X-symmetric : 35 34 33 32 31 30 29 28 27 26Y-symmetric : 26 25 24 23 22 21 20 19 36 35Strand: 31 32 33 34 35 36X-symmetric : 25 24 23 22 21 20Y-symmetric : 34 33 32 31 30 29A.3. Finding Symmetric Strand Pairs (37PAIR.EXE)This program uses the 37xy.dat binary file created by 37xy.exe. It then outputs all x, y, and originsymmetric strand pairs on the screen. A structure array D is defined to store symmetric pairs of strandswith an unknown dimension at the beginning. After running 37PAIR.EXE, the dimension of D is foundto be 185. The first pair of each of the 185 symmetric groups of D is now stored in the structure array Dl,and DI is saved as the binary file 37pair.dat on disk.Program 37pair.exe finds the pairs of strands in a 37-strand conductor that are symmetric with resp-ect to the x, y, and xy axes. First, the pair i—j, where 0 <= i <= 35 and i+1 <=j <= 36 is found. If thispair is not in any of the previously found pairs of D, a new group (dgroup) in D is created and this pair isput as the first element of that group. Then, the program finds the x-symmetric pair of i—j, say a —b, andsearches to see if this pair already exists. If it is already found, the program does not do anything; and if itis not already found, the program puts this a—b pair in the same group as i—j. It then finds the y-sym-metric pair of a—b, say c—d, and it checks if c—d is already found, if not, it puts c—d in the sante group asi—j. Finally, the program gets the x-symmetric pair of c—d, say e—f, and if it is not already found, it putse—f in the i—j group. The program now goes to the next i—j and repeats the above procedure for all i andall j. At the end, it outputs all distinct groups and their elements. Obviously, in each group the, e can be amaximum of 4 pairs (like 2 - 11, 6- 15, 5- 17, 3-9) and a minimum of 1 pair like 1 -4 (see Figure 11 onpage 21). Finally, the first strand pairs from each group of D are put into the array DI.The importance of this program is that considering any pair of strands (among 666 possibility in a37-strand conductor), either that pair itself, or its x-symmetric pair, or its y-symmetric pair, or its x—y-symmetric pair will be exactly one of the pairs in the structure array DI.59A.3.1. 37PA1R Sample OutputHere is a sample screen output from the program 37PAIR.EXE.This is the 37PAIR program that finds the pairs of strands symmetric with respect to x and y axes andputs them into groups that have no common pairs. At the end all groups with their members will beshown. The miracle of this program is that any pair of strands you consider (among 666 possibility in a30/7 ACSR), either that pair itself or its x-symmetric pair of its y-symmetric pair or its x—y-symmetricpair will be exactly the FIRST pair of one of the groups in structure array D. Then we put all of the firstpairs of D into another structure array Dl and save Dl on disk to be used by 37DSPR program.Dgroup # 1 has symmetric pairs: 0-1^0-4 total pairs = 2Dgroup #2 has symmetric pairs: 0-2^0-6^0-5^0-3 total pairs = 6Dgroup #3 has symmetric pairs: 0-7 0-13 total pairs = 8Dgroup #4 has symmetric pairs: 0-8 0-180-14 0-12 total pairs = 12Dgroup # 12 has symmetric pairs: 1-2^1-6^4-5^3-4 total pairs = 40Dgroup # 13 has symmetric pairs: 1-3^1-5^4-6^2-4 total pairs = 44Dgroup # 14 has symmetric pairs:^1-4 total pairs = 45Dgroup # 15 has symmetric pairs: 1-7 4-13 total pairs = 47Dgroup #36 has symmetric pairs: 2-8 6-18 5-14 3-12 total pairs = 119Dgroup # 37 has synunetric pairs: 2-9 6-17 5-15 3-11 total pairs = 123Dgroup #38 has symmetric pairs: 2-10 6-16 5-16 3-10 total pairs = 127Dgroup #39 has symmetric pairs:^2-11 6-15 5-17 3-9 total pairs = 131Dgroup # 40 has symmetric pairs: 2-12 6-14 5-18 3-8 total pairs = 135Dgroup #41 has symmetric pairs: 2-13 6-13 5-7 3-7 total pairs = 139Dgroup #49 has symmetric pairs: 2-21 6-35 5-30 3-26 total pairs = 171Dgroup #50 has symmetric pairs:^2-22 6-34 5-31 3-25 total pairs = 175Dgroup # 51 has symmetric pairs: 2-23 6-33 5-32 3-24 total pairs = 179Dgyoup # 52 has symmetric pairs: 2-24 6-32 5-33 3-23 total pairs = 183Dgroup # 105 has symmetric pairs: 8-34 18-22 14-25 12-31 total pairs = 382Dgroup # 106 has symmetric pairs: 8-35 18-21 14-26 12-30 total pairs = 386Dgroup # 107 has symmetric pairs: 8-36 18-20 14-27 12-29 total pairs = 390Dgroup # 108 has symmetric pairs: 9-10 16-17 15-16 10-11 total pairs = 394Dgroup # 182 has symmetric pairs: 22-34 25-31 total pairs = 660Dgroup # 183 has symmetric pairs: 23-24 32-33 total pairs = 662Dgroup # 184 has symmetric pairs: 23-32 24-33 total pairs = 664Dgroup # 185 has symmetric pairs: 23-33 24-32 total pairs = 666These pairs are put into DI array 0-1 0-2 0-7 0-8 0-9 0-10 0-19 0-20 0-21 0-22 0-23 1-2 1-3 1-4^1-7 1-81-9 1-10 1-11^1-12 1-13^1-19 1-20 1-21^1-22 1-23 1-24 1-25 1-26 1-27 1-28 2-3 2-5 2-6 2-7 2-8 2-9 2-10602-11 2-12 2-13 2-142-15 2-16 2-17 2-18 2-19 2-20 2-21 2-22 2-23 2-24 2-25 2-26 2-27 2-28 2-29 2-30 2-312-32 2-33 2-34 2-35 2-36 7-8 7-9 7-10 7-11 7-12 7-13 7-19 7-20 7-21 7-22 7-23 7-24 7-25 7-26 7-27 7-288-9 8-10 8-11 8-12 8-14 8-15 8-16 8-17 8-18 8-19 8-20 8-21 8-22 8-23 8-24 8-25 8-26 8-27 8-28 8-29 8-308-31 8-32 8-33 8-34 8-35 8-36 9-10 9-11 9-15 9-16 9-17 9-19 9-20 9-21 9-22 9-23 9-24 9-25 9-26 9-27 9-289-29 9-30 9-31 9-32 9-33 9-34 9-35 9-36 10-16 10-19 10-20 10-21 10-22 10-23 10-29 10-30 10-31 10-32 19-2019-21 19-22 19-23 19-24 19-25 19-26 19-27 19-28 20-21 20-22 20-23 20-24 20-25 20-26 20-27 20-29 20-3020-31 20-32 20-33 20-34 20-35 20-36 21-22 21-23 21-24 21-25 21-26 21-30 21-31 21-32 21-33 21-34 21-3522-23 22-24 22-25 22-31 22-32 22-33 22-34 23-24 23-32 23-33A.4. Combining Equal-Distance-Angle Pairs with Symmetric Pairs(37DSPR.EXE)The program 37DSPR.EXE reads the structure array C (C contains all strand pairs with equal dis-tance and angle from each other) from the output of 37dist.exe (file 37dist.dat), and it reads the structurearray DI (D1 contains the first pair of each group in structure array D) from the output of 37pan..exe (file37pair.dat) from the hard disk. Then for each pair of each group in C, say a—b in group g, it looksthrough all pairs of Dl. If it finds a—b in Dl, it puts a—b in a new e—group, say el, in structure array E,and it records the distance between a and b in group el.Then the program looks at the second pair in group g of C, say c—d, and it searches to find if c—d is inD1 or not. If yes, it puts c—d beside a—b in group el in E, if not, it ignores c—d and goes to the next pair ofstrands in g. When all pairs of group g are done, it goes to group g+1 in C and for any pair in group g+1that is in DI, it creates a new e—group in E that is group el+1.If any pair of a group in C is not found in Dl, the program does not do anything and goes to the nextpair in C like group 15 in C (see the output of 37DSPR program in Appendix A.4.1. on page 61). Thepairs in each group of E are not symmetric with each other, but they have the same distance and anglefrom each other.For example, if 6 out of 8 pairs of group gl in C are in Dl, then this program puts all of these 6 pairsinto group say e3 of E. Next, if none of pairs of g2 in C is in Dl, it does not do anything. Next. if 3 out of4 pairs of g3 are in D1, it puts these 3 pairs in group e4 of E.The important result from this program is that any pair of strands such as a—b will be either exactlythe same as one of the pairs in E (say c—d) or a—b will be the same as an x-synunetric pair of c—d, or the y-symmetric pair of c—d, or xy-symmetric pair of c—d. Therefore, not all GMD's have to be calculated.Only the GMD values for the first pair of each group in E have to be calculated. The other pairs in thesame group will have the same GMD values because these pairs have equal distance and angle from eachother (they are derived from the array C that contains the strand pairs with equal distance and angle).A.4.1. 37DSPR Sample OutputHere is the screen output of 37DSPR.EXE.Egroup #^pair distance pairs^ total pairs1^2r^0-1^1-7^2-3^7-19^ 42 2r 0-2^2-9^9-22 73^2r^1-2 84 4r 0-7^1-4^1-19^9-11^ 125^4r^0-8 136133343536...969798991001581591601615.291503r2.478627r2.478627r4.472136r6.928203r3.902546r2.625786r2.625786r3.902546r10.392305r2.083778r12r11.817693r2-172-82-102-129-179-209-219-239-2422-3423-2423-3223-337-22 55565758120121122123124182183184185A.5. Before Constructing the Huge Z Matrix (37HZR13.EXE)All arrays and structures in the 37HZR13.EXE program are defined dynamically, so that the memorytaken by some arrays can be freed when appropriate to be used by other arrays.The program 37HZR13.EXE reads the binary files 37xy.dat and 37dspr.dat created by previous pro-grams into structure arrays A and E respectively (the same structure arrays as in 37xy.exe and37dspr.exe). It then asks the user to input the frequency of operation, and if he/she would like the com-puter to work in the automatic mode or in the manual mode.In the automatic mode, for frequencies less than or equal to 100 kHz, the user will have four optionsfor percent tolerances, namely 4%, 10%, 13%, and 21%. For frequencies above 100 kHz, the user willhave no option, and the computer will run with 21% tolerance. In the automatic mode, all variables in-cluding M the number of straight line divisions and N the number of circular divisions within each strandare chosen by the computer. In the manual mode, the user has to enter M and N.In both the automatic, and the manual modes, the computer chooses the number of points in the angu-lar direction within each elemental (THETPoints), and the number of points in radial direction withineach elemental (RPoints) to calculate the GMD between the two elements.To find the GMD between any two elements that do not belong to the same strand, (THETPoints *RPoints)2 number of distances are found and multiplied together, and the (THETPoints * R Poi nts)2 throot of the product is taken (in function Find_One_GMD). The program asks the user to enter the ra-dius of one strand (all strands have equal radii) and the average height of the transmission line fromground The program gives the option to the user to change the values for Aluminum conductivity ,Steel conductivity Cr Steel (for ACSR conductors), and Aluminum relative permeability pr(Al)•The program 3711ZR13.EXE allocates memory for Aluminum and Steel Resistance matricei,. Each ofthese matrices contains N double precision numbers (for any M). For example, if M_N = 8_3, as in Fig-ure 5 on page 5, the program stores the resistance of the central circle , with radius R[o], the resistance ofone of the elements between the central circle and circle R[1], and the resistance of one of the elementsbetween circle R[1] and the strand body (circle R[2]).62B. Technical Details of 37HZR13B.1.Locate_Points FunctionThe Locate_Points function takes the element number, \"el\", the strand number, \"st,\" RPoints, THET-Points, and the address of points! or points2 complex arrays. If the element number \"el\" turns out to bezero, that means the element is a circle, and this function puts all of the RPoints and THETPoints at thecentre of the circle. For elemental shape elements, it positions RPoints in the radial direction and THET-Points in the angular direction within the specific element \"el, st\", as shown in Figure 15 on page 28 andFigure 16 on page 28.B.2.Find_One_GMD FunctionThis function receives two complex arrays and RP an integer for the number of points in the radial di-rection (equivalent to RPoints) and TP an integer for the number of points in the angular direction(equivalent to THETPoints). It checks the locations of the points in the two complex arrays to identifywhether the element is a circle or an elemental.If none of the two regions is a circle, it calculates n distances, where n = RP * TP, and returns the(n2 ) th root of the product of n2 distances.If only one of the regions is a circle, it puts one point at the centre of that region and uses n = RP *TP points in the elemental shape region and returns the nth root of the product of the n distances as theGMD. By writing a separate computer program to find the GMD between a circle and an elemental inmany different ways, it was discovered that one point at the centre of the circle gives exactly the same re-sult as all the other methods.If both regions are circles, this function returns the distance between the centres of the two circles.To find the GMD within one strand, the program positions SelfR_Times times RPoints andSelfT_Times times THETPoints number of points in the radial and angular directions within each ele-mental, and then it finds the GMD. To find a correct value for the SelfR_Times = SelfT_Times constants,the final resistance and inductance values of SelfR_Times = SelfT_Times = 2 were compared withSelfR_Times = SelfT_Times = 3 (where M_N_R_O = 4_2_1_4, and GMRR_Times = GMRT_Times = 4while a 7 strand conductor was used). The difference was found to be less than 0.12% for resistance andless than 0.11% for inductance at all frequencies. Therefore, SelfR_Times = SelfT_Times = 2 is chosenfor all runs.The exact locations of the points within each element are stored inside each of the complex arrays:pointsl and points2. These two complex arrays, as well as the values of RPoints and THETPoints, aregiven to a function called Find_One_GMD. Inside this function, TOTPoints = RPoints * THETPointsnumber of distances are calculated, multiplied together, and the TOTPoints root of the product is taken.This function returns the resultant as a double precision number. This double precision number is storedin the GMD matrix as the GMD between two elements, but it is immediately replaced by the mutual in-ductance between the two elements.The mutual inductance is utua = -in ( 2h ) . The above procedure is repeated for all of the27GMDstrand elements of each of the first strand pairs in the structure array E.63B.3.Find_GMR FunctionTo find the GMR of one elemental, the program positions GMRR_Times times RPoints andGMRT_Times times THETPoints number of points in the radial and angular directions, respectively,within one elemental, and then it finds the GMR. To find a correct value for the GMRR_Times =GMRT_Times constants (where M_N_R_I3 = 4_2_1_4, and SelfR_Times = SelfT_Times = 2, while a 7strand conductor was used), the final resistance and inductance values of GMRR_Times = GMRT_Times= 2 were compared with 3, and 3 with 4 and 4 with 5. For the case of comparing 4 and 5, the differencewas found to be less than 1.2% for the resistance and less than 0.1% for the inductance at all frequencies.Therefore, GMRR_Times = GMRT_Times = 4 was chosen for all runs. In Figure 19 on page 32, RPoints= 1, and THETPoints = 2 are multiplied by 4 for GMR calculations.B.4.Find_SelfGMD_Ls_And_Resistances FunctionThis function finds the self and mutual GMD's for elements within the same strand and stores themin the SelfGMD matrix. The self GMD of a circle = GMR = radius * exp(-0.25) = radius * exp(—Circu-lar_GMRconst). Self GMD of an elemental shape is calculated numerically by positioning points withinone sample elemental and finding their GMD's.The data stored in the SelfGMD matrix will be used over and over when constructing the huge Z ma-trix because for each strand, the SelfGMD values are the same. Because the GMD's calculated all belongto the same strand, it is a good idea to use more number of points within each elemental for GMD calcu-lations. That is why for GMD calculations within a single strand, twice the THETPoints and RPoints aresent to the Locate_Points function.In this function, the resistance of each of the elements within one strand is defined using the con-ductivity and the element area.B.5. The GMD and SelfGMD MatricesA dynamic array called GMD is introduced that stores E_Dimension * (strnd_els)2 dot& le preces-sion numbers. For example, for Figure 12 on page 22 where M=4, and N=2, strnd_els = M(N-1)+1 = 5,the GM]) array stores 161 * 25 = 4025 double precision numbers.There is another dynamic array called SelfGMD that stores the GM]) between elements within thesame strand. Each element has a GM]) by itself (called GMR) and a GMD with each of the other ele-ments within the same strand. Therefore, the SelfGMD matrix has to store strnd_el * (strnd_els + 1)12double precision numbers. For example, if strnd_els = 5, then SelfGMD has to have space for 5 * 6 / 2 =15 double precision numbers.Two dynamic complex arrays are defined, pointsl and points2, such that each one stores TOTPoints= THETPoints * RPoints complex numbers. The program goes through the first strand pair of the struc-ture array E for which it has to find all of the GMD's between different elements of the two strands.64B.5.1. How the GMD Matrix is ConstructedHow the GMD matrix is filled in is the key for accessing the right value of mutual inductance at thetime the huge Z matrix is being constructed. The first pair of the first group in E is referred to as E[0](from Appendix A.4.1. on page 61, this is the strand pair 0-1), and the first pair of the second group in Eis referred to as E[1] (which is 0-2), etc. In Figure 12 on page 22, where M=4, and N=2, the GMD ma-trix will have elements from GMD[0] to GMD[4024] with mutual inductances between elements accord-ing to Table 14.Table 14. Contents of the GMD Matrix for M_N = 4_2.E[0], corresponding to strand pair 0-1 contributes to:GMD 0 1 2 3 40,0-0,1 1,0-0,1 2,0-0,1 3,0-0,1 4,0-0,1LmutualGMD 5 6 ... 10 ...Lmutual0,0-1,1 1,0-1,1 ... 0,0-2,1 ...GMD^15^20^24Lmutual^0,0-3,1 0,0-4,1 4,0-4,1E 1 , corresponding to strand pair 0-2, contributes to:GMD 25 26 27 28 290,0-0,2 1,0-0,2 2,0-0,2 3,0-0,2 4,0-0,2'mutualGMD^30^35^49L mutual^0,0-1,2 0,0-2,2 4,0-4,2E[160], corresponding to strand pair 23-33, contributes to:GMD 4000 4001 4002 4003 4004L mutual0,23-0,33 1,23-0,33 2,23-0,33 3,23-0,33 4,23-0,33 GMD 4005 4006 ... 4010 ...L mutual0,23-1,33 1,23-1,33 ... 0,23-2,33 ...GMD 4015 4020 4024L mutual0,23-3,33 ... 0,23-4,33 ... 4,23-4,33In general, each E[i] contributes to (strnd_els) 2 number of mutual inductances starting at GMD[i *(stmd_els) 2 I up to and including GMDRi+1) * (strnd_els) 2 — 1]. In the above example, each E[i] con-tributes to (strnd_els)2 = 25 values of mutual L in GMD matrix. These 25 values are divided into(strnd_els) = 5 sections each having (strnd_els) = 5 elements. Within each section, the element number ofthe second strand stays unchanged (such as 0,33 for GMD[4000] to GMD[4004]) and the element numberof the first strand increases from 0 to strnd_els — 1 = 4.65B.5.2. How the SelfGMD Matrix is ConstructedThe SelfGMD matrix has (strnd_els)*(strnd_els + 1)! 2 = 5 * 6 / 2 = 15 (for M_N = 4_2) doubleprecision numbers. GMD between el,st — e2,st is stored in SelfGMD[el + e2(e2+1)/2]. This matrix isstored as a lower triangular matrix. For example for M_N = 4_2 in Table 15, Se1fGMD[4] = 1,a-2,a isshown as [4]1,a-2,a (this means GM[) between element 1 strand a and element 2 strand a, taking \"a\" as ageneral strand number).Table 15. Lower Triangular SelfGMD Matrix Saved as a One Dimensional Matrix.[010,a-0,a[1]0,a-1,a [211,a-1,a[3]0,a-2,a [4]1,a-2,a [512,a-2,a[6]0,a-3,a [7]1,a-3,a [8]2,a-3,a [9]3,a-3,a[10]0,a-4,a [11]1,a-4,a [1212,a-4,a [13]3,a-4,a [1414,a-4,aB.6. Constructing the Huge Z MatrixThe huge Z matrix is a symmetric complex matrix because mutual inductances L and Li, areequal. Only the diagonal elements of huge Z have real parts (resistances) and all non diagonal elementsare purely imaginary (inductive reactances).Here in Figure 36, a lower triangular matrix is shown that represents HugeZ where diagonal ele-ments are triangular blocks and non diagonal elements are rectangular blocks.Figure 36 Lower Triangular HugeZ Matrix.If M_N = 4_2, the total number of elements within each strand will be M(N-1)+1 = 5, and therefore,each rectangular block of huge Z will be a 5 x 5 square matrix, and each triangular block will be a 5 x 5lower triangular matrix as shown in Figure 37.66R(0,36)+j roL(0,36 - 0,36)joa,(0, 36-1,36) R(1,36)+jm L(1,36 - 1,36)joiL(0, 36 - 2, 36) j w L(1, 36 - 2, 36)j^L(0, 36 -3, 36) j w L(1, 36 -3, 36)j wL(0, 36-4, 36) j^L(1, 36 - 4, 36)MN = 4_2R(4,36)+j to L(4,36 - 4,36)1Inside the Triangular Block 36 -36j wL(0, 32 - 0, 35) i w L(1, 32 - 0, 35) . • i 0 L(4, 32 - 0, 35)j 64(0, 32 -1, 35) j w L(1, 32 - 1, 35) - • j w L(4, 32 - 1, 35)j (01(O, 32 - 2, 35) j co L(1, 32 - 2, 35) • - j w L(4, 32 - 2, 35)j wL(O, 32 - 3, 35) j w L(1, 32 - 3, 35) - . j wL(4, 32 - 3, 35)j wL(0, 32 - 4, 35) i w L(1, 32 - 4, 35) • • j w L(4, 32 - 4, 35)Inside the Rectangular Block 32 - 35Figure 37. Inside a Rectangular and a Triangular Block of HugeZ (see Fig. 36).Notice that in the computer program, the HugeZ matrix is stored as a one dimensional matrix thesame way that GNLD and SelfGMD matrices (see Table 15 on page 66) are stored.8.7. Find_R_L_For HUGEZ FunctionFour integers are transferred to this function, and a complex number representing R + j coL is re-turned. The four integers are el (element number of first strand), sl (first strand number), e2 (elementnumber of second strand), and s2 (second strand number).If s 1 is equal to s2, a triangular block of Huge Z (Fig. 36) is being processed. The function uses theSelfGMD array that contains the self and mutual inductances between the elements of a single strand andel and e2 to obtain the appropriate inductance values.If sl is different from s2, a rectangular block of Huge Z (Fig. 36) is being processed. The functionuses the GMD matrix that contains the mutual inductances between elements of different strands and thespmnetry property to find the correct inductance. The E_Array that was created by the 37DSPR.EXEprogram contains the strand pairs with x, y, and origin symmetry. For example, the strand pairs 2-14 isin E_Array, and all mutual inductances between the elements of these two strands are stored in the GMDmatrix. In Figure 12 on page 22, GMD(1,2-4,14) is the GMD between element 1 of strand 2 and ele-ment 4 of strand 14. This GMD is equal to (x symmetry) GMD(4,6-1,12) and equal to (y s.ymmetry)GMD(3,5-2,8) and equal to (x symmetry again) GMD(2,3-3,18).The program goes through the above steps by first checking if the given strand numbers sl and s2 arein E_Array. if not, the program checks for x-symmetric pairs of s 1 and s2, if not, it checks for originsymmetric pairs of s 1 and s2, and finally, it may check for y-symmetric pairs of s 1 and s2. In ally step, if67the pair is found, the program refers to the GMD matrix and extracts the appropriate mutual inductanceand returns it to the calling function.B.8.Row-Column Subtraction and SymmetryAs it is explained in section 1.1. on page 5., after the Huge Z matrix is constructed, the first row issubtracted from all other rows. This in effect, replaces the [V] vector with a one non-zero element only.Then the first column of the resulting Z matrix is subtracted from all other columns, which replaces the IIof the [I] vector by 'total. After the row-column subtraction, the final matrix will be symmetric; there is,therefore, no need to use a square matrix, and a lower triangular one-dimensional matrix is sufficient.The row-column subtraction is performed at the same time that the Huge Z matrix is constructed, oneelement at a time from top to bottom and left to right.B.9.Reducing the Huge Z Matrix and Saving the ResultsIt is explained in section 1.1. on page 5 how, in general, the elements of the voltage and current vec-tors are exchanged [4]. Here, additional details of the procedure are explained. For simplicity, taking a 3x3 symmetric impedance matrix, the equation [V] = [Z] [I] is wntten as:_ _v1 z11 z12 z13 -11v2 = z12 z22 z23 12v3_- -z13 z23 z33 13In the reduct procedure, if v2 to vn are substituted by their negative values, the impedance matrix re-mains symmetric at each step of the reduction,. On the other hand, after row column subtraction, voltagev2 to vn will become zero. Replacing v2 to vn by their negative values:vi z12-v2[z11= z12 z22z131z23 12-v3 z13 z23 z33 13Expanding,vl = zll + z12 i2 + z13 i3;v2 = -z12 - z22 i2 - z23 i3;v3 = -z13 ii - z23 i2 - z33 i3.Using this last equation to obtain i3,i3 = -v3/z33 - z13 i11z33 - z23 i2/z33Substituting i3 into the first two equations:vi = zll ii + z12 i2 + z13 (-v31z33 - z13 il/z33 - z23 i2/z33) = (z11z23/z33) i2 - (z13/z33) v3and-v2 = z12 il + z22 i2 + z23 (-v3/z33 - z13 il/z33 - z23 i21z33) = (z12z23/z33) i2 - (z23/z33) v3.Looking at the last three equations, one can write:v1^z11 - (z13)2 /z33 z12 -z13 z23 /z33z12 -z23 z13 / z33 z22 +(z23)2 /z33-z13 / z33^-z23/z33-v2 =i3- z13 z13/z33)^+ (z12 - z13- z23 z13/z33)^+ (z22 + z23- z13 / z33-z231z33 /2 =-l/z33 v3_ _68A B CI1D EC E F v3To simplify the explanation, the contents of the new impedance matrix is replaced by A, B, C, etc.Following the same procedure for finding i2 in terms of v2 and substituting into the vi and i3 equations,i2 will be exchanged with v2 and the above equation will become:[vil [A -B2 /D -B /D C - BE /11112 = -B/D -1/D -E /D v2113 C - EB /D -E /D F - E2 /D v 3Since the final goal is to obtain element (1,1) of the reduced impedance matrix, there is no need tocalculate the values of the last rows of the impedance matrix. That is, while exchanging v3 by i3, there isno need to find the values of the third row of the impedance matrix, only the first two rows are important.Similarly, while exchanging v2 by i2, there is no need to find the values of the second row of the imped-ance matrix; only the first row values are important. Indeed only element (1,1) is important and not theother elements of the first row. However, the impedance matrix is stored as a one dimensional lower tri-angular matrix, and the only element to be calculated when reaching row one is element (1,1).The program saves the final result on disk in a text file. A sample of the output from 37HZR13 .EXEis shown in Table 16.B.9.1. 37HZR13 Sample OutputHere is a sample of the output produced by the program 37HZR13.EXE with the A option of theautomatic mode. Note that all inductance values shown here correspond to the internal inductance of theconductor, but the inductance curves in the Results section, for comparison with Mtline results, are thetotal inductance of the conductor with ground return. The run times are measured on a 80486-33 MHzpersonal computer.Table 16. Sam le Text Output of 37hzr13.exe.Frequency (Hz) 0.000e+00 1.000e-02 5.000e-02 1.000e-01MNRO 1111 1^1^1_1 1^1^1_1 1111 Resistance (ohms/km) 8.527588650e-02 8.527588654e-02 8.527588744e-02 8.527589024c-02Inductance (H/kin) 5.285104995e-05 5.285104994e-05 5.285104967e-05 5.285104882e-05Run Time (hrinin:sec) 0: 0:3 0: 0:3 0: 0:3 0: 0:3Radii (mm) 1.729740 1.729740 1.729740 1.729740Line Height (m) 20.000000 20.000000 20.000000 20.000000Aluminum^Conductivity3.46620e+07(S/m)3.46620e+07 3.46620e+07 3.46620e+07Steel Conductivity (S/m) 3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07Aluminum^Relative^Per-1.00000e+00meability1.00000e+00 1.00000e+00 1.00000e+00Minimum RAM (bytes) 19252 19252 19252 19252Local Date and Time Tue Jan 1923:17:53 1993 Mon^Jan^18^20:16:45Mon1993Jan^18^20:16:50.Mon1993Jan^18^20:16:561993Frequency (Hz) 5.000e-01 1.000e+00 5.000e+00 1.000e+01MNRO 1111 1111 1111 1111Resistance (ohms/km) 8.527597995e-02 8.527626028e-02 8.528522994e-02 8.531324911e-0269Inductance (H/km) 5.285102157e-05 5.285093643e-05 5.284821217e-05 5.283970237e-05Run Titne (hr:min:sec) 0: 0:3 0: 0:3 0: 0:3 0: 0:3Radii (mm) 1.729740 1.729740 1.729740 1.729740Line Height (m) 20.000000 20.000000 20.000000 20.000000Aluminum Conductiv-ity (S/m)3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07Steel^Conductivity 3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07(S/m)Aluminum^Relative 1.00000e+00 1.00000e+00 1.00000e+00 1.00000e+00PermeabilityMinimum RAM (bytes) 19252 19252 19252 19252Local Date and Time Mon Jan 18 20:17:02 Mon Jan 18 20:17:08 Mon Jan 18 20:17:14 Mon Jan 18 20:17:201993 1993 1993 1993Frequency (Hz) 5.000e+01 6.000e+01 1.000e+02 5.000e+02MNRO 1111 1111 1111 1111Resistance (ohms/km) 8.620112384e-02 8.660248624e-02 8.887089029e-02 1.332864441e-01Inductance (H/km) 5.257015801e-05 5.244838516e-05 5.176105277e-05 3.882569495e-05Run Time (hrmin:sec) 0: 0: 3 0: 0:3 0: 0: 3 0: 0:3Radii (mm) 1.729740 1.729740 1.729740 1.729740Line Height (m) 20.000000 20.000000 20.000000 20.000000Aluminum^Conductiv-ity (S/m)3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07Steel^Conductivity(S/m)3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07Aluminum^RelativePermeability1.00000e+00 1.00000e+00 1.00000e+00 1.00000e+00Minimum RAM (bytes) 19252 19252 19252 19252Local Date and Time Mon Jan 18199320:17:26 Mon Jan 18199320:17:32 Mon Jan 18199320:17:38 Mon Jan 18 20:17:441993Frequency (Hz) 1.000e+03 5.000e+03 1.000e+04 5.000e+04MNRO 1111 2313 2315 4413Resistance (ohms/km) 1.701295549e-01 3.248302953e-01 5.498413733e-01 4.376091981e+00Inductance (H/km) 2.999090483e-05 2.132589891e-05 1.989521306e-05 2.296007304e-05Run Time (hrmin:sec) 0: 0:3 0:^1:16 0:^1:23 0:14:33Radii (mm) 1.729740 1.178779,1.7297401.509823, 1.238906,1.7297401.537791,^1.330550,^1.526183,1.644385,^1.729740Line Height (m) 20.000000 20.000000 20.000000 20.000000Aluminum^Conductiv-ity (S/m)3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07Steel^Conductivity(S/m)3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07Aluminum^RelativePermeability1.00000e+00 1.00000e+00 1.00000e+00 1.00000e+00Minimum RAM (bytes) 19252 316724 316724 2087020Local Date and Time Mon Jan 18199320:17:50 Mon Jan 18199320:19:08 Mon Jan 18199320:20:35 Mon Jan 18 20:35:111993Frequency (Hz) 1.000e+05 5.000e+05 1.000e+06 5.000e+06MNRO 4513 6617 6619 6 6 1^1170Resistance (ohms/kin) 7.845298575e+00 2.272808254e+01 4.103950308e+01 2.383459102e+02Inductance (H/lan) 2.425358324e-05 2.196185705e-05 2.050536831e-05 1.821123543e-05Run Time (hrmin:sec) 0:30: 5 2:53:22 3: 3:2 3: 14:50Radii (mm) 1.370672,^1.523172, 1.529218,^1.606554, 1.584477,^1.640590, 1.662784,^1.688679,1.613942,^1.678969, 1.651935,^1.684194, 1.673463,^1.696808, 1.703831,^1.714583,1.729740 1.709248,^1.729740 1.714928,^1.729740 1.722924,^1.729740Line Height (m) 20.000000 20.000000 20.000000 20.000000Aluminum Conductiv-ity (S/m)3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07Steel^Conductivity 3.46620e+07 3.46620e+07 3.46620e+07 3.46620e+07(S/m)Aluminum^Relative 1.00000e+00 1.00000e+00 1.00000e+00 1.00000e+00PermeabilityMinimum RAM (bytes) 3559892 11800372 11800372 11800372Local Date and Time Mon Jan 18 21:05:20 Mon Jan 18 23:58:45 Tue Jan 19 03:01:50 Tue Jan 19 06:16:431993 1993 1993 1993Frequency (Hz) 1.000e+07MNRO 6 6 1^11Resistance (ohms/km) 5.215313610e+02Inductance (H/km) 1.896013710e-05Run Time (hrmin:sec) 3: 14:51Radii (mm)^1.682071,^1.700510,1.711297,^1.718951,1.724888,^1.729740Line Height (m) 20.000000Aluminum Conductiv-ity (S/m)3.46620e+07Steel^Conductivity(S/m)3.46620e+07Aluminum^RelativePermeability1.00000e+00Minimum RAM (bytes) 11800372Local Date and Time Tue Jan^19 09:31:37199371C. Analytic GMD FormulasFigure 38 shows two rectangles in Cartesian coordinates.A YaFigure 38. Geometric Mean Distance (GMD) Analytic Formula between Two Rectangles.The natural logarithm of the GMD between the two rectangles is derived as:In (GMDre„angie)=1 k egb 2^ 5$ fiIn [( x2 - x1) +( Y2 -yr) 21cfridYidx2dY22(b - a)(g - f)(e - c)(k - h) h c if aIt is possible to evaluate this integral analytically. According to [14], ifF(x,y)=(x4 - 6x 22 + y4 )^2^224^In (x2 +y)— —XY x arctan(1^2^x)+ y arctan(-)3 I^x ; then,a4F(xi-x2 Y ' -Y2) = -ln [(x2 -xl)2+ (y, - yi)21 - —265x, yi ax2 y,To find the Geometric Mean Radius (GMR) of a rectangle, the two rectangles have to coincide, that isthe limits of integration have to be changed such that a = c, b = e, f = h, and g = k.Figure 39 shows two elementals that belong to the same strand.The natural logarithm of the GMD between the two elementals of Figure 39 is derived as:in (GMDelemental) =1^ 1.1'2^it9 2^fr4^1.0 41^j^In I 4Za - 12Z/3 I da dl, dig d12=^(04 - 03 )(r4 - r3 )( 02 - A X r2 - ri ) '112=4 ill 1 411.---r3 cr=8 3where In 14La -443 1=1n 14 cos a + j 4 sin a -12 cos 13 - j 12 sin 13 I == -1 In [(4 cos a - /2 cos 13) 2 + (4 sin a - /2 sin f3 ) 2 1=2lin r42 + .22 2 4 /2 cos (ot -13 )1272Figure 39. Geometric Mean Distance Analytic Formula between Two Elementals within the Same Strand.To solve this integral, the author has used a number of possible methods, but unfortunately has beenunsuccessful. One method that comes to mind is to convert the integral from Polar to Cartesian coordi-nates by the following substitutions:Y= VX22 +y22 , a = arctan( 2—) , 12 = Aix12 + y12 ,13 = arctan(—Y1)x2^ X1therefore,1E12^n [42 +l2 -2l /2 cos (a — (3. )] dot d4 d13 d/2Siff In {(x2 xi)2 ± (Y2 —Y0212 1 dx, dyi d x2 dy2X1 2 + yi2 •X22 + y2 2It was not possible to solve this integral in Cartesian coordinates either. That is why thc computerprogram uses points within each elemental to find the GMD between two elementals. Because of com-plexity, the analytic formula for the GMD between elementals that do not have the same centre has notbeen derived.To obtain the GMR of an elemental, the following parameters have to be set equal: r 1 = r3, r2 = r4,01 = 03, and 02 = 04.73"@en ; edm:hasType "Thesis/Dissertation"@en ; vivo:dateIssued "1993-05"@en ; edm:isShownAt "10.14288/1.0065231"@en ; dcterms:language "eng"@en ; ns0:degreeDiscipline "Electrical and Computer Engineering"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "University of British Columbia"@en ; dcterms:rights "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en ; ns0:scholarLevel "Graduate"@en ; dcterms:title "Frequency dependent impedance of stranded conductors using the subdivision method"@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/2548"@en .